1. Spatial Frequency Analysis - Optical Transfer Function (OTF)
Ofer Hadar
Communication Systems Engineering Dept., BGU
URL:
O. Hadar
LABRI – University of Bordeaux ,

2. System classification
Continuous time and discrete time systems
Linear systems: These apply superposition in delay, addition and multiplication
Time-invariant systems F1:
Linear, Time-Invariant (LTI) system combines the previous
Non-linear systems, examples
diodes, transistors, modulators, mixers
Note that a system might appear to be nonlinear in a domain but linear in some other domain. Thus nonlinear systems can often be linearized
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3. Signal transmission and filtering
Linear systems
apply superposition, hence
delay:
addition:
multiplication:
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4. Linear systems are characterized by impulse response or transfer function
System F[ ] response to impulse:
When the impulse response h(t) is known, the response y(t) to some other excitation x(t) is obtained via convolution:
Here H(f) is the impulse response in frequency domain, that is the transfer function:
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5. System response in frequency and time domain
Note that system response is easy to calculate in frequency domain. In time domain: but in frequency domain convolution replaced by multiplication
Note that the transfer function H(f) has both magnitude and phase components. For real h(t) transfer function magnitude has even and phase has odd-symmetry, (the conjugate symmetry):
Transfer function and impulse response are easily determined for electrical circuits by Laplace transforms or Fourier techniques
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*prove left to tutorials

6. Linear Systems Approach to Imagingx x’
Any Optical System
Exit
Window
Entrance
Window
Isoplanatic
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7. Convolution
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8. Terminology h is called the point spread function (PSF)
H is called the optical transfer function (OTF)
Magnitude is called Modulation Transfer Function (MTF)
Phase is Phase Transfer Function (PTF)
fx and fy are spatial frequencies
Uobject
Uimage
Uobject
Uimage
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9. The Fourier Domain Gaussian Fourier Modulus (also Gaussian)
These Fourier modulus of a Gaussian produces another Gaussian. A large object comprised of low spatial frequencies produces a compact Fourier modulus and a smaller object with higher spatial frequencies produces a larger Fourier modulus.
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10. f(x,y)=(x) g(x,y) = LSF system Fourier T. Fourier T. S(f) F system 1
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11. Point spread function (PSF)
Describes the unsharpness that results when a point in the object is not reproduced as a true point in the image. The unsharpness is a blurring effect (i.e., a spreading out of the point image to form a measurable circle). PSF is the impulse response of the imaging system.
object
image
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12. If the system is space-invariant: PSF0,0(x,y) = PSFx0,y0(x-x0, y-y0)
f(x,y)=(x,y)
system
g(x,y) = PSF
If the system is space-invariant:
PSF0,0(x,y) = PSFx0,y0(x-x0, y-y0)
If the system is also linear:
G(u,v)=F(u,v) F (PSF)= F(u,v) OTF(u,v)
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13. Spatial Frequency
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14. Illustration of Spatial Frequency
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15. Angular Frequency
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16. Spatial Frequency Analysis
When we look at a visual scene, we specify it in terms of spatial locations of light, dark, contour, and color.
Visual scenes, however, can be broken down into much smaller components.
According to spatial frequency analysis, or Fourier analysis, a visual scene can be broken down into a series of sine waves.
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17. Sine Waves
A sine wave is a continuous waveform that undulates in a smooth regular fashion.
In vision, a sine wave refers to a pattern in which luminance undulates in a smooth regular fashion.
The sine wave contains a number of important components.
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18. Sine Waves (cont’d)
1) Spatial frequency refers to the size of the image of an object.
Spatial frequency tells us the number of times a cycle of a sine wave (one light stripe and one dark stripe) repeat in 1 degree of visual space.
Thus, it is measured in cycles per degree (c/deg).
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19. Sine Waves (cont’d)
The spatial frequency is inversely related to the wavelength of sine wave.
Thus, a low spatial frequency sine wave has a large wavelength and thus, has large stripes .
Conversely, a high spatial frequency sine wave has a small wavelength and thus, thin stripes.
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20. Sine Waves (cont’d)
2) Contrast: Refers to the difference in light intensity between an object and its surroundings.
In terms of sine waves, it refers to the difference in light intensity between a light stripe and a dark stripe.
3) Phase: Refers to the difference in timing between two or more waves.
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21. Sine Waves (cont’d)
In other words, it refers to the relative position of the crests and troughs of two or more waves.
4) Orientation: Refers to the angle at which an object is viewed.
Thus, any visual scene can be broken down into a series of sine waves of various spatial frequencies, contrasts, orientations, and phases. This applies even to square waves.
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22. Spatial Frequency Spectra
It’s easiest to think about Fourier’s analysis in terms of waves. However, these sine waves can also be specified as equations or as spectra.
The location of the spike depends on the frequency, and the height depends on the contrast.
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23. Spatial Frequency Spectra
1/
contrast
Spatial Frequency
low
high
1/
contrast
Spatial Frequency
low
high
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24. MTF Modulation Transfer Function
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25. The Modulation Transfer Function
Any visual scene can be broken down into component sine waves.
Why is this important?
We can predict how good an image an optical system can form of any visual scene by seeing how well it “perceives”…
...Sine waves.
This is the purpose of the modulation transfer function (MTF).
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26. Modulation Transfer Function (cont’d)
An optical system (e.g., lens) is presented with a series of sine wave gratings of different spatial frequencies of a specific contrast level.
The performance of the system is evaluated by comparing the contrast of the sine wave gratings to the contrast of the image that is produced.
The MTF can then be plotted.
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27. Modulation Transfer Function (cont’d)
Relative
Contrast
0.1
1.0
Spatial Frequency
low
high
Low SFs are
reproduced well. High
SFs are not. Note the
drop-off.
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28. Modulation Transfer Function (cont’d)
Relative
Contrast
0.1
1.0
Spatial Frequency
low
high
Here is a lens of slightly
poorer quality. The
drop-off is at a lower
SF.
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29. Some Resolution Charts (1)
Edge
Point and
Lines
Sinusoidal Chart
Bar Charts
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30. Contrast Resolution:
Describes ability to distinguish small value differences. b1 b2
How much is the contrast?
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31. Some Resolution Charts (2)
Air Force
ISO
Bar Charts
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32. Radial Target and Image
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
Radial Target and Image
Colorbar for all 20 40 60 80
100
120
140
160
180
Object
Image 20 40 60 80
100
120
140
160
180 20 40 60 80
100
120
140
160
180
Point-Spread
Function of
System
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33. low medium high object: 100% contrast image contrast spatial frequency
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spatial frequency

