A COMPUTATIONAL STUDY OF
GABOR ZONE PLATE
GAMMA RAY HOLOGRAPHY
by
CLARE ESTELLE JACKSON
A thesis submitted to
The University of Birmingham
for the degree of
DOCTOR OF PHILOSOPHY
Scho ol of Physics and Astronomy
The University of Birmingham
Decemb er
University of Birmingham Research Archive
e-theses repository
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation.
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ABSTRACT
Gamma ray zone plate holography is a new technique with applications to Nuclear
Medicine Unlike other tomographic techniques three dimensional images can b e recon
structed from just one pro jection
The history of zone plate holography is reviewed and the di erences b etween this
technique and conventional holography are outlined Sources of error in the recorded
hologram are reviewed and metho ds for the assessment of image quality are given
Three image reconstruction techniques are describ ed and compared These techniques
are convolution deconvolution and the CLEAN algorithm
Simulated di raction is the main image reconstruction metho d which has previously
b een used to reconstruct images from zone plate holograms This metho d is a form of
convolution reconstruction Several variations on this technique are intro duced and com
pared Matched ltering is also investigated and compared with the simulated di raction
based metho ds
An approximate Fourier Wiener lter is used to reconstruct the images bydeconvolu
tion Several di erentversions of this lter are discussed and compared
The CLEAN algorithm reconstruction is an iterative metho d which is based on either
convolution or deconvolution
The three metho ds are tested using b oth computer generated and real gamma ray zone plate holograms
CONTENTS
LIST OF FIGURES
LIST OF TABLES
INTRODUCTION
Background to the Thesis
Original Contribution
LITERATURE REVIEW
History of Holography
History of Zone Plate Holography
General History
The Zone Plate Holography Pro ject at the University of Birmingham
Summary
COMPARISON OF ZONE PLATE HOLOGRAPHY WITH CONVEN
TIONAL HOLOGRAPHY
Derivation of the Kirchho Fresnel Di raction Formula Pro of of the Huygens
Fresnel Principle
Formation of a General Hologram
Gab or In Line Holography of a Two Dimensional Ob ject
The Central Fresnel Zone is much larger than the Ob ject
The Central Fresnel Zone is much smaller than the Smallest Detail
in the Ob ject
The Central Fresnel Zone is Larger than the Smallest Detail in the
Ob ject but Smaller than the Ob ject Itself
Zone Plate HolographyofTwo Dimensional Sources
Comparison of Gab or and Zone Plate Holograms of Single Points
Contents
Zone Plate Hologram of a Point Gamma Ray Source
Gab or Hologram of an Opaque Point in the Middle of an Otherwise
TransparentTransparency
Optical Convolution Image Reconstruction
Matched Convolution
Unmatched Convolution
Comparison of Holograms of more than One Point
Zone Plate Hologram of twoPoint Gamma Ray Sources
Gab or Hologram of two Opaque Points
Hologram of a Three Dimensional Ob ject
Zone Plate Holography
Gab or In Line Holography
Summary
INHERENT ERRORS ASSOCIATED WITH GAMMA RAY ZONE
PLATE HOLOGRAPHY
Gamma Camera
Floating Point Arithmetic
RadioactiveDecay
Sampling Theory
Summary
ASSESSMENT OF RECONSTRUCTED IMAGE QUALITY
System Transfer Function
Relationship b etween the Source Distribution and the Output Image
Optical and Mo dulation Transfer Functions OTF MTF
Spatial Resolution
Point Spread Function PSF
Line Spread Function LSF and Edge Spread Function ESF
Noise
Variance of an Image of a Uniform Source
Wiener Sp ectrum W
Contents
Di erence Metrics
Summary
IMAGE RECONSTRUCTIONBYCONVOLUTION
Optical Reconstruction how Convolution Algorithms can b e used to Sim
ulate Di raction
Using the Zone Plate Pattern as a MatchedFilter
Hybrid Metho d using R to improveSimulated Di raction
Normalised Convolution
Comparison of Reconstructed Points using Error Metrics
Comparison of Mo dulation Transfer Functions
Comparison of Point Spread Functions
Comparison of Images of Uniform Sources
Summary
IMAGE RECONSTRUCTION BY DECONVOLUTION
Expression of the hologram formation pro cess using Fourier transforms and
using matrix notation
Deconvolution using Fourier and standard matrix Wiener lters
Application of Wiener lters to zone plate holography
Using the p ower sp ectral density
Finding a constantwhichgives the minimum error
Normalisation
on the Magnitude of the Re To Account for the E ect of K K
constructed Point
To account for part of the zone plate hologram falling outside the
detector
E ect of a mis aligned zone plate
Using a truncated Gab or zone plate with a Wiener lter
Comparison of reconstructed p oints using error metrics
Comparison of Mo dulation Transfer Functions
Comparison of p oint spread functions
Contents
Comparison of Images of Uniform Sources
Summary
IMAGE RECONSTRUCTION USING THE CLEAN ALGORITHM
Assessment of Image Quality
Summary
RECONSTRUCTION OF COMPUTER GENERATED HOLOGRAMS
Hologram of a Simple Two Dimensional Cross Source Placed Parallel to
the Zone Plate
Reconstruction of Images from an Ideal Hologram
Reconstruction of Images from Holograms with Added Errors
Reconstruction of Images from Holograms of Ob jects of Di erent Sizes with
the Same Parameters
Reconstruction of Images from Holograms of the Same Ob ject with Di er
entParameters
Reconstruction of Two Cross Sources in Di erent Planes
Summary
RECONSTRUCTION OF REAL HOLOGRAMS
Cross Source Placed Parallel to the Zone Plate
Three Dimensional Heart Phantom
Summary
DISCUSSION CONCLUSIONS AND FURTHER WORK
Discussion of Results
Conclusions
Suggestions for Further Work
A COHERENT AND INCOHERENT LIGHT
B ELECTROMAGNETIC WAVES
Contents
D DERIVATION OF THE OPTIMAL WIENER FILTER FOR DE
CONVOLUTION
E OTHER RECONSTRUCTION METHODS
E Matrix Wiener Filtering
E To eplitz Blo ckTo eplitz Structure of the Hologram Forming Matrix
E Inverse Hessian and Steep est Descent Metho ds of Iterative Reconstruction
E Comparison of Reconstruction by Deconvolution with that by Matrix Wiener
Filtering and by Steep est Descent Iteration
E Chirp Functions and Wavelets
F CONFERENCE PAPER
F INTRODUCTION
F BINARY GABOR ZONE PLATE
F COMPARISON BETWEEN ZONE PLATE HOLOGRAPHY AND CON
VENTIONAL HOLOGRAPHY
F Conventional Holography
F Zone Plate Holography
F RECONSTRUCTION METHODS
F Reconstruction by Correlation
F Reconstruction by Deconvolution
F COMPARISON BETWEEN CORRELATION AND DECONVOLUTION
EXPERIMENTAL RESULTS
F Two Dimensional Ob jects
F Three Dimensional Ob jects
F CONCLUSIONS
GLOSSARY
REFERENCES
LIST OF FIGURES
Equipment used to record a hologram
Gaborin linehologram
Zone plates and shadow casting
Binary zone plate patterns
Region of integration
Di raction b ehind an ap erture surface
Di raction b ehind an ap erture surface
Relative p ositions of source transparency and hologram
Three consecutive op erations forming a general hologram
Variation in the values of a and b with hologram to detector distance
Variation of on axis intensity
Relative p ositions of the transparencies and the hologram
Diagram of a gamma camera detector head
Example pulse height sp ectrum from a NaI Tl scintillation detector
Errors asso ciated with basic arithmetic op erations
Errors in example calculations p erformed using NAG routines
E ect of a nite detector area on a Fourier transform
Fourier transforms of Gab or zone plates
E ect of the nite detector size
E ect of truncating the zone plate at a maximum radius
Comparison of actual and calculated Fourier transforms hologram
Comparison of actual and calculated Fourier transforms hologram
Field of view of detector with zone plate
Illustration of the useful hologram area
Theoretical reconstruction of a p oint source bysimulated di raction
List of Figures
Calculation of the binary zone plate pattern
Theoretical reconstruction of a p oint source using a matched lter
Balanced convolution and mo di ed mismatch metho ds
Point spread functions in one image plane
Truncated Fresnel function reconstruction
Point spread functions matched lters
Illustration of the zone plate pattern overlapping the edge of the detector
Unnormalised and normalised convolution imageamplitude
Mo dulation transfer functions
Point spread functions in x y andz directions
Unnormalised images of uniform sources
Normalised images of uniform sources
Fitting a signal mo del and a noise mo del to the p ower sp ectrum
Power sp ectra of ideal and noisy holograms
E ect of varying K on the reconstructed image
E ect of varying K on the reconstructed image
Magnitudes of theoretical reconstructed p oints in di erent p ositions
Mis alignment of the zone plate
E ect of a mis aligned zone plate on the p oint spread function
Point spread functions for Gab or Wiener ltering
Mo dulation transfer functions
Unnormalised images of uniform sources
Normalised images of uniform sources
Normalised reconstruction by convolution and K Wiener lter
Normalised reconstruction by convolution
Normalised reconstruction by K Wiener lter
Partially normalised reconstruction by convolution and K Wiener lter
Partially normalised reconstruction by convolution
Partially normalised reconstruction by K Wiener lter
Partially normalised reconstruction using the CLEAN algorithm
List of Figures
Partially normalised reconstruction using the CLEAN algorithm
➁ convolution deconvolution Images from hologram
Images from hologram ➁ CLEAN algorithm
➃ convolution deconvolution Images from hologram
Images from hologram ➃ CLEAN algorithm
Images from hologram convolution deconvolution
CLEAN algorithm Images from hologram
Images from hologram CLEAN algorithm
Images from hologram convolution deconvolution
Images from hologram convolution deconvolution
CLEAN algorithm Images from hologram
Images from hologram CLEAN algorithm
Images from hologram convolution deconvolution
➊ convolution deconvolution Images from hologram
Images from hologram ➊ CLEAN algorithm
Images from hologram ➋ CLEAN algorithm
➋ convolution deconvolution Images from hologram
Images from hologram ➌ convolution deconvolution
Images from hologram ➌ CLEAN algorithm
➍ CLEAN algorithm Images from hologram
Images from hologram ➍ convolution deconvolution
➎ convolution deconvolution Images from hologram
Images from hologram ➎ CLEAN algorithm
Images of two cross sources convolution deconvolution
Images of two cross sources CLEAN algorithm
Cross source recorded using the collimator
Images of rst cross source in one plane
First cross source in ve planes convolution deconvolution
First cross source in veplanes CLEANalgorithm
Images of second cross source in one plane
List of Figures
Second cross source in ve planes convolution deconvolution
Second cross source in ve planes CLEAN algorithm
Heart phantom
End view of heart phantom in veplanes convolution deconvolution
End view of heart phantom in ve planes CLEAN algorithm
Side view of heart phantom in four planes convolution deconvolution
Side view of heart phantom in four planes CLEAN algorithm
C Characteristic Hurter Dri eld curve
E Reconstruction of a pixel hologram
E Reconstruction of a pixel hologram
E Real time and pro cessor time taken to solve random matrix equation
E Reconstruction of a p oint source bytheinverse Hessian metho d
E Reconstruction of a p oint source by the steep est descent metho d
E Comparison of reconstruction of two p ointsources
E Two dimensional windowed Fourier transform of a Fresnel function
F Equipment used to record a hologram
F Zone plates
F Hologram formation
F Comparison b etween correlation and deconvolution cross source
F Heart phantom
F Comparison b etween correlation and deconvolution heart phantom
LIST OF TABLES
Mass attenuation co e cients
Linear attenuation co e cients
Spatial resolution of example gamma cameras
IEEE standard for single and double precision oating p ointnumbers
Limiting values of di erentnumb er formats
Equations t to errors from basic arithmetic op erations
Equations t to errors from Fourier transforms and matrix inversion
Hologram parameters
Comparison of error metric values
Comparison of MTF error metric values
Error metric values for p oint spread functions
Error metric values for uniform sources
Comparison of error metric values
Comparison of MTF error metric values
Comparison of p oint spread functions
Error metric values for uniform sources
Error metrics for normalised images of cross source
Error metrics for partially normalised images of cross source
Error metrics for cross source CLEAN algorithm
Error metrics for noisy holograms convolution deconvolution
Error metrics for noisy holograms CLEAN algorithm
Error metrics for images of text ob ject
Hologram parameters for holograms of star
Error metrics for images of star ob ject
Error metrics for images of t wo cross sources
List of Tables
Parameters for the rst hologram of the cross source
Errormetricsfor rstcrosssource
Parameters for the second hologram of the cross source
Error metrics for second cross source
Parameters for the holograms of the heart phantom
Error metrics for end view of heart phantom
Error metrics for side view of heart phantom
Chapter
INTRODUCTION
Background to the Thesis
Gab or zone plate holography can b e used to image gamma rays emitted from radiophar
maceuticals A three dimensional image of the source can b e reconstructed from just one
pro jection Tomography is commonly used for three dimensional imaging for example
x ray CT Computed Tomography PET Positron Emission Tomography or SPET Sin
gle Photon Emission Tomography In tomography a large numb er for example
readings are taken at di erent angles around the patient In general readings are taken
at as many pro jection as there are pixels p er side in the image The patient therefore
receives a large dose of radiation and it takes a long time to reconstruct the image As
only one reading at one angle needs to b e taken for zone plate holography this technique
could provide faster image acquisition at a reduced dose to the patient
A zone plate is placed in front of and parallel to a gamma ray detector The gamma ray
detector is a gamma Anger camera with its collimator removed Figure shows this
arrangement Gamma rays emitted by a radiopharaceutical cast a shadow of the zone plate
on the detector The shadow will b e called a hologram as it contains three dimensional
information ab out the distribution of the radiopharmaceutical i e the source An image of the source can then b e reconstructed from the hologram
detector (gamma camera)
(γ -ray) source
zone plate computer
Figure Equipment used to record a hologram
If the hologram is reduced to an appropriate size recorded on photographic lm and
Intro duction
develop ed it can b e reconstructed optically by shining lightback through it This re
construction can b e simulated using a computer Computer based reconstruction has the
advantage that other techniques not based on simulated di raction can b e used Zone
plate holography research has taken place at the University of Birmingham since
but the image reconstruction techniques still need improvement
The aim of this work was to nd new reconstruction techniques whichwillimprove
the image quality Techniques were applied to a range of computer generated holograms
to test their e ectiveness for ideal error free data The reconstruction metho ds were then
tested on computer generated holograms to which di erenttyp es of noise were added
Images were then nally reconstructed from real gamma ray zone plate holograms
In Chapter a short summary of the history of holographyisgiven from its invention
by Gab or in to the presentday The history of zone plate holography is discussed in
more detail with particular emphasis on the work that has b een done at the Universityof
Birmingham Reconstruction techniques used to image zone plate holograms of neutrons
white light and infra red sources can also b e used for holograms of gamma ray sources so
the work that has b een done on these pro jects is also covered in depth
In Chapter expressions are derived for hologram formation for b oth zone plate and
conventional holography The similarities and di erences b etween the two techniques
are discussed Algorithms describing standard optical image reconstruction are given
and equations for the reconstructed image of a p oint source are derived for eachtyp e of
hologram Issues sp eci c to imaging gamma ray sources using zone plate holographyare
highlighted
In Chapter other sources of error inherent in the recording and pro cessing of zone
plate holograms of gamma ray sources are discussed Metho ds for the assessmentof
reconstructed image quality are listed in Chapter
Three di erent reconstruction techniques are outlined in Chapters and Theo
retical zone plate holograms of p oint sources are calculated using the formulae derived in
Chapter The quality of the images pro duced byeach technique are assessed using the
techniques given in Chapter
The three techniques are compared further in Chapters and Chapter Images
reconstructed from theoretical holograms of extended ob jects are shown in Chapter and
Intro duction
from real holograms of gamma ray sources in Chapter
All of this work is discussed in Chapter and conclusions are drawn Suggestions for
further work are given
Original Contribution
Gamma ray zone plate holography has b een compared with one typ e of conventional
holography and the di erences have b een highlighted
Most of the work done on zone plate holography in the past has concentrated on
reconstruction metho ds based on simulated di raction
Simulated di raction is a convolution reconstruction metho d Di erentconvolution
and matched lter image reconstruction metho ds have b een compared Convolution with
a truncated Fresnel function was found to give b etter results than simulated di raction for
computer generated holograms of p oint sources This metho d was then used to reconstruct
images from a selection of computer generated and real holograms
Fourier Wiener ltering and the CLEAN algorithm are commonly used to recon
struct two dimensional images These techniques have b een used to reconstruct three
dimensional images from zone plate holograms Di erent approximate Wiener lters have
b een tested on computer generated p oint sources and the most e ective one has then b een
used to reconstruct images from a selection of computer generated and real holograms
These images were then compared with those obtained by convolution
The advantages and disadvantages of the three techniques convolution deconvolution
and the CLEAN algorithm have b een discussed
Chapter
LITERATURE REVIEW
History of Holography
Holographyisaway of recording information from a three dimensional ob ject in two
dimensions The idea is based on the Huygens Fresnel Principle which states that
Every p ointofa wave frontmay b e considered as a centre of secondary distur
bance whichgives rise to secondary wavelets and the wave frontatany later
instance may b e regarded as the envelop e of those wavelets The secondary
wavelets mutually interfere
This means that at anypoint in space a wave contains the whole of the information
ab out its source If the phase and amplitude of a wave could b e recorded the source
could b e reconstructed in three dimensions Most photographic pro cesses just record the
intensity or amplitude of a wave Holographyprovides a way of recording the phase as
well
In the British Hungarian scientist Gab or was awarded the Nob el Prize in
Physics for his discovery of holography It is sometimes argued that the Russian scientist
Denisyuk should have also b een awarded the Nob el Prize for his invention of re ection
holograms which turned holographyinto a useable technique His work will b e dis
cussed in more detail later
Gab or had invented holographytwenty four years b eforehand when he was investigat
ing ways to improve the resolving p ower of electron microscop es He built on work that
had b een done by others developing two stage microscopytechniques for example Bo er
sch The resolution of electron microscop es was limited by the spherical ab erration of
electron lenses whichwere provided by magnetic elds Gab or s pap er in Nature in
entitled a new microscopic principle describ ed how micrographs could b e obtained
without electron lenses Literature Review
plane waves coherent waves plane waves (scattered from a point object)
virtual image hologram real image
point object detector
-> hologram
a Hologram formation b Hologram reconstruction
Figure Gab or in line hologram
Gab or s metho d had two stages In the rst stage the ob ject was illuminated byan
electron b eam This b eam was partially scattered from the ob ject The scattered wave
then interfered with the primary b eam and the interference pattern was recorded on a
photographic plate In the second stage coherentlightwas shone through the develop ed
photograph A real and a virtual image of the ob ject were obtained This metho d is
illustrated in Figure As it was not p ossible at the time to pro duce suitable b eams
of electrons Gab or demonstrated the feasibility of his technique by using visible light
instead of electrons He used a mercury arc lamp as his source The light pro duced by
this lamp was coherent This meant that Gab or could only use very small ob jects
for his holograms An explanation of the di erence b etween coherent and incoherentlight
is given in App endix A
In his next pap er Gab or develop ed the theory b ehind his metho d He also
discussed the impurities in the reconstructed image The main problem was the presence
of the real image in line with the virtual image The real image was referred to as the
twin image Only p o or quality holograms of two dimensional ob jects were pro duced
Gab or later called his metho d holography after the Greek word holos meaning whole
and gramma meaning message He had invented a metho d of recording the phase
as well as the amplitude of a wave the whole of the wavewas preserved
Literature Review
In Rogers noticed the similaritybetween a Fresnel zone plate and the hologram
of a p oint source and said that a hologram could b e considered as a zone plate with
a complex pattern
Zone plate holographywas rst suggested in by Mertz and Young Fresnel
zone plate imaging was suggested as a technique for use in x ray cameras Neither refract
ing nor re ecting materials were available to provide fo cusing at x raywavelengths and
zone plate imaging provided a way round this problem They illustrated their technique
by replacing the lens of a normal camera with a Fresnel zone plate
Fresnel Zone plate
Light / x-rays from 4 point sources
Shadow
a Gab or zone plate b Fresnel zone plate c Shadow casting
Figure Zone plates and shadow casting The shade of gray indicates the transmission
of the plate Black transmission and white transmission
AFresnel zone plate is a binary zone plate This means that its transmission function
at anypoint is either zero or one As is discussed in Chapter a Gab or hologram of a
p oint source has the following transmission function T r
r
T r cos
r
where r is the radial distance from the centre of the plate and r is a constant
A zone plate with the transmission function t r is called a Gab or zone plate A Fresnel
zone plate is a binary approximation of a Gab or zone plate It consists of alternating
opaque and transparent concentric rings which are called zones The outer radii of the
transparent concentric zones are prop ortional to the square ro ot of etc A Fresnel
Literature Review
zone plate and a Gab or zone plate of the same dimensions can b e seen in Figures b
and a The hologram of a p oint source would b e as large as the detector The Fresnel
and Gab or zone plates however are truncated at a maximum radius Light or x rays
would cast a shadow of the zone plate on the detector as is illustrated in Figure c
If the light source is considered to consist of manypoint sources the pattern recorded
on the photographic lm can b e seen as the sup erp osition of the shadows from eachof
these sources The photograph would then b e an approximation of a Gab or hologram
of an equivalent source This zone plate hologram could then b e reconstructed in the
same way as a Gab or hologram by shining coherent light through it The advantage
of this technique is that the hologram is formed by a shadow casting pro cess and so is
indep endentofwavelength or coherence length
As was explained in Chapter the aim of the work presented here was to nd new
image reconstruction techniques for zone plate holograms of gamma ray sources The
history of zone plate holography will b e discussed in detail in Section For the pur
p ose of this rep ort this general history of holography could end here For the sakeof
completeness however the ma jor advances made in holography since will nowbe
discussed
In Leith and Uptanieks saw similarities b etween Gab or s technique and the
research they had b een doing into radar They describ ed Gab or s two step pro cess from
a communication theory viewp ointandsawthetwin image as an aliasing problem They
suggested that a carrier frequency in the form of an o axis reference b eam could b e used
to improve image quality The laser had b een invented in and in Leith and
Upatnieks made the rst laser transmission hologram of D ob jects using their o axis
metho d Laser transmission holograms pro duced clear images with realistic depth but
could only b e viewed with laser light
Also in Denisyuk found a waytomake white light re ection holograms He
combined holography with a form of colour photographyinvented in by the Nob el
Laureate Lippmann He placed a sheet of photosensitiveemulsion b etween the reference
b eam and the ob ject The emulsion had a thickness large compared to the wavelength
of visible light Interference b etween the reference b eam and the b eam re ected from the
ob ject caused standing waves inside the emulsion When the emulsion was develop ed
Literature Review
its density followed the intensity pattern of the standing wave Denisyuk holograms are
often termed volume holograms since they are recorded over the thickness of the emulsion
and not just in two dimensions If mono chromatic light is used to pro duce the hologram
white light can b e used in the reconstruction The hologram will only re ect the fraction
of the light with the same wavelength as the original mono chromatic source to form the
holographic image The remaining wavelengths will pass through the hologram having no
e ect on the image If more than one wavelength is used colour images are obtained
Another signi cant advance was holographic interferometry Powell and Stetson pub
lished the rst pap er on this in Small distortions b etween two holographic
exp osures are seen as contours on the reconstructed image Holographic interferograms
are particularly useful in non destructive testing uid ow analysis and qualitycontrol
Benton invented the rainbow hologram in This was a transmission hologram
which could b e viewed with white light Hologram formation was bya two step o
axis technique The real image from a rst hologram was used as the ob ject for a second
hologram The rst hologram was however masked down to a narrow horizontal slit This
resulted in a reconstructed image in one or several colours dep ending on the viewp oint
The rst moving three dimensional images were made by Cross in Cross
combined white light transmission holography with conv entional cinematography He
recorded a sequence of holograms in narrow strips along the photographic lm As the
hologram was viewed from left to right di erent images were seen giving the illusion of
movement
In Foster invested a way for duplicating holograms mechanically He used an
emb ossing pro cess similar to the one used to make audio disks Rainbow holograms were
turned into re ection holograms by an aluminium backing and could b e mass pro duced
at a very low cost
Holography has many applications Holographic lenses are now used in sup ermarkets
to direct laser lightonto the bar co des during checkout Holograms are also used on some
credit cards to make them di cult to copy Holography can b e used for archival recording
of fragile museum pieces A pulsed laser hologram was even used to record Lindow Moss
a year old Iron Age man unearthed from Lindow Moss a p eat b og in Cheshire
Literature Review
History of Zone Plate Holography
General History
As mentioned in section the rst pap er on zone plate holographywas written by Mertz
and Young in Incoherent radiation sources can b e imaged using this technique so
it has found many applications
In Barret used a lead Fresnel zone plate to image gamma rays emitted from a
thyroid phantom The detector he used was a gamma camera Gamma cameras are
used in nuclear medicine and are usually tted with a pinhole ap erture or a parallel hole
collimator which only lets a tiny fraction of the incident gamma rays
through The advantage of a zone plate is that only half of the gamma rays are stopp ed
The holograms were reconstructed optically
Fresnel zone plates were again used for gamma ray imaging by Rogers et al in
An image intensi er was used instead of a gamma camera Zone plate holog
raphy is basically an on axis metho d and Rogers discovered that the virtual image in line
with the real image pro duced artifacts in the reconstruction He suggested that an o axis
zone plate could b e used to improve the image
In Ceglio used zone plates to study the x ray emission of laser pro duced plas
mas and Other applications have included imaging of ssion gamma rays pro
duced by reactor fuel and fo cusing x rays in eV synchrotron sources
One problem with zone plates is that Moir e zone plate patterns are pro duced when
several zone plate patterns overlap Moir e patterns are zone plates themselves and fo cus
light Jaroszewicz wrote a pap er on this e ect
Jones and Kirz intro duced phase zone plates Instead of opaque zones phase
zone plates have zones which cause a phase shift This meant that the intensityofthe
light fo cussed by the plate was increased by a factor of
In McDonald et al showed how gamma rayFresnel zone plate holograms could
b e reconstructed computationally He used a two dimensional fast Fourier transform
algorithm One of the advantages of computer reconstruction is that the image can b e
pro cessed
Literature Review
The Zone Plate Holography Pro ject at the Universityof
Birmingham
Researchinto zone plate holography has b een undertaken at the University of Birmingham
since Pro jects have b een completed in neutron holography white light holography
and infra red holography Work is now concentrated on gamma ray holography The
work presented here concerns the computer reconstruction of gamma ray holograms The
algorithms however could b e applied to holograms formed by other typ es of radiation
A brief description of current and completed work follows
Neutrons
The rst work on neutron zone plate holographywas done by Pink in as part of his
MSc under the sup ervision of Professor Beynon Neutron holographywas investigated
as an alternative to neutron tomography for use in non destructive testing The zone plate
was made of mm thick aluminium The opaque zones were made by using FeCl to etch
gro oves in the aluminium These gro oves were then lled with a gadolinium containing
paint Cold neutrons with energies less than eV from a research reactor were
scattered from test ob jects The hologram was recorded using a gadolinium converter
screen with x ray lm b ehind it The holograms were reconstructed optically by reducing
them in size and shining laser light through them This work was published in a pap er in
Mast did further work in this eld and submitted for his PhD in He used
thermal neutrons from a radiotherap y b eam but the rest of the exp erimental setup was
the same as that used by Pink The holograms were digitised and reconstructed using
a computer The rst algorithm used was an unmo di ed two dimensional fast Fourier
transform DFFT as describ ed by Singleton Another single DFFT metho d
based on the Rayleigh Sommerfeld equation was then used Results from b oth com
puter reconstruction metho ds were compared with those from the optical reconstruction
The unmo di ed DFFT metho d pro duced a p o or quality reconstruction The Rayleigh
Sommerfeld metho d however pro duced images of a similar quality to those obtained
by optical reconstruction Mast also used the Monte Carlo co de MORSE to make
simulated holograms This work was published in Another article was written
Literature Review
by Professor Beynon in
The next studenttowork on this pro ject was Mathews who completed his PhD in
The equipment he used was the same as that used by Mast He used two
zone plates of opp osite p olaritytoformtwo holograms Opp osite p olarity means that
one plate had opaque zones where the other had transparent zones One hologram was
