ECE 425 CLASS NOTES – 2000
DIFFRACTION OPTICS
Physical basis
• Considers the wave nature of light, unlike geometrical optics
• Optical system apertures limit the extent of the wavefronts
• Even a “perfect” system, from a geometrical optics viewpoint, will not form a point image of a point source
• Such a system is called diffraction-limited Diffraction as a linear system
• Without proof here, we state that the impulse response of diffraction is the Fourier transform (squared) of the exit pupil of the optical system (see Gaskill for derivation)
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ECE 425 CLASS NOTES – 2000
• impulse response called the Point Spread Function (PSF)
• image of a point source
• The transfer function of diffraction is the Fourier transform of the PSF • called the Optical Transfer Function (OTF) Diffraction-Limited PSF
• Incoherent light, circular aperture
()2 J1 r' PSF() r' = 2------r'
where J1 is the Bessel function of the first kind and the normalized radius r’ is given by,
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ECE 425 CLASS NOTES – 2000
D = aperture diameter πD πr f = focal length r' ==------r ------where λf λN N = f-number λ = wavelength of light
λf • The first zero occurs at r = 1.22 ------= 1.22λN , or r'= 1.22π D Compare to the course notes on 2-D Fourier transforms
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ECE 425 CLASS NOTES – 2000
2-D view (contrast-enhanced)
“Airy pattern”
central bright region, to first- zero ring, is called the “Airy disk”
1
0.8
0.6 PSF 0.4 radial profile 0.2
0 0246810 normalized radius r'
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ECE 425 CLASS NOTES – 2000
• Example calculation of PSF size • system specs: D = 1cm f = 50mm N = f/D = 5 λ = 0.55µm (green)
• radius of PSF = 1.22λN = 3.36µm
• very small!
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ECE 425 CLASS NOTES – 2000
Diffraction-Limited OTF
• Incoherent light, circular aperture
()ρ 2 ()ρ ρ ρ 2 OTF ' = π--- acos ' – '1– '
where the normalized radial spatial frequency is given by, ρ ρρ⁄ ' = c
and the cutoff frequency is given by,
D = aperture diameter D 1 f = focal length ρ ==------where c λf λN N = f-number λ = wavelength of light
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ECE 425 CLASS NOTES – 2000
1
0.8
0.6 OTF 0.4
0.2
0 0 0.2 0.4 0.6 0.8 1 ρ/ρ normalized spatial frequency ( c)
In 2-D spatial frequency space, the OTF is nearly a “cone” function
OTF OTF is a low-pass filter of spatial frequencies
v
ρ ρ u = c
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ECE 425 CLASS NOTES – 2000
Example calculation of OTF cutoff frequency
• system specs: D = 1cm f = 50mm N = f/D = 5 λ = 0.55µm (green)
• cutoff frequency = 1/λN = 0.363cycles/µm = 363cycles/mm
• very high! (the human vision system has a cutoff frequency of about 10cycles/mm (object scale) at normal viewing distance)
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ECE 425 CLASS NOTES – 2000
Modulation Transfer Function (MTF)
• Amplitude part of complex OTF output signal modulation MTF() u, v ==OTF() u, v ------input signal modulation
• signal modulation max– min signal modulation = ------max+ min
• modulation is a measure of signal contrast
()π • for sinewave, AB+ sin 2 uox , modulation = B/A
• Use MTF to predict image contrast, given object and system • output modulation(u,v) = input modulation(u,v) x MTF(u,v)
() ()π • for sinewave input (object) to LSI system input x = AB+ sin 2 uox
() () ()π the output (image) is output x = A+ MTF uo Bsin 2 uox 215
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B and the image modulation at spatial frequency u is image modulation = MTF() u ⋅ --- o o A
MTF
1 “typical” MTF 0.8 for imaging system 0.5
0.1 u
u0 2u0 4u0
u = u0 u = 2u0 u = 4u0
input
output
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DIFFRACTION IMAGING EXAMPLES
Sine-wave imaged by circular aperture, diffraction- limited
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optics spatial frequency, u
Input pattern (object)
System response convolution (PSF)
Output pattern (image)
cutoff-frequency, uc
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Square-wave imaged by circular aperture, diffraction-
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limited optics spatial frequency, u
Input pattern (object)
System response convolution (PSF)
Output pattern (image)
cutoff-frequency, uc
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Line and Edge Spread Functions
• PSF is often difficult to measure because of insufficient energy
• Use a line source (slit) to increase energy –>
Line Spread Function (LSF)
• Integrate PSF along direction of the line source
∞ () (), LSFx x = ∫ PSF x y dy –∞ ∞ () (), LSFy y = ∫ PSF x y dx –∞
• Use edge source (knife edge) to further increase energy –>
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Edge Spread Function (ESF)
• Integrate LSF up to position of ESF measurement
x () ()αα ESFx x = ∫ LSFx d –∞ y () ()αα ESFy y = ∫ LSFy d –∞
• Equivalent to step response in electronic system
• ESF is a monotonic function of position
• LSF is the derivative of the ESF,
d LSF ()x = ESF ()x x dx x
• Both LSF and ESF are orientation-dependent, if the PSF is not isotropic
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• Must measure LSF and ESF at multiple orientations to reconstruct full 2-D PSF
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RELATIONSHIPS BETWEEN PSF, LSF, ESF AND OTF
• For example, the relationships in the x-direction are:
PSF(x,y) ESFx(x) one-sided 1-D integration
one-sided 2-D 1-D 1-D integration Fourier derivative Transform 1-D integration
OTF(u,v) OTF(u,0) LSFx(x) profile 1-D Fourier Transform
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DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000 SECTION III – INTEGRATED IMAGING SYSTEM ANALYSIS
Scanning Image Quality System Simulation
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SCANNING
Instantaneous image on focal plane detector(s) Scan the field-of-view
• Object-space scanning (mirror moves) θ detector
2θ
object
Show that the object-space scan angle is twice that of the mirror scan angle
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• Image-space scanning (detector moves) detector
object
• Translation scanning (camera or object moves) detector
object
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Examples from airborne or satellite remote sensing systems whiskbroom scanner 1-D array line scanner pushbroom scanner
in-track FOV cross-track
GFOV FOV: Field-Of-View (radians) 2-D array wavelength dispersion pushbroom scanner GFOV: Ground- projected FOV (km) (aka “swath width”)
What types of scanning are these examples?
