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The Beginnings of Visual Perception: The Retinal Image and its Initial Encoding. Appendix: Fourier Transforms and Shift‐Invariant Linear Operators

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Abstract

The sections in this article are:

1 Scope and Organization
2 Physiological Explanation in Visual Science: Three Classes of Perceptual Phenomena
3 Visual Acuity
3.1 Overview
3.2 The Retinal Image
3.3 Spatial Modulation Detection
3.4 Neural Limits of Visual Acuity
4 Color Vision
4.1 Scotopic Spectral Sensitivity
4.2 Monochromacy
4.3 Dichromacy
4.4 Trichromacy
4.5 Color Blindness
4.6 Seeing Colors
5 Visual Adaptation
5.1 Overview
5.2 Experimental Paradigms
5.3 Identifying Neural Substrates
5.4 Single‐ Variable Theories
5.5 Concluding Comment
6 Note Added in Proof
Figure 1. Figure 1.

Blur circles in a schematic misfocused eye for two different pupil sizes.

From Cornsweet
Figure 2. Figure 2.

Chromatic aberration of the human eye. λ, Wavelength; D, magnitude of chromatic aberration in diopters (assuming the eye is in perfect focus for 578 nm, D gives the power of the spectacle lens required to focus other wavelengths). Solid line and solid dots, average from 12 observers of Bedford and Wyszecki ; crosses, average from 14 observers of Wald and Griffin ; dashed lines, total range for all observations of Bedford and Wyszecki.

Adapted from Bedford and Wyszecki
Figure 3. Figure 3.

Irregularities in refractive power across the pupil of a normal eye during relaxed accommodation (pupil diam, 7.2 mm). Contours join points of equal excess power (diopters).

From van den Brink , with permission from Pergamon Press, Ltd
Figure 4. Figure 4.

Pointspread function of an optical system. Σo is the object plane, with coordinates y,z; Σi, is the image plane; S(y,z), irradiance distribution produced in Σi by point source located at origin of Σo. Illustration is schematic.

From Hecht and Zajac
Figure 5. Figure 5.

Linespread function of an optical system. S(Z) is the irradiance distribution in the image plane produced by an infinitely long thin line source in object plane. (The absence of secondary ripples here like those in Fig. has no significance; both illustrations are schematic.)

From Hecht and Zajac
Figure 6. Figure 6.

Linespread functions of the human eye for white light at various pupil diameters. Heavy dotted curves, measured intensity (along horizontal axis) of retinal image of a line target as measured ophthalmoscopically. Narrower dashed curves, theoretical spread resulting from diffraction at the pupil alone. (All curves have been normalized to equal 1.0 at origin.)

From Campbell and Gubisch , with permission from Cambridge University Press
Figure 7. Figure 7.

Pointspread functions of the human eye for white light at various pupil diameters. Solid curves, radial profiles of theoretical pointspread functions calculated from linespread curves in Fig. . Narrower dashed curves, the point spread resulting from diffraction alone.

From Gubisch
Figure 8. Figure 8.

Modulation transfer functions (MTFs) of the human eye for various pupil diameters. Each curve shows contrast reduction (i.e., retinal image contrast divided by the object contrast) imposed on sinusoidal targets as function of their spatial frequency. Solid curves

From Gubisch . from top to bottom) correspond to pupil diameters of 2.4, 3.8, and 6.6 mm; dashed curve corresponds to 1.5 mm. MTFs calculated from linespread data of Fig.
Figure 9. Figure 9.

Sinusoidal gratings.

From Cornsweet
Figure 10. Figure 10.

Effects of defocus on human spatial contrast sensitivity (2‐mm pupil). Data points show retinal image contrast reduction (inferred from threshold data of one observer) imposed on each spatial frequency as a function of defocus. Upper panel: •, 1.5 diopters (myopic); ⋄, 2.5 diopters. Lower panel: ▪, 2.0 diopters, ◯, 3.5 diopters. Spatial frequency expressed as fraction of highest frequency passed by the optics of the eye, i.e., 1.0 ∼ 60 cycles/deg. Solid lines drawn by hand through data points. Dashed lines show results predicted from diffraction alone.