subtracted from the other and the result was reconstructed This improved the image He
again used the Rayleigh Sommerfeld DFFT algorithm to reconstruct the holograms He
compared a p olar co ordinate r version of this technique with a Cartesian co ordinate
x y version He also applied zone plate holographytoPositron Emission Tomography
PET This is gamma ray holography and will therefore b e discussed later
a Moir e zone plate pattern b Hampton court c Rotated binary
pro duced bytwooverlapping binary Gab or zone Gab or zone plate
Fresnel zone plates plate
Figure Binary zone plate patterns
Kirk nished his PhD in He started by analysing Moir e zone plate patterns
which app ear when Fresnel zone plates overlap Moir e patterns act as zone plates and
fo cus light themselves A Moir e pattern is shown in Figure a As Moir e patterns
have a similar e ect to zone plates themselves Kirk investigated the p ossibility of making
a pure Gab or zone plate byoverlapping several Fresnel zone plates
Fresnel zone plates also contribute to noise in the image A Gab or zone plate has one
fo cal p oint a distance f from the plate An equivalentFresnel zone plate would havea
f f f
etc If the hologram was reconstructed optically the series of fo cal p oints at f
presence of these extra fo cal p oints would reduce the quality of the image Fresnel zone
plates had always b een used b ecause of the di culty of creating a Gab or zone plate with
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a smo othly varying transmission function
Kirk argued that the width of the zones in a binary zone plate could b e varied az
imuthally as well as radially so that the average transmission at anyradiuswould b e
the same as for a Gab or zone plate The result was the Hampton court zone plate so
called b ecause it resembled the Hampton Court clo ck Mast and Mathews showed that
the binary Gab or plate only had one fo cal p oint and hence put muchofthiswork on a
rmer theoretical fo oting A pap er was later published on this zone plate design In
this pap er the plate was called a binary Gab or zone plate The design whichisshown
in Figure b was patented The binary Gab or zone plate lead to artifacts in the
image like the sp okes of a bicycle wheel The rotated binary Gab or zone plate shown in
Figure c was an attempt to solve this problem
White Light
Wo o dgate researched white light zone plate holography for his PhD which he nished
in Just as Gab or used light from a mercury arc lamp to demonstrate his
technique for obtaining electron micrographs the exp eriments with white lightwere done
with the aim of nding new techniques to improve neutron holography Neutron zone
plate holograms contain errors due to the limited numb er of neutrons whichformthe
hologram Such statistical errors do not apply when white light is used
Many of the exp eriments were p erformed jointly byWo o dgate and Caplan another
PhD student who submitted her thesis in At the same time Strange was
researching di erenttyp es of co ded ap erture imaging and antennas In his PhD thesis
whichwas completed in he discussed metho ds for the computational reconstruction
of zone plate holograms
A CCD camera was placed in a light tightbox with a translucent screen at one end
The camera was fo cussed on the opp osite side of the screen to which the hologram w as
formed The hologram was digitised by a SUN video card and stored on a computer
Wo o dgate wrote p ostscript programs to draw the zone plate patterns Zone plates pat
terns could then b e laser printed and photographed onto lm The nal zone plate was
pro duced on the develop ed lm Lightwas re ected o ob jects of which holograms were
formed
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All the holograms were reconstructed computationally Wo o dgate used Mast s Rayleigh
Sommerfeld simulated di raction reconstruction technique He p ointed out the similarity
between this technique and correlating the hologram with a suitably scaled Gab or zone
plate pattern This correlation technique was shown to pro duce images nearly the same
as those pro duced by the di raction technique A Welch window was used with the
correlation technique to improve image quality He also tried to reconstruct the holo
grams by deconvolution but this gave p o or results He suggested that further work could
b e to pro duce an iterative reconstruction algorithm such as the one used by Helstrom in
or Fleming in Exp erimental holograms were compared to simulated
holograms pro duced using the Monte Carlo co de MCNP
Strange lo oked at a numb er of other reconstruction metho ds including matrix inver
sion matrix Wiener ltering and single value decomp osition Single value decomp osition
was used in order to remove artifacts from images byremoving certain frequencies Strange
used these techniques to reconstruct images from small simulated holograms They could
however not b e used for larger exp erimentally pro duced holograms b ecause the amount
of computer memory RAM that would b e needed to p erform the reconstruction was
incredibly high Strange also designed an approximate binary Gab or zone plate made
from a series of lines instead of smo oth curves This was needed so that the plate could
easily b e cut from a piece of metal and used for future work in gamma ray holography
Caplan used the correlation technique to reconstruct her holograms She investigated
the e ects of varying the numb er of zones in the zone plate and the frequency of their
azimuthal variation on the quality of the reconstructed image She also investigated the
v ariation of the p oint spread function with p osition If the detector is small not all of
the zone plate shadow is recorded Caplan p erformed exp eriments to lo ok at this e ect
Infra Red
Infra red holography could b e used to image the temp erature of the b o dy in three dimen
sions Regions of cancerous tissue often have a higher temp erature and therefore could
b e lo cated in this way The rst studenttowork on this pro ject was Kuo who nished
his PhD in Shen also worked on this pro ject and nished his PhD in
Zone plates for infra red holography are pro duced in the same way as the ones used
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in the white light exp eriments Neutron and white light holography detected radiation
scattered from ob jects Infra red holography images heat pro ducing sources for example
a tungsten lament lamp or an electric heater The hologram is recorded by using a small
detector attached to a X Y plotter whichmoves it across the plane in which the hologram
is formed The human b o dy emits radiation in the far infra red region A suitable detector
could however not b e found to record far infra red so exp eriments were p erformed in the
near infra red The detector used was a InGaAs detector of mm diameter A system
was develop ed to read the detector at intervals so the two dimensional hologram could b e
recorded as the plotter moved across the plane
Kuo used Mast s di raction algorithm to reconstruct the holograms He reconstructed
a uniform hologram and then subtracted this from the reconstruction of the real hologram
to remove artifacts due to digitisation He also used a mo di ed Welchwindow to remove
noise from the image Thresholding was used to remove the background noise from the
image
Shen used correlation to reconstruct the holograms He has also reconstructed the
images by correlating it with the appropriate binary Gab or zone plate pattern He used a
sine as well as a cosine zone plate to form two holograms which can b e combined to give
a hologram pro duced by complex real and imaginary zone plate The reconstruction of
this complex hologram showed improvementover those from a single zone plate
Gamma Rays
Mathews applied zone plate holography to PET In PET two gamma rays from p ositron
annihilation are detected on opp osite sides of the p ositron source bytwo PET cameras
A path through the source along which the annihilation must have taken place can then
b e constructed Mathews imagined there were Fresnel zone plates infront of eachcamera
and subtracted every gamma ray countwhic hwould have hit an opaque zone of one of
the plates The resulting holograms were then reconstructed as normal The images
pro duced using this technique were of comparable quality to those pro duced by standard
back pro jection The holography based technique pro duced the images in a fraction of
the time This work was published in a pap er in
Caplan discussed the di erence b etween white light and gamma ray holographyin
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her PhD thesis In order to attenuate the gamma rays zone plates used for gamma ray
holographyhavetobeseveral millimetres thick This leads to the additional problem
of vignetting where gamma rays travel through the zone plate at an angle and are only
partially absorb ed This means that gamma ray holograms are generally of p o orer quality
than those pro duced by white light
Perks pro ject was to image gamma rays in the energy range keV to keV and he
m
submitted his thesis in He used the gamma emitting isotop es Tc F and
Na which are used in nuclear medicine The detector he used was a gamma camera with
the collimator removed The rst zone plate was a cm lead approximate binary Gab or
zone plate whichwas commissioned by British Nuclear Fuels Ltd for studies into gamma
ray imaging This was later replaced bya mmsintered tungsten plate for imaging low
energy gammas keV An Ohio Nuclear series gamma camera was rst used to
record holograms These holograms were then compared to ones taken using a state of
the art Toshiba GCA A gamma camera at St Thomas Hospital London and using
an ADAC gamma camera at the Queen Elizab eth Medical Centre Birmingham The
Toshiba GCA A camera holograms gave b etter images
Exp erimental holograms were again compared with holograms simulated using MCNP
The MCNP le for the simulation was written by Dhaba an a research assistantinthe
applied physics group It included a representation of the approximate binary Gab or zone
plate Perks used Mast s di raction algorithm for all his reconstructions
Rew nished her PhD in gamma ray zone plate holography in She con
centrated on gamma ray energies used in Nuclear Medicine in general and in BNCT in
particular Research is b eing done into BNCT at the Universit y of Birmingham BNCT
is a treatmentfora typ e of brain tumour called Glioblastoma multiforme The patient
is injected with a b oron comp ound This comp ound concentrates to a higher degree in
brain tumour cells than in normal brain tissue When the head is irradiated with thermal
neutrons the following reaction o ccurs
n B Li
The alpha particles pro duced in the reaction kill the tumour cells in whichtheyare
concentrated The gamma rays haveanenergyof keV and can p otentially b e
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imaged using zone plate holography Thiswould p otentially provide a metho d to monitor
the treatment while it is taking place
A mm thick sintered tungsten zone plate would b e needed to e ectively stop
keV gamma rays Rew used a series of mm plates placed together A mm plate for
example would have a minimum transmission of The tungsten plate was cut by
a laser and so the original design can b e accurately repro duced She set up a gamma ray
holography lab at the University of Birmingham with a GE Maxi Camera which has
a higher e ciency than the Ohio camera used byPerks but a lower e ciency than the
Toshiba GCA A camera which he also used BNCT involves b eams of neutrons so the
e ects of neutron activation in the camera and scattering o the walls in the treatment
ro om were also investigated
Takhar and George are now b oth working on the gamma ray holography
pro ject They will b e using an ADACForte gamma camera as well as the GE Maxi
Camera to record further zone plate holograms of gamma ray sources It is hop ed
that zone plate holograms will b e recorded during BNCT patient trials and that the
reconstruction techniques presented here will b e tested using this data
Summary
Manyadvances have b een made in holography since Gab or s rst pap er in One
advance was Mertz and Young s invention of zone plate holography in Zone plate
holography is a metho d of imaging incoherent radiation sources in three dimensions and so
has found many applications One application is radioisotop e imaging in nuclear medicine
It could also b e used to image the keV gammas emitted during Boron Cancer Therapy
Reconstruction algorithms develop ed for zone plate imaging of neutrons lightand
infra red can also b e used for gamma ray imaging All the previous PhD pro jects on
zone plate holography completed at the University of Birmingham have therefore b een
reviewed Although manytechniques have already b een tried the reconstruction still
needs improvement
This work suggests some new reconstruction techniques and tries them on computer
generated holograms and some of the limited numb er of actual gamma ray holograms
available at this p oint Takhar and George will continue the work on this pro ject using
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an ADACForte gamma camera and a GE Maxi Camera to record more holograms
The aim of their work will b e to develop gamma ray zone plate holography to the stage
where it can b e used to image patients undergoing BNCT
Chapter
COMPARISON OF ZONE PLATE
HOLOGRAPHY WITH
CONVENTIONAL HOLOGRAPHY
Holographyisavery broad eld and it is b eyond the limits of this pro ject to analyse
all di erenttyp es of hologram The typ e of hologram which is most similar to zone
plate holograms is an in line Gab or hologram and this is the typ e which will therefore b e
considered
In order to derive the equations for hologram formation and reconstruction approxi
mations from di raction theory must b e used These approximations will b e reviewed in
this Chapter As in Chapter the starting p oint of this analysis will b e the Huygens
Fresnel principle The Huygens Fresnel principle is as Born and Wolf state the
most p owerful approximate metho d and is adequate for the treatment of the ma jorityof
problems encountered in instrumental optics and can b e stated as follo ws
Every p ointonawavefrontserves as a source of spherical secondary wavelets
of the same frequency as the primary wave The optical eld at any p oint
beyond an obstruction is the sup erp osition of all suchwavelets reaching that
p oint
The Kirchho Fresnel di raction formula is a mathematical expression of the Huygens
Fresnel principle and is the formula used to analyse a large numb er of holographic systems
This formula is based on the Kirchho b oundary conditions which will b e given later in
equations and These b oundary conditions ignore edge e ects at an ap erture
and thus are only valid for ap ertures which are larger than a wavelength This equation
can therefore not b e used to describ e very near eld di raction In this case the Green s
function solution should b e used
Comparison of Zone Plate Holography with Conventional Holography
The Kirchho Fresnel di raction theory can b e applied to Gab or in line holography
and so is the theory which will b e discussed here An outline of the derivation of this
formula will rst b e given to show where the approximations are made and therefore
under which conditions the formula is valid The formation of a hologram of an opaque
pattern on a transparent sheet will then b e analysed This transparency will b e assumed
to b e illuminated by a planar coherentlight source and that it is parallel to b oth the
source and the lm on which the hologram is recorded This system will b e referred to as
a general holographic system as the sp eci c cases of Gab or in line holography and zone
plate holography can b e analysed using the relationships found for this more general
case Both these cases will b e analysed and nally factors a ecting the qualityofthe
reconstructed image will b e discussed
Derivation of the Kirchho Fresnel Di raction
Formula Pro of of the Huygens Fresnel Princi
ple
The Huygens Fresnel principle can b e proven byshowing that secondary waves from every
p oint on a primary wavefrontgive the same optical eld as the primary wavefrontwould
have done This is done in Born and Wolf by dividing the primary waveinto zones
making assumptions ab out the directional variation of the amplitude of the secondary
waves from each zone and approximately summing the contributions from the secondary
waves
A more thorough pro of of the Kirchho Fresnel di raction formula is also given by
Born and Wolf The physical signi cance of the assumptions made in this derivation is
easier to understand and so this is the metho d which will b e discussed here The waves
will b e assumed to b e mono chromatic
As describ ed in app endix B the amplitude of an electromagnetic wave x y z t
satis es the scalar wave equation
x y z t
r x y z t
c t
where c is the sp eed of light x y z t can b e separated into a time dep endent and a
space dep endent part If wehavemonochromatic light the time dep endent part of the
Comparison of Zone Plate Holography with Conventional Holography
wave just has one frequency and x y z t can b e written as
i t
x y z t x y z e
where x y z is the spatial part of the variation and is the frequency of the wave
Substituting this into equation gives
x y z where c r x y z
p
c
is the p ermeability of free space and is the p ermittivity of free space This is the
time indep endentwave equation Helmholtz equation
Consider a wave with space dep endentpart x y z It will satisfy the Helmholtz
equation
r k where k
c
Let V be a volume b ounded by a surface S and let P b e a p oint within it Let x y z
have continuous rst and second order partial derivatives within the volume and on the
surface If x y z isany other function which satis es the same continuity requirements
as x y z Green s theorem would give
ZZ Z ZZ
r r dV dS
n n
nn nn
V S
where denotes di erentiation along the inward normal to the surface S
n
nn
In particular if x y z also satis es the Helmholtz equation i e
r k
then substituting equations and into equation the left hand side of equation
vanishes giving
ZZ
dS
n n
nn nn
S
s
iks
e
One solution to the Helmholtz equation is x y z where s is the distance from
s
s
p oint P to the p oint x y z This gives a singularityats so P has to b e excluded
from the region of integration This is done by surrounding P by a sphere of radius The
integration is then b etween surface S and the surface S of the sphere This is illustrated
Comparison of Zone Plate Holography with Conventional Holography
n
V n P S’
S
Figure Region of integration
in Figure
Equation then b ecomes
ZZ ZZ
s s s s
iks iks iks iks
e e e e
dS dS
n s s n n s s n
nn s s nn nn s s nn
S S
and then
ZZ ZZ
s s s s
iks iks iks iks
e e e e
dS ik dS
n s s n s s s n
nn s s nn s s s nn
S S
ZZ
ik ik
e e
ik d
s
ss
where d isanelement of the solid angle and is the small radial displacementof S
from p oint P The integral over S is indep endentof and so the right hand side can b e
replaced by its limit as This limit is x y z This gives
ZZ
s s
iks iks
e e
x y z dS
n s s n
nn s s nn
S
which is one form of the integral theorem of Helmholtz and Kirchho
Kirchho considered the propagation of a wave from a p oint source P through an
opaque screen with an op ening in it and determined the light disturbance at a p oint
P x y z b ehind the screen This is illustrated in Figure The surface in Kirchho s
integral is formed by the op ening A a p ortion of the non illuminated side of the screen
B and a large sphere of radius R centred on P C
ZZ ZZ ZZ
s s
iks iks
e e
x y z dS
n s s n
nn s s nn
A B C
Kirchho then applied the following b oundary conditions
i
i
on A
n n
nn nn
on B
n
nn
Comparison of Zone Plate Holography with Conventional Holography
B
n R Q r s A P
P0 BC
Figure Di raction b ehind an ap erture surface used to derive the Kirchho Fresnel
di raction formula
i
where is the value relating to the incident eld These b oundary conditions arise from
the continuity of the electric and magnetic elds and the gradients of these elds across
op ening A and the opaque nature of surface B The following expressions are known for
i
i
r
as the p oint source at P emits spherical waves and the distance P Q r and
n
nn
r r
ikr i ikr
Ae Ae
i
n r
ik cos n r
r n r r
r nn r r
n r r n
cos n r is the cosine of the angle b etween the vector r and the normal to the op ening n
and A is a constant The wave eld on the spherical p ortion of the surface C remains
to b e considered If it is assumed that the source b egins to radiate at some time t t
at a time t the light will have reached a maximum distance from the source of c t t
If the radius R is chosen to b e further away from the source than this the light will not
havereached the surface region C and the integral over this region will vanish As the
integral over B also vanishes only the integral over A need therefore b e considered The
integral b ecomes
ZZ
r s
ik r s
e iA
n r n s
cos n r cos n s dS x y z
r s
rrss
A
which is the Kirchho Fresnel di raction formula is the wavelength of the waves
k
r s
As the integration moves over all the p oints in the ap erture r s will change bymany
r s
r s
r s
wavelengths so the factor e will oscillate rapidly If the distances r and s are however
n r n s
large compared to the dimensions of the ap erture then the factor cos n r cos n s
will not vary appreciably This factor may b e replaced by cos where is the angle
between the line P P and the normal to the screen If the distances b etween P and P
Comparison of Zone Plate Holography with Conventional Holography
r s
This may b e replaced by and the origin are called r and s resp ectively the factor
rs r s
rs r s
gives
ZZ
iA cos
r s
ik r s
x y z e dS
r s
r s r s
A
Any other surface which approximately lls the ap erture could havebeenchosen in
stead of A ab ove If a p ortion W of the incidentwave front from P is chosen instead
a p ortion C of a cone with its vertex at P can b e used to complete the surface If the
radius of curvature of the wave is su ciently large the integral over C can b e neglected
n r
n r If a new variable is This surface is illustrated in Figure On W cos n r
Q χ r 0 s P0 W C
P
Figure Di raction b ehind an ap erture surface used to prove the Huygens Fresnel
principle
r s
intro duced where r s equation b ecomes
ZZ
ik r ik s
i Ae e
x y z cos dS
r s
W
where r is the radius of curvature of the wave front This can b e written as
ZZ
ik r ik s
Ae e i
x y z K dS where K cos
r s
W
It can b een seen that every elementofthewave front W acts as a source of secondary
waves contributing to the eld at P The amplitude of these waves is determined by
K the inclination factor which describ es the variation with direction of the amplitude
of the secondary waves K is a maximum in the original direction of propagation and
decreases to zero when The factor i included in K can b e also written as
i
e and may b e accounted for by assuming that the secondary waves oscillate a quarter
also included in this of a p erio d out of phase with the primary wave The other factor
Comparison of Zone Plate Holography with Conventional Holography
times term may b e explained if the secondary waves are assumed to have an amplitude
that of the primary wave
To conclude the Huygens Fresnel principle can b e mathematically obtained from the
di erential wave equation when some assumptions are made These assumptions are that
Kirchho s b oundary conditions for the optical eld b ehind an ap erture apply and that
the dimensions of the ap erture are small compared to the distance of the source and
the p oint where the eld is to b e calculated from the ap erture Assumptions must also
b e made ab out the amplitude and phase of the secondary waves The amplitude of each
secondary wave is assumed to dep end up on the angle b etween the direction of propagation
of the primary wave and that of the secondary wave
Formation of a General Hologram
L M Soroko derived a formula for the di raction pattern pro duced by a coherent
planar light source a distance p b ehind a transparency in a plane a distance q in frontof
the transparency The transparency is a thin sheet of material for example photographic
lm with a varying transmittance function This variation or pattern could takethe
form of a picture or some writing and is the two dimensional ob ject of which the hologram
is to b e made Gab or s early holograms were of such ob jects Soroko s derivation will
b e summarised here
x
The amplitude of the di raction pattern u x will dep end up on the amplitude trans
mittance function of the transparency t as well as the source amplitude distribution
x
function s x x y and are vectors whichgive the p osition in the holo
gram source and transparency planes resp ectively These planes are p erp endicular to the
z axis This is illustrated in Figure
The extended source can b e considered as the sum of many coherent p oint sources
The waves from eachpoint source will b e di racted through the ap erture as describ ed by
the simpli ed version of the Kirchho Fresnel di raction formula given in equation
As the waves pass through a transparency rather than an ap erture the wave amplitude
will b e multiplied by the amplitude transmittance of the transparency t at each p oint
Comparison of Zone Plate Holography with Conventional Holography
η
ε β
s α y
t x p z u
q
Figure Relative p ositions of source transparency and hologram
The distances r s r and s are given by
p p r r
s x y q s x y q
For equation to hold the distances p and q must b e much greater than the distances
In this case the following approximations can b e used
r p p r p
p p p
x
x y x
s q q s q
q q q
Substituting into equation and using k each p oint in the source therefore has
the contribution
Z
p q
s cos
x
i i x
i
p q
t e x y q e e d
i pq
A
is the angle b etween the line b etween the source p oint and the p oint in the di raction
x
pattern and the normal to the transparency Sincep and q are much larger than and x
the approximation cos can b e used for all p oints in the source Integration over all
p oints in the source gives
Z Z
p q
s t
x
i x i
i
p q
x
e d d e e u x
i p q
and this can b e rewritten as
x x
x x p q
i i
i
p q
x x x
u x t x e e s x e
i pq
where denotes a convolution The amplitude of the light in the hologram plane is
therefore given by the following three consecutive op erations
Comparison of Zone Plate Holography with Conventional Holography
x
x
i
p
x
Convolution of the light source amplitude distribution and a function F x e
Multiplication of this result by the amplitude transmittance function of the trans
parency
x
x
i
q
x
Convolution of the result with a second function F x e
This is illustrated in Figure
source s(x) object detector incident waves emerging waves waves recorded by the detector s(x) x F1 (x) [s(x) x F1 (x)] x t(x) {[s(x) x F 1 (x)] x t(x)} x F2 (x)
t(x) = amplitude transmittance function of the transparency (object) F (x), F (x) = Fresnel functions
1 2
Figure Three consecutive op erations forming a general hologram
Gab or In Line Holography of a Two Dimensional
Ob ject
Gab or made a hologram of a transparency which had a clear background and opaque
writing The writing was the two dimensional ob ject of which the hologram was made He
illuminated the transparency with a collimated b eam of mono chromatic light and recorded
its Fresnel di raction pattern on a photographic plate The lm was develop ed and
the image reconstructed by shining the mono chromatic lightback through it The writing
was easily readable in the reconstructed image Gab or won the Nob el Prize for this work
which he published in a pap er in The relationship b etween the transmittance
of a develop ed hologram and the intensityoflight the lm was exp osed to is given in
app endix C
In the case of Gab or in line holography the light source is collimated and so emits
term can b e replaced by plane waves therefore r p and s over all of The
p
Comparison of Zone Plate Holography with Conventional Holography
A the amplitude of the mono chromatic plane waves Equation then b ecomes
Z
q p
x
x i
i i
q
x
e u x Ae d t e
i q
which can b e written as
Z
p q
C
x
i x i
x
u x and t e d where C Ae
i q
The amplitude of the light in the hologram plane is therefore given bytheconvolution of
x
the amplitude transmittance function of the transparency with the Fresnel function z x
C
x
i x
x x x x
e u x t x z x where z x
i
F
where The Fresnel function has the Fourier transform Z
i
F
Z ie
and Sorokoshowed that equation can b e written in four forms as
x y
follows
C
x x x
u x t x z x rst form
i
Z
C
x
F F i xx
x
T Z e d second form u x
i
Z
C
F
x
i x
x x
x t z e d third form u x Z
C
F
F F
x x x x
x fourth form x Z u x x T Z
F
x
where T istheFourier transform of t x These di erent expressions can b e used
to analyse the di raction pattern First however some characteristic lengths need to b e
intro duced The Fresnel function can b e approximated as a series of concentric circles
AFresnel zone plate can b e used to di ract light and the size of the rst Fresnel zone
is a measure of di raction AFresnel zone plate is a series of alternating transparent
and opaque concentric rings which are called zones Rays travelling to the fo cal p ointa
distance q b ehind the plate via a p oint on the outer edge of the mth zone will reachthe
m
fo cal p oint a out of phase by with a ray that passed through the centre of the plate
The radius of the mth zone R can b e shown to b e
m
p
R m q
m
Comparison of Zone Plate Holography with Conventional Holography
The radius of the central Fresnel zone l is therefore given by
p
l q
The size of the nest detail in the ob ject is related to the maximum spatial frequency
in the spatial structure of the ob ject by
The following limiting cases will b e considered
The size of the ob ject opaque regions on the transparency is much less than the
size of the central Fresnel zone
The size of the central Fresnel zone is much less than the size of the smallest detail
in the ob ject
The size of the rst Fresnel zone exceeds the size of the smallest detail in the ob ject
but this zone is less than the ob ject itself
The Central Fresnel Zone is much larger than the Ob ject
If the size of the ob ject is a the following relationship can b e used
a l so a and z
The third representation of equation then b ecomes
Z
C
F
x
i x
x x
u x x t e Z x d
which can b e written as
F
F
x x x
x T x Z u x C
F
x
i x
x
x ie wehave and as Z
C
F
x x x
u x z x T x
i
In this case therefore the di raction pattern corresp onds to the Fourier transform of the
transmittance function of the transparency multiplied bytheFresnel function Another
Comparison of Zone Plate Holography with Conventional Holography
x
notably simpler metho d to obtain this formula is to rst expand the terms in x in
equation giving
Z
C
x x
i x i x i
x
u x t e e e d
i
Born and Wolf state that when the size of the ap erture ob ject is so small that
quadratic and higher order terms in can b e neglected This is referred to as Fraunhofer
di raction Fresnel di raction describ es the case when the quadratic terms cannot b e
neglected Neglecting the quadratic terms in in equation gives
Z
C
x x
i x i x
x
e t e d u x
i
It can b e clearly seen that this is the same result as that in equation Fraunhofer
di raction is also called far eld di raction and the resulting hologram is called a Fourier
hologram Fourier holograms are mainly used to holographically store information and in
digital optical computing
The Central Fresnel Zone is much smaller than the Small
est Detail in the Ob ject
The following relationship applies here
l or
F
This means that the maximum frequency in Z ismuch smaller than and so
F
Z i The second representation of equation can therefore b e simpli ed
Z Z
i C
x x
F i x F i xx
x x
u x x C T e d Ct x T e d
i
The di raction pattern is simply the amplitude transmittance function of the transparency
multiplied by a factor C which dep ends up on the amplitude and phase of the light source
This is known as the geometric shadow region Holograms recorded in this region are called
shadowgrams As will b e discussed in Section zone plate holograms are shadowgrams
Shadowgrams are also recorded in co ded mask imaging techniques
Comparison of Zone Plate Holography with Conventional Holography
The Central Fresnel Zone is Larger than the Smallest De
tail in the Ob ject but Smaller than the Ob ject Itself
The following relationship applies in this case
l a
and equation is still valid This is the general Fresnel di raction case Gab or recorded
his holograms in this region As was stated in section this is called Fresnel di raction