34. The modulation transfer function (MTF) indicates the ability of an optical system to reproduce (transfer) various levels of detail (spatial frequencies) from the object to the image.
Its units are the ratio of image contrast over the object contrast as a function of spatial frequency.
It is the optical contribution to the contrast sensitivity function (CSF).
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35. MTF: Cutoff Frequency 1 mm 1 2 mm 4 mm 6 mm 8 mm 0.5 50 100 150 200
Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size
8 mm
modulation transfer
0.5 50
100
150
200
250
300
spatial frequency (c/deg)
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36. Effect of Defocus on the MTF
450 nm
650 nm
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Charman and Jennings, 1976

37. How is sharpness measured?
A property of a lens, film, sensor, or system defined by boundaries between zones
Bar pattern: Sharpness defined by 10-90% risetime.
Patterns of increasing spatial frequency (Log scale)
Sine pattern: Sharpness defined by contrast at a given spatial frequency.
The top half of each pattern is sharp; the bottom is less sharp.
How is sharpness measured?
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38. Sharpness and spatial..frequency response
MTF50
Spatial frequency LP/mm
Sine and bar patterns are shown with and without rolloff for a high quality 35mm lens
(on a 0.5 mm virtual target)
Amplitude response of bar pattern
Rise distance (10-90%) difficult to calculate for compound systems.
The relative contrast of a sine pattern (pure tone) is called
Spatial Frequency Response (SFR) or
Modulation Transfer Function (MTF);
Multiplicative for compound systems.
Perceived image sharpness strongly correlates with MTF50, the spatial frequency where contrast is half its low frequency value. MTF50 is a close approximation to bandwidth W in Shannon capacity calculations.
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39. PTF
Phase Transfer
Function
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40. low medium high object image phase shift 180 -180 spatial frequency
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-180
spatial frequency

41. Imatest Sharpness calculation derived from ISO-12233 standard
Results are derived from a slanted-edge image in a test chart.
Algorithm: Find average edge location. Put each line into one of four “bins” based on avg. edge. Find mean 4x oversampled edge, then take Fourier transform of the spatial derivative.
Upper (black) curve is the average edge.
Lower (black) curve is the Spatial Frequency Response (SFR or MTF).
These results are strongly affected by sharpening, The dashed red curves are the edge and MTF response with standardized sharpening that corrects for oversharpening.
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42. Liming resolution of optical system
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All hi-res, hi-contrast images are not created equal.

43. Contrast Sensitivity Function
The visual system has its own MTF, it’s known as the contrast sensitivity function (CSF).
Psychophysical techniques are used to measure contrast threshold at several different spatial frequencies.
i.e., how much contrast you need to see stripes of a certain spatial frequency.
The inverse of each threshold is then calculated (contrast sensitivity) and plotted to form the CSF.
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44. CSF (cont’d)
Thus, if you have a low contrast threshold, you have a high contrast sensitivity. It takes little contrast for you to see the stripes.
If you have a high contrast threshold, you have a low contrast sensitivity. It takes a lot of contrast for you to see the stripes.
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45. CSF (cont’d) high Contrast Sensitivity low 0.1 1.0 10 100
Spatial Frequency
low
high     
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46. Spatial Response Fourier transform over multidimensional space
- Continuous space FT (CSFT)
Discrete space FT (DSFT)
Sampled space FT (SSFT)
Frequency domain characterization of video signals
Spatial frequency
Temporal frequency
Temporal frequency caused by motion
Frequency response of the HVS
Spatial frequency response
Temporal frequency response and flicker
Spatio-temporal response
Smooth pursuit eye movement
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47. CSF (cont’d) The CSF possesses a characteristic inverted U shape
The CSF is in essence, a window of visibility.
That is, we can detect all combinations of contrasts and spatial frequencies under the curve, but not those above the curve.
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48. CSF (cont’d)
The CSF is similar to the MTF in that there is high frequency attenuation.
i.e., we don’t see small objects particularly well.
However, unlike the MTF, the CSF shows low frequency attenuation.
We don’t see large objects particularly well.
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49. Perceived Image Content
MTF Area (MTFA) Method first proposed in 1965 by Charman and Olin
MTF can greatly effect perceived resolution, as well as perceived content
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