From Campbell and Green , with permission from Cambridge University Press
Figure 11. Figure 11.

Top: spoke target for demonstrating spurious resolution in human vision. See text for directions. Bottom: out‐of‐focus picture of spoke pattern illustrating spurious resolution in a photographic image.

Figure 12. Figure 12.

Human spatial contrast thresholds for vertical sinusoidal gratings at various mean intensity levels (2‐mm pupil, monochromatic light, wavelength 525 nm). Each curve plots threshold contrast as a function of spatial frequency for mean retinal illumination level (trolands) indicated by curve parameter (100% corresponds to contrast of 1.0).

Data from Van Ness and Bouman ; figure from Westheimer
Figure 13. Figure 13.

Cone density as a function of distance from the center (0°) of the fovea.

Data from Osterberg ; figure adapted from Ripps and Weale
Figure 14. Figure 14.

Spatial contrast‐sensitivity functions for sinusoidal interference targets formed with a coherent source; wavelength, 633 nm. Data for two observers; smooth curves drawn by hand through data points. Mean illumination level, 500 trolands.

From Campbell and Green , with permission from Cambridge University Press
Figure 15. Figure 15.

Spatial contrast‐sensitivity functions for interference fringes and normal targets at same mean retinal illuminance (500 trolands). Continuous smooth curve in lower portion of figure taken from data (closed circles) of Fig. (i.e., these are interference target thresholds). Open circles show contrast thresholds for targets viewed normally, i.e., through the optics of the eye. Closed circles in the top portion show the ratio between the two curves below: this is an inferred measure of the modulation transfer function of the optics of this observer's eye.

From Campbell and Green , with permission from Cambridge University Press
Figure 16. Figure 16.

Comparison between local spatial acuity for interference targets (633 nm) and intercone spacing at various distances from the center of the fovea (0° eccentricity). Closed circles show center‐to‐center intercone distances (1/distance in min). Open points show local spatial acuity for two observers. Visual acuity here is arbitrarily defined as twice the highest resolvable spatial frequency in cycles/min. This definition allows a direct comparison between local spatial acuity and local cone density: if acuity and cone density were matched according to the sampling theorem, both should fall on a common curve—as they do out to 2°. Targets subtended 32.4 min; mean intensity level was 1,200 trolands.

From Green , with permission from Cambridge University Press
Figure 17. Figure 17.

A: spatial contrast‐sensitivity functions for three observers (dashed lines) with fitted function s2es (solid curve), which can be regarded as the overall “modulation transfer function” of the human visual system. B: pointspread function obtained by Fourier inversion of the smooth curve in Figure A.

From Kelly
Figure 18. Figure 18.

The top and middle rows illustrate aliasing by an array of photoreceptors. Top left: photomicrograph of a 12′ × 13′ patch of human retina near the center of the fovea. The mean center‐to‐center distance between receptors is 0.53′, implying an aliasing cutoff of 57 cycles/deg. Top right: sampling grid made by punching a pinhole through the center of each receptor in the 12′ × 13′ patch. Middle left: 30‐cycle/deg square wave grating seen through the pinhole grid. Middle right: 80‐cycle/deg square wave grating seen through the pinhole grid. Note the appearance of broad curved line segments due to aliasing. The bottom row illustrates the effect of a regularly spaced (checkerboard) array of sampling points having roughly the same sample‐point distance as the average value for the retinal array. Bottom left: 30‐cycle/deg square wave grating seen through the regular sampling grid. Bottom right: 80‐cycle/deg square wave grating seen through the regular sampling grid.

Top left from Polyak , p. 268
Figure 19. Figure 19.

The scotopic visual action spectrum plotted two ways. Horizontal axis is linear with wave number (number of waves per cm). See text for explanation of data collection procedure.

From Cornsweet
Figure 20. Figure 20.

Absorption spectrum of human ocular media.

From Ludvigh and McCarthy
Figure 21. Figure 21.

Scotopic visual action spectrum corrected for absorption by ocular media (X) compared with absorption of rhodopsin (dashed curve).