and is also called near eld di raction Holograms recorded in this region are called Fres
nel holograms Most conventional holograms used to record and display three dimensional
images are of this typ e
Gab or also recorded his holograms in this region The light source he used wasamer
cury arc lamp This lamp pro duces green lightofwavelength nm as well as blue light
of wavelength nm If for example the green lightwas used and the photographic
plate was placed metre b ehind the transparency the size of the central Fresnel zone
would b e
p
m mm l
So the smallest detail in the ob ject would have to b e less than mm and the ob ject
itself would have to b e bigger than this If for example the transparency was ab out cm
square and had writing on it it would satisfy these conditions The exact parameters
used for Gab or s exp eriment are not known here but with this wavelength of light it can
b e seen that the hologram was probably in this region
Zone Plate HolographyofTwo Dimensional Sources
Figure illustrates the equipment needed for gamma ray holography if the transparency
was replaced with a sintered tungsten zone plate and the photographic lm with a gamma
camera with the collimator removed The source would of course b e a gamma ray
source instead of a coherentlight source Gamma rays have frequencies b etween and
Hz This means that the longest wavelength would b e nm Ifthe
gamma camera was metre b ehind the zone plate for example the size of the central
Comparison of Zone Plate Holography with Conventional Holography
Fresnel zone of the waves would b e
p
m m l
One p ossible application of this technique is to nuclear medicine Here the gamma ray
sources would b e several centimetres in size So even if the sources pro duced coherent
rays the wavelength of gamma rays would de nitely put any exp eriment in the geometric
shadow region as describ ed in Section The fact that gamma ray sources do not
pro duce coherentwaves further ensures that no di raction is seen
Each p oint in the source casts a shadow of the zone plate on the detector The
hologram is the sum of these shadows First a planar source will b e considered This
source will b e a distance p b ehind the zone plate as in Figure It will b e assumed that
the shadow is the same size and shap e for p oint sources at the same distance p b ehind the
zone plate i e the shadow is shift invariantforpoints in the same plane For this to b e
the case the distances p and q have to b e large compared with the size of the zone plate
and the distance of the p oint source from the central axis of the imaging system which
passes through the centre of the zone plate and of the detector The zone plate also must
b e assumed to b e in nitely thin Using the same notation as b efore each shadowwould
b e magni ed by a factor M where
q
M
p
A p oint source with intensity S at p osition will give a shadow the centre of
i i
i i ii i i
x
whichisatx x y where
i
ii i i
q q
x L y L
i i i i i i
p p
A de nition of the intensity of a source of electromagnetic waves is given in app endix B
If the zone plate has an intensity transmittance function T the detector is of size D
x
wide by D high and p q D the source will cast a shadowofintensity U x given by
i
x y D
S T L L U x y
i i i i i i
p q M M
D
The factor is an approximation of the fraction of the radiation which is emitted
p q
by the isotropic source which is recorded by the detector As p q D the curvature of
a sphere of radius p q over the detector area is small enough to b e neglected
Comparison of Zone Plate Holography with Conventional Holography
x
If the extended source is seen as the sum of many p oint sources the shadow U x
cast by the extended source of intensity S is the integral over all these p oint sources
Z
x
x D
x
S T U x L d
p q M
Two new variables and will nowbeintro duced where L and L Substituting
L into equation gives
Z
x
D x
x
S U x T d
p q L M
The resulting shadow is therefore a convolution b etween the source distribution magni ed
by a factor L and the intensity transmittance of the zone plate magni ed by a factor M
x x
x D x
x
S U x T
p q L M
Lo oking at equation note that as p and q are b oth p ositivenumbers L is negative
This means that the source distribution is ipp ed horizontally and vertically b efore it is
scaled and convolved with the zone plate pattern
Comparison of Gab or and Zone Plate Holograms
of Single Points
Comparing equation to it can b e seen that the amplitude of a Gab or holo
gram and the intensity of a zone plate hologram of two dimensional sources can b oth
b e approximated as the convolution b etween two functions For simplicity the constant
multiplication factors will b e ignored here The amplitude of a Gab or hologram is the
convolution b etween the amplitude transmittance of the transparency the ob ject and
aFresnel function whereas the intensity of a zone plate hologram is the convolution b e
tween the source distribution and the intensity transmittance of the zone plate If the
intensity transmittance of the zone plate was also a Fresnel function the resulting zone
plate hologram would b e similar to a Gab or hologram of an equivalent coherentlight
source As is stated in app endix C intensity transmittance is the square of the amplitude
transmittance
If conventional holographic reconstruction techniques are to b e used the ideal zone
plate to use would give a hologram whichwas as similar to a conventional hologram as
Comparison of Zone Plate Holography with Conventional Holography
p ossible It would therefore haveanintensity transmittance whichwas a Fresnel function
x
x i
zp
x
T x e
zp
Where is a constant This function however has b oth real and imaginary parts The
zp
intensity transmittance function can only b e real and havea value b etween opaque
and transparent The b est p ossible alternative is therefore
h n oi
x
x i
zp
x x
cos e T x x
zp
p p
x x
where ff g is the real part of a function f cos x has zeros at x
zp
p
p
p p
p
n
r r n r and so the transmittance function can r
b e written as
x
xx
x
T x cos
r
a zone plate with this transmittance function is called a Gab or zone plate
Zone Plate Hologram of a Point Gamma Ray Source
If the source is a p oint source of unit intensity at p oint the source
distribution can b e describ ed by a delta function S If a Gab or zone plate is
h i
x
xx
x
used T x cos and equation gives the intensity of the hologram as
r
x
xx D
x
cos U x
p q M r
This can b e written as
p
D M r
x x
where cos x U x
zp
zp zp zp
p q Mr
A zone plate hologram of a p oint source therefore contains a constantbackground term
minus the real part of a Fresnel function of spatial frequency
zp
Gab or Hologram of an Opaque Point in the Middle of an
Otherwise TransparentTransparency
In equation the amplitude of the hologram was given as
Z
p q
C
x
i x i
x
and u x t e d where C Ae
i q
Comparison of Zone Plate Holography with Conventional Holography
If the transparency were completely transparent i e t over all of equation
would b ecome
Z
p q
C
x
i x i
x
u x e d C Ae
i
If the transparency contains a single opaque p ointat the amplitude of the hologram
would no longer contain the contribution from this p oint The p oint will b e assumed to
b e square and of size and that is so small that the waves from one side of the
p p p
source are the same as those from the other side This gives
p
x
i x
x
e u x C
i
The intensity of the hologram is therefore
C
p p
x x x x
U x u x u x sin x
p
A Gab or hologram of a p oint therefore contains a constant background term plus the
imaginary part of a Fresnel function of spatial frequency w
Optical Convolution Image Reconstruction
Both typ es of hologram can b e reconstructed in the wayinwhich a Gab or hologram is
formed A Gab or hologram is already recorded on photographic lm the lm then has to
b e develop ed The resulting amplitude transmittance function of the lm will b e assumed
to b e linearly related to the intensity recorded If is a constantwhich dep ends up on
the way the lm was develop ed the amplitude transmittance function will b e
where sin t G G
G
C
p p
and G G C
The relationship b etween the amplitude transmittance of photographic lm and the in
tensity of radiation the lm was exp osed to is discussed in app endix C In zone plate
holography the hologram is recorded using a gamma camera It is then stored as two
dimensional pixel array on a computer This can b e printed out onto a transparency
Again it will b e assumed that the resulting amplitude transmittance function is linearly
Comparison of Zone Plate Holography with Conventional Holography
related to the intensity recorded will b e intro duced as the constant of linearity
t Z Z cos where
Z
zp zp
D D M r
Z and Z
p q p q
As was shown in Section shining a b eam of plane coherent lightwaves through a
transparency gives a pattern in the hologram plane equivalent to the convolution b etween
the amplitude transmittance of the transparency and a Fresnel function The rst form
of equation gives
C C
r r
x x
i i x x
r r
x x x x
and u x t x e t x e u x
Z G Z G
r r
i i
where C and are constants dep endent on the wavelength of the light and the separation
r r
of the hologram and the detector
p q
r r
i
r
C A e
r r
r
q
r r
is the wavelength of the light used to reconstruct the image q is the hologram to
r r
detector distance and p is the source to hologram distance As plane waves are used p
r r
just gives the phase of the waves as they reach the hologram and have no e ect on the
intensity of the reconstructed image A is the amplitude of the plane waves
r
The convolution theorem states that the Fourier transform of the convolution of two
functions is equal to the pro duct of their individual Fourier transforms The Fresnel
function of spatial frequency has the following Fourier transform
r
i
F
r
Z ie
where From this the following Fourier transform pairs can b e deduced
x y
n o
i
x
i x
r
r
x
therefore cos x sin e ie
r r r
r
n o
i
x
i x
r
r
x
e ie therefore sin x cos
r r r
r
where ff g is the real part and ff g the imaginary part of a function f
To summarise expressions for the amplitude transmittance of b oth Gab or and zone
plate holograms of on axis p oint ob jects have b een derived It has b een shown that if
Comparison of Zone Plate Holography with Conventional Holography
the image is reconstructed by shining lightback through the hologram the image seen
at a given distance b ehind the transparency is given by the convolution of the amplitude
x
i x
r
x
transmittance function of the hologram with a Fresnel function z x e
r
The Fourier transform of the convolution of two functions is equal to the pro duct of
their individual Fourier transforms The Fourier transforms of the real and imaginary
parts of a Fresnel function have b een found and can b e used to evaluate the reconstructed
image from b oth typ es of hologram Expressions for the reconstructed image will b e found
rst when and andthenwhen and In the rst case
r r zp r r zp
which will b e called matched convolution an image of the p oint should b e obtained In
the second case unmatched convolution there should b e no discernible image
Matched Convolution
Gab or Hologram
r
First a Gab or hologram of a p ointwillbeinvestigated It will b e assumed that the
hes to in the x and y directions The constant term will b e ignored hologram stretc
as this will just give a constant term in the reconstructed image An expression will b e
found for the contribution to the reconstructed image from the sin term when
r
x
The aim is to nd the amplitude of the reconstructed image u x
G
C
r
x
i x
x x
where C u x C sin x e G
g G g
i
Taking the Fourier transform of each side gives
F
sin i cos U C cos
g
G
sin i cos C
g
then taking the inverse Fourier transform gives the following expression for the amplitude
of the reconstructed image
x x x
sin u x C cos x i x
G g
x
where x is the Dirac delta function
x
The imaginary part of the reconstructed image therefore contains a p ointatx
p
as well as the imaginary part of a Fresnel function of frequency The real part just
Comparison of Zone Plate Holography with Conventional Holography
p
If an optical reconstruction contains a real part of a Fresnel function also of frequency
is used only the intensity will b e recorded This is given by
x x x
x x U x C sin cos
G
g
C
g
x
The intensity of the reconstructed image therefore contains just a p ointatx aswell
as a constantbackground term as would b e exp ected
Zone Plate Hologram
r zp
The analysis here is very similar to that for the Gab or hologram Again the constant
background term will b e ignored The aim is to nd the amplitude of the reconstructed
x
image u x
Z
C
r
x
x i
zp
x x
Z where C u x C cos x e
z Z z
zp zp zp
i
Taking the Fourier transform of each side gives
F
sin C U i sin cos
z
Z
zp zp zp
C cos i sin
z
zp zp
then taking the inverse Fourier transform gives the following expression for the amplitude
of the reconstructed image
zp zp
x x x
x i x sin cos u x C
Z z
zp zp
The intensity is therefore given by
x
U x
Z
zp zp
x x
C cos sin x x
z zp zp
C
z zp
x
Again the intensity of the reconstructed image therefore contains just a p ointatx
as well as a constantbackground term
Comparison of Zone Plate Holography with Conventional Holography
Unmatched Convolution
Gab or Hologram
r
The frequency of the Fresnel function in a plane a distance q b ehind the transparency is
r
q
As discussed in the previous Section when the reconstructed given by
r r
q
r
image has a p oint in the correct p osition and a constantbackground Here a formula will
b e derived for the reconstructed image when that is in a plane other than the
r
fo cal plane As in Section the aim is to calculate the amplitude of the reconstructed
x
image u x
G
x
x i
r
x x
u x C sin x e
G g
r
Taking Fourier transforms of b oth sides gives
F
U C cos sin i cos cos
g
G
r r
sin C sin
g
r r
i cos cos
r r
Taking the inverse Fourier transform gives the following expression for the amplitude of
the reconstructed image
x
u x
G
r r r r
x x
C cos x cos x
g
r r r r
r r r r
x x
sin x sin x i
r r r r
i h
r r x x
ib x ia x
g g
b where a b e C a e
g g g g g
r r
The intensity of the image is therefore given by
x
U C a b a b cos a b x
G g g g g g
g g
r r r r
x
cos x C
g
r r r r
which as exp ected has a singularitywhen
r
Comparison of Zone Plate Holography with Conventional Holography
Zone Plate Hologram
r zp
The same metho d as b efore gives the following expression for the amplitude of the recon
x
structed image u x
Z
x
u x
Z
r zp r zp r zp r zp
x x
sin x sin x C
z
r zp r zp r zp r zp
r zp r zp r zp r zp
x x
i cos x cos x
r zp r zp r zp r zp
i h
r zp r zp
x x
ib x ia x
z z
ib e where a b C ia e
z z z z z
r zp r zp
and the intensity is therefore
x
U C a b a b cos a b x
G z z z z z
z z
r r r zp zp zp r zp
x
cos x C
z
zp zp zp zp r r r r
which again has a singularity when
zp r
To compare the variation in intensity of the reconstructed image with hologram to
detector image an example Gab or hologram will b e analysed The hologram will b e
assumed to have b een recorded with a transparency to photographic lm separation of
q m The wavelength of light will b e set as nm which is the wavelength of light
emitted by the mercury arc lamp Gab or used in his exp eriments These twovalues give
m Thewavelength of light is used in the reconstruction will also b e set to
nm The constant C will b e set to C To compare a zone plate hologram
g g
to this the parameters and C will b e set equal to the equivalent parameters for the
zp z
Gab or hologram that is and C C These may b e arti cial values for a zone
zp z g
plate hologram but the recorded hologram w ould have to b e reduced in size to di ract
light if the image were to b e reconstructed optically This would give the same values for
the constants a and b in b oth cases i e a a a and b b b
g z g z
Figure shows the variation of the constants a and b as the hologram to detector
distance varies from m to m in intervals of m b tends to in nityas q m
r
x
Figure shows howtheintensityatx varies with hologram to detector distance
for b oth typ es of hologram The intensity tends to in nity in b oth cases as q m
r
Comparison of Zone Plate Holography with Conventional Holography
a b 2.0 200.0
100.0 1.5
0.0
1.0 -100.0
0.5 -200.0 0.0 0.5 1.0 1.5 2.0
Hologram to detector distance (m)
Figure Variation in the values of a and b with hologram to detector distance
4e+02 4e+04 4e+06 Gabor Gabor Gabor zone plate zone plate zone plate
3e+02 3e+04 3e+06
2e+02 2e+04 2e+06 Intensity Intensity Intensity
1e+02 1e+04 1e+06
0e+00 0e+00 0e+00 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
Hologram to detector distance (m) Hologram to detector distance (m) Hologram to detector distance (m)
a interval m b interval m c interval m
Figure Variation of on axis intensity with hologram to detector distance for ideal
Gab or and zone plate holograms of a single p oint
Values have b een calculated at intervals of m Figure a m Figure b
and m Figure c The largest values of intensity are therefore at m and
m m and m and m and m resp ectively A real hologram do es not
stretch to in nityinthe x and y directions however so the intensity in the fo cal plane
would not b e in nity in this case
Comparison of Holograms of more than One Point
Zone Plate Hologram of twoPoint Gamma Ray Sources
Referring back to equations and a Gab or zone plate will give a hologram with
the following intensity
Z
M r D
x x
U x x LM d cos S
zp zp zp
p q
Comparison of Zone Plate Holography with Conventional Holography
o o
o and b oth at the same o and o A hologram of two p oint gamma ray sources at o
distance p from the zone plate will b e given by
D M r
x x o x o
U x cos x LM o cos x LM o
zp zp zp zp zp
p q
If this hologram is again printed out onto a transparency its transmittance function will
be
o o
Z o o t Z Z cos LM o cos LM o
Z
zp zp zp zp
where Z and Z are as in equation This is simply the sup erp osition of holograms
from each individual p oint
Gab or Hologram of two Opaque Points
o o
Referring back to Section a hologram of two opaque p oints at o and o
in an otherwise transparent transparency would have the following amplitude
p p
o o x x
i x o i x o
x
u x C e e
i i
where is the size of each p oint as b efore The corresp onding intensitywould b e
p
C
p
o o x x x
sin x o sin x o U x
p p
p
o o x x
cos x o x o
The develop ed lm would therefore have the following transmittance function
o o
t G G sin o G sin o
G
C
p
o o x o o
G cos o o o o x o o o o C where G
G and G are as in equation The second and third terms corresp ond to holograms
of the individual p oints but the fourth term comes from interference b etween the contri
butions from b oth p oints The relative magnitude of this contribution compared to those
from the individual p oints is given by
C
G
p p p
F
G C q
p
Comparison of Zone Plate Holography with Conventional Holography
If as in Section the wavelength of the light is set to nm and the distance
of the image b ehind the hologram to q m this gives F Therefore as
p
long as eachpoint has mm this contribution will b e smaller than that from the
p
individual p oints
The interference term has a cosine variation The frequency of this variation is given
o o o o
o m So o o o m Using the same value for this gives o o o by o
o o
y m The size o m for example zeros in this term will b e separated b o o if o
p
of the central Fresnel zone l q for the terms from individual p oints will b e m
Unlike a zone plate hologram which is simply the sup erp osition of the holograms
from eachpoint in the source a Gab or hologram also contains terms from the mutual
interference of waves from di erent p oints in the source This Section showed the terms for
two p oints but holograms of more than two p oints or extended ob jects will contain further
terms The magnitude and frequency of these contributions compared to the magnitudes
of the Fresnel functions from the individual p oints will dep end up on the separation of the
p oints as well as the frequency of light used and the distance of the transparency from
the photographic plate
Investigations could b e made to determine whether adding some of the mutual inter
ference terms to a zone plate hologram to make it more likeanequivalentconventional
hologram would improve the image reconstruction As these terms dep end up on the ob
ject of which the hologram was made this would have to b e an iterative pro cedure A rst
approximation of the ob ject would have to b e found initially the corresp onding mutual
interference terms could then b e calculated and used to improve the image This pro cess
could then b e rep eated until the image quality reached an acceptable level
Hologram of a Three Dimensional Ob ject
Zone Plate Holography
The factors M and L whichwere given in equations and dep end up on the source
to zone plate distance p as well as the xed zone plate to detector distance q A zone plate
direction as well as in directions p erp endicular hologram of a source extended in the z
to this axis will havevarying values of M and L Instead of M and L M p and
Comparison of Zone Plate Holography with Conventional Holography
L p will b e used as these values are functions of p
An extra complication is that a gamma ray from a p ointwhich is not on the surface
nearest to the zone plate will have to travel through the rest of the source b efore travelling
through the air to the zone plate The attenuation of this gamma rayby the source must
therefore b e considered The linear attenuation co e cient for gamma rays is de ned
l
by
t
l
I I e
where I is the numb er of transmitted photons I is the numb er without an absorb er and t
is the thickness of the absorb er This can also b e given as the mass attenuation co e cient
or the mean free path where
m mf
l
and
mf m
l
and is the density of the medium Intensity of radiation is the rate of transfer of energy
across unit areas by the radiation For a mono energetic source this is prop ortional to the
numb er of photons b eing emitted p er second I in equation can therefore b e replaced
by s the intensity after travelling through the source and I by s theintensity
a b
b efore travelling through the source
m
One commonly used radiopharmaceutical is Tc keV gamma rays The
m
e ect of this attenuation can b e estimated using this example Tc is usually obtained
from a Mo Tc generator as a salt dissolved in water In nuclear medicine this is then
injected into a p erson for imaging purp oses Mass attenuation co e cients for keV
gamma rays in tissue as well as in water are therefore given in Table The densityof
water is kgm of air kgm and the density of tissue will b e assumed
tobethesameasthatofwater Linear attenuation co e cients were calculated using
these values and can b e seen in Table
m
Radioisotop e Tc
Gamma ray energy keV
Air
H O
Tissue
kg Table Mass attenuation co e cients m
Comparison of Zone Plate Holography with Conventional Holography
m
Radioisotop e Tc
Gamma ray energy keV
Air
H O
Tissue
Table Linear attenuation co e cients m
m
If a phantom contains Tc and is m thick for example keV gamma rays will
b e attenuated to a factor e times the original intensity in travelling from
the back to the front of the phantom This means that of the gamma rayintensity
will b e lost If the phantom is cm thick is lost If keV gamma rays travel
through m of air only will b e lost
An expression for the resulting hologram is given by equation but integrated
over all distances p in the z direction of sources from the zone plate The attenuation of
gamma rays which pass through the source also need to b e considered and this will b e rep
resented by a factor p S p which dep ends up on the entire source distribution
S p as well as the p osition of the p oint p
Z Z
x
D x
x
U x L p d dp p S p S p T
p q M p
The attenuation term adds an extra degree of complexity to the problem of reconstructing
an image of the source It implies that the whole of the source distribution needs to b e
known b efore an expression for the contribution from one p oint in the source can b e found
In practice an image of the source could b e calculated rst neglecting attenuation so
the lo cation of the source could b e found The attenuation could b e included in a second
stage to correct the amplitude of the image This approach is often used in SPECT which
is a well established imaging technique for nuclear medicine
Another factor to consider is the scaling of images from zone plate holograms in the z
direction if they are reconstructed optically Aswas shown in Section should b e
r
set equal to for an image of a p oint to b e obtained If a Gab or zone plate is used
zp r
and are given by
zp
p
r
p
and
r zp
q p q r
r r
Comparison of Zone Plate Holography with Conventional Holography
where is the wavelength of the light used in the reconstruction r is the rst zone
r
radius of the zone plate and q is the p oint where the reconstructed image will b e found
r
So when q is given by
r zp r
p q r
q
r
p
r
q is therefore not a linear function of q This will pro duce an undesirable scaling e ect
r
in the z direction in the reconstructed image
Gab or In Line Holography
Gab or used the exp erimental arrangement describ ed in Section to make holograms of
two dimensional ob jects In order to compare this technique to zone plate holographyfor
three dimensional imaging a series of transparencies will have to b e used as the mo del of
the ob ject to b e imaged The transparencies will b e assumed not to refract or attenuate
the lightwaves travelling through it and the opaque regions will completely absorb any
light incident on them
The hologram from a series of N thin transparencies will b e considered Each trans
parency will b e separated a distance q from its neighb ours with the closest one a
distance q from a sheet of photographic lm and the furthest one at a distance
min
q q q N The transparencies are illuminated by coherentplanewaves
max min
This is illustrated in Figure In Section the amplitude of the light in the hologram
plane for a general hologram was shown to b e given by the convolution of the light source
distribution with one Fresnel function multiplication of this by the amplitude transmit
tance function of the transparency and then convolution of this with a second Fresnel
function Consider the case when N i e there are only two transparencies If the
p
i
amplitude of the plane waves at the rst transparency is given by Ae and the waves
emerging from the rst transparency are seen as a source for a hologram of the second
transparency referring back to equation the amplitude of the light in the hologram
plane will b e given by
x
x
q q
x
x
p
min
i
i
i i
q
q
min
x x x
t x q e e t x q q e u x Ae
min min
i qq
min
Comparison of Zone Plate Holography with Conventional Holography
plane waves N transparencies each separated from its neighbours by distanceΔ q
β
α y
x
z Δq u Δq Δ
q qmin
Figure Relative p ositions of the transparencies and the hologram
For N transparencies this b ecomes
q q N
p
min
i i
x x
u x Ae e t x q q N
min
N
i q q
min
x x x
x x x
i i i
q q q
x x
e t x q q N e t xx q q N e
min min
x
x
x
x
i
i
q
q
min
x x
t x q e t x q q e
min min
It would b e interesting to see what happ ens as q andN but in this case
Kirchho s di raction formula would no longer b e valid The intensity whichwillbe
x x x
recorded on the photographic lm is again given by U x fu x g fu x g Itis
obvious that unlike zone plate holography this is not simply the sum of the contributions
from each plane parallel to the detector
As mentioned b efore however this is not a typ e of hologram which has b een used It
is impractical as lightwould be stopped by the ob ject pattern on one transparency b efore
it reached the next transparency and little detail ab out the side nearest the detector
would b e obtained Leith and Upatnieks o axis holograms whichgive go o d quality
images involved scattering an ob ject b eam and an angled reference b eam o the front
of a solid ob ject Information from the side of the ob ject viewed from the direction of
the photographic plate is then recorded This technique however only gives holograms
which contain information ab out the surface of the ob ject Zone plate holographyistruly
three dimensional with p oints throughout the volume of a three dimensional radiation
source con tributing to the hologram
Comparison of Zone Plate Holography with Conventional Holography
Summary
In this Chapter zone plate and conventional holographywere discussed Born and Wolf s
derivation of the Kirchho Fresnel di raction formula was summarised and Soroko s deriva
tion of the amplitude of a general hologram was reviewed Using these formulae a par
ticular typ e of conventional holography Gab or in line holography was analysed in detail
and compared to zone plate holography Holograms of single p oints were shown to b e
similar Equations for the optical image reconstruction of holograms of single p oints were
derived and the origins of the signal and noise terms were shown The similarities and
di erences b etween zone plate and Gab or holograms of two p oints were derived and an
attempt was made to quantify these di erences It was concluded that these di erences
will haveaneven greater e ect in holograms of extended ob jects sources Holograms of
three dimensional ob jects sources were compared and the di erences were indicated The
e ect of attenuation of gamma rays by the source material was discussed and an attempt
was made to nd the magnitude of this e ect An undesirable scaling e ect in the z
direction of an optically reconstructed zone plate hologram was found
Chapter
INHERENT ERRORS
ASSOCIATED WITH GAMMA
RAY ZONE PLATE
HOLOGRAPHY
The equipment used for zone plate holographywas describ ed in Section A gamma
camera is used to record the hologram and this a ects the quality of the hologram The
pixel size must also b e small enough for the frequencies in the hologram to b e correctly
recorded The image is reconstructed using a computer The accuracy of the digitisation
and of the oating p oint arithmetic will also have an e ect Radioactivedecay is a random
pro cess and therefore the quality of the hologram will improve with the time for which
it is recorded In this Chapter an attempt will b e made to analyse and quantify the e ect
of each of these factors on the quality of a hologram
Gamma Camera
The comp onents of a gamma camera detector head can b e seen in Figure A lead
collimator is usually attached The most common typ e of collimator is a parallel hole
X -> x-coordinate Y -> y-coordinate XYZZ -> pulse height logic signal protective shield or bowl
pulse arithmetic circuit
array of photomultiplier tubes
transparent light guide
sodium iodide [Na(Tl)] crystal
collimator
Figure Diagram of a gamma camera detector head
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
collimator and this ensures that only gamma rays incidentatright angles to the detector
head will reach the crystal The crystal is thallium activated so dium io dide NaI Tl
When a gamma ray is absorb ed by the crystal molecules in the crystal are excited and
then decay back to their ground state by emitting visible light The total light emitted
by NaI Tl dep ends up on the gamma ray energy absorb ed This relationship is close to
linear over most of the signi cant energy range
The crystal is optically coupled to an array of photomultiplier PM tub es EachPM
tub e converts the scintillation light from the crystal into an electrical pulse The total
charge in the pulse is prop ortional to the mean numb er of photons incident on the tub e
A pulse arithmetic circuit then calculates the p osition X Y and energy Z of the gamma
ray which caused the pulse using the signals from all the photomultiplier tub es
The Ohio nuclear inc series radioisotop e camera for example contains pho
tomultiplier tub es and has a cm diameter crystal which is cm thick Despite the
small numb er of photomultiplier tub es the pulse arithmetic circuit can calculate the p o
sition of a gamma ray hitting the crystal to a high degree of accuracy The FWHM of the
line spread function at the surface of the crystal is cm for keV gammas The
spatial resolution for the cameras whichhave b een used in work on gamma ray holography
at the University of Birmingham are shown in Table
manufacturer mo del spatial resolution FWHM m
Ohio Nuclear inc Series radioisotop e camera
GE Maxicamera
ADAC Vertex gamma camera
Forte gamma camera
Toshiba GCA A
Table Spatial resolution of example gamma cameras
Gamma rays react with the crystal molecules in twomainways photo electric absorp
tion and Compton scattering Photo electric absorption results in the whole of the gamma
ray energy b eing absorb ed by the crystal The emitted light and hence the pulse amplitude
will b e prop ortional to the gamma ray energy Pulses from this interaction contribute to
