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and Astronomy The University of Birmingham Decemb er University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties.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. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. ABSTRACT Gamma ray zone plate holography is a new technique with applications to Nuclear Medicine Unlike other tomographic techniques threedimensional images can b e recon structed from just one pro jection The history of zone plate holography is reviewed and the dierences 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 diraction 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 diraction based metho ds An approximate Fourier Wiener lter is used to reconstruct the images bydeconvolu tion Several dierentversions 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 KirchhoFresnel Diraction Formula Pro of of the Huygens Fresnel Principle Formation of a General Hologram Gab or InLine Holography of a TwoDimensional 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 HolographyofTwoDimensional 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 OpticalConvolution 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 ThreeDimensional Ob ject Zone Plate Holography Gab or InLine 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 OTFMTF 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 Dierence Metrics Summary IMAGE RECONSTRUCTIONBYCONVOLUTION Optical Reconstruction how Convolution Algorithms can b e used to Sim ulate Diraction Using the Zone Plate Pattern as a MatchedFilter Hybrid Metho d using R to improveSimulated Diraction 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 Eect of K K constructed Point To account for part of the zone plate hologram falling outside the detector Eect of a misaligned 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 TwoDimensional 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 Dierent Sizes with the Same Parameters Reconstruction of Images from Holograms of the Same Ob ject with Dier entParameters Reconstruction of Two Cross Sources in Dierent Planes Summary RECONSTRUCTION OF REAL HOLOGRAMS Cross Source Placed Parallel to the Zone Plate ThreeDimensional Heart Phantom Summary DISCUSSION CONCLUSIONS AND FURTHER WORK Discussion of Results Conclusions Suggestions for Further Work A COHERENT AND INCOHERENT LIGHT B ELECTROMAGNETIC WAVES C PHOTOGRAPHIC FILM Contents D DERIVATION OF THE OPTIMAL WIENER FILTER FOR DE CONVOLUTION E OTHER RECONSTRUCTION METHODS E Matrix Wiener Filtering E To eplitzBlo ckTo eplitz Structure of the Hologram Forming Matrix E InverseHessian and Steep est Descent Metho ds of Iterative Reconstruction E Comparison of Reconstruction by Deconvolution with that by Matrix Wiener Filtering and by Steep estDescent 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 Gaborinlinehologram Zone plates and shadow casting Binary zone plate patterns Region of integration Diraction b ehind an ap erture surface Diraction 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 onaxis intensity Relative p ositions of the transparencies and the hologram Diagram of a gamma camera detector head Example pulse height sp ectrum from a NaITl scintillation detector Errors asso ciated with basic arithmetic op erations Errors in example calculations p erformed using NAG routines Eect of a nite detector area on a Fourier transform Fourier transforms of Gab or zone plates Eect of the nite detector size Eect of truncating the zone plate at a maximum radius Comparison of actual and calculated