From Cornsweet
Figure 22. Figure 22.

Spectral sensitivity curve of rods (upper curve) compared with spectral sensitivity curve for a hypothetical set of cones (lower curve). Note that this lower curve does not represent any real set of cones and is strictly for explanatory purposes.

Figure 23. Figure 23.

A: absorption spectra for two classes of cones. (These are two of the three classes present in the normal human eye.) B: two‐dimensional visual action space of a person whose eyes contain only the two kinds of cones plotted in A. See text for derivation of system B from A.

From Cornsweet
Figure 24. Figure 24.

A representation in two‐dimensional action space of the effects of adding lights together. The result can be represented simply as the vector sum of the effects of the components.

Figure 25. Figure 25.

The effects of a mixture of lights of wavelengths 500 nm and 620 nm and of appropriately chosen intensities (2,050 and 910 quanta · s−1 · mm−2, respectively, in this example) will be identical to the effects of a light of 560 nm (at an intensity of 1,000 quanta · s−1 · mm−2).

Figure 26. Figure 26.

If a mixture of two wavelengths is to match a third whose effects do not lie on a line between the two to be mixed, one of the two components of the mixture must have a negative intensity, as shown by the downward direction of the 500‐nm vector. The physical realization of this negative vector requires the addition of 500 nm light of that magnitude to the 620‐nm component.

Figure 27. Figure 27.

Estimates of the absorption spectra of the three normal human cone pigments.

From Wald . Copyright 1964 by the American Association for the Advancement of Science
Figure 28. Figure 28.

Representation of the three‐dimensional action space of the normal human visual system. The curved figure is the locus of the effect of the three systems of all wavelengths at a fixed intensity.

From Cornsweet
Figure 29. Figure 29.

A plane through trichromatic action space, showing the hue names of some of the stimuli it represents. The slightly curved triangular line in the shaded plane is the locus of the intersections of lights of all wavelengths with the plane. Therefore, it encloses a space representing the effects of all possible mixtures of wavelengths.

From Cornsweet
Figure 30. Figure 30.

A representation of the effects of mixtures of lights at 640 nm and 490 nm. As the relative intensities of the two components change, the effect of the mixtures move in a plane indicated by the straight lines emerging from the origin. The intersections of these lines with the shaded plane are indicated by dots, and form a straight line between the points representing 640 nm and 490 nm.

Figure 31. Figure 31.

The hue names of mixtures of 490‐nm and 640‐nm light change as the ratio of the intensities of the two components change, as indicated along the line between 640 nm and 490 nm. There is a particular mixture of 440‐nm and 578‐nm lights that will produce an identical hue, as indicated where the two straight lines cross.

Figure 32. Figure 32.

Increment‐threshold curves for rod‐initiated detection. Test stimulus, 9° diam, 0.2 s, centered 9° from fovea. Conditioning field, 20° diam, exposed steadily. Means from four observers are shown. Lowest curve correctly placed with respect to both axes. Other curves displaced upwards by 0.5, 1.0, and 1.5 log units, respectively.

From Aguilar and Stiles
Figure 33. Figure 33.

Dark‐adaptation curves follow adaptation to conditioning fields of various intensities. Filled symbols indicate thresholds where the test flash had a colored appearance

From Hecht et al.
Figure 34. Figure 34.

Panel A: recovery curves for 5′ (top) and 6° (bottom) test lights following offset of 3,600′ conditioning field. Panel B: increment‐threshold curves for the same test lights superimposed on a 3,600′ steady‐background field.

From Blakemore and Rushton
Figure 35. Figure 35.

Dark‐adaptation curves following adaptation to conditioning fields of various intensities.

From Winsor and Clark
Figure 36. Figure 36.

Time constant of best‐fitting exponential curve to A: psychophysical threshold recovery during dark adaptation, and B: rhodopsin regeneration in the living human eye. Both are plotted as a function of energy in the conditioning field. Open symbols in B are individual runs, filled symbols are the arithmetic means of individual runs.

Data from Pugh
Figure 37. Figure 37.

Fourier analytic concepts involved in one‐dimensional spurious resolution. See text for description.