a photop eak in the pulse height sp ectrum as can b e seen in Figure Compton scat
tering results in the partial absorption of gamma ray energy dep ending up on the angle
of scatter The pulse amplitude in this case will dep end up on the fraction of the energy
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
absorb ed This gives rise to the Compton continuum in the pulse height sp ectrum The
p ositions at which the photons whichwere absorb ed directly dep end up on the p osition of
the source The p ositions at which photons were scattered and the fraction of energy that
was absorb ed in eachcaseprovides little useful information The gamma camera allows
a window to b e placed over the photop eak Only the pulses within this window are then
accepted and recorded Rew studied the e ect of this window on reconstructed image
quality in her PhD thesis
15000
noise
10000 energy window
photopeak
Number of pulses 5000 backscatter peak Compton continuum Compton edge 0 0 128 256 384 512 640 768 896 1024
Channel number (pulse height)
Figure Example pulse height sp ectrum from a NaI Tl scintillation detector
The pulses from the photop eak are output as a list of X Y p ositions The data
is stored in a square pixel arraywhich usually contains b etween and
pixels When a gamma ray is detected the closest pixel to its calculated p osition is
incremented by one Each arrayelement therefore contains the numb er of gamma rays
which hit the detector crystal at that pixel p osition The pixel array forms an image of
the gamma rayintensity incident on the detector
Floating Point Arithmetic
A digital computer is only capable of representing a nite set of numb ers The computers
used in this pro ject are Sun Solaris SPARC Ultra and use the IEEE binary standard
to store real numb ers This standard sp eci es oating p oint representations for single and
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
double precision numb ers In general the oating p oint representation can b e written as
follows
p e
d d d d
p i
d d d is called the signi cand and has p digits is the base and is assumed
p
to b e even and e is the exp onent The values of these parameters for single and double
precision IEEE binary standard numb ers are shown in Table The numb er of bits
required to store each part of the numb er are also shown An extra bit is needed to store
the sign p ositive or negative of the numb er The exp onent is biased In the case of
single precision this means that the exp onentisgiven by the stored number minus
In double precision numb ers is subtracted Floating p ointnumb ers are usually
normalised This means that the most signi cant bit of the signi cand d is non zero
is for normalised numb ers and there is no The IEEE binary standard has so d
need to waste a bit of memory storing it Only p bits are therefore needed to store
the signi cand not p
Parameter Single precision Double precision
p
e
max
e
min
Exp onent width bits
Signi cand width bits
Format width bits
Table IEEE standard for single and double precision oating p ointnumbers
The C header les oat h and limits h were used to acquire sp eci c information ab out
the limiting values of real and integer numb ers using Sun workstations These values can
b e seen in Table It can b e seen that the values for oating p ointnumb ers corresp ond
to the IEEE standard
The minimum and maximum p ositive normalised oating p ointnumb ers for given
values of p and e are resp ectively
p e e
minimum and
e e p
maximum
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
min value max value
Single precision oat e e
Double precision double e e
Character char
Integer int
Long long
Unsigned integer unsigned
Table Limiting values of di erentnumb er formats
When a real numb er is rounded to one of these numb ers the maximum p ossible di erence
p e
between the real numb er and the oating p ointnumber is Away of measuring
the error is to nd the di erence b etween the real numb er and the oating p ointnumber
divided by the real numb er This is called the relative error This error therefore lies in
the following range
p p
relative error
p
The maximum value of the error is called the machine precision and
p for single precision so e For double precision and p so
e
NAG routines are a useful to ol for manipulating data These routines can for
example p erform fast Fourier transforms invert matrices and solvesetsofsimultaneous
equations NAG fortran double precision numerical library mark A was the version in
stalled on the computers used in this pro ject so it was decided to p erform all computations
using double precision numb ers
Toinvestigate the errors involved in arithmetic op erations i e addition subtraction
multiplication and division two square arrays of size N N were lled with random
double precision numb ers The numb ers had the largest p ossible numb er of signi cant
gures Eachnumb er in the second arraywas added to the corresp onding number in
the rst array and then was subtracted from it The di erences b etween the resulting
numb ers and the numb ers in the rst arraywere found These errors were then divided
by the corresp onding numb ers from the rst arraytogive relative errors The sum of
the magnitudes was found for arra y sizes from N to N was found
m m
to b e the same when a NAG routine was used as when the program was written in C
The average relative error should b e the same for each array size so the sum of the errors
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
would b e exp ected to increase as N The op erations needed to calculate from the
m
relative errors will intro duce errors into its value
This pro cedure was also p erformed for multiplication and division That is random
numb ers were rst multiplied and then divided by other random numb ers and then the
B
relative errors in the results were found Functions of the form y Ax were tted to
the data using DWPOL where y is and x is N This can b e seen in Figure m
10-5
10-6
10-7
10-8
10-9 addition/subtraction 10-10 multiplication/division y = 1.261475e-12*x^1.996005 y = 2.928085e-13*x^1.990108 10-11 Sum of magnitudes relative errors (log scale) 10-12 0 128 256 384 512 640 768 896 1024
Size of side of square array
Figure Errors asso ciated with basic arithmetic op erations
fitted
i
max
ln A B y y
i
i i
min
addition subtraction
multiplication division
Table Equations t to errors from basic arithmetic op erations
The values of B given in Table are very close to so the errors vary as N
as predicted Using the tted functions to predict the relative errors involved in one
addition and one subtraction the error of for an addition followed by a subtraction was
calculated to b e e and the error for a multiplication followed by a division was
e As stated b efore the machine precision for double precision numb ers is
e These results show that arithmetic op erations reduce this accuracy
TwoNAG routines which are used in this pro ject are two dimensional fast Fourier
transforms D FFT and matrix inversion If the two dimensional arrayofwhichthe
Fourier transform is to b e found has size N N and the matrix also has size N N the
errors in these op erations could again b e exp ected to vary as N The array and matrix
were again lled with random numb ers The Fourier transform of the arraywas found
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
and then the inverse Fourier transform The di erence b etween the result and the original
arraywas then calculated The matrix was inverted and the result and the original matrix
multiplied The di erence b etween this and an identitymatrixwas found The results can b e seen in Figure
10-7 10-5
-6 10-8 10
-7 -9 10 10 10-8 10-10 10-9 10-11 10-10 10-12 10-11 -13 Error in calculation Error in calculation 10 y=4.506521e-16*x^2.322880 10-12 y = 3.312368e-17*x^3.159830 -14 10 10-13
-15 10 10-14 Sum of magnitudes relative errors (log scale) Sum of magnitudes relative errors (log scale) 10-16 10-15 0 128 256 384 512 640 768 896 1024 0 128 256 384 512 640 768 896 1024
Size of side of square array Size of side of square matrix
a D FFT inverse D FFT b matrix inversion multiplication
Figure Errors in example calculations p erformed using NAG routines
f itted
i
max
ln A B y y
i
i i
min
D FFT inverse D FFT
matrix inversion multiplication
Table Equations t to errors from Fourier transforms and matrix inversion
As can b e seen from Table due to D FFTs increases as N whereas due
m m
to a matrix inversion followed byamatrixmultiplication increases as N With b oth of
these op erations therefore the relative error p er numb er will increase as the size of the
array or matrix increases
Hologram pixel values are often stored as integers b etween and In this case
the accuracy of the pixel values is much less than double precision so double precision
arithmetic will not noticeably decrease the accuracy Exp ected holograms for given source
distributions are used in the reconstruction algorithms however and these can b e calcu
lated to double precision It is here that the accuracy of oating p oint arithmetic should
b e considered
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
Radioactive Decay
The fundamental law of radioactive decay gives the activity of a radioactive source
as A t with half life T
dN t ln
A t N t N t
dt T
decay
where N t is the numb er of radioactive atoms at time t and is called the decay constant
If N is the numb er of atoms at time t N t isgiven by
t
N t N e
The probability p that a single atom will decay in a time interval t to t t is
therefore
t
p e
Each atom has the same probability of decay The exp ected counts in time t is given by
the numb er of atoms in the source multiplied by the probability p The numb er of atoms
at t N can b e calculated using equation
A t
t
Exp ected countin t N t N t t N e
t
Usually the counting interval t is much shorter than the half life of the radioactive
source In this case p andthePoisson distribution can b e used to describ e the
numb er of counts recorded exp erimentally The probability P N that a number
actual
of counts N are recorded is given by
actual
N
actual
N
t
e N
t
P N
actual
N
actual
The standard deviation of this distribution is given by
q
N
t
The higher the numb er of counts recorded the less the e ect these statistical uctuations
will have on a hologram
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
Sampling Theory
A signal is called bandlimited if its Fourier transform is zero outside a b ounded region in
the frequency plane If the Fourier transform is F it is bandlimited if
x y
F for j j j j
x y x y
x y
s
The Nyquist theory states that if you sample a signal at a rate in the x direction and
x
s
a rate in the y direction the signal will b e undersampled if these frequencies are less
y
than twice the maximum frequency in the signal i e
s s
x x y y
In this Section a more thorough investigation of the e ects of sampling will b e given
The e ects of sampling and undersampling on the Fourier transform of the data stored
in a hologram will b e demonstrated
The gamma camera is used to record the hologram as an arrayofN N pixels This
pro cess can b e thought of as sampling the hologram at a series of p oints separated by
a distance equal to the pixel size If the intensity of the radiation incidentonthe
p
alues whichwould ideally b e recorded would b e detector is given by h x y the series of v
where h
h h
j j
y x
h h h h
h h x y x j y j h j j h h
p p p p
j j
x y x y
y x
N N
h h
where j and j vary b etween and The sampled signal h x y can therefore
x y
b e describ ed by
N N
X X
h h
h x y h x y x j y j
p p
x y
N N
h h
j j
y x
The two dimensional comb or shah x x y a b function is given by
X X
h h
x j a y j b x x y a b jajjbj
x y
h h
j j
x y
N N
p p
N N
x and y h x y is therefore In the region
p p
describ ed by
h x y x x y h x y
p p
p
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
Atwo dimensional Fourier transform D FT can b e used to nd the spatial frequencies
in the hologram and to discover if it is under sampled or not The D FT of x x y a b
X isgiven by
x y
a b
X X
x y
F fx x y a b g X
x y x y
a b a b
y x
Taking the limit as N the Fourier transform of the sampled hologram H
x y
will b e given by
H H X
x y x y x y
p p
where denotes a convolution The Fourier transform of the sampled signal is therefore
equivalent to the sum of a series of Fourier transform patterns each shifted by a distance
in the or direction with resp ect to each other If the hologram is under sampled
x y
p
these sp ectra overlap For this to happ en the highest frequency in the x or y direction
in the hologram would have to b e greater than This is called the Nyquist
max
p
limit
The continuous two dimensional Fourier transform D FT F of a function
x y
f x y isgiven by
Z Z
i x y
x y
F f x y e dxdy
x y
The discrete Fourier transform of the N N array of hologram pixels can b e calculated
using a NAG routine The two dimensional discrete Fourier transform D D DFT
k k
x y
of the hologram pixels h is given by
j j
x y
h i
N N
X X
j k
j k y y
x x
i
N N
e D h
N N
k k
x y
j j
y x
N
j j
y x
where k k j and j vary b etween and N The NAG routine calculates this using
x y x y
the Fast Fourier Transform metho d As can b e seen in equation the continuous
Fourier transform is given byanintegral over all space x y
Equation can b e re arranged so it can b e compared with the D FT k k j and
x y x
N N
and as in equation Using this range of indices j need to vary b etween
y
re ects the fact that x y and can b e negativeaswell as p ositive Substituting
x y
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
N N N N
k k and j j j j into equation gives k k
y x y x
y x y x
N N
j k
j k
y y
X X x x
i
N N
k j i k i j
y y x x
D e h e e
N N
j j
k k x y
y x
N
N N
j j
y x
The size of a side of the detector P isgiven by P N The D DFT ab ove can b e
p
written as
N N
j k
p j k
p
y y
x x
X X
i
p
P P
j j k k
x y x y
D k k h e
j j
x y
x y
P
N N
j j
y x
If the size of the detector P remains constant As N the ab ove equation will
b ecome
j k
p
j k
p y y
x x
X X
i
p
P P
j j k k
x y x y
k h e D k
j j
y x
x y
P
j j
x y
Comparing this to equation and noting that x j and y j the following
p p
x y
sequence for calculating the two dimensional Fourier transform can b e devised
Algorithm for calculating D Fourier transform
N N
j j
x y
Multiply the hologram arrayvalues h by where j
j j
x
x y
N N
j and
y
Calculate the D DFT
N N N
k k
x y
Multiply the result by where k and k
x y
N
convolved with X k k k k The result D k k tends to F
x y x y x y
P P P P
p p
as N where F is the D FT of a function f x y and
x y
P P P P
h x y if x and y
p p
f x y
P P P P
if x x y or y
p p
k As the D FT is convolved with the shah function it will b e rep eated at
x
x
P
p
where and are integers This means that the separation and k
y x y
y
P
p
of adjacent copies of the D FT in b oth directions k will b e N Aliasing will
therefore o ccur if F k k has non zero values when jk j or jk j is greater
x y x y
P P
N
than
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
As discussed in Section a Gab or zone plate hologram of an on axis p oint source has
the following variation
p
p
v x y cos x y where
zp
zp zp
r p q
and r is the rst zone radius p is the separation of the source and zone plate and q is
the separation of the zone plate and detector We aim to discover when a hologram is
under sampled The exact magnitude of this function in the hologram and the constant
term which is also present can b e neglected for the purp ose of this analysis
Referring back to equation v x y has the following Fourier transform
x y
V sin
x y
zp
As in equation only the part of v x y whichisincident on the detector is recorded
This is similar to multiplying v x y bya box function which truncates it at the edges
of the detector The Fourier transform of a b ox function is a sinc function The Fourier
transform of v x y is therefore convolved with this sinc function An expression for the
resulting Fourier transform will b e derived here This e ect is often called ringing and
can b e lessened by using a window function Kuo investigated the application of
several di erentwindows to images from zone plate holograms as part of his PhD
This part of v x y recorded by the detector is f x y The Fourier transform of
f x y isgiven by
F V E
x y x y x y
i P
x y
p
sin Erf
p x
zp
zp
zp
i i P P
p p
Erf Erf
p x x
zp zp
zp zp
P i
p
Erf
y
zp
zp
where Erf x is the error function and
p
i
Z Z
p
K K
i
t i u
p p
e dt i Erf K e du
p p
Z Z
K K
u u u u cos sin sin du i cos du
r r r r
K K K K FresnelS i FresnelS FresnelC FresnelC
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
FresnelC x and FresnelS x are the cosine and sine Fresnel integrals and can b e calculated
using NAG routines Using the relationship in equation the real and imaginary parts
k and k Figure shows of E can b e calculated As b efore
y x y x
x y
P P
the real part of E forN p q and r
x y p
a x pixel ar b x y pixel numb er z amplitude
ray black e
white e
Figure Calculated Fourier transform of the e ect of having a nite detector area
E k k real part for k and k
x y x y
P P
k k and the real Figure on page shows the corresp onding values of V
x y
P P
j are j and jk part of F k k In this case E k k b ecomes zero when jk
y x y x y x
P P P P
N
less than so the pixel size is su ciently small for the frequencies in the hologram to b e
correctly recorded
Figure on page shows the real and imaginary parts of E k over values
x
P
of k from to This function is shown for the ab ovevalues and also when is
p
x
increased by a factor The gure illustrates that when the hologram is
p
under sampled
The zone plates used exp erimentally however haveavarying transmission up to a
maximum radius R after which the transmission has a minimum value Considering the
case where the minimum transmission is zero the hologram will havea variation v x y
R
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
a x pixel ar b x y pixel numb er z amplitude
ray black e
white e
c x pixel ar d x y pixel numb er z amplitude
ray black e
white e
Figure Calculated Fourier transform of an in nite Gab or zone plate V k k
x y
P P
and calculated Fourier transform of the p ortion of the zone plate recorded by the detector
F k k for k and k
x y x y
P P
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
5.0 5.0 real part real part imaginary part imaginary part 4.0 4.0
3.0 3.0
2.0 2.0 amplitude amplitude
1.0 1.0
0.0 0.0
-1.0 -1.0 -256 -192 -128 -64 0 64 128 192 256 -256 -192 -128 -64 0 64 128 192 256
pixel number pixel number
a b
p p
k for k Figure E ect of the nite detector size E
x x
P
where
p
v x y x y R
p
v x y
R
x y R
Considering v x y tobe f x y multiplied by a circular ap erture function a x y
R
v x y f x y and a x y are all radially symmetric so they can b e written as func
R
tions of the radius r
r R
v r f r a r where a r
R
r R
The DFT F s of a radially symmetric function f r can b e written in terms of r rather
than x and y as follows
Z
F s sr rdr f r J
J x is a zeroth order Bessel function of the rst kind This way of writing the DFT is
called a Fourier Bessel Transform or Hankel Transform The Hankel transform of a r
A s is given by
q
R
A s J Rs where s
y x
s
and J x is a rst order Bessel function of the rst kind This function can b e calculated
using a NAG routine The Fourier transform of v x y V is therefore given by
R R x y
V F A
R x y x y x y
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
a x pixel array b x y pixel numb er z amplitude
black e
white e
c x pixel ar d x y pixel numb er z amplitude
ray black e
white e
Figure Fourier transform of a circular ap erture of radius equal to the maximum radius
k k and of the truncated Gab or zone plate V k k of the zone plate A
R
x y x y
P P P P
real part for k and k
x y
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
k k With the parameters used for Figure and R thevalues for A
p
x y
P P
and V k k are as shown in Figure
R
x y
P P
k k makes the Fourier In this case convolution with the circular ap erture A
x y
P P
transform of the hologram decrease to near zero at an even lower frequency
In summary the Fourier transform of a zone plate hologram of a single on axis p oint
source can b e analysed to see whether the pixel size is su ciently small for the frequencies
to b e correctly recorded Using equations and the exp ected Fourier
transform for a hologram sampled at a high enough frequency can b e calculated The
Fourier transform obtained using the algorithm outlined on page can then b e compared
with the calculated transform If the hologram is under sampled the two transforms
will b e obviously di erent To illustrate this the Fourier transforms calculated using
the equations and the algorithm are compared in Figures and The hologram
in Figure has the parameters used in Figure a and Figure those used in
Figure b
Summary
In this Chapter sources of error due to the equipment used to detect store and pro cess
zone plate holograms of gamma ray sources have b een listed The way that the gamma
camera oating p oint arithmetic and the statistics of radioactive decaycontribute to
errors in the recorded hologram have b een explained
The di erence b etween continuous and discrete Fourier transforms has b een outlined
and an expression for the calculation of the discrete Fourier transform from the equivalent
continuous transform has b een derived Formulae describing the e ect of the limited
detector size and the truncation of the zone plate pattern at a maximum radius havealso
b een found These expressions can b e used to test if the hologram is under sampled
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
a b
f j j v j j
p p R p p
x y x y
black e black e
white white
e e
c d D DFT of e D DFT of f D DFT of
j j E k k f j j a j j v
p p p p p p p p R
x y x y x y x y
black e black e black e black e
white e white e white e white e
g h i j
k k k k k V k F k A k V
p p p p p p p p R
y y y y x x x x
black e black e black e black
white e white e white e e
white e
Figure Comparison of actual and calculated Fourier transforms for a hologram with
N p q r and R
p
Inherent Errors Asso ciated with Gamma Ray Zone Plate Holography
a b
f j j v j j
p p R p p
x y x y
black e black e
white white
e e
c d D DFT of e D DFT of f D DFT of
j j E k k f j j a j j v
p p p p p p p p R
x y x y x y x y
black e black e black e black e
white e white e white e white e
g h i j
k k k k k V k F k A k V
p p p p p p p p R
y y y y x x x x
black e black e black e black
white e white e white e e
white e
Figure Comparison of actual and calculated Fourier transforms for a hologram with
N p q r and R
p
Chapter
ASSESSMENT OF
RECONSTRUCTED IMAGE
QUALITY
The aim of this work is to discover and evaluate metho ds for reconstruction of images
from zone plate holograms Reconstruction of a p erfect image is made di cult bythe
sources of error outlined in Chapter and these errors will result in di erent artifacts in
the reconstructed image for each metho d Sub jective judgement of image quality that
is I think image A lo oks b etter than image B maygive a reasonable assessmentof
the e ectiveness of a reconstruction metho d but is sub ject to the p ersonal preferences
of the human observer A metho d for measurement of image quality is needed so the
reconstruction metho ds can b e compared
ICRUreport entitled Medical Imaging The Assessment of Image Quality
states that there are three kinds of physical parameters which are fundamental to image
system sp eci cation
Large scale macro system transfer function characteristic curve which
measures the relationship b etween system input e g exp osure quanta
and the output image e g optical density
Spatial resolution prop erties
Noise prop erties
Each of these parameters will b e considered in turn
System Transfer Function
This gives a measure of the resp onse of the imaging system to source distributions con
taining di erent spatial frequencies
Assessment of Reconstructed Image Quality
Relationship b etween the Source Distribution and the Out
put Image
How is the input signal determined so the output signal can b e compared with it To
answer this question the equations which describ e the formation of zone plate holograms
will rst b e reviewed
Equation gave the relationship b etween source distribution S p and hologram
x
intensity U x as
Z Z
x
D x
x
U x p S p S p T L p d dp
p q M p
where T istheintensity transmittance of the zone plate p S p is a factor which
dep ends up on the source distribution and as was explained in Chapter the other factors
and constants dep end up on the arrangement of the equipment
To simplify matters a simple two dimensional source will b e used to illustrate the
relationship b etween the source distribution and the output image The source will b e a
distance p from the zone plate As the source is two dimensional the emitted gamma
rays will not have to travel through other parts of the source If gamma ray attenuation
due to air is assumed to b e negligible the approximation p S p can b e
x
x
x x x
used If S x S p and T x T M p x are scaled versions of the source
L p
x
distribution and zone plate pattern resp ectively the hologram intensity U x can b e
written as a convolution
x x x
x x T x x C p q D S x U x
C p q D is a constantcontaining the factors in equation
The four sources of error discussed in Chapter were the gamma camera oating
p ointnumb ers the uctuations due to radioactive decay and under sampling The uc
tuations due to radioactive decay can b e considered as noise added to the source strength
term this noise will b e called N The gamma camera and oating p ointnumb ers con
s
tribute to errors or noise in the recorded hologram intensity this noise will b e called N
a
Equation then b ecomes
x x x x x
U x C p q D S x N x T x N x
s a
Assessment of Reconstructed Image Quality
If the hologram is under sampled this will lead to artifacts in the reconstructed image
This e ect is di erent from a random uctuation however and so cannot b e included in
a noise term
This work do es not involve the improvement of zone plate holograms bychanging the
equipment used The aim of this pro ject is to nd and compare di erent reconstruction
metho ds This can b e done using real zone plate holograms or computer generated holo
grams Computer generated holograms can b e used to test if a reconstruction metho d
would work if no noise was present and if so if it still works when di erenttyp es of noise
are added In this case the size shap e and p osition of the source are precisely known and
so reconstructed images can b e compared with a theoretical p erfect image of the source
It is however extremely di cult if not imp ossible to accurately simulate all of the
e ects describ ed ab ove Future and currentwork on the zone plate holography pro ject
must aim to pro duce an imaging system that can b e used clinically and therefore metho ds
must work for zone plate holograms of sources in vivo The quality of an image from a real
hologram can not b e compared with a theoretical p erfect image as this is unknown The
hologram whichwould have b een generated by the reconstructed source can however b e
estimated using equation This estimated hologram can then b e compared with the
actual hologram
If a metho d do es not work for a computer generated hologram it most probably will
not work for a real one This work will therefore rst concentrate on computer generated
holograms Techniques which are seen to work well will then b e applied to the few real
holograms that are presently available
Optical and Mo dulation Transfer Functions OTF MTF
x x
If a reconstruction technique reconstructs a p oint source S xx p x x p asaspot
x
H x p and the sp ot has the same shap e no matter where the p oint source is i e
x
what x p is the system is spatially invariant In zone plate holography the sp ot is
x
approximately the same for a constantvalue of p and anyvalue of x x
A commonly used measure of the relationship b etween system input and the output
image for a spatially invariant imaging system is the optical transfer function OTF
Assessment of Reconstructed Image Quality
and its magnitude the mo dulation transfer function MTF The MTF gives the relative
amplitude of the output signal as a function of the spatial frequency of a sinusoidal
F
x
x p the optical transfer is the two dimensional Fourier transform of H x input If H
p
function is the normalised frequency resp onse
F
H
p
OTF
F
H
p
The mo dulation transfer function is therefore
F
jH j
p
MTF
F
jH j
p
Spatial Resolution
The spatial resolution can b e investigated by nding the output image when the source
is a known simple ob ject This information can then b e used to predict the resolution
for more complex sources
Point Spread Function PSF
x x x
As was mentioned in Section a p oint source S xx p x p atpoint x p will
x
have a contribution H x p to the reconstructed image If the reconstruction technique
x x
works well H x p should b e lo cated at x p but will have probably b een blurred
ween by the imaging and reconstruction pro cesses The degree of blurring mayvary b et
di erent p ositions and di erent reconstruction techniques The reconstruction techniques
used in this work assume that the shadow of the zone plate is the same size and shap e for
p oint sources in the same plane same value of p It could therefore b e assumed that the
p oint spread function is the same for sources in the same plane Indeed the work done
by Caplan showed that this was a good approximation
The degree of blurring can b e measured by tting a function to the p oint spread func
tion The central limit theorem states that the mean X of n measurements follows a
n
Gaussian distribution whatever the distribution of the individual measurements A Gaus
sian curve is therefore often tted to exp erimental measurements to nd the variance of
the measurements ab out their mean The p oint spread function is a result of a non ideal
imaging system and is due to many factors Random measurement errors play their part
Assessment of Reconstructed Image Quality
but so do es the image reconstruction technique used In fact this work is concerned with
the di erences in blurring b etween image reconstruction techniques
It is di cult to decide on a b etter function to t to the p oint spread functions so
Gaussian functions will b e used A Gaussian curve can b e tted to a line of data through
the middle of the reconstructed p oint in each of the three directions x y p separately
A one dimensional Gaussian curveisgiven by
jx j
p
exp p x
where p x is the amplitude of the p oint at p osition x is the p osition of the centre of
the p ointand is the variance The variance of this tted curve is a measure of the
spatial resolution
Line Spread Function LSF and Edge Spread Function
ESF
The line spread function and the edge spread function are the resp onses of the system to
a line and an edge resp ectively The LSF could b e investigated by placing a line source in
a plane parallel to the detector but the ESF would b e more di cult to determine b ecause
an uniform source whichentirely lled a p ortion of the eld of view would b e needed
The edge would b e the edge of this extended source
Noise
For simplicity the di erence b etween a reconstructed image and the ideal image will b e
referred to as noise As discussed b efore noise comes from three main sources statistical
uctuations of radioactive decay the spatial resolution of the gamma camera and errors
intrinsic to the reconstruction pro cess and storage of the hologram using a computer
Variance of an Image of a Uniform Source
The image of a uniform source should b e uniform The variance of the intensity of this
image gives a measure of output noise This however do es not take the spatial correlation
of the noise into account As was shown in equation there are twotyp es of noise
source noise N and recording noise N Thevariance of an image of a uniform source
s a
gives no indication of the relative e ects of these two di erenttyp es of noise
Assessment of Reconstructed Image Quality
Wiener Sp ectrum W
The Wiener sp ectrum measures the noise p ower as a function of spatial frequency It
is given by the Fourier transform of the noise auto correlation function The total noise
x
N x x isgiven by the image minus the ideal image If Rfg is a function describing the
t
reconstruction pro cess the total noise for the simpli ed case in equation will b e
x x x x x x
N x R C p q D S x N x T x N x S x
t s a
x
x is given by The noise auto correlation function A x
N
Z
x x
x N N x d A x
t t N
x
The Wiener Khintchine theorem states that the auto correlation of a function F x is
F F
simply given by the Fourier transform of the absolute square of F where F isthe
F
x x
