Abstract

The inherent structure of the encoding in early stages of the visual system is investigated from a combined information-theoretical, psychophysical, and neurophysiological perspective. We argue that the classical modeling in terms of linear spatial filters is equivalent to the assumption of a Cartesian organization of the feature space of early vision. We show that such a linear Cartesian feature space would be suboptimal for the exploitation of the statistical redundancies of natural images since these have a radially separable probability-density function. Therefore a more efficient representation can be obtained by a nonlinear encoding that yields a feature space with polar organization. This prediction of the information-theoretical approach regarding the organization of the feature space of early vision is confirmed by our psychophysical measurements of basic discrimination capabilities for elementary Gabor patches, and the necessary nonlinear operations are shown to be closely related to cortical gain control and to the phase invariance of complex cells. Finally, we point out some striking similarities between the polar representation in visual cortex and basic image-coding strategies pursued in shape-gain vector quantization schemes.

© 1999 Optical Society of America

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1998

J. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
[CrossRef]

C. Zetzsche, G. Krieger, K. Schill, B. Treutwein, “Natural image statistics and cortical gain control,” Invest. Ophthalmol. Visual Sci. 39 (Suppl.), S424 (1998).

C. Zetzsche, G. Krieger, “Exploitation of natural scene statistics by orientation selectivity and cortical gain control,” Perception 27 (Suppl.), 154 (1998).

E. Simoncelli, O. Schwartz, “Derivation of a cortical normalization model from the statistics of natural images,” Invest. Ophthalmol. Visual Sci. 39 (Suppl.), S424 (1998).

1997

C. Zetzsche, E. Barth, G. Krieger, B. Wegmann, “Neural network models and the visual cortex: the missing link between cortical orientation selectivity and the natural environment,” Neurosci. Lett. 228(3), 155–158 (1997).
[CrossRef] [PubMed]

A. Bell, T. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

1996

B. Wegmann, C. Zetzsche, “Feature-specific vector quantization of images,” in Special issue on Vector Quantization, IEEE Trans. Image Process. 5, 274–288 (1996).
[CrossRef]

B. Olshausen, D. Field, “Wavelet-like receptive fields emerge from a network that learns sparse codes for natural images,” Nature (London) 381, 607–609 (1996).
[CrossRef]

1995

W. S. Geisler, D. G. Albrecht, “Bayesian analysis of identification performance in monkey visual cortex: nonlinear mechanisms and stimulus certainty,” Vision Res. 35, 2723–2730 (1995).
[CrossRef] [PubMed]

B. Treutwein, “Adaptive psychophysical procedures: a review,” Vision Res. 35, 2503–2522 (1995).
[CrossRef] [PubMed]

1994

M. Carandini, D. G. Heeger, “Summation and division by neurons in primate visual cortex,” Science 264, 1333–1336 (1994).
[CrossRef] [PubMed]

1992

J. H. van Hateren, “Real and optimal neural images in early vision,” Nature (London) 360, 68–69 (1992).
[CrossRef]

J. J. Atick, “Could information theory provide an ecological theory of sensory processing?” Network 3, 213–251 (1992).
[CrossRef]

I. Fujita, M. Ito, M. I. K. Cheng, “Columns for visual features of objects in monkey inferotemporal cortex,” Nature (London) 360, 343–346 (1992).
[CrossRef]

M. Stryker, “Elements of visual perception,” Nature (London) 360, 301–302 (1992).
[CrossRef]

1991

D. Albrecht, W. Geisler, “Motion selectivity and the contrast-response function of simple cells in the visual cortex,” Visual Neurosci. 7, 531–546 (1991).
[CrossRef]

P. J. Bennett, M. S. Banks, “The effects of contrast, spatial scale, and orientation on foveal and peripheral phase discrimination,” Vision Res. 31, 1759–1786 (1991).
[CrossRef] [PubMed]

1990

R. Vogels, “Population coding of stimulus orientation by striate cortical cells,” Biol. Cybern. 64, 25–31 (1990).
[CrossRef] [PubMed]

S. Bowne, “Contrast discrimination cannot explain spatial frequency, orientation or temporal frequency discrimination,” Vision Res. 30