Figure 38. Figure 38.

Two‐dimensional disk function (top) and its Fourier transform (bottom). Up to a constant factor π, the latter is the graph of Equation .

From Goodman
Figure 39. Figure 39.

Concepts involved in the one‐dimensional sampling theorem; f(x) is an input signal with transform F(s) which vanishes for s > sc. III (x/t) is an infinite row of delta functions (i.e., sample points) spaced t units apart; its transform is a row of delta functions spaced t−1 units apart. The third row shows the sampled version of f (i.e., the product f(x) III (x/t)) and its transform. Note that here the sampling rate is inefficiently high: successive replicas of F(s) are separated by empty intervals. The fourth row illustrates the optimal sampling scheme, where t = (2Sc)−1: Here the replicas of F(s) in the transform of the sampled signal are precisely adjacent. The fifth row shows the effect of sampling too coarsely: In the transform of the sampled signal the replicas of F(s) overlap (aliasing).

From Bracewell . The Fourier Transform and Its Applications (2nd ed.), by R. Bracewell. Copyright © 1978, McGraw‐Hill Book Company. Used with permission of McGraw‐Hill Book Company
Figure 40. Figure 40.

One‐dimensional aliasing effects. Row a is a linearly increasing frequency pattern. Row b shows the sampled appearance of pattern a viewed through a raster plate consisting of fine vertical slits spaced at a frequency equal to nr on the scale. (This sampling rate guarantees perfect reconstruction for frequencies up to 0.5 nr, with aliasing distortion for higher frequencies.) Row c shows the effect of spatially postfiltering the sampled image b (in this case by adjusting the scanning spot size in a television camera which views pattern b. The visual analogue would be lateral neural interactions at the receptors or beyond.) Notice that this removes the raster lines but does not eliminate aliasing: Frequencies higher than 0.5 nr still appear as lower frequencies. Row d shows the effect of the spatial postfilter alone (i.e., on pattern a viewed without the raster plate).

From Schade
Figure 41. Figure 41.

Configurations for producing optical Fourier transforms. The focal length of the lens is represented by f; thin vertical lines represent the illumination (a monochromatic plane wave).

From Goodman
Figure 42. Figure 42.

Optical Fourier transforms. Each panel in the left block (three left‐hand columns) shows the optical transform (i.e., amplitude spectrum) of the corresponding image in the right block (three right‐hand columns).

From Harburn et al. . Reprinted from G. Harburn, C. A. Taylor, and T. R. Welberry: Atlas of Optical Transforms. Copyright © 1975 by G. Bell and Sons, Ltd. Used by permission of the publisher, Cornell University Press


Figure 1.

Blur circles in a schematic misfocused eye for two different pupil sizes.

From Cornsweet


Figure 2.

Chromatic aberration of the human eye. λ, Wavelength; D, magnitude of chromatic aberration in diopters (assuming the eye is in perfect focus for 578 nm, D gives the power of the spectacle lens required to focus other wavelengths). Solid line and solid dots, average from 12 observers of Bedford and Wyszecki ; crosses, average from 14 observers of Wald and Griffin ; dashed lines, total range for all observations of Bedford and Wyszecki.

Adapted from Bedford and Wyszecki


Figure 3.

Irregularities in refractive power across the pupil of a normal eye during relaxed accommodation (pupil diam, 7.2 mm). Contours join points of equal excess power (diopters).

From van den Brink , with permission from Pergamon Press, Ltd


Figure 4.

Pointspread function of an optical system. Σo is the object plane, with coordinates y,z; Σi, is the image plane; S(y,z), irradiance distribution produced in Σi by point source located at origin of Σo. Illustration is schematic.

From Hecht and Zajac


Figure 5.

Linespread function of an optical system. S(Z) is the irradiance distribution in the image plane produced by an infinitely long thin line source in object plane. (The absence of secondary ripples here like those in Fig. has no significance; both illustrations are schematic.)

From Hecht and Zajac


Figure 6.