Fourier transform of F x If N is the Fourier transform of N x x equation
t
t
can b e written as
F
x
x F jN j A x
N
t
where Ffg denotes a Fourier transform The Wiener sp ectrum W is simply given by
F
W jN j
t
The amount of noise at di erent frequencies can b e seen This sp ectrum is easy to calculate
when the exact source distribution and therefore the ideal image is known Noise due to
statistical uctuations as well as that due to the gamma camera can b e simulated and
added to a computer generated hologram The relative contribution of the twotyp es of
noise can therefore b e compared for images reconstructed using di erent metho ds
Di erence Metrics
Mean Squared Error MSE and Signal to Noise Ratio SNR
A commonly used measure of the error b etween an output image and the ideal image
x
x as b efore and the is given by the mean squared error If the ideal image is S x
x
x reconstructed image of the source is S x the mean squared error is given by
R
X
x x
S x S x MSE
R
N
p
Assessment of Reconstructed Image Quality
where N is the total numb er of pixels and the sum is over all of them This metric
p
x
x you will get a larger error for a larger source activity dep ends up on the scale of S x
It is therefore b etter to use a normalised error NMSE
P
x x
jS x S x j
R
NMSE
P
x
jS x j
this is called the normalised mean square error metric Its square ro ot E is called the
normalised ro ot mean square error NRMSE
The signal to noise ratio SNR is de ned in decib els as
SNR log
NMSE
Fienup Error Metric FEM
If the reconstructed image is shifted by a few pixels with resp ect to the ideal image or
is multipliedby a constant factor the image will lo ok go o d but the MSE will b e huge
This is a ma jor limitation of mean squared error metrics they fail to di erentiate b etween
di erenttyp es of error
Fienup derived an expression for a metric whichisinvariant to a shift and mul
tiplication by a constant factor This error metric will b e referred to as the Fienup error
metric FEM
x
jr x j max
s x
x S S s x
s
s s
R
FEM
r r
S S S S
R R
x x x x
x is the auto correlation of r x is the cross correlation of S x x andS x x r
s s
s S S s R S S
R
x x x x
x j is the maximum x jr x is the auto correlation of S x max x x and r S
s x s
s s x S S s R x s S S
s
s s
R R R
x x
value of r x x forallvalues of x x
s s
S S s s s s
R
Summary
In this Chapter metho ds for assessing image qualitywere reviewed Measures of image
quality based on the system transfer function as well as spatial resolution and signal to
noise ratios were given These metho ds dep end up on having an ideal reference image with
which the reconstructed image can b e compared It is therefore easier to rst assess the
e ectiveness of image reconstruction metho ds by using computer generated holograms
Assessment of Reconstructed Image Quality
The OTF MTF can b e calculated to investigate the frequency resp onse of zone plate
holography using theoretical p oint sources The PSF can b e calculated to compare the
resolution of the di erent reconstruction techniques It is felt that LSF and ESF are more
suited to exp erimental measurement MSE NMSE and FEM error metrics can b e used
to measure the amount of error in the images
When images are reconstructed from real holograms an ideal reference image is not
available The reason for recording the hologram is to determine the distribution of
gamma ray sources from the reconstructed image An estimation of the hologram that
would have b een generated by a reconstructed source distribution can however b e cal
culated and this can b e compared with the recorded hologram
Chapter
IMAGE RECONSTRUCTION BY
CONVOLUTION
In this Chapter the metho d of reconstruction by convolution will b e discussed Recon
struction byconvolution with a Gab or zone plate pattern has b een used bymany p eople
at the University of Birmingham to reconstruct images from binary Gab or zone plate
holograms Mast Mathews Kirk Wo o dgate Strange Kuo
Caplan Perks Rew and Shen have all used this technique and they
referred to it as the correlation reconstruction metho d This metho d involved correlating
the hologram with an appropriately scaled zone plate pattern to reconstruct the image
in one plane parallel to the zone plate The same e ect is obtained if the hologram is
convolved with a version of the same zone plate pattern which has b een ipp ed ab out lines
through the centre in b oth the x and y directions As the zone plate is symmetrical in
b oth the x and y directions the ipp ed plate is identical to the original plate Correlation
and convolution therefore pro duce the same result Thinking of the technique as con
volution rather than correlation makes the relationship b etween the Fourier transforms
of the hologram zone plate and image more straightforward The form of convolution
reconstruction which has b een used the most is simulated di raction This metho d will
b e outlined and di erentconvolution reconstruction techniques will b e suggested An
improvement to the convolution technique which will b e called Normalised Convolution
Reconstruction will also b e presented here
An expression for the relationship b etween source distribution S p and hologram
x
intensity U x was given in equation as follows
Z Z
x
D x
x
U x L p q d dp p S p S p T
p q M p q
x
T x was the intensity transmittance of the zone plate As was explained in Chapter
Image Reconstruction byConvolution
p S p is a factor which dep ends up on the p osition of a p oint in the source distri
bution p aswell as the rest of the source distribution in relation to it This work will
concentrate on metho ds that do not include this factor That is it will b e assumed that
p S p over all of p This approximation can b e used if the source is twodi
mensional or su ciently small so that the gamma rays do not have to pass through other
parts of the source to reach the detector This factor can also b e neglected if the source
has a low linear attenuation co e cient Linear attenuation co e cients were discussed in
Section If a large dense source was used this factor would have to b e consid
ered Equally ifthistechnique was used in nuclear medicine to image brain tumours the
gamma rays would have to pass through the brain tissue and the skull b efore reaching the
detector and this would also havetobeaccounted for For the simple computer generated
sources and real sources whichhave b een used in this work however this approximation
is su cient
The gamma camera records the hologram as a square arrayofN N pixels where
N is usually a p ower of that is or The hologram is also to b e
planes at distances p p p p from the zone plate reconstructed in a series of N
p N
p
and parallel to the zone plate and gamma camera The space in which the ob ject lies
can b e thought of not only as a series of planes but also as voxels which increase in size
with distance from the camera Eachvoxel corresp onds to a region of space from whicha
gamma ray passing through the centre of the zone plate will b e detected in one particular
pixel by the gamma camera The source distribution can b e represented byaverage values
for eachvoxel and the zone plate transmission function can b e represented by pixels of
the appropriate size This is illustrated in Figure
The images generally have pixels in each x y plane To reconstruct a truly
three dimensional image the voxels would havetohave a similar size in the z direction
as in each x y plane This would mean reconstructing planes in the z direction or
some other similar number and would require a lot of computation In the work presented
here the image is reconstructed in a small numb er of parallel planes rather than a set of
voxels which completely ll the eld of view of the detector This is a sensible approach
as most of the sources which will b e used are two dimensional and placed parallel to the
detector We can then see if the image just app ears in the correct plane or if there are
Image Reconstruction byConvolution artifacts in the surrounding planes
, z1
z1 b z2
Shadow from second source
Central axis
Shadow from first source Plane containing second source Zone Detector plate (records superposition of shadows) Plane containing
first source
Figure Illustration of eld of view of detector with zone plate attached
Equation can b e written as a summation rather than a continuous integral This
helps us to understand the value stored in each hologram pixel
N
N
p
X X
D
k k k k
U C where U U i j C T S
i j ij p p
i i j j ij
p q
k
N
k
i j
p p p p
k k
j p and T T i j S S i
k
ij ij
L p q L p q M p q M p q
k k k k
N N
The factors D q L p q and M p q are as de ned in Chapter As i j
N N
k
values of T outside i j are needed This problem is overcome by padding
ij
k
T with zeros to twice its original size that is
ij
N N
k
i j T
ij
k
N
padded T
N i j
ij
N
i j N
In practice three fast Fourier transforms are used to p erform a convolution The Fourier
transforms are p erformed in the way describ ed in Section When a theoretical holo
k k
and S are padded as ab ove If this is not done the convolu gram is calculated b oth T
ij ij
N
tion b ecomes circular In this case if i for example this value do es not exist
N
so the value at i is taken instead
Image Reconstruction byConvolution
Optical Reconstruction howConvolution Algo
rithms can b e used to Simulate Di raction
In Section the di raction pattern pro duced when plane waves are shone through a
x
transparency with transmittance t x was given as
q
C
r
q
r
x x x
x where u x q t x z
r
i
p q
r r
i
x
q i x
r
r
r
x
z x e C A e and
r r
r r
q
r r
when l a Aswas stated b efore where is the maximum spatial frequency
p
x
q is the size of the central Fresnel zone and a is the total size of the in t x l
r r
x
pattern represented by t x As b efore q is the distance from the transparency to the
r
detector is the wavelength of the light used in the reconstruction and A and p are
r r r
constants representing the amplitude and phase of the light
If the hologram were recorded using an ideal Gab or zone plate it will contain a
p
constant term plus a Fresnel function of frequency M p q r Aswas shown in
zp
Section if l a the ob ject distribution that was in plane p will b e reconstructed
a distance q b ehind the hologram where
r
p q r
q
r
p
r
If the image was reconstructed using a ruby laser the wavelength of lightwould b e
nm Values which are commonly used for gamma ray zone plate holograms
r
are p m q m and r m If the hologram was printed out on lm the same
size as the detector the distance of the image b ehind the lm would b e q m and
r
the radius of the central Fresnel zone would b e l m The hologram would haveto
b e reduced in size to fo cus the image in a more reasonable p osition If the hologram was
reduced by a factor R thiswould give
s
p q r
q
r
p R
r
s
If di raction is simulated using a computer by convolving the hologram with the appropri
q
r
x
x pattern for one x y plane at a time the value of is not imp ortant In this ate z
r
case is set equal to the exp ected value of for an ob ject in the appropriate image
r zp
Image Reconstruction byConvolution
plane Equations for the calculation of were given in Section As in equation
zp
q
r
we can write the equation for the reconstructed image in plane q V as a summation
r
ij
N
X
q q
r r
U Z V
ij
i j
i i j j
N
i j
q
r
q
r
x
U represents the hologram pixel values and Z x represents the Fresnel function z
ij
ij
q
r
are padded as in equation used for the reconstruction Both U and Z
ij
ij
To illustrate reconstruction b e simulated di raction a computer generated hologram
of a p oint source with unit strength p er unit area will b e used The zone plate hologram
values will b e those in Table These are typical parameters for a zone plate hologram
The parameters of the appropriate reconstructing Fresnel function can b e calculated the
using the equations given in Section
First zone radius r e m Source zone plate distance z m
Detector pixel size e m Zone plate detector distance b m
q
R
Numb er of pixels n n Numb er of zones n
x y
r
Table Hologram parameters used for the theoretical hologram used in most of this
Chapter
These parameters are typical for actual gamma ray zone plate holograms R is the
maximum radius of the zone plate it is totally opaque outside this radius Although
the detector stores data in a pixel array it is only the central which
represent the active region of the detector A real exp erimental hologram is shown in
Figure to illustrate this The p oint can b e reconstructed byconvolving the zone plate
pattern with the Fresnel function For simplicity the parameters of the reconstructing
q
C
r
The reconstructed p oint can b e seen in Figure light are set so
r
i
Using the Zone Plate Pattern as a Matched Filter
A pure Gab or zone plate GZP is not used to record zone plate holograms a binary plate
is used instead As was mentioned in Chapter this plate has the same average intensity
transmittance at each radius as the pure Gab or plate but the plate is either transparent
or opaque at eachpoint The advantage that this has over the pure plate is that it is much
easier to manufacture Holes just have to b e cut in a metal sheet of constant thickness
Image Reconstruction byConvolution
max
min
a a hologram as b the central
recorded by the pixels
gammacamera
Figure Illustration of the useful hologram area
a simulated holo b Fresnel function c Fresnel function
gram of p oint real part pixel imaginary part
source pixel value pixel value
value e
max
min
d reconstructed e reconstructed
point real part point intensity
e pixel e pixel
value e value e
Figure Theoretical reconstruction of a p oint source bysimulated di raction
Image Reconstruction byConvolution
Another advantage of a binary Gab or zone plate over a pure Gab or zone plate is that
it transmits more light if it is used as a lens If the zone plate patterns are recorded on
photographic lm so that the amplitude transmittance of the lm is the same as the zone
plate the total intensity transmittance will b e greater for the binary plate than for the
pure Gab or plate lens
The amplitude transmittance of a pure Gab or zone plate t r is
GZP
r
t r cos
GZP
r
where as in Chapter r is the radial distance from the centre of the plate and r is a
constant A binary Gab or zone plate BGZP has the same average transmittance for
each radius as the the pure plate The amplitude transmittance for a binary Gab or zone
plate is either zero or one at any one p osition Whether it is zero or one dep ends up on the
numb er of sectors in the plate This is illustrated in Figure The average amplitude
and intensity transmittance of the plate is una ected bythenumb er of zones To simplify
matters a binary plate with just one zone will b e considered In this case the amplitude
transmittance of a binary Gab or zone plate t r is
BGZP
t r
GZP
t r
BGZP
t r
GZP
The average intensity transmittance AIT of a pure Gab or zone plate is given by
h h ii
R R
R
R R
r
R
cos rd dr
t r rd dr
r
GZP
AIT
GZP
R R
h ii h
R
R
r
cos rdr
r
R
where R is the maximum radius of the lens The average intensity transmittance of a
binary Gab or zone plate is given by
R R R R
R R t r
GZP
t r rd dr rd dr
BGZP
AIT
BGZP
R R
h i
R
R
r
cos rdr
r
R
p
If the maximum radius of the lens R is set to R nr where n is an integer the
integrals in equations and simplify to the following numb ers
AIT AIT
BGZP GZP
Image Reconstruction byConvolution
Both zone plate patterns would fo cus light at the same p osition As the binary zone plate
pattern has a higher average intensity transmittance the intensityoflight at the fo cal
p oint could b e exp ected to b e brighter than the intensity if a pure Gab or plate was used
Further prop erties of binary zone plates are discussed by Beynon and Strange
x
A linear lter L x can b e used to reconstruct an image from the hologram by
convolving the hologram with the lter Ob jects whichwere at di erent distances p
b ehind in front of the zone plate can b e reconstructed by using a zone plate pattern
scaled to the size of the shadowcastbya point at a distance p
N
X
p p
V U L
ij
i j
i i j j
N
i j
p
Again U represents the hologram values and V the reconstructed image values at a
ij
ij
distance p in front of the zone plate The similarity of this equation to equation may
b e noted As Jain proves the linear lter which will pro duce an image with the
highest signal to noise ratio is simply the hologram of a single p oint In this case this will
b e a magni ed version of the binary plate pattern Jain calls the optimum linear lter a
matched lter
If the binary plate pattern is to b e used in a reconstruction algorithm it needs to
b e generated using an algorithm An example of a binary zone plate can b e seen in
Figure Figure illustrates how the pattern is generated If the exp ected hologram
centre of sector which pixel is in
pixel
radius = r angle = θ
angle of half a sector α = π/s maximum radius zone plate divided into s sectors R = r*(n^(1/2)) (12 in this case) required intensity transmittance at radius r = t(r) if θ/α <=1-t(r), pixel is opaque.
otherwise, pixel is transparent.
Figure Calculation of the binary zone plate pattern
of a p oint source is convolved with itself the p oint should b e reconstructed The hologram
Image Reconstruction byConvolution
from Figure was reconstructed using a matched lter and the result can b e seen in
Figure The reconstructed p oint in Figure c sits on top of an error term which
max
min
a simulated holo b matched lter c recon
gram of point pixel value structed point
source pixel e pixel
value e value e
Figure Theoretical reconstruction of a p oint source using a matched lter
steadily increases towards the centre of the image Fenimore and Cannon discovered a
similar e ect when using a matched lter to reconstruct images generated using uniformly
redundant arrays URAs URAs are binary patterns which like zone plates can b e
used to obtain extra information ab out light sources by detecting the shadow cast bya
source through the pattern Again like zone plates they have an auto correlation which
approximates a delta function Fenimore and Cannon removed the pyramid error term
from their images bychanging the values assigned to white and black pixels in their
p
matched lter This new lter was said to b e balanced The balanced lter new L
ij
p
in the following way can b e calculated from the old lter old L
ij
p
if old L
p
ij
new L
ij
p
if old L
ij
is the ratio of transparent white pixels to total pixels Fenimore and Cannon compared
p
this to Brown s mismatch metho d where the new value of L is set to if its old value
ij
was zero Fenimore and Cannon s metho d is the same as this if the numb er of white and
black pixels are equal A binary zone plate has approximately equal numb ers of white and
p
x y R but all the pixels are opaque black outside this region black pixels for
A mo di ed mismatch metho d is therefore suggested This metho d uses Brown s metho d
Image Reconstruction byConvolution
p
for x y R but outside R all pixels are set to zero That is
p
p
if old L
ij
if x y R
p
p
new L
if old L
ij
ij
p
x y R if
Reconstructed p oints using b oth these metho ds are shown in Figure Values taken in a
a simulated holo b lter for bal c lter for mo di ed
gram of p oint anced convolution mismatch pixel
source pixel pixel value
value e value
max
min
d reconstructed e reconstructed
point using bal point using mo d
anced convolution i ed mismatch
e pixel e pixel
value e value e
Figure Theoretical reconstruction of a p oint source using balanced convolution and
mo di ed mismatch metho ds
diagonal line through the centre of p oints reconstructed using the matched lter balanced
convolution and mo di ed mismatch metho ds are compared in Figure The mo di ed
mismatch metho d is closest to zero around the p oint and therefore app ears to b e the
b est one of these lters These metho ds will b e compared using the image assessment
metho ds from Chapter in Sections and
Image Reconstruction byConvolution
4e-09 matched filter balanced convolution modified mismatch 3e-09
2e-09
1e-09
result of convolution (real part) 0e+00
-1e-09 -0.20 -0.10 0.00 0.10 0.20
distance from centre of image plane (m)
Figure Diagonal line taken through images of p oints reconstructed using lters based
up on the binary zone plate pattern
Hybrid Metho d using R to improve Simulated
Di raction
As the zone plate pattern is truncated at a maximum radius R itmakes sense to trun
cate the Fresnel function used in simulated di raction at the same radius The results
are shown in Figure A diagonal line of data taken through the real part of the
reconstructed p ointisshown together with the same data from the standard simulated
di raction reconstruction in Figure Lo oking at this gure it app ears that the e ect
is negligible for a p oint source Again these metho ds will b e compared using the image
assessment metho ds from Chapter in Sections
Normalised Convolution
If a p oint is reconstructed in one plane using a linear lter the value of the convolution
where the p oint is reconstructed dep ends up on the p ortion of the zone plate shadow which
is actually seen by the detector
D
In Section the term was used to approximate the fraction of the emitted
p q
radiation whichwas recorded by the detector As b efore D is the area of the detector If
the detector had N N pixels and the hologram was recorded in each of these pixels the
p p
D
is the appropriate source intensity where S S countineach pixel would b e
ij ij
p q N
and is the data acquisition time However the size of the pixels varies with the source
to zone plate distance p It therefore makes sense to calculate source strength p er unit
area instead of total source strength in each pixel That is the value reconstructed in
Image Reconstruction byConvolution
a simulated holo b Fresnel function c Fresnel function
gram of p oint real part pixel imaginary part
source pixel value pixel value
value e
max
min
d reconstructed e reconstructed
point real part point intensity
e pixel e pixel
value e value e
Figure Theoretical reconstruction of a p oint source byconvolution with a truncated
Fresnel function
2.0e-09 simulated diffraction truncated Fresnel function 1.5e-09
1.0e-09
5.0e-10
result of convolution (real part) 0.0e+00
-5.0e-10 -0.2 -0.1 0.0 0.1 0.2
distance from centre of image plane (m)
Figure Diagonal line taken through images of p oints reconstructed using lters based
up on the binary zone plate pattern
Image Reconstruction byConvolution
p
S
ij
each pixel will b e where is the size of a pixel in plane p
p
p
If the p oint is reconstructed using a lter based up on the binary zone plate pattern
p
the value of the convolution at the p oint V will b e the numb er of the pixels which
ij
recorded a value i e the numb er of transparent pixels in the correctly scaled zone plate
p
pattern times the count in each pixel The aim is to recover S from this information
ij
The hologram is however only recorded in pixels corresp onding to holes in the zone plate
pattern The zone plate pattern could b e totally recorded by the detector overlap in the
x or y direction or b oth This is illustrated in Figure We will assume that p and q
havebeenchosen so that the whole shadow ts on the detector when it is in the centre i e
that it can t overlap two opp osite edges of the detector at once If x and y are distances
3 y’ 2y x’ x 2x y
1 2
a all the shadowis b the shadowoverlaps c the shadowoverlaps
recorded in one direction in b oth directions
Figure Illustration of the zone plate pattern overlapping the edge of the detector
from the centre of the detector x and y which are shown in Figure are given by
s s
D D D D
x R R y x y x y
➀ ➁ ➁ ➂ can b e calculated as follows The areas and
x y
➀ R
s
D
x
D D
R arccos ➁ x R x
x
R
s
D
y
D D
➁ R arccos y R y
y
R
p
q q
x y y x
➂ R arcsin x R x y y
Image Reconstruction byConvolution
p
The area of the zone plate shadow seen by the detector A is then calculated as follows
ij
➀ ➁ ➁ ➂ if x and y are b oth real and p ositive
x y
if x and y are b oth real and negative ➀ ➁ ➁
x y
p
A
if x is a complex number and y is real and negative ➀ ➁
x
ij
➀ ➁ if y is a complex number and x is real and negative
y
➀ if x and y are b oth complex numb ers
A hologram generated using a binary plate has approximately the same numb er of white
as black pixels inside the maximum radius of the plate shadow so will assume that the
area of the detector which the radiation hits is approximately half the ab ovevalue Using
this approximation the value of the convolution for the reconstructed p oint can b e written
as
p p p
p q V S A
ij ij ij
p p
S and so V
p
ij ij
p q A
p p
ij
The ab ove equation dep ends up on p and therefore corrects for di erent distances in the z
p direction Figure shows the e ect of the normalisation in the z as well as the x y
p
S
ij
direction As b efore holograms were simulated for p oint source with and the
p
p oints were reconstructed using the mo di ed mismatch metho d When the reconstruction
was not normalised the reconstructed p oints had varying amplitude Normalisation made
the amplitude of each p oint within a small margin of error The errors result from
the numb er of black and white pixels in the zone plate shadow not b eing exactly equal
for all p ositions Normalisation of simulated di raction or convolution with a truncated
Fresnel function is however more complicated Simulated di raction relies up on the
Fresnel function extending to in nityinthex and y directions As it is truncated by
either the size of the detector or the maximum radius of the zone plate the calculated
amplitudes are a ected As the Fresnel function is smo othly varying b etween and
so wewould exp ect a reconstructed p ointtohave a smaller amplitude than if it were
reconstructed using a binary lter What this amplitude would b e is however di cult
to calculate
Toovercome this problem the theoretical reconstructed image amplitude of an on
axis p oint will rst b e calculated This will b e done in the appropriate plane using the
p sd
appropriate reconstruction metho d This amplitude V will b e used to calibrate the
p
can b e replaced with a correctly scaled version for this metho d Using this value A
ij
Image Reconstruction byConvolution
4e-09 1.1 point at x=y=0 point at x=y=0 point at x=y=0.059 point at x=y=0.059 0.9 3e-09
0.7 2e-09
0.5
1e-09 0.3
0e+00
value of reconstructed image pixel value of reconstructed image pixel 0.1
-1e-09 -0.1 -0.2 -0.1 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2
distance from centre of plane along diagonal (m) distance from centre of plane along diagonal (m)
a p oints separated in x y unnor b p oints separated in x y nor
malised convolution malised convolution
4e-09
point at p=0.22m 1.0 point at p=0.22m point at p=0.28m point at p=0.28m
3e-09 0.8
0.6 2e-09
0.4
1e-09 0.2 value of reconstructed image in central pixel value of reconstructed image in central pixel 0e+00 0.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.10 0.15 0.20 0.25 0.30 0.35 0.40
position p (m) position p (m)
c p oints separated in z unnor d p oints separated in z nor
malised convolution malised convolution
Figure Magnitudes of theoretical reconstructed p oints in di erent p ositions for b oth
unnormalised and normalised convolution
p
metho d A as follows
ij
p sd p sd
p q V V
p
p p
A A A
p
ij ij ij
V R
Comparison of Reconstructed Points using Error
Metrics
The values of the di erence metrics MSE NMSE and FEM whichwere de ned in Chap
p
S
ij
ter are b e given in Table for p oints with reconstructed using techniques
p
in this Chapter The parameters for the theoretical hologram are those given in Ta
ble The ideal reconstructed image would b e a single p oint with amplitude
The sum of squares of exp ected image values is therefore also This means that
Image Reconstruction byConvolution
NMSE MSE in this case As only a single p oint has b een reconstructed the
signal dep ends up on this single value whereas any non zero value in another pixel will
contribute to the noise For this reason the errors are high The signal to noise ratio
SNR is negative as the errors are so high and therefore will not b e considered here
Normalisation increases the error for two reasons First it increases all the values in the
MSE NMSE FEM
simulated di raction real part e
simulated di raction real part normalised e
simulated di raction intensity e
simulated di raction intensity normalised e
matched lter e
matched lter normalised
balanced convolution e
balanced convolution normalised
mo di ed mismatch e
mo di ed mismatch normalised
truncated Fresnel real part e
truncated Fresnel real part normalised
truncated Fresnel intensity e
truncated Fresnel intensity normalised e
Table Error metric values for the di erentconvolution reconstruction techniques
reconstructed image and second it increases the values towards the edges more than those
in the centre Giving the reconstructed p oint the correct amplitude improves the signal
but it also increases the unwanted error values in the other pixels As the number of pixels
containing the signal is much less than those only containing only errors this increases
the MSE and NMSE values FEM should b e constant when the image is multiplied bya
constantvalue but the normalisation is not constant As the values in the unnormalised
images are so low they are practically zero in comparison to the exp ected value whichis
equal to one This is why the NMSE values are all equal to one and the MSE values are all
for the unnormalised images The central p ointsignalismultipliedby a smaller
numb er than the edges and so FEM also decreases when the image is normalised Despite
all of this convolution with a truncated Fresnel function app ears to b e the b est metho d
for p oint sources and if the intensity is taken rather than the real part this provides an
obvious improvement Later results in Chapter however show that the real part gives
b etter results than the intensity for other source ob jects Wo o dgate also found this
Image Reconstruction byConvolution
to b e the case
Comparison of Mo dulation Transfer Functions
The mo dulation transfer functions MTF for the images of p oint sources from Section
are shown in Figure The MTFs are only shown for the unnormalised images as there
is little visible di erence b etween the normalised and unnormalised cases
a simulated b simulated c matched lter d balanced
di raction di raction e pixel convolution
real part intensity value e pixel
e pixel e pixel value
value value
max
min
e mo di f truncated g truncated
ed mismatch Fresnel real part Fresnel intensity
e pixel e pixel e pixel
value value value
Figure Mo dulation transfer functions
The MTF for an ideal image of a p ointwould b e uniform and equal to one for all fre
quencies Finding the error metric for the di erence b etween the MTFs and this uniform
array is another measure of the qualityofeach image The greater the error the worse
the image is As the MTF dep ends only on the frequencies in the reconstructed image
and not on its amplitude the di erence b etween the normalised and unnormalised cases is
Image Reconstruction byConvolution
small The unnormalised images have slightly lower error metric values As in Section
convolution with the truncated Fresnel function app ears to b e the most e ective metho d
overall Mo di ed mismatch is the b est matched lter metho d
NMSE FEM
simulated di raction real part
simulated di raction real part normalised
simulated di raction intensity
simulated di raction intensity normalised
matched lter
matched lter normalised
balanced convolution
balanced convolution normalised
mo di ed mismatch
mo di ed mismatch normalised
truncated Fresnel real part
truncated Fresnel real part normalised
truncated Fresnel intensity
truncated Fresnel intensity normalised
Table Comparison of mo dulation transfer function error metric values for the di erent
convolution reconstruction techniques
Image Reconstruction byConvolution
Comparison of Point Spread Functions
In Chapter it was suggested that a p oint spread function could b e measured by tting
a one dimensional Gaussian function to the reconstructed p ointineach of the three di
rections separately As usual the theoretical hologram used had the parameters given in
Table The function to b e tted in direction x p x is given by
jx j
p x A exp
As mo di ed mismatch ltering and convolution with the truncated Fresnel function are
the b est binary and di raction like metho ds resp ectively itwas decided to just analyse
these twotechniques The p oint spread function PSF should b e the same for b oth
x and y directions so it was decided to just t the function along the diagonal of the
reconstructed image plane for the p oint spread function in the x and y directions will
b e the same as for the diagonal if it is assumed that the PSF is circularly symmetric