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Scanning is equivalent to convolution
• At each instant, the detector integrates a small area of the image
• Integration time at each sample determines number of photons collected
• Generally, a scanned image is sampled at an interval equal to detector size • undersampled Why is one sample/detector element undersampled?
• PSF of scanning detector is rect(x/W,y/W), where W is the detector size
• If sample integration time is not negligible, there is also image smear 229
DR. ROBERT A. SCHOWENGERDT [email protected] 520 621-2706 (voice), 520 621-8076 (fax) ECE 425 CLASS NOTES – 2000
• PSF of linear image smear is rect(x/S), where S is the amount of smear during the integration time: S = image velocity x integration time
= vimgtint
• Image smear sometimes used to increase integration time/ pixel, and therefore SNR (even though image is blurred)
• Total scanning PSF • Combine scanning detector and image smear during pixel integration (normalize by volume PSF) rect() x⁄ W, yW⁄ * rect() x⁄ S PSF ()xy, = ------scan volumePSF []rect() x⁄ W * rect() x⁄ S rect() y⁄ W = ------volumePSF
• if S = W, i.e. image smear is equal to one detector width:
tri() x⁄ W rect() y⁄ W PSF ()xy, = ------scan volumePSF
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IMAGE QUALITY
An imaging system consists of several subsystems
image light transmission display human scene acquisition vision source subsystem subsystem* subsystem subsystem
coder decoder optics neural network optics detector* electronics
retina* brain
* points of signal transduction, optical <—> electronic Want to maintain the highest possible image quality through this process
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Spatial Resolution
• “Resolution” refers to the ability to detect two closely spaced objects in an image
• Many different ways to measure resolution • Rayleigh Criterion
• quantitative; does not depend on visual examination
• appropriate for diffraction-limited optical image
• two point sources —> two Airy patterns
• resolution is the separation of the two images when the first zero of one pattern coincides with the peak of the other pattern
• does not incorporate noise into resolution measure
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Airy Airy pattern pattern 1 2
λ R Rayleigh = 1.22 N
• Target-Specific Criteria
• separation of two particular objects when they are just “resolved”
• depends on visual examination
• incorporates noise into resolution measure
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imaging system PSF and resulting imagery target
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• Usually have fixed system PSF; vary target frequency to evaluate resolution
target
1234 5 6
imaging system PSF
image of target
123456
profile of image
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• Define a line pair = one dark bar & one white bar • Define resolution to be measured by the target which is “just resolved” (correct number of bars can be visually distinguished along their entire length)
• Rbar = 1/d4 (line pairs/mm)
where d4 = mm/line pair for target 4
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• image noise lowers resolution
123456
• Rbar = 1/d3 (line pairs/mm)
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SENSOR SIMULATION
Landsat Thematic Mapper (TM) satellite system Whiskbroom scanner
• side-to-side scanning, cross-track
whiskbroom scanner • satellite orbit motion, in- track
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• Three major spatial cross-track 30m GIFOV response components
optics detector electronics in-track cross-track only
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• Simulate input scene with high resolution aerial photography
rotated to align with TM orbit and scan direction - 2m pixels
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• Apply each component PSF at 2m and downsample to 30m
optics optics and detector
downsample 2m —> 30m GSI
optics, detector and electronics
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Compare to real TM of same area, acquired 4 months later
simulated TM real TM (magnified) (magnified)
real TM (contrast-adjusted)
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PHOTOGRAPHIC GRANULARITY
• photographic density is aggregated “grains” of silver halide in developed film
• causes random noise at every pixel in scanned image
• Ex: aerial photograph of Dulles Airport, Virginia
enlargement
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• Signal-dependent noise • Standard deviation in Digital Numbers (DNs) versus DN
16
14
12 DN σ 10
8
6 0 50 100 150 200 µ DN
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ELECTRO-OPTICAL SCANNER NOISE
• Due to • line scan calibration (gain and offset) changes • detector-to-detector calibration differences • interference with other electronics • saturation hysteresis
• Common problem
• Can be removed effectively by image processing
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aerial video aperiodic scanline noise Landsat–4 MSS coherent noise
Landsat–1 MSS bad scanline noise AVIRIS scanline and random pixel noise
Landsat–4 TM banding noise
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Signal-to-Noise Ratio (SNR)
• Measure of image quality
• Several common definitions: σ signal SNRstd = ------σ noise σ2 SNR = ------signal var σ2 noise () SNRdB = 10log SNR
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• Generally correlated with visual quality
SNR = 2 SNRstd = 1 std SNR = 4 SNRvar = 1 var SNR = 6 SNRdB = 0 dB
SNR = 5 std SNRstd = 10 SNR = 25 var SNRvar = 100 SNR = 14 dB SNRdB = 20
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• Visual quality depends on noise type
noiseless
global random noise detector striping noise
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