Linespread functions of the human eye for white light at various pupil diameters. Heavy dotted curves, measured intensity (along horizontal axis) of retinal image of a line target as measured ophthalmoscopically. Narrower dashed curves, theoretical spread resulting from diffraction at the pupil alone. (All curves have been normalized to equal 1.0 at origin.)

From Campbell and Gubisch , with permission from Cambridge University Press


Figure 7.

Pointspread functions of the human eye for white light at various pupil diameters. Solid curves, radial profiles of theoretical pointspread functions calculated from linespread curves in Fig. . Narrower dashed curves, the point spread resulting from diffraction alone.

From Gubisch


Figure 8.

Modulation transfer functions (MTFs) of the human eye for various pupil diameters. Each curve shows contrast reduction (i.e., retinal image contrast divided by the object contrast) imposed on sinusoidal targets as function of their spatial frequency. Solid curves

From Gubisch . from top to bottom) correspond to pupil diameters of 2.4, 3.8, and 6.6 mm; dashed curve corresponds to 1.5 mm. MTFs calculated from linespread data of Fig.


Figure 9.

Sinusoidal gratings.

From Cornsweet


Figure 10.

Effects of defocus on human spatial contrast sensitivity (2‐mm pupil). Data points show retinal image contrast reduction (inferred from threshold data of one observer) imposed on each spatial frequency as a function of defocus. Upper panel: •, 1.5 diopters (myopic); ⋄, 2.5 diopters. Lower panel: ▪, 2.0 diopters, ◯, 3.5 diopters. Spatial frequency expressed as fraction of highest frequency passed by the optics of the eye, i.e., 1.0 ∼ 60 cycles/deg. Solid lines drawn by hand through data points. Dashed lines show results predicted from diffraction alone.

From Campbell and Green , with permission from Cambridge University Press


Figure 11.

Top: spoke target for demonstrating spurious resolution in human vision. See text for directions. Bottom: out‐of‐focus picture of spoke pattern illustrating spurious resolution in a photographic image.



Figure 12.

Human spatial contrast thresholds for vertical sinusoidal gratings at various mean intensity levels (2‐mm pupil, monochromatic light, wavelength 525 nm). Each curve plots threshold contrast as a function of spatial frequency for mean retinal illumination level (trolands) indicated by curve parameter (100% corresponds to contrast of 1.0).

Data from Van Ness and Bouman ; figure from Westheimer


Figure 13.

Cone density as a function of distance from the center (0°) of the fovea.

Data from Osterberg ; figure adapted from Ripps and Weale


Figure 14.

Spatial contrast‐sensitivity functions for sinusoidal interference targets formed with a coherent source; wavelength, 633 nm. Data for two observers; smooth curves drawn by hand through data points. Mean illumination level, 500 trolands.

From Campbell and Green , with permission from Cambridge University Press


Figure 15.

Spatial contrast‐sensitivity functions for interference fringes and normal targets at same mean retinal illuminance (500 trolands). Continuous smooth curve in lower portion of figure taken from data (closed circles) of Fig. (i.e., these are interference target thresholds). Open circles show contrast thresholds for targets viewed normally, i.e., through the optics of the eye. Closed circles in the top portion show the ratio between the two curves below: this is an inferred measure of the modulation transfer function of the optics of this observer's eye.

From Campbell and Green , with permission from Cambridge University Press


Figure 16.

Comparison between local spatial acuity for interference targets (633 nm) and intercone spacing at various distances from the center of the fovea (0° eccentricity). Closed circles show center‐to‐center intercone distances (1/distance in min). Open points show local spatial acuity for two observers. Visual acuity here is arbitrarily defined as twice the highest resolvable spatial frequency in cycles/min. This definition allows a direct comparison between local spatial acuity and local cone density: if acuity and cone density were matched according to the sampling theorem, both should fall on a common curve—as they do out to 2°. Targets subtended 32.4 min; mean intensity level was 1,200 trolands.

From Green , with permission from Cambridge University Press


Figure 17.

A: spatial contrast‐sensitivity functions for three observers (dashed lines) with fitted function s2es (solid curve), which can be regarded as the overall “modulation transfer function” of the human visual system. B: pointspread function obtained by Fourier inversion of the smooth curve in Figure A.