The reconstructed p oints are normalised and examples can b e seen in Figure The
1.0 truncated Fresnel (real part) 1.0 truncated Fresnel (real part) truncated Fresnel (intensity) truncated Fresnel (intensity) modified mismatch modified mismatch 0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2 0.0
result of normalised convolution 0.0 result of normalised convolution -0.2
-0.2 -0.4 -0.010 -0.005 0.000 0.005 0.010 0.20 0.21 0.22 0.23 0.24
distance from centre of image plane (m) position p (m)
a PSF along diagonal x y b PSF in z direction
plane
Figure Comparison of mo di ed mismatch reconstruction with reconstruction by
convolution with a truncated Fresnel function
Gaussian was tted to the p eaks using the nonlinear regression routine from the statistical
package SPSS The go o dness of t is indicated by the R squared value R squared is the
ratio of the variation predicted by the tted equation to the total variation An ideal t
would therefore have an R squared value of one and if the equation did not predict the
variation at all the R squared value would b e zero The values of A and R squared
Image Reconstruction byConvolution
can b e seen in Table Mo di ed mismatch and the real part of the truncated Fresnel
A R squared
x y diagonal
truncated Fresnel real part e
truncated Fresnel intensity e
mo di ed mismatch e e
z direction
truncated Fresnel real part e e
truncated Fresnel intensity e e e
mo di ed mismatch e e e
Table Point spread functions for the mo di ed mismatch and truncated Fresnel tech
niques
reconstruction have similar values of with mm in the x and y directions and
mm in the z direction Taking the intensity rather than the real part of the truncated
Fresnel reconstruction improves the resolution in the x and y directions but makes it worse
in the z direction The values of the resolution will obviously b e di erent for di erent
hologram parameters For example as the eld of view of the detector decreases for
distances closer to the zone plate the resolution increases This technique will b e tried
for di erent sources in Chapters and From these data the indication is that the z
resolution is worse than the x and y resolution
Comparison of Images of Uniform Sources
In Section images of p oint sources were compared In Chapter the variance of
an image of an uniform source was given as a measure of the quality of an imaging
system A hologram of a two dimensional uniform source with unit strength p er unit area
and the parameters from Table was generated The image from this hologram was
then reconstructed in the correct plane using all the di erentconvolution reconstruction
techniques The unnormalised images can b e seen in Figure and the normalised
images in Figure
As the Fienup Error Metric FEM is invarianttoa multiplication by a constant factor
the FEM value for images of uniform source is directly related to the v ariance The FEM
values are given in Table The NMSE values have b een included for completeness
The NMSE values for the unnormalised images are again mostly equal to one This
Image Reconstruction byConvolution
a simulated b simulated c matched lter d balanced
di raction di raction e pixel convolution
real part intensity value e e pixel
e pixel e pixel value e
value e value e
max
min
e mo di ed f truncated g truncated
mismatch Fresnel real part Fresnel intensity
e pixel e pixel e pixel
value e value e value e
Figure Unnormalised images of uniform sources
is b ecause the amplitudes of the unnormalised images are muchlower than the exp ected
amplitudes The FEM value for matched ltering is very low compared to the other values
but this is b ecause the image do es not have a negative minimum pixel value whereas all
the other images do Lo oking at the FEM values for the other metho ds the matched lter
metho ds give a b etter reconstruction than either the real part of the simulated di raction
or the truncated Fresnel function reconstructions
Image Reconstruction byConvolution
a simulated b simulated c matched lter d balanced
di raction di raction e pixel convolution
real part intensity value e pixel
pixel e pixel value
value value e
max
min
e mo di ed f truncated g truncated
mismatch Fresnel real part Fresnel intensity
pixel pixel pixel
value value value e
Figure Normalised images of uniform sources
NMSE FEM
simulated di raction real part
simulated di raction real part normalised e
simulated di raction intensity
simulated di raction intensity normalised e
matched lter
matched lter normalised e
balanced convolution
balanced convolution normalised e
mo di ed mismatch
mo di ed mismatch normalised e
truncated Fresnel real part
truncated Fresnel real part normalised e
truncated Fresnel intensity
truncated Fresnel intensity normalised e
Table Comparison of error metric values for images of uniform sources for the di erent
convolution reconstruction techniques
Image Reconstruction byConvolution
Summary
Metho ds to p erform simulated di raction using convolution and reconstruction using lin
ear lters have b een outlined Three di erent lters based on the binary zone plate pattern
were given and tested on a computer generated hologram of a p oint source Mo di ed mis
matchwas found to b e the b est metho d for this hologram but the matched lter metho d
gave b etter results for a hologram of a two dimensional uniform source
Ahybrid metho d combining the maximum radius of the zone plate pattern with
simulated di raction was intro duced This metho d was found to give the b est results for
the hologram of a p oint source
A metho d for normalising the reconstructed image was also derived enabling p oints
to b e reconstructed with the correct amplitude
Although convolution with the actual zone plate pattern could b e exp ected to give
b etter images than convolution with a di erent ideal zone plate simulated di raction
pro duced b etter results for a p oint in a single plane than convolution with the binary
plate pattern
Chapter
IMAGE RECONSTRUCTION BY
DECONVOLUTION
In this Chapter the metho d of reconstruction by deconvolution using Fourier transforms
will b e discussed Reconstruction by deconvolution was used byWo o dgate but as he
did not use a Wiener lter his reconstructed images were of p o or quality Deconvolution
using a matrix Wiener lter was used by Strange on holograms generated using
MCNP The reconstructed images were go o d but the technique could only b e used
on holograms with a maximum of pixels due to the limitations of the computers
available The use of a parametric Wiener lter will b e explained here and techniques for
optimising the output will b e given As in the Chapter on reconstruction byconvolution
Chapter an attempt will b e made to normalise the reconstructed image
Expression of the hologram formation pro cess us
ing Fourier transforms and using matrix notation
The equation for hologram formation was written as the following summation in Chap
ter
N
N
p
X X
D
k k k k
S T where U U i j C U C
ij p p i j
ij i i j j
p q
k
N
k
i j
p p p p
k k
S S i j p and T T i j
k
ij ij
L p q L p q M p q M p q
k k k k
x
S p represented the source distribution and T x the intensity transmittance of the
N N
i j The contribution of zone plate pattern U are pixel values and
i j
p
ob jects in a single plane at a distance p from the zone plate U to the hologram is given
p p p
by the convolution of S with T multiplied by a constant C
p p p p p p p p p
U C S T H S where H C T
Image Reconstruction by Deconvolution
p p p
From the convolution theorem states that the Fourier transforms of U H and T
are related in the following way
F F F
p p p
S H U
F
where A is the Fourier transform of a variable A
p p
If U and S were written as column vectors instead of two dimensional arrays the
hologram formation pro cess could b e written in matrix form This will b e explained
using the following simple example A source distribution and a zone plate
transmittance function will b e used
p p p p p
C T S so H and U
p p p
If U and S are written as column vectors the convolution by T and multiplication by
p p
C can b e combined into a hologram formation matrix H
C C B C B B
p p
p p p p p
C C B C B B
S U
S U H S U
A A A
Deconvolution using Fourier and standard ma
trix Wiener lters
Deconvolution involves the inversion of the convolution equation it tries to undo the
action of convolution If Fourier transforms were used equation could b e inverted and
p
S could b e calculated in the following way
F
p
U
F
p p
S inverse Fourier transform of S
F
p
H
p
p p
S
Similarly if matrix notation were used H could b e inverted to calculate S
p p
p p p p p
S U
S S H U U
This will only work however for an ideal noise free imaging system As b oth reconstruc
tion techniques involve division of one set of data by another small errors in the data
will lead to large errors in the output This is particulary the case when the problem is
p p
ill conditioned Ill conditioning means that small p erturbations in H or U lead to large
Image Reconstruction by Deconvolution
p
di erences in S Thiswas b een shown to b e the case for zone plate holographywhen
Wo o dgate used deconvolution without a Wiener lter and obtained p o or results A
Wiener lter is used to optimise the output for a noisy imaging system
Andrews and Hunt state that there are two di erenttyp es of Wiener lter the
Fourier Wiener Filter whichusesFourier transforms and the standard Wiener lter
which uses the matrix representation Each metho d assumes that the signal is not p erfect
but has noise added to it and that the signal hologram and noise are uncorrelated The
sources of error in a zone plate hologram were discussed in Chapter In Chapter these
errors were classi ed as source noise N and recording noise N The signal recorded from
s a
p
p
ob ject plane p is not U but a noisy version U given by
p p p p p p p p p p
p
U C S N T N H S H N N
s a s a
As the Wiener lter assumes that the signal and noise are uncorrelated wehave to replace
p
this by a single error term E
p p p
p
U H S E
p
p
The reconstructed image S should b e as close to S as p ossible Fourier transforms
and a lter can b e used to achieve this The lter is designed to minimise the sum of
p
p
squared di erences b etween S and S in the following way
F
p
p
U
F
p
S
F
p
H
p
The Fourier Wiener lter is the lter which optimises S It is derived in app endix D
F
p
U
p
Fourier Wiener lter
F F
p p
E U
Inserting this into equation gives
F F
F
F p
p
p p
U
H U U
F
p
S
F
p
F F
F
p
p p
H
F
E
E U p
H
F
p
S
F
p p
is the complex conjugate of the Fourier transform of H The parametric Fourier H
Wiener lter includes a parameter whichcanbevaried to increase or decrease the e ect
of ltering
F
p
U
p
parametric Fourier Wiener lter
F F
p p
U E
Image Reconstruction by Deconvolution
p
The standard matrix Wiener lter is the lter which minimises the sum of squared
p
p p p
p
p
S S
S when it is calculated using the inverse of matrix H di erences b etween S and S
p p
p p
p p
S U
S H U
p
Using a similar derivation to that used for the Fourier Wiener lter is found to b e
p p T p p T p
p p
Wiener lter H H H H
S E
p p
and are the signal and noise covariance matrices If eachpixelvalue in the
S E
signal is assumed to b e uncorrelated with other values in the signal and the same can
b e said for the noise the o diagonal elements in these matrices can b e ignored The
diagonal elements are the exp ected variances of each pixel value
There is also a parametric Wiener lter whichisgiven by
p T p p p T p
p p
H H parametric Wiener lter H H
E S
where again is a variable which can b e changed to increase or decrease the e ect of
the lter
The matrix Wiener lter has the advantage that more than one plane of data could
b e included in the matrix equation If a Fourier Wiener lter is used the image planes
have to b e reconstructed one at a time and information from sources in other planes is
just treated as a contribution to the noise term The following simple example shows how
two planes could b e reconstructed simultaneously using a matrix Wiener lter One of
the sets of data is the one whichwas used in equation
p p p p
S H S H
p p p p
U U U
This can b e written in matrix form
C B
C B
C B
C B
B C C B B C
B C C B B C
C B
A A
C B
C B
C B
A
Image Reconstruction by Deconvolution
The two planes could b e reconstructed by simply inverting the matrix In the example
ab ove the matrix happ ens to b e singular but this is due to the numb ers whichwere used
The matrix Wiener lter provides a way to use a singular matrix for reconstruction H
T
may b e singular but H H can b e inverted if is large enough
S E
Application of Wiener lters to zone plate holog
raphy
If the hologram parameters given in Table are used the hologram is a pixel
array If matrix representation were used this would require inversion of a
matrix or larger As the Fourier Wiener lter is far more practical it is the one which
will b e investigated in this Chapter
The Fourier Wiener lter relies up on knowing the magnitude of the ratio of the Fourier
F
p
E
transform of the error term to the Fourier transform of the signal from the source
F
p
U
for all frequency values Two p ossible ways of estimating this will b e presented
Using the p ower sp ectral density
F
p
Press et al suggest that the p ower sp ectral density U should b e plotted to give
an indication of the relativecontribution of errors to the hologram This can b e done by
assuming that the p ower sp ectral density of the hologram is approximately equal to the
power sp ectral density of the signal from the source plus the p ower sp ectral densityofthe
noise
F
F F
p p
p
U E U
The p ower sp ectral density of the noise is also called the Wiener sp ectrum The Wiener
sp ectrum was mentioned as a to ol for analysing the errors in an image in Chapter Press
pF
E
is found using the graph whichisshown in Figure The et al illustrate how
F
p
U
contribution of the signal to the p ower sp ectrum should rst b e estimated A mathemat
ical mo del should then b e tted to this contribution Subtracting this from the p ower
sp ectrum will give an estimation of the noise term
Figure shows a theoretical ideal hologram with the parameters given in Table
and the same hologram with additive random noise taken from an uniform distri
Image Reconstruction by Deconvolution
power spectral density
|N^k|^2 - extrapolated log scale
|U^k|^2 - deduced
1 0.0
frequency ->
Figure Fitting a signal mo del and a noise mo del to the p ower sp ectrum taken
from Numerical Recip es by Press et al
bution This is arti cial a hologram would not have negative pixel values but it is just
designed to illustrate the e ect of adding noise The p ower sp ectra whichwere taken
as lines at y out from the centre of the two dimensional Fourier transforms of the
holograms are shown The e ect of adding noise to the hologram can b e clearly seen
but it would b e very di cult to nd a function which tted the sp ectrum of the ideal
F
p
E
hologram It would therefore b e impractical to apply this metho d for nding to
F
p
U
zone plate holography
10-4 ideal hologram hologram with 50% noise 10-6
max 10-8
10-10
10-12 amplitude of Fourier transform (log scale)
10-14 0.0 1000.0 2000.0 3000.0 frequency (w)
min
a ideal holo b hologram c p ower sp ectra
gram pixel with noise
value e e pixel
value e
Figure A theoretical ideal hologram and the same hologram with noise added
together with their p ower sp ectra
Image Reconstruction by Deconvolution
Finding a constantwhich gives the minimum error
As it is not obvious how to separate the signal from the noise an approximate Wiener
lter must b e found Two p ossibilities are given b elow
F
p
Assuming is constant E
One kind of noise in recorded signals is band limited white Gaussian noise It has equal
power p er hertz in some band of frequencies and a Gaussian distribution of amplitudes if
a large numb er of measurements of its amplitude are made In the discussion of sources
of noise in the hologram in Chapter some sources of error w ere listed which de nitely
F
p
could not b e describ ed as white noise However assuming E is constant for all
frequencies is a useful approximation If it is assumed that equation can b e used
F
p
equation for the reconstructed signal S b ecomes
F
F p
U K
p
U
F
B C
p
S
A
F
p
F
H
p
U
F
p
where K E is a constant K canbevaried b etween zero and the maximum
F F
F
p
p p
U U value of If K b ecomes negativeforany frequency S should b e set to
zero for that frequency The e ect of this lter is illustrated in Figure The hologram
used had the parameters given in Table and additive random noise taken from an
uniform distribution The variation of the NMSE and FEM values with K are also shown
in Figure These error metric values were de ned in Chapter and are a measure of
the di erence b etween the reconstructed image and the source ob ject This Wiener lter
will b e referred to as a K Wiener lter
pF
E
is constant Assuming
F
p
S
F
p
E
If were constant this would mean that the errors would b e highly correlated with
F
p
S
the source term As a large prop ortion of the errors arise from statistical uctuations
F
p
E
K into due to radioactive decay this is a sensible approximation Substituting
F
p
S
equation gives
F
F
p
p
U H
F
p
S
F
p
H K
Image Reconstruction by Deconvolution
max
min
a reconstruction b reconstruction c reconstruction
of hologram with of hologram with of hologram with
noise K istoo noise optimal noise K istoo
small pixel K pixel large pixel
value value value
107 102 NMSE (10% noise) NMSE (10% noise) 106 FEM (10% noise) FEM (10% noise)
105 101 104
103
102 100 101 error metric value (log scale) error metric value (log scale)
100
10-1 10-1 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-10 10-9
K1 (log scale) K1 (log scale)
d variation of NMSE and FEM e variation of NMSE and FEM
metric values with K for holo metric values with K for holo
grams with noise grams with noise close up of
minima
Figure E ect of varying K on the reconstructed image from a theoretical noisy
hologram
K is a constant which can b e varied b etween zero and in nity However it only makes
F
p
sense to increase it up to the maximum value of H The e ect of this lter is
illustrated in Figure Again the hologram used had the parameters given in Table
and additive random noise taken from an uniform distribution The variation of the
NMSE and FEM values with K isalsoshown in Figure
Comparing Figures and it can b e seen that this metho d gives lower error values
than the previous metho d This metho d will b e referred to as K Wiener ltering
Image Reconstruction by Deconvolution
max
min
a reconstruction b reconstruction c reconstruction
of hologram with of hologram with of hologram with
noise K istoo noise optimal noise K istoo
small pixel K pixel large pixel
value value value
104 NMSE (10% noise) NMSE (10% noise) FEM (10% noise) FEM (10% noise) 103
102
101
error metric value (log scale) 100 error metric value (log scale)
10-1 10-1 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-11
K2 (log scale) K2 (log scale)
d variation of NMSE and FEM e variation of NMSE and FEM
metric values with K for holo metric values with K for holo
grams with noise grams with noise close up of
minima
Figure E ect of varying K on the reconstructed image from a theoretical noisy
holograms
Normalisation
To Account for the E ect of K K on the Magnitude of
the Reconstructed Point
If the image from an ideal hologram of a p ointsourcewas reconstructed using deconvolu
p
tion a Wiener lter would not b e needed as there would b e no errors As H is equivalent
to the hologram of a central p oint source with unit magnitude in plane p reconstructing
p
a hologram of such a source would involve dividing the Fourier transform of H by itself
This will pro duce an array full of ones Dep ending up on the way that the inverse discrete
Image Reconstruction by Deconvolution
Fourier transform is de ned in the subroutine used taking the inverse Fourier transform
of an array full of ones will reconstruct the p oint source with either unit amplitude or
with amplitude N So in this one case the amplitude will either b e correct or can b e
corrected by dividing the output pixel values by N
If a Wiener lter is used the lter value will have a large e ect on the amplitude
For example if an ideal hologram with the parameters given in Table is reconstructed
without using a Wiener lter the reconstructed p oint has an amplitude If a K
Wiener lter is used and with K the reconstructed p oint has an amplitude e
Away to normalise the reconstructed images to the correct amplitude is to calculate
the ideal hologram from a unit source in the centre of the reconstruction plane and then
reconstruct this using the same Wiener lter as was used to reconstruct the image from
the hologram The pixel values in the reconstructed image can then b e normalised by
dividing them by the amplitude of the p oint reconstructed from the ideal hologram The
two Wiener lters are then given by the following equations
F
p
F
K U
pF
p
U H
F
F
F
p
p
H
j j
U
p
normalised S
F
p
K H
j j
maximum of F
F
p
H
j j
F
F
p
p
U H
F
F
p
K H
j j
p
or normalised S
F
p
H
j j
maximum of F
F
p
K H
j j
where F denotes an inverse Fourier transform
To account for part of the zone plate hologram falling
outside the detector
p
which accounted for the fact Equation in Chapter gave normalisation constants A
ij
that for p oints sources in some p ositions the zone plate shadow is not wholly recorded by
the detector but overlaps the edge This means that for some frequencies the amplitude
of the Fourier transform is reduced This in turn decreases the amplitude of the recon
structed p oint Although the relationship b etween reconstructed amplitude and overlap
Image Reconstruction by Deconvolution
is not as straightforward as that for convolution it has b een found that multiplying the
p
A
is an e ectiveway to normalise the output image amplitude by
p
A
ij
If the image from an ideal noiseless hologram of a p oint source is reconstructed using
deconvolution no Wiener lter will b e needed to reconstruct the plane containing the
source However if the image is reconstructed in di erent planes the errors due to the
di erence b etween the hologram and the theoretical shadow used for the reconstruction
will give rise to large errors in the image For this reason a theoretical hologram with
additive noise taken from a uniform distribution was used to investigate the e ect of
normalisation Again the hologram had the parameters given in Table The optimum
K Wiener lter was found for the plane containing the source and then this was used
for all the planes
p
A
Multiplying the image amplitude by do es help normalise the image amplitude for
p
A
ij
di erent x y p ositions However by comparing Figure to Figure it can b e seen
that the normalisation metho d used in convolution is more e ective The normalised K
Wiener lter app ears to have b etter z resolution than the unnormalised lter but this may
b e b ecause it had a higher optimal value of K The values for the normalised case were
found to b e K e for the p ointatz m and K e for the p ointat
z m For the unnormalised case these values were K e and K e
resp ectively Better results mayhave b een obtained by nding the optimal Wiener lter
for each plane individually but this would have b een very time consuming
E ect of a mis aligned zone plate
In this Chapter so far only metho ds using the binary zone plate pattern havebeen
discussed In Chapter techniques using b oth the binary plate and Fresnel functions
were discussed The techniques which used the binary plates were not found to work as
well as the ones which used Fresnel functions Although the binary plate is the obvious
pattern to use for deconvolution other p ossibilities should also b e investigated
One disadvantage of using the binary zone plate pattern is that it needs to b e aligned
correctly When a zone plate hologram is recorded it is p ossible that the zone plate may
not b e correctly aligned with the detector The zone plate may b e rotated by an angle
up to that of half a sector with resp ect to the exp ected alignment This is illustrated
Image Reconstruction by Deconvolution
point at x=y=0 point at x=y=0 point at x=y=0.059 point at x=y=0.059 1.0 1.0
0.5 0.5
value of reconstructed image pixel 0.0 value of reconstructed image pixel 0.0
-0.20 -0.10 0.00 0.10 0.20 -0.20 -0.10 0.00 0.10 0.20
distance from centre of plane along diagonal (m) distance from centre of plane along diagonal (m)
a p oints separated in x y unnor b p oints separated in x y nor
malised K Wiener lter malised K Wiener lter
point at p=0.22m point at p=0.22m point at p=0.28m point at p=0.28m 1.0 1.0
0.5 0.5
0.0
0.0 value of reconstructed image in central pixel value of reconstructed image in central pixel
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.10 0.15 0.20 0.25 0.30 0.35 0.40
position p (m) position p (m)
c p oints separated in z unnor d p oints separated in z nor
malised K Wiener lter malised K Wiener lter
Figure Magnitudes of theoretical reconstructed p oints in di erent p ositions for b oth
unnormalised and normalised K Wiener ltering of a hologram containing additive
noise taken from a uniform distribution
in Figure The hologram with noise used in the previous Section is aligned with
Figure a It was reconstructed using a optimal K Wiener lter and the zone plate
shadows in each of Figures a b and c A line of data was then taken
across the diagonal of each reconstructed image and this can b e seen in Figure From
this Figure it is clear that the zone plate b eing mis aligned has a large e ect on the
reconstructed image
Image Reconstruction by Deconvolution
a alignment of holo b zone plate pattern c zone plate pattern
gram rotated by a quarter of rotated by half a sec
a sector tor
Figure How the exp ected zone plate can b e mis aligned with the actual zone plate
1.0 correctly aligned zone plate rotated by half a sector rotated by a full sector 0.8
0.6
0.4
0.2
value of reconstructed image pixel 0.0
-0.2 -0.010 -0.005 0.000 0.005 0.010
distance from centre of image plane (m)
Figure E ect of a mis aligned zone plate on the p oint spread function
Using a truncated Gab or zone plate with a Wiener
lter
Pure Gab or zone plates were discussed in Chapter They are the typ e of zone plate
whichwould make a zone plate hologram as muchlikea conventional optical hologram as
p ossible Binary zone plates are an approximation of this typ e of plate Gab or zone plates
are radially symmetric so the alignment of the plate is not an issue For this reason it is
suggested that a Gab or plate could b e used for the reconstruction as well as the binary
plate pattern If p o or results were obtained when the binary plate was used this could
b e used to check that this was not due to mis alignment of the zone plate The zone
plate should b e truncated at the same radius as the binary plate and have all the same
parameters This metho d will b e referred to as Gab or Wiener ltering As b efore the
Image Reconstruction by Deconvolution
optimal lter could b e a K oraK lter Again a K Wiener lter is used here as it
was found to pro duce b etter results
Point spread functions for this metho d and a hologram with additive noise and
the parameters in Table are shown in Figure
1.0 unnormalised 1.0 unnormalised normalised normalised 0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2 0.0
value of reconstructed image pixel 0.0 value of reconstructed image pixel -0.2
-0.2 -0.4 -0.010 -0.005 0.000 0.005 0.010 0.20 0.21 0.22 0.23 0.24
distance from centre of image plane (m) position p (m)
a PSF along diagonal x y b PSF in z direction
plane
Figure Point spread functions for b oth unnormalised and normalised K Gab or
Wiener ltering of a hologram containing additive noise taken from a uniform distri
bution
Comparison of reconstructed points using error
metrics
This Section can b e compared with Section for convolution reconstruction For a single
p oint source the MSE value is just a constantmultiple of the NMSE value Therefore
only the NMSE and FEM values are given here The image from the hologram with
noise used in the previous Sections was reconstructed using an optimal K Wiener lter
and an optimal K Gab or Wiener lter The NMSE and FEM values for b oth normalised
and unnormalised reconstructions can b e seen in Table Comparing this to Table
for the convolution reconstruction the K andK binary Wiener lter metho ds have
far b etter FEM and NMSE values than anyoftheconvolution metho ds The Gab or K
Wiener lter has b etter FEM values than all the matched lter metho ds but the Gab or
K Wiener lter has worse FEM valuesthanmostoftheconvolution techniques
Image Reconstruction by Deconvolution
NMSE FEM
K Wiener lter
K Wiener lter normalised
K Wiener lter
K Wiener lter normalised
Gab or K Wiener lter
Gab or K Wiener lter normalised
Gab or K Wiener lter
Gab or K Wiener lter normalised
Table Error metric values for the di erent Wiener lter reconstruction techniques
Comparison of Mo dulation Transfer Functions
This Section can b e compare with Section for the convolution metho ds The mo du
lation transfer functions MTF are shown in Figure for the di erent deconvolution
reconstruction metho ds As in Section the NMSE and FEM values for the di erence
between the MTFs and an array lled with ones have b een calculated Again the K
binary Wiener lter has the lowest NMSE and FEM values The values are lower for the
unnormalised images than for the normalised images All the NMSE and FEM values
for the binary Wiener lters are lower than those for all the convolution techniques The
Gab or K Wiener lter has lower values than most of the convolution metho ds and the
K Wiener lter has higher values than most
max
min
a K b K c Gab or K d Gab or K
Wiener lter Wiener lter Wiener lter Wiener lter
pixel pixel e pixel e pixel
value value value value
Figure Mo dulation transfer functions
Image Reconstruction by Deconvolution
NMSE FEM
K Wiener lter
K Wiener lter normalised
K Wiener lter
K Wiener lter normalised
Gab or K Wiener lter
Gab or K Wiener lter normalised
Gab or K Wiener lter
Gab or K Wiener lter normalised
Table Comparison of mo dulation transfer function error metric values for the di erent
deconvolution reconstruction techniques
Image Reconstruction by Deconvolution
Comparison of p oint spread functions
As in Section for the convolution metho ds one dimensional Gaussian functions were
tted to the reconstructed p oints The function tted in direction x p x was given by
jx j
p x A exp
This was tted along the x y diagonal of the plane containing the image of the p oint
source and also in the z direction The p oints whichwere analysed were reconstructed by
using an optimal normalised K Wiener lter and Gab or Wiener lter The images have
to b e reconstructed in a series of planes to nd the z resolution and it would havebeen
very time consuming to do this for each deconvolution metho d The results can b e seen
in Table Comparing these results to the equivalent ones for the convolution metho ds
given in Table b oth Wiener lter metho ds app ear to have b etter resolution in b oth
the x y and z directions This will b e investigated later with di erent holograms
A R squared
x y diagonal
K Wiener lter e
Gab or K Wiener lter e
z direction
K Wiener lter e e e
Gab or K Wiener lter e e
Table Point spread functions for the di erent Wiener lter techniques
Comparison of Images of Uniform Sources
Images of two dimensional uniform sources were reconstructed using the deconvolution
metho ds This was done for the convolution metho ds in Section The same hologram
was used here as in Section Unlike the hologram of the p oint source no noise was
added Part of the shadowcastbya two dimensional uniform source would not havebeen
recorded by the hologram The hologram therefore already contained inherent errors
and required Wiener ltering
The unnormalised images can b e seen in Figure and the normalised images in
Figure The NMSE and FEM values are given in Table The K Wiener lter
has lower values than th K Wiener lter and unexp ectedly the Gab or K Wiener lter
Image Reconstruction by Deconvolution
has lower values than the standard K Wiener lter In general the NMSE and FEM
values are comparable to those for the convolution metho ds
max
min
a K Wiener l b K Wiener l c Gab or K d Gab or K
ter pixel ter pixel Wiener lter Wiener lter
value value pixel pixel
value value Figure Unnormalised images of uniform sources
max
min
a K b K Wiener l c Gab or K d Gab or K
Wiener lter ter pixel Wiener lter Wiener lter
e pixel value pixel pixel
value e value value
Figure Normalised images of uniform sources
NMSE FEM
K Wiener lter e
K Wiener lter normalised e
K Wiener lter
K Wiener lter normalised e
Gab or K Wiener lter e
Gab or K Wiener lter normalised e
Gab or K Wiener lter
Gab or K Wiener lter normalised e
Table Comparison of error metric values for images of uniform sources for the di erent
deconvolution reconstruction techniques
Image Reconstruction by Deconvolution
Summary
Metho ds of reconstruction by deconvolution have b een outlined The need for a Wiener
lter has b een explained and di erent metho ds for nding the optimal lter havebeen
given A way to normalise the image amplitude has also b een given A metho d using a
Gab or zone plate pattern has b een intro duced to overcome e ects caused by mis alignment
of the zone plate Error metric values mo dulation transfer functions and p oint spread
functions have b een calculated and shown to b e b etter than those for convolution metho d
This technique however has the disadvantage of having to nd an optimal value of K
or K The metho d giving the most favourable results for theoretical p oint sources was
the K Wiener lter
Chapter
IMAGE RECONSTRUCTION
USING THE CLEAN ALGORITHM
The CLEAN algorithm was invented byH ogb om in for use in high resolution radio
interferometry The CLEAN algorithm wasoneofanumb er of techniques which
H ogb om presented in his pap er To quote H ogb om
The pro cedure that app ears to work the b est at the time of writing has b ecome
known as CLEAN It is an iterative pro cedure that op erates in the map plane
and which uses the known shap e of the dirty b eam to distinguish b etween real
structure and sidelob e disturbances in the dirty map
No explanation is given for the pro cedure b eing called CLEAN In this case the dirty
map would b e the hologram and the dirtybeamwould b e the zone plate pattern that is
the shadow cast by a p oint source The metho d works by subtracting the contribution of
each p oint to the hologram a p ointatatimeuntil very little remains
The CLEAN algorithm has b een used in microscopy and radioastronomy
for image reconstruction There are manyvariations on this basic algorithm and these
are outlined on the Mathworld web site This algorithm is however usually used
to reconstruct two dimensional images in one plane The version of the algorithm used
here p erforms the following iteration pro cedure to reconstruct an image in anynumber of
planes in the z direction
Set all pixel values in the reconstructed image planes to zero
Reconstruct the image in all the planes by normalised convolution The resulting
image will b e referred to as the hologram image As the reconstruction is normalised
the hologram image pixel values should roughly corresp ond to the source intensity
p er unit area for each p osition
Image reconstruction using the CLEAN algorithm
Find the maximum hologram image pixel value and then increment the corresp ond
ing reconstructed image pixel by a given amount
c
Subtract the contribution that a source in this p osition with strength would have
c
had from the hologram data
Go back to step numb er and rep eat this pro cess until the maximum value of the
hologram image is less than
c
An alternative to this sequence would b e to replace the word convolution by deconvolution
in the ab ove sequence Potentially any of the metho ds outlined in Chapters and could
b e used As the truncated Fresnel function and K Wiener lter metho ds were found to
b e the most promising these are the ones that will b e used in this work
The convolution or deconvolution reconstruction may not reconstruct all the image
pixels with the correct amplitude If it did this technique would not b e needed What is
imp ortant however is that the image pixel with the maximum amplitude corresp onds to
a p osition where there is a source If is to o high there is a risk that the pixel will b e
c
given to o high an amplitude and to o much will b e subtracted from the hologram for that
p osition If this happ ens the b ehaviour of the algorithm will b e unpredictable A higher
value of will mean the image is reconstructed in less time but a lower value should
c
give a more accurate image
Assessment of Image Quality
To compare this technique with convolution and deconvolution holograms of p oint sources
with the same parameters as those in Chapters and were reconstructed using b oth
the convolution based and the decon volution based CLEAN algorithms In each case the
reconstructed p oints were in the correct pixels and all the other pixel values were still
zero This was still the case when the reconstruction planes were placed m apart
The accuracy of the amplitude of the reconstructed p ointdependedonthevalue of
c
with a lower value of giving a more accurate value Finding the FEM and NMSE error
c
metric values is therefore meaningless here A Gaussian can also not b e tted to a single
pixel value surrounded by zeros
Image reconstruction using the CLEAN algorithm
As only one pixel value is changed byeach iteration the time this reconstruction
metho d takes is much longer than convolution and deconvolution metho ds The time it
takes is the numb er of iterations multiplied by the time for one iteration The number
of iterations dep ends up on the value of and the size and strength of the source The
c
time taken for one iteration is slightly more than the time taken to reconstruct the same
image by either convolution or deconvolution whichever one is used
Summary
The application of the CLEAN algorithm to the reconstruction of three dimensional im
ages has b een explained This algorithm can b e based on either a convolution or a decon
volution reconstruction metho d The results for theoretical p oint sources are incredibly
go o d but the CLEAN algorithm reconstruction takes many times longer to run than con
volution or deconvolution reconstruction The time needed dep ends up on the size and
strength of the source as well as the numb er of pixels in the hologram the number of
image planes and the amountbywhich an image pixel is incrementedby one iteration
This algorithm app ears to work well for p oint sources This could have b een exp ected
as it was originally designed to image p oint sources It has only b e shown to work well for
extended sources when given a reasonable starting mo del but no a priori knowledge
is used to reconstruct the images in Chapters and The algorithm used in this work
is similar to the basic H ogb om CLEAN algorithm but is used for images in more than
one plane There are manyvariations of the CLEAN algorithm whichhave b een shown
to achieve greater sp eed and pro duce more realistic images but investigating all these
ted here di erent p ossibilities is b eyond the scop e of the work presen
Chapter
RECONSTRUCTION OF
COMPUTER GENERATED
HOLOGRAMS
In this Chapter some more theoretical holograms will b e reconstructed using the three
reconstruction metho ds whichwere explained in Chapters and The convolution
metho d which will b e used is convolution with a truncated Gab or zone plate as this was
found to give the b est results for the p oint sources in Chapter K Wiener ltering
will b e the deconvolution metho d used here as this was the b est metho d for p oint sources
in Chapter First the reconstruction techniques will b e tested on images from an
ideal theoretical hologram of a simple one pixel thick cross source The parameters of
this hologram will again b e those given in Table The zone plate dimensions and
the pixel size in this table are the same as those of the real exp erimental holograms in
Chapter
The theoretical holograms will b e calculated using the relationship b etween source
distribution given by equation As in equation this relationship assumes that
the source to zone plate p and zone plate to detector q distances are much larger than
the dimensions of the detector In this case the distance a gamma raytravels from a
p oint in the source to the detector is approximately equal to p q no matter where
the p oint is is the eld of view In practise for the parameters commonly used for zone
plate holography p and q would b e to o small for this to b e a go o d approximation The
work in this Chapter however will just attempt to give an indication of whether the
metho ds should work or not and will not include all of the factors whicha ectazone
plate hologram The e ciency of the gamma camera and the fact that the zone plate is
several millimetresthick for example are other factors which will not b e included here
The e ect of adding other sorts of errors to an ideal hologram will however b e investi
Reconstruction of Computer Generated Holograms
gated These are the errors whichwere mentioned in Chapter and they will b e measured
using the error metrics given in Chapter Images from a hologram of two di erent cross
sources in two di erent planes will then b e reconstructed Next the reconstruction meth
o ds will b e used to reconstruct images from ideal computer generated holograms of the
same ob ject but where the size of the ob ject is di erent in di erent holograms Finally
the metho ds will b e used to reconstruct images from holograms of the same ob ject but
with di erent parameters
Hologram of a Simple Two Dimensional Cross
Source Placed Parallel to the Zone Plate
Reconstruction of Images from an Ideal Hologram
An ideal hologram of the cross source shown in Figure a was calculated using the
parameters in Table This hologram was reconstructed using the di erent meth
ods in ve planes each separated by cm where the image should app ear in the middle
plane First the image was reconstructed using normalised convolution and normalised
K Wiener ltering As was explained in Chapter the CLEAN algorithm needs to b e
based on another image reconstruction technique As reconstruction using the CLEAN
algorithm takes many times longer it seemed sensible to reconstruct the image using the
other metho ds and then decide which one of these metho ds the CLEAN algorithm should
b e based on
convolution real part convolution intensity K Wiener lter
FEM e e e
NMSE e e e
Table Error metrics for the normalised image reconstruction of a hologram of the
cross source
To nd the optimum value of K K was rst set equal to the maximum value of the
square of the magnitude of the Fourier transforms of ideal holograms of p oints in each
plane This was used as a starting p ointaswe can b e fairly certain that the errors do
not have a greater contribution than the maximum frequency contribution of the signal
The image was reconstructed using this value of K The hologram that would havebeen
generated by this reconstructed image was then calculated The FEM error metric for
Reconstruction of Computer Generated Holograms
a source ob ject b theoreti c convolution real
pixel value cal hologram part pixel
e pixel value
value e
max
min
d convolu e K Wiener l
tion intensity ter pixel
e pixel value
value
Figure Normalised reconstruction byconvolution and K Wiener lter
the di erence b etween this calculated hologram and the actual hologram were then found
Next K was divided by ten and the whole pro cess was rep eated and the new FEM value
was found K continued to b e divided bytenuntil the FEM value stopp ed decreasing
and increased The previous value of K was then assumed to b e the optimal value
First images were reconstructed using normalised convolution and deconvolution The
images reconstructed in the middle plane can b e seen in Figure and all the recon
structed image planes can b e seen in Figures and Table gives the FEM and
NMSE error metric values for the di erence b etween the source ob ject and the image
over all ve planes The error metric values con rm what is obvious from lo oking at the
Figures K Wiener ltering gives a b etter image than convolution In the calculations
p erformed in Chapter it was predicted that the intensity of a pure Gab or zone plate
Reconstruction of Computer Generated Holograms
hologram of a p oint source should givea point image but that the real part would have
other terms in the background The intensitywas also found to give a b etter reconstruc
tion for binary Gab or zone plate holograms of p oint sources in Chapter For this cross
source however the real part gives a b etter reconstruction
convolution real part convolution intensity K Wiener lter
FEM e e e
NMSE e e e
Table Error metrics for the partially normalised image reconstruction of a hologram
of the cross source
As was explained in Chapter for some p oints the whole of the zone plate pattern is
not recorded by the gamma camera This was accounted for in the normalisation of the
convolution reconstruction Although this helps p oints towards the edge of the hologram
have the correct magnitude it also ampli es errors increasing with distance from the
centre In Chapter an attempt was also made to account for this overlap problem
in the Wiener ltering metho ds The results were however not as satisfactory as it was
more di cult to assess the e ect that overlapping has in this case Partially normalised
reconstructions where the overlap factor was not included were therefore calculated
The images reconstructed in the middle plane can b e seen in Figure and all the
reconstructed image planes can b e seen in Figures and Table gives the FEM
and NMSE error metric values for the di erence b etween the source ob ject and the image
over all ve planes The Figures showanobvious di erence for the intensityofthe
convolution but the real part of the convolution and the K Wiener lter lo ok very
similar The FEM and NMSE error metric values are however all lower than those for
the fully normalised reconstruction The partial normalisation metho d therefore app ears
to b e b etter
CLEAN convolution real part CLEAN K Wiener lter
FEM e e
NMSE e e
Table Error metrics for the partially normalised image reconstruction of a hologram
of the cross source using the CLEAN algorithm
The partially normalised convolution real part and K Wiener lter metho ds were
used as the basis for the CLEAN algorithm the incrementby which a pixel value
Reconstruction of Computer Generated Holograms
is increased by one iteration was set to the maximum value of the image reconstructed
by convolution or deconvolution divided by ten for convolution was greater than that
for deconvolution so more iterations would have b een needed Deconvolution also takes
longer b ecause there are more arithmetic op erations involved The convolution based
CLEAN algorithm reconstruction ran overnightandwas nished by the morning When
the reconstruction using the deconvolution based CLEAN algorithm had not nished two
days later was decreased to half of the maximum value of the image reconstructed by
deconvolution
The results for the ob ject plane can b e seen in Figure and in all ve planes in
Figure The CLEAN algorithms mainly reconstructed p oints in the correct plane
most of the values in the other three planes remained equal to zero When straightforward
convolution and deconvolution were used this was not the case The corresp onding FEM
and NMSE error metric values are given in Table
Reconstruction of Computer Generated Holograms
a real part
b intensity
Figure Normalised reconstruction byconvolution
Reconstruction of Computer Generated Holograms
Figure Normalised reconstruction by K Wiener lter
max
min
a convolution real b convolu c K Wiener l
part pixel tion intensity ter pixel
value e pixel value
value
Figure Partially normalised reconstruction byconvolution and K Wiener lter
Reconstruction of Computer Generated Holograms
a real part
b intensity
Figure Partially normalised reconstruction by convolution
Reconstruction of Computer Generated Holograms
Figure Partially normalised reconstruction by K Wiener lter
max
min
a CLEAN algorithm b CLEAN algo
convolution pixel rithm deconvolution
value pixel value
Figure Partially normalised reconstruction using the CLEAN algorithm
Reconstruction of Computer Generated Holograms
a algorithm based on convolution
b algorithm based on deconvolution
Figure Partially normalised reconstruction using the CLEAN algorithm
Reconstruction of Computer Generated Holograms
Reconstruction of Images from Holograms with Added Er
rors
Four di erenttyp es of error were added to the ideal hologram of the cross source to see
the e ect it had on the image These errors will b e numb ered ➀ ➁ ➂ and ➃ The errors
included in each one are listed b elow
➀ The source magnitude was replaced bya number taken from a Poisson distribution
with a mean of for each pixel containing the source As a Poisson distribution has
a standard deviation equal to the square ro ot of the mean the standard deviation of
this distribution would have b een This was intended to simulate a source of
magnitude with ten p ercent noise due to statistical uctuations The theoretical
hologram was then calculated using this source
➁ The ideal hologram was convolved with a two dimensional Gaussian function with
FWHM equal to cm which is the spatial resolution of a Toshiba GCA A
gamma camera given in Table This was in order to simulate the fact that the
gamma camera stores the data in pixels which are smaller than the resolution of the
camera
➂ Noise taken from a uniform distribution b etween zero and ten p ercent of the maxi
mum ideal hologram value was added to the ideal hologram This was to simulate
errors due to oating p oint arithmetic and counts from background radiation The
minimum value of the noise was zero so that none of the hologram values would
b ecome negative This is sensible b ecause the output of the gamma camera do es
not contain negativevalues
➃ The thickness of the cross source was increased from one to ve pixels This was
to see if the reconstruction techniques worked as well for a source whichcovered a
larger numb er of pixels next to each other
Images were reconstructed from these holograms in the same ve planes as for the ideal
hologram Section using convolution deconvolution and the CLEAN algorithm
Corresp onding error metric values to those given for the ideal hologram can b e seen
Reconstruction of Computer Generated Holograms
in Tables and Values for the intensity of the convolution reconstruction are not
included as this was found to givea worse reconstruction than the real part in Section
➀ and All of the reconstructed images will not b e shown here as those from holograms
➂ do not vary noticeably from those for the ideal hologram In these cases the CLEAN
reconstructions also only reconstructed p oints in the correct plane When images were
reconstructed from holograms ➁ and ➃ using the CLEAN algorithms p oints were also
reconstructed in the planes next to the image plane The partially normalised recon
structed image in all ve planes can b e seen in Figures and for hologram ➁ and
Figures and for hologram ➃
convolution real part K Wiener lter
➀ FEM e e
NMSE e e
➁ FEM e e
NMSE e
➂ FEM e e
NMSE e e
➃ FEM e e
NMSE e e
Table Error metrics for the partially normalised image reconstruction of noisy holo
grams of the cross source using convolution and deconvolution
CLEAN convolution real part CLEAN K Wiener lter
➀ FEM e e
NMSE e e
➁ FEM e e
NMSE e e
➂ FEM e e
NMSE e e
➃ FEM e e
NMSE e
Table Error metrics for the partially normalised image reconstruction of noisy holo
grams of the cross source using CLEAN algorithms
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction for hologram ➁ byconvolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
a algorithm based on convolution
b algorithm based on deconvolution
Figure Partially normalised reconstruction for hologram ➁ using CLEAN algorithms
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction for hologram ➃ byconvolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
In summary errors representing statistical uctuations in the source strength holo
➀ and recording noise or counts due to background radiation hologram ➂ had gram
less of an e ect on the reconstructed image than the limited spatial resolution of the
gamma camera hologram ➁ This was just done for one set of hologram parameters and
one theoretical source ob ject The magnitudes of each of these errors would dep end up on
the exact details of the exp eriment for example source strength These images however
give an indication of the sort of e ect these errors have
In the case of K Wiener ltering this e ect can b e lessened by deconvolving the
hologram with a zone plate pattern whichisalsoblurredby the same Gaussian function
instead of an ideal zone plate pattern For a partially normalised reconstruction this
reduced the FEM error metric value to e but strangely increases the NMSE
error to e If convolution is used the blurred zone plate pattern can b e used
to calculate the magnitude of the reconstruction of an ideal p oint source instead of the
pure zone plate pattern and can hence improve the normalisation Again this reduces
the FEM error value to e but increases the NMSE value to e
From Table it can b e seen that the CLEAN algorithm based on deconvolution gave
consistently lower error metric values than the CLEAN algorithm based on convolution
The algorithm based on deconvolution will therefore b e used for the rest of the holograms
in this Chapter
Reconstruction of Images from Holograms of Ob
jects of Di erent Sizes with the Same Parameters
The same ob ject was used for each of these four holograms It was some text CEJ in the
centre on the central plane This was chosen b ecause it is only slightly more complicated
than a cross source and b ecause its size can easily b e changed bychanging the font size
The parameters for the hologram were again those in Table Four holograms were
generated with increasing font size and will b e represented by and The
size of the ob ject of which the holograms were calculated increase from hologram to
hologram
Figures and show the images from hologram in veplaneseach separated
by cm The ob ject should app ear in the middle plane Figures and showthe
Reconstruction of Computer Generated Holograms
images from hologram Figures and those from hologram and Figures
The corresp onding error metric values are and show the image from hologram
given in Table
The error metric values for images reconstructed using the CLEAN algorithm increase
as the size of the ob ject increases The NMSE values also increase with ob ject size
for convolution and deconvolution Thereisnoobvious trend in the FEM values for
convolution and the FEM values decrease with ob ject size for deconvolution
Reconstruction of Computer Generated Holograms
a algorithm based on convolution
b algorithm based on deconvolution
Figure Partially normalised reconstruction for hologram ➃ using CLEAN algorithms
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction of hologram by convolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
Figure Partially normalised reconstruction of hologram using the deconvolution
based CLEAN algorithm
Figure Partially normalised reconstruction of hologram using the deconvolution
based CLEAN algorithm
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction of hologram by convolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction of hologram by convolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
Figure Partially normalised reconstruction of hologram using the deconvolution
based CLEAN algorithm
Figure Partially normalised reconstruction of hologram using the deconvolution
based CLEAN algorithm
Reconstruction of Computer Generated Holograms
convolution real part K Wiener lter CLEAN algorithm
FEM e e e
NMSE e e e
FEM e e e
NMSE e e e
FEM e e e
NMSE e e e
FEM e e e
NMSE e e e
Table Error metrics for the partially normalised image reconstruction of holograms
of the text source ob ject using convolution deconvolution and the CLEAN algorithm
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
by convolution and K Figure Partially normalised reconstruction of hologram
Wiener lter
Reconstruction of Computer Generated Holograms
Reconstruction of Images from Holograms of the
Same Ob ject with Di erentParameters
A ve p ointed star with a hole in the middle was used here This was chosenasitisa
more extended ob ject than the ones used in the previous sections Five holograms were
generated of the ob ject The rst hologram had the parameters from Table in the
other four holograms either the zone plate to detector distance or the source to zone plate
distance was changed The hologram parameters for eachofthe ve holograms are given
in Table
Parameter holograms value Parameter holograms value
First zone radius all e m Detector pixel size all e m
Source zone plate ➊➍➎ m Zone plate detector ➊➋➌ m
➋ m ➍ m
➌ m ➎ m
Numb er of pixels all Numb er of zones all
Table Hologram parameters used for the holograms of the star ob ject
Figures and show the images reconstructed from hologram ➊ and Figures
➋ Figures and show the images from hologram and those from hologram
➌ Figures and those from hologram ➍ and Figures and those from
hologram ➎ The NMSE and FEM values are given in Table
The only conclusion that can b e drawn from these results is that the images lo ok
b etter when the ob ject is smaller in the eld of view of the detector holograms ➌ and
➎ than when it is larger holograms ➋ and ➍ Holograms ➌ and ➎ havelower error
metric values for the images reconstructed using the CLEAN algorithm However there
is no obvious trend in the results for convolution or deconvolution
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction of hologram ➊ byconvolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
Figure Partially normalised reconstruction of hologram ➊ using the deconvolution
based CLEAN algorithm
Figure Partially normalised reconstruction of hologram ➋ using the deconvolution
based CLEAN algorithm
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction of hologram ➋ byconvolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction of hologram ➌ byconvolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
Figure Partially normalised reconstruction of hologram ➌ using the deconvolution
based CLEAN algorithm
Figure Partially normalised reconstruction of hologram ➍ using the deconvolution
based CLEAN algorithm
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction of hologram ➍ byconvolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction of hologram ➎ byconvolution and K
Wiener lter
Reconstruction of Computer Generated Holograms
Figure Partially normalised reconstruction of hologram ➎ using the deconvolution
based CLEAN algorithm
convolution real part K Wiener lter CLEAN algorithm
➊ FEM e e e
NMSE e e e
➋ FEM e e e
NMSE e e e
➌ FEM e e e
NMSE e e e
➍ FEM e e e
NMSE e e e
➎ FEM e e e
NMSE e e e
Table Error metrics for the partially normalised image reconstruction of holograms
of the star shap ed source using convolution deconvolution and the CLEAN algorithm
Reconstruction of Two Cross Sources in Di erent
Planes
The rst one of the two cross sources used here is the same as the one in Section
and has the same parameters it is p ositioned at m b ehind the zone plate The other
Reconstruction of Computer Generated Holograms
cross is a diagonal one and is p ositioned m from the zone plate Image planes were
reconstructed at m m m and m The reconstructed images in all ve
planes can b e seen in Figures and and the corresp onding error metric values are
in Table
The image from the CLEAN algorithm showed the two crosses mainly in the correct
planes Very little was reconstructed in the surrounding planes Convolution and decon
volution gave almost as much in the surrounding planes as in the planes which actually
contained the sources
Reconstruction of Computer Generated Holograms
a convolution real part
b K Wiener lter
Figure Partially normalised reconstruction for hologram of two cross sources by
convolution and K Wiener lter
Reconstruction of Computer Generated Holograms
Figure Partially normalised reconstruction for hologram of two cross sources using
the CLEAN algorithm
convolution deconvolution CLEAN deconvolution
FEM e e e
NMSE e e e
Table Error metrics for the partially normalised image reconstruction of a hologram
of two cross sources
Summary
The three reconstruction techniques have b een tested for a variety of computer generated
holograms Thee reconstructed images had lower error metric values when the overlap
factor describ ed in Section was not included These partially normalised reconstruc
tion metho ds were then used to reconstruct the rest of the images in the Chapter The
CLEAN algorithm wasfoundtowork b etter when it was based on deconvolution than
when it was based on convolution so this metho d was used
The e ect of di erenttyp es of error increasing ob ject size and di erent hologram
parameters were investigated Although a more thorough investigation would b e required
to draw de nite conclusions these results gave some indications of the sources of error in
Reconstruction of Computer Generated Holograms
the reconstructed images These results suggest that the nite resolution of the gamma
camera is a ma jor contributing factor to errors in the reconstructed image for these
reconstruction metho ds They also indicate that the errors in the image are generally
larger for ob jects which are larger in the eld of view of the detector
Chapter
RECONSTRUCTION OF REAL
HOLOGRAMS
The zone plate holograms which will b e used in this Chapter were recorded byPerks
as part of his PhD pro ject Perks recorded holograms using a Toshiba GCA A
gamma camera at St Thomas hospital London Since then Rew has recorded many
holograms using a GE Maxicamera in the Physics and Astronomy departmentofthe
University of Birmingham The holograms recorded using the GE camera are however
of p o orer quality than those recorded on the Toshiba camera The spatial resolution was
given in Table for all these cameras The GE camera has a spatial resolution of mm
and the Toshiba camera has a spatial resolution of mm Although the spatial resolution
of the GE camera is only slightly worse than that of the Toshiba camera the GE camera
needs to re adjusted and this has made the images worse than could b e exp ected
Takhar and George are nowworking on their PhDs on this pro ject and have
started taking images using an ADACForte gamma camera whichisalsointhePhysics
and Astronomy department The images from this camera are comparable to those from
the Toshiba camera It will b e left to them to test these reconstruction techniques further
on real holograms taken using the ADAC camera Suggestions for further work they can
do to test the reconstruction techniques are given in Chapter
Cross Source Placed Parallel to the Zone Plate
Perks recorded two holograms of a cross source made of four capillary tub es lled with
technicium m The bungs holding the Technicium in the tub es were all placed together
in the centre of the cross so there was a region of no activity here The two holograms
had the same source to zone plate distance m but di erent zone plate to detector
distances m and m The other parameters can b e seen in Tables and
Reconstruction of Real Holograms
Note that the pixel size rst zone radius and numb er of zones in the zone plate are all
the same as those used for the computer generated holograms
Figure Cross source recorded using the collimator taken from Perks thesis
Figure shows an image of the cross source taken using the gamma camera with
the collimator still attached This picture was taken from Perks thesis The region of no
activity in the centre of the cross can not b e seen here
First zone radius r e m Source zone plate distance z m
Detector pixel size e m Zone plate detector distance b m
q
R
Numb er of pixels n n Numb er of zones n
x y
r
Table Parameters for the rst hologram of the cross source
a convolution b K c CLEAN d CLEAN
real part Wiener lter algorithm algorithm
e pixel e pixel convolution pixel deconvolution pixel
value e value e value e value e
Figure Partially normalised reconstruction for the rst real hologram of the cross
source in the ob ject plane white minimum black maximum pixel value
Figures and show the partially normalised reconstructions of the image
of the cross for the rst hologram This technique was shown to b e b etter than the fully
normalised reconstruction in Chapter The deconvolution based CLEAN algorithm gave
b etter results than the convolution based CLEAN algorithm in Chapter This metho d
was then used for the remaining computer generated holograms It takes a long time to
Reconstruction of Real Holograms
reconstruct images using the CLEAN algorithm so it did not seem sensible to try b oth
techniques for all the computer generated holograms However as there are only a limited
numb er of real holograms available b oth techniques will b e tried here
Figure shows the image in the ob ject plane and Figures and show
the image in all ve planes Table gives the corresp onding error metric values
As the ob ject is not exactly know the error values were calculated by comparing the
actual hologram to the theoretical hologram whichwould have b een pro duced bythe
reconstructed source The deconvolution reconstruction gives the b est FEM error metric
value and the convolution based CLEAN reconstruction gives the b est NMSE error metric
value in this case The convolution based CLEAN reconstruction only reconstructed
p oints in the images plane but the p oints were incremented by a large value in each
iteration to make the reconstruction b e completed in a reasonable amount of time The
p oints were incrementedby the maximum value of the image reconstructed byconvolution
or deconvolution divided by ten If a smaller value had b een chosen the reconstruction
would havetaken longer but the image mayhave b een b etter
Reconstruction of Real Holograms
a convolution real part
b K Wiener lter
Figure Reconstruction by convolution and K Wiener lter of the rst real hologram
of the cross source in all ve planes
Reconstruction of Real Holograms
a algorithm based on convolution
b algorithm based on deconvolution
Figure Reconstruction using the CLEAN algorithm of the rst real hologram of the
cross source in all ve planes
Reconstruction of Real Holograms
convolution deconvolution CLEAN convolution CLEAN deconvolution
FEM e e e e
NMSE e e e e
Table Error metrics for the partially normalised image reconstruction of the rst
real hologram of the cross source
The second hologram of this source has a larger zone plate to detector distance see
Table This means that the eld of view of the camera through the zone plate is
smaller for any given distance This is why the cross app ears to b e larger in the image
planes Figure shows the image in the ob ject plane and Figures and show
the image in all ve planes Table gives the corresp onding error metric values
The convolution based CLEAN metho d again only reconstructed the image in the
correct plane whereas the deconvolution based CLEAN metho d gave a few stray p oints
in other planes For this hologram b oth CLEAN metho ds gave almost equally low error
values whichwere b etter than either the straightconvolution or deconvolution
First zone radius r e m Source zone plate distance z m
Detector pixel size e m Zone plate detector distance b m
q
R
Numb er of pixels n n Numb er of zones n
x y
r
Table Parameters for the second hologram of the cross source
a convolution b K c CLEAN d CLEAN
real part Wiener lter algorithm algorithm
e pixel e pixel convolution pixel deconvolution pixel
value e value e value value
e e
Figure Partially normalised reconstruction for the second real hologram of the cross
source in the ob ject plane white minimum black maximum pixel value
Reconstruction of Real Holograms
a convolution real part
b K Wiener lter
Figure Reconstruction byconvolution and K Wiener lter for the second real
hologram of the cross source in all ve planes
Reconstruction of Real Holograms
a algorithm based on convolution
b algorithm based on deconvolution
Figure Reconstruction using the CLEAN algorithm for the second real hologram of
the cross source in all ve planes
Reconstruction of Real Holograms
convolution deconvolution CLEAN convolution CLEAN deconvolution
FEM e e e e
NMSE e e e e
Table Error metrics for the partially normalised image reconstruction of the second
real hologram of the cross source
Three Dimensional Heart Phantom
Figure shows a diagram of the heart phantom This again is taken from Perks
thesis The phantom is a cylinder which is joined at one end Its walls are hollowand
were lled with Technicium m via a hole in the joined end whichwas then plugged
3.5mm Screw for filling
8mm
60mm Perspex inner and walls
Water and Tc 99m
42mm
Figure Heart phantom taken from Perks thesis
Holograms of this phantom were recorded from two directions an end view and a side
view The parameters are given in Table
First zone radius r e m Source zone plate distance z m m
Detector pixel size e m Zone plate detector distance b m
q
R
Numb er of pixels n n Numb er of zones n
x y
r
Table Parameters for the holograms of the heart phantom b oth end and side views
Figures and show the reconstructed images from the end view hologram in
planes at m m m m and m Both CLEAN reconstructions gave p oints
in the planes at m and m but only the o ccasional p oint in the other planes As
the phantom was present in these planes this is go o d but it should have also app eared in
the plane at m As usual the straightconvolution and deconvolution metho ds gave
Reconstruction of Real Holograms
an image in all the planes Lines have b een drawn on the reconstructed images to indicate
where the heart phantom should b e The error metric values are given in Table As
with the rst hologram of the cross source deconvolution gave the b est FEM value but
the CLEAN algorithms gave the b est NMSE values
Reconstruction of Real Holograms
a convolution real part
b K Wiener lter
Figure Reconstruction byconvolution and K Wiener lter of the end view of the
heart phantom
Reconstruction of Real Holograms
a algorithm based on convolution
b algorithm based on deconvolution
Figure Reconstruction using the CLEAN algorithm of the end view of the heart
phantom
Reconstruction of Real Holograms
convolution deconvolution CLEAN convolution CLEAN deconvolution
FEM e e e e
NMSE e e e e
Table Error metrics for the partially normalised image reconstruction of the end
view of the heart phantom
Figures and show the reconstructed images from the side view hologram
in planes at m m m and m Again lines have b een drawn on the recon
structed image to indicate where the heart phantom should b e Both CLEAN algorithms
give the image in just the plane at m but this is not where the heart phantom is The
FEM values are worse for the CLEAN algorithms than for the straightdeconvolution but
the NMSE values are surprisingly b etter
Reconstruction of Real Holograms
a convolution real part
b K Wiener lter
Figure Reconstruction byconvolution and K Wiener lter of the side view of the
heart phantom
Reconstruction of Real Holograms
a algorithm based on convolution
b algorithm based on deconvolution
Figure Reconstruction using the CLEAN algorithm of the side view of the heart
phantom
Reconstruction of Real Holograms
convolution deconvolution CLEAN convolution CLEAN deconvolution
FEM e e e e
NMSE e e e e
Table Error metrics for the partially normalised image reconstruction of the side
view of the heart phantom
Summary
The K Wiener lter deconvolution metho d and the CLEAN reconstruction metho d
have b een shown to work b etter for real holograms than convolution which is the metho d
which has b een used previously The metho ds however work much b etter for a simple
two dimensional cross than for a three dimensional heart phantom
The CLEAN algorithm maybeimproved byimproving the mo del for hologram forma
tion This theoretical work should b e combined with exp erimental work to ensure that the
normalisation is correct A p oint source should b e reconstructed with the same amplitude
no matter where it is placed in the eld of view of the detector The CLEAN algorithm
relies up on b eing able to accurately predict the hologram whichwould b e recorded from
a p oint source in any p osition This predicted hologram is subtracted from the actual
hologram when the corresp onding p oint is added to the image E ects such as vignetting
due to the nite thickness of the zone plate i e it is not in nitely thin could proveto
b e imp ortant If holograms of p oint sources in all p ossible p ositions can b e accurately
predicted the CLEAN algorithm maywork b etter for real three dimensional sources
Chapter
DISCUSSION CONCLUSIONS
AND FURTHER WORK
Discussion of Results
Holography is a technique where information from a three dimensional ob ject is recorded
using a two dimensional detector A three dimensional image can then b e reconstructed
from this information The history b ehind this technique has b een summarised in this
rep ort The history of zone plate holography has b een discussed in more detail
Zone plate holographywas compared and contrasted with Gab or in line holography
Expressions for the hologram formation were found for each technique It was shown that
if a pure Gab or zone plate were used the resulting hologram would contain similar terms
to a Gab or in line hologram of an equivalentcoherentlight source Gab or in line holo
grams were shown to contain extra self interference terms If images were reconstructed
optically from b oth typ es of hologram these di erences would result in di erences in the
images
Other sources of error in a gamma ray zone plate hologram were discussed These
include errors due to the limited resolution of the gamma camera statistical uctuations
in the activity of the gamma ray source oating p oint arithmetic and under sampling
Metho ds for the assessment of reconstructed image qualitywere listed
Image reconstruction metho ds based on convolution were outlined These techniques
included simulated di raction and convolution with a matched lter A combination
of these two metho ds convolution with a truncated Fresnel function was intro duced
This technique was based on simulated di raction and hence the Fresnel function was
complex The reconstructed images therefore had b oth real and imaginary parts The
sum of the squares of these two parts gave the intensity of the image The matched lter
Discussion Conclusions and Further Work
metho ds pro duced images with just a real part as b oth the lter and the hologram only
had real parts
Holograms of p oint sources were compared using the image assessment metho ds men
tioned b efore Mo di ed mismatchwas found to b e the b est matched lter metho d and
convolution with a truncated Fresnel function the b est overall metho d for computer gener
ated holograms of p oint sources Convolution with a truncated Fresnel function pro duced
go o d results when either the real part or the intensitywas used It was decided to use
this metho d as the convolution reconstruction metho d for the rest of the work presented
here One dimensional Gaussian curves were tted to the image whichwas reconstructed
from the hologram of a p oint source with the parameters given in Table The standard
deviation of the Gaussian was mm for the real part and mm for the intensity
in the x and y directions In the z direction was mm for the real part and mm
for the intensity
Metho ds of reconstruction based on deconvolution using Wiener lters were then in
vestigated These techniques attempted to reverse the hologram formation pro cess to
recover the source ob ject A Wiener lter was needed to reduce the e ect of errors in
the hologram Two di erent approximate Wiener lters were given These were called
K and K Wiener lters These lters were based on the binary zone plate pattern
The image qualitywas shown to decrease dramatically if the zone plate was misaligned
Deconvolution with a pure Gab or zone plate pattern was intro duced as a waytocheckif
the zone plate was aligned correctly Deconvolution using the binary zone plate pattern
with a K Wiener lter was found to b e the most promising metho d It was decided
to use this metho d as the deconvolution reconstruction metho d for the rest of the work
presented here One dimensional Gaussian curves were tted to the image whichwas re
constructed from the hologram of a p oint source with the parameters given in Table
The standard deviation of the Gaussian was mm in the x and y directions and
mm in the z direction
The third reconstruction metho d whichwas used was the CLEAN algorithm which
was an iterative technique and could b e based on convolution or deconvolution The
same computer generated holograms were used as for the convolution and deconvolution
metho ds Even when planes were placed mm either side of the image plane the
Discussion Conclusions and Further Work
image was still reconstructed in a single pixel in the correct place in the correct plane
with b oth versions of the metho d For p oint sources therefore this technique seemed
very go o d although it to ok many times longer to p erform the reconstruction than for
straightforward convolution or deconvolution
All three techniques were tested further using theoretical holograms of extended ob
jects The real part of the convolution reconstruction proved to b e of sup erior qualityto
the intensity so only the real part was then used In general deconvolution gave a b etter
reconstruction to convolution The CLEAN algorithm gave b etter results than either of
the other two techniques for holograms of simple ob jects but did not worksowell for
larger or more complicated ob jects Although a gamma camera can pro duce images in
mm mm pixels the intrinsic spatial resolution of the camera is approximately mm
This limited resolution of the gamma camera was found to degrade the image more than
statistical uctuations of the activity of the source or counts due to background radiation
When images were reconstructed from real gamma ray zone plate holograms a similar
e ect was seen to that for the theoretical holograms Images of the simple two dimensional
cross source were reconstructed well but images of the extended three dimensional heart
phantom were not
Other reconstruction techniques were investigated but these proved to b e imprac
tical for reconstructing real zone plate holograms These techniques are discussed in
app endix F
Conclusions
Di erent forms of reconstruction byconvolution and deconvolution havebeengiven The
metho d of convolution to b e used was chosen as convolution with a truncated Fresnel
function The metho d of deconvolution was chosen as K Wiener ltering with the
binary zone plate pattern
Reconstruction byconvolution deconvolution and using the CLEAN algorithm give
images of varying quality Partial normalisation gave b etter results than full normalisa
tion If the images are fully normalised they are normalised in each x y plane as well
as in the z direction Partial normalisation just normalises the images in the z direc
tion Partial normalisation was therefore the metho d whichwas used Reconstruction by
Discussion Conclusions and Further Work
deconvolution takes longer than that byconvolution as the optimum Wiener lter value
needs to rst b e found The CLEAN algorithm takes many times longer than either of
the other two metho ds
Deconvolution gives a b etter reconstruction that is a lower Fienup error metric FEM
value for the ma jority of the theoretical holograms and for all the real holograms which
were investigated The CLEAN algorithms gavelower FEM error metric values for some
holograms but not others Theoretical and real holograms of two dimensional cross sources
were reconstructed mainly in the image plane by the CLEAN algorithm and not in planes
placed cm either side The more complicated and the larger the ob ject was the more
likely the CLEAN algorithm was to reconstruct the image in the wrong plane This was
not totally unexp ected as this technique is known to work well for p oint ob jects but not
for extended ob jects unless a priori knowledge ab out the source in incorp orated into the
algorithm
When some of the p ossible errors in gamma ray zone plate holograms were simulated
errors due to the limited resolution of the gamma camera were found to have the largest
e ect on the reconstructed image This e ect can b e lessened by using a binary zone plate
pattern which has b een blurred by the same p oint spread function for the deconvolution
reconstruction For the simple theoretical holograms whichwere investigated the resolu
tion of the gamma camera did not app ear to haveasmuch of a detrimental e ect on the
image as increasing the size of the ob ject in the eld of view of the camera or complexity
thickness of the ob ject in pixels Further investigations would however b e needed to
prove this
Images reconstructed using the CLEAN algorithm improv e if the images are incre
mented by a smaller amount byeach iteration The algorithm has to run overnight
with a large value of to obtain images of cross sources It to ok several days to obtain
the images of the star ob ject in Chapter This numb er of results that could b e generated
has b een limited by this
In summary the achievements of this pro ject have b een as follows
The formation of zone plate holograms has b een studied in depth and compared to
an example of conventional holography The similarities and di erences havebeenhigh
lighted Other sources of error in gamma ray holographyhave b een listed and metho ds
Discussion Conclusions and Further Work
for image quality assessmenthave b een discussed
Three reconstruction metho ds have b een describ ed Images pro duced using these
metho ds have b een compared using the quality assessment metho ds The errors in gamma
ray holographyhavebeensimulated and their relative e ects measured
The existing convolution reconstruction metho d has b een improved byintro ducing
normalisation Mo di ed mismatch has b een shown to give a b etter p oint spread function
for the matched lter metho d
Reconstruction by deconvolution using Fourier transforms has b een shown to b e ef
fective for real holograms A metho d for nding the optimum lter value automatically
has b een intro duced
The CLEAN algorithm has b een used to reconstruct images in three dimensions A
version of this algorithm based on deconvolution rather than convolution has also b een
used
Suggestions for Further Work
This work has fo cussed on the development of new image reconstruction techniques For
most of the time during whichthiswork was b eing done very few gamma ray zone plate
holograms were available The holograms recorded by Rew were of p o orer quality
than the ones previously recorded byPerks whichhave b een used here This work
has therefore concentrated on reconstructing computer generated holograms This has
shown that many of the artifacts in the images are due to the reconstruction metho ds
themselves rather than errors in the exp erimental holograms
Takhar and George are now recording holograms of comparable quality to those
hniques whichhave b een develop ed here need to now b e tested recorded byPerks The tec
and improved using real holograms The normalisation needs to b e checked Holograms
of p oint gamma ray sources should b e recorded These sources should have a long half
life so that the activity do es not decrease appreciably over the p erio d during whichthe
measurements are taken The amplitude of the reconstructed p oint should b e the same
no matter where the source is placed
The versions of the CLEAN algorithm used in this work were based up on convolution
with a truncated Fresnel function and K deconvolution with the binary zone plate pat
Discussion Conclusions and Further Work
tern The algorithm maywork b etter when it is based up on a di erentconvolution or
deconvolution technique
If the convolution or deconvolution could b e sp eeded up lower values of could b e
used for the CLEAN algorithm In Chapter a metho d for calculating the discrete Fourier
transform of a Fresnel function was derived This Fourier transform could b e calculated
directly instead of b eing p erformed bya NAG routine to sp eed it up Alternatively
the p ossibility of p erforming Fourier transforms electronically instead of computationally
could b e investigated as a way to sp eed the pro cess up
The optimum Wiener lter was found automatically using the metho d given in Sec
tion Work could b e done to discover if the optimum values were in fact found or
if another value would have pro duced b etter results If the values were not the optimum
ones the technique for nding the value could p erhaps b e improved
Holograms of simple ob jects could b e recorded where the exact p osition and size of
the ob ject are known This was not known exactly for Perks holograms The exp ected
hologram could then b e calculated for an equivalent theoretical source ob ject This could
then b e compared with the actual hologram using error metrics Errors for example the
limited resolution of the gamma camera could then b e added to the calculated hologram
to make it more like the actual hologram When all the ma jor e ects had b een accounted
for the two holograms would b e very similar Once all the prop erties and amplitudes
of the errors were known metho ds could b e develop ed for removing the e ects of these
errors from the reconstructed image
In Chapter formulae were derived for the optically reconstructed image from an
ideal pure Gab or zone plate hologram of a p oint source The same could b e done for a
hologram whichcontained errors taken from a certain statistical distribution The e ect
of say detection noise w ould then b e b etter understo o d
Random noise may b e removed from a real hologram by recording two holograms of
the same source one after the other If the rst hologram is correlated with the second
only the parts of the signal which are the same in b oth holograms will remain This means
that any random noise will b e minimised as the correlation b etween the two sets of noise
will b e small if not zero
To summarise this work has intro duced some new reconstruction techniques and has
Discussion Conclusions and Further Work
tried to investigate their e ectiveness using computer generated holograms These tech
niques need to now b e tested on real data The results obtained from real holograms can
then b e used to further improve these metho ds byimproving the mo del of hologram for
mation Once the mo del of hologram formation has b een improved the CLEAN algorithm
maywork b etter for real three dimensional sources
App endix A
COHERENT AND INCOHERENT
LIGHT
This explanation is based on lecture notes from a short course on imaging and digital
pro cessing whichwas given in September at De Montfort University
Consider an ob ject which is illuminated by p erfectly spatially coherent light in the
form of a plane or spherical wave If u x y represents the complex amplitude of the
in
wave emerging from the ob ject and u x y is the complex amplitude of the waveinthe
out
image plane the two are related by
Z Z
u x y u x y p x y x y dx dy A
out in
where p x y x y represents the complex amplitude at x y in the output due to a unit
strength p ointat x y in the input
If the optical system is isoplantic i e a simple shift in the ob ject results in a simple
shift in the image scaled by the magni cation of the optical system this reduces to
Z Z
u x y p x x y y dx dy A u x y
in out
If we consider the case of narrowband light its complex amplitude varies with time
Equation A then b ecomes
Z Z
u x y t u x y t p x y x y dx dy A
out in
The instantaneous intensity is de ned as
I x y t ju x y t j A
whereas the time averaged intensity I x y whichwould b e recorded by an optical de
tector is equal to
Z Z
T T
I x y t dt lim ju x y t j dt A I x y lim
T T
T T
T T
Coherent and IncoherentLight
The intensity of the image is given by
Z
T
I x y lim ju x y t j dt
out out
T
T
T
Z Z
p x y x y p x y x y
Z
T
lim u x y t u x y t dt A
in
in
T
T
T
The term in is called the mutual intensity of the narrowband light J x y x y
Incoherent lightisde nedtobesuch that
J x y x y I x y x x y y A
y have uncorrelated elds for any That is two neighbouring points at x y and x
x y x y Equation A then b ecomes
Z Z
I x y I x y dx dy A jp x y x y j
out in
The quantity jp x y x y j is the intensitypoint spread function
For p erfectly incoherent illumination an optical system is linear in intensity and if
isoplanicity holds the output image intensity is equal to the input ob ject intensity
convolved with the intensity p oint spread function This simpli cation cannot b e made
y term is still imp ortant for coherent light and the mutual intensit
App endix B
ELECTROMAGNETIC WAVES
Maxwell s equations can b e simpli ed for an in nite vacuum containing no charges or
currents to the following
H
HH
E
Faraday s induction law B r E
t
E
EE
H
r H Amp ere s circuital law B
t
E
r E Gauss s law electric B
H
r H Gauss s law magnetic B
E H
where denotes a cross pro duct a dot pro duct E is the electric eld H is the magnetic
eld is the p ermeability of free space and is the p ermittivity of free space These
E H
equations can b e re arranged to give the following vector wave equations for E and H
E H
E H
E H
r E r H B
c t c t
where c is the sp eed of lightand
B c
p
E
These equations are valid for each rectangular comp onentofE E x E y E z and
x y z
H
H H x H y H z Therefore the scalar wave equation
x y z
V
B r V
c t
is valid for V E E E H H or H
x y z x y z
For a plane wavetravelling in thez direction appropriate solutions to the vector wave
equations are
i t kz i t kz
E H E H
c B E E e H H e where
p
k
Electromagnetic Waves
H E
H H x H y H z These solutions must E E x E y E z and H and E
x y z x y z
also satisfy equations B and B Substituting equation B into these equations gives
E and H and
z z
i t kz i t kz i t kz i t kz
E H
E E e x E e y H H e x H e y
x y x y
B
substituting this into equation B gives
r r
E and H E B H
y y x x
S
The time averaged Poynting vector SS gives the time averaged p ower ow due to the
wave p er unit area and also gives the direction of this ow In this case it can b e
shown that
r
r
S E H
SS E H E E z H H z B
x y x y
In his b o ok entitled Optics W elford states that
It is found that all kinds of waves havetobecharacterised bytwo di erent
quantities These are of widely di ering physical natures dep ending up on the
kind of wave but in all cases there is an amplitude whichvaries in time and
space and gives interference e ects and an intensity which represents the rate
of energy transp ort With suitable interpretations the complex amplitude and
its squared mo dulus the intensity can b e used in all cases All interference
exp eriments and many di raction exp eriments can b e describ ed in these terms
Intensityofwaves radiation is de ned as the rate of transfer of energy across unit areas
by the radiation This is given by the magnitude of the time averaged Poynting
vector Intensity I is therefore given by
r r
r r
E H
I E H E H jE j jH j
y y x x
B
The wave amplitude isgiven by
p
E H
I jE j jH j B
App endix C
PHOTOGRAPHIC FILM
The intensity transmittance of a develop ed photographic lm T dep ends up on the inten
sity of light the lm was exp osed to An illustration of the characteristic Hurter Dri eld
curve relating intensity transmittance to exp osure is shown in gure C
ln T −γ 0 gradient of line =
ln Tmin
ln I
Figure C Characteristic Hurter Dri eld curve plotted using co ordinates convenientin
holography I exp osure energy incident p er unit area T intensity transmittance
When the transmittance has its maximum value i e the photographic lm is
transparent ln T The curve is not linear for low exp osures At high exp osures
max
the curve is also not linear as the lm reaches a maximum optical density which corre
sp onds to a minimum transmittance T The linear part of the curve can b e describ ed
min
by the following relationship
T I C
It is convenient if the transmittance of a hologram recorded of photographic lm increases
as the exp osure increases and therefore must b e negative Thisisachieved bytakinga
p ositiveprint of the negative hologram If this simpli es matters considerably as
p
T I and t x y T x y I x y C
where t x y is the amplitude transmittance of the hologram
App endix D
DERIVATION OF THE OPTIMAL
WIENER FILTER FOR
DECONVOLUTION
This derivation is taken from Numerical Recip es by Press et al Equation gave
the equation for hologram formation as
N
N N
p p
X X X
k k k k k
C U C U S T D
i j
i j ij i i j j
N
k k
i j
p
The contribution of ob jects in a single plane a distance p from the zone plate U tothe
p p p
hologram is given by the convolution of S with T multipliedby a constant C
p p p p
U C S T D
Taking Fourier transforms of b oth sides and rearranging gives
F
p
U
F
p
S D
F
p
p
C
T
The imaging pro cess is however not p erfect for the reasons outlined in chapter Instead
p p p
p
of U a noisy version U is recorded which includes the error term E For simplicity E
will b e considered to b e an additive error
p p p p p p
p
U U E C S T E D
The convolution theorem gives the following relationship b etween the Fourier transforms
p p p
of U S and T
F
F F F
p p p p
p
D E T U C S
F p
is the Fourier transform of variable A The aim is to nd an optimal lter Where A
p
p
as p ossible as close to S that will give a reconstructed image S
F
p
p
U
F
p
S D
F
p
C
T
k
Derivation of the Optimal Wiener Filter for Deconvolution
p p
p
must minimise the sum of squared di erences b etween S and S From Parseval s
theorem
X X
F
F
p p
p p
S S S S D
k k
this gives
F F
F p p p
p
X X
U E
U
p
p
D S S
F F
p p
C
T T
k
k k
The Wiener lter assumes that the signal and noise are uncorrelated In this case ex
panding the ab oveformula gives
n o
X
F F F
p p p p p
T U E j j D j j
p
C
k
p
Di erentiating with resp ect to and setting the result equal to zero gives
F
p
U
p
D
F F
p p
E U
App endix E
OTHER RECONSTRUCTION
METHODS
E Matrix Wiener Filtering
This metho d was describ ed in chapter The standard matrix Wiener lter was given
by equation as
p T p p p T p
p p
H H E Wiener lter H H
E S
p
p p
are the signal and noise co and H is the hologram formation matrix and
E S
variance matrices The parametric Wiener lter was given by equation as
p p T p p T p
p p
parametric Wiener lter H H H H
S E
E
Poisson noise was added to simulated holograms so that the matrix Wiener lter could
b e tested for holograms with known amounts of noise For a Poisson distribution the
probability of obtaining r events if the mean exp ected number is isgiven by
r
e
E P r
r
Using a NAG routine to generate random numb ers in a Poisson distribution each pixel
value g in the hologram was replaced by a random numb er from a Poisson distribution
ij
with g
ij
Figures E and E show the reconstruction by parametric Wiener ltering with
di erentvalues of for two holograms of the same ob ject The hologram in gure E
has pixels and the hologram in gure E Before the Poisson noise was
added each hologram was scaled so all the values laybetween and e When
equation b ecomes a matrix inversion It can b e seen from gure E that
matrix Wiener ltering provides an improvementover matrix inversion
GLOSSARY
D length of one size of the detector
E di erence b etween Fourier transform of f x y andFourier transform of v x y
x y
K constant used in rst approximate Wiener lter
K constant used in second approximate Wiener lter
L factor source distribution is magni ed by
x
L x a linear lter
M factor zone plate shadow is magni ed by
S intensity of incoherent gamma ray source
T r Intensity transmittance function of the zone plate
F
x
T Fourier transform of t x
x
U x intensityofshadowcastby an incoherent gamma ray source
U hologram pixel values
ij
p
V reconstructed image values at a distance p in front of the zone plate
ij
F
Z Fourier transform of the Fresnel function
constant relating the amplitude transmittance of photographic lm to the total intensity
it was exp osed to
ff g imaginary part of a function f
maximum spatial frequency in the ob ject
Glossary
x y z t amplitude of an electromagnetic wave
ff g real part of a function f
constant relating the amplitude transmittance of photographic lm to the total inten
sity recorded by the gamma camera
size of the nest detail in the ob ject
D convolution op eration
x y z spatial variation of the amplitude of an electromagnetic wave
a size of the ob ject
a x y circular ap erture function
f x y partofv x y recorded by the detector
h x y p erfect recorded signal
l radius of the central Fresnel zone
p source to ob ject zone plate distance
q ob ject zone plate to detector distance
s source amplitude distribution function
t amplitude transmittance function of the transparency
x
u x amplitude of the waves incident on the detector
v x y signal from Gab or zone plate hologram of an on axis p oint source
x
z x Fresnel function
D DFT discrete two dimensional Fourier transform
D FFT t wo dimensional fast Fourier transform
D FT continuous two dimensional Fourier transform
Glossary
BGZP binary Gab or zone plate
BNCT Boron Neutron Capture Therapy
CT Computed Tomography
ESF edge spread function
FEM Fienup error metric
GZP pure Gab or zone plate
LSF line spread function
MSE mean squared error
MTF mo dulation transfer function
NMSE normalised mean squared error
OTF optical transfer function
PET Positron Emission Tomography
PM tub e photomultiplier tub e
PSF point spread function
SNR signal to noise ratio
SPET Single Photon Emission Tomography
W Wiener sp ectrum
X D FT of x x y a b
x y
a b
x x y a b sampling function Appendix F – this conference paper has been removed from the electronic version of this thesis due to copyright restrictions.
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