From Kelly


Figure 18.

The top and middle rows illustrate aliasing by an array of photoreceptors. Top left: photomicrograph of a 12′ × 13′ patch of human retina near the center of the fovea. The mean center‐to‐center distance between receptors is 0.53′, implying an aliasing cutoff of 57 cycles/deg. Top right: sampling grid made by punching a pinhole through the center of each receptor in the 12′ × 13′ patch. Middle left: 30‐cycle/deg square wave grating seen through the pinhole grid. Middle right: 80‐cycle/deg square wave grating seen through the pinhole grid. Note the appearance of broad curved line segments due to aliasing. The bottom row illustrates the effect of a regularly spaced (checkerboard) array of sampling points having roughly the same sample‐point distance as the average value for the retinal array. Bottom left: 30‐cycle/deg square wave grating seen through the regular sampling grid. Bottom right: 80‐cycle/deg square wave grating seen through the regular sampling grid.

Top left from Polyak , p. 268


Figure 19.

The scotopic visual action spectrum plotted two ways. Horizontal axis is linear with wave number (number of waves per cm). See text for explanation of data collection procedure.

From Cornsweet


Figure 20.

Absorption spectrum of human ocular media.

From Ludvigh and McCarthy


Figure 21.

Scotopic visual action spectrum corrected for absorption by ocular media (X) compared with absorption of rhodopsin (dashed curve).

From Cornsweet


Figure 22.

Spectral sensitivity curve of rods (upper curve) compared with spectral sensitivity curve for a hypothetical set of cones (lower curve). Note that this lower curve does not represent any real set of cones and is strictly for explanatory purposes.



Figure 23.

A: absorption spectra for two classes of cones. (These are two of the three classes present in the normal human eye.) B: two‐dimensional visual action space of a person whose eyes contain only the two kinds of cones plotted in A. See text for derivation of system B from A.

From Cornsweet


Figure 24.

A representation in two‐dimensional action space of the effects of adding lights together. The result can be represented simply as the vector sum of the effects of the components.



Figure 25.

The effects of a mixture of lights of wavelengths 500 nm and 620 nm and of appropriately chosen intensities (2,050 and 910 quanta · s−1 · mm−2, respectively, in this example) will be identical to the effects of a light of 560 nm (at an intensity of 1,000 quanta · s−1 · mm−2).



Figure 26.

If a mixture of two wavelengths is to match a third whose effects do not lie on a line between the two to be mixed, one of the two components of the mixture must have a negative intensity, as shown by the downward direction of the 500‐nm vector. The physical realization of this negative vector requires the addition of 500 nm light of that magnitude to the 620‐nm component.



Figure 27.

Estimates of the absorption spectra of the three normal human cone pigments.

From Wald . Copyright 1964 by the American Association for the Advancement of Science


Figure 28.

Representation of the three‐dimensional action space of the normal human visual system. The curved figure is the locus of the effect of the three systems of all wavelengths at a fixed intensity.

From Cornsweet


Figure 29.

A plane through trichromatic action space, showing the hue names of some of the stimuli it represents. The slightly curved triangular line in the shaded plane is the locus of the intersections of lights of all wavelengths with the plane. Therefore, it encloses a space representing the effects of all possible mixtures of wavelengths.

From Cornsweet


Figure 30.

A representation of the effects of mixtures of lights at 640 nm and 490 nm. As the relative intensities of the two components change, the effect of the mixtures move in a plane indicated by the straight lines emerging from the origin. The intersections of these lines with the shaded plane are indicated by dots, and form a straight line between the points representing 640 nm and 490 nm.



Figure 31.

The hue names of mixtures of 490‐nm and 640‐nm light change as the ratio of the intensities of the two components change, as indicated along the line between 640 nm and 490 nm. There is a particular mixture of 440‐nm and 578‐nm lights that will produce an identical hue, as indicated where the two straight lines cross.



Figure 32.

Increment‐threshold curves for rod‐initiated detection. Test stimulus, 9° diam, 0.2 s, centered 9° from fovea. Conditioning field, 20° diam, exposed steadily. Means from four observers are shown. Lowest curve correctly placed with respect to both axes. Other curves displaced upwards by 0.5, 1.0, and 1.5 log units, respectively.

From Aguilar and Stiles


Figure 33.

Dark‐adaptation curves follow adaptation to conditioning fields of various intensities. Filled symbols indicate thresholds where the test flash had a colored appearance

From Hecht et al.


Figure 34.

Panel A: recovery curves for 5′ (top) and 6° (bottom) test lights following offset of 3,600′ conditioning field. Panel B: increment‐threshold curves for the same test lights superimposed on a 3,600′ steady‐background field.

From Blakemore and Rushton


Figure 35.

Dark‐adaptation curves following adaptation to conditioning fields of various intensities.

From Winsor and Clark


Figure 36.

Time constant of best‐fitting exponential curve to A: psychophysical threshold recovery during dark adaptation, and B: rhodopsin regeneration in the living human eye. Both are plotted as a function of energy in the conditioning field. Open symbols in B are individual runs, filled symbols are the arithmetic means of individual runs.

Data from Pugh


Figure 37.

Fourier analytic concepts involved in one‐dimensional spurious resolution. See text for description.



Figure 38.

Two‐dimensional disk function (top) and its Fourier transform (bottom). Up to a constant factor π, the latter is the graph of Equation .

From Goodman


Figure 39.

Concepts involved in the one‐dimensional sampling theorem; f(x) is an input signal with transform F(s) which vanishes for s > sc. III (x/t) is an infinite row of delta functions (i.e., sample points) spaced t units apart; its transform is a row of delta functions spaced t−1 units apart. The third row shows the sampled version of f (i.e., the product f(x) III (x/t)) and its transform. Note that here the sampling rate is inefficiently high: successive replicas of F(s) are separated by empty intervals. The fourth row illustrates the optimal sampling scheme, where t = (2Sc)−1: Here the replicas of F(s) in the transform of the sampled signal are precisely adjacent. The fifth row shows the effect of sampling too coarsely: In the transform of the sampled signal the replicas of F(s) overlap (aliasing).

From Bracewell . The Fourier Transform and Its Applications (2nd ed.), by R. Bracewell. Copyright © 1978, McGraw‐Hill Book Company. Used with permission of McGraw‐Hill Book Company


Figure 40.

One‐dimensional aliasing effects. Row a is a linearly increasing frequency pattern. Row b shows the sampled appearance of pattern a viewed through a raster plate consisting of fine vertical slits spaced at a frequency equal to nr on the scale. (This sampling rate guarantees perfect reconstruction for frequencies up to 0.5 nr, with aliasing distortion for higher frequencies.) Row c shows the effect of spatially postfiltering the sampled image b (in this case by adjusting the scanning spot size in a television camera which views pattern b. The visual analogue would be lateral neural interactions at the receptors or beyond.) Notice that this removes the raster lines but does not eliminate aliasing: Frequencies higher than 0.5 nr still appear as lower frequencies. Row d shows the effect of the spatial postfilter alone (i.e., on pattern a viewed without the raster plate).

From Schade


Figure 41.

Configurations for producing optical Fourier transforms. The focal length of the lens is represented by f; thin vertical lines represent the illumination (a monochromatic plane wave).

From Goodman


Figure 42.

Optical Fourier transforms. Each panel in the left block (three left‐hand columns) shows the optical transform (i.e., amplitude spectrum) of the corresponding image in the right block (three right‐hand columns).

From Harburn et al. . Reprinted from G. Harburn, C. A. Taylor, and T. R. Welberry: Atlas of Optical Transforms. Copyright © 1975 by G. Bell and Sons, Ltd. Used by permission of the publisher, Cornell University Press
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John I. Yellott, Brian A. Wandell, Tom N. Cornsweet. The Beginnings of Visual Perception: The Retinal Image and its Initial Encoding. Appendix: Fourier Transforms and Shift‐Invariant Linear Operators. Compr Physiol 2011, Supplement 3: Handbook of Physiology, The Nervous System, Sensory Processes: 257-316. First published in print 1984. doi: 10.1002/cphy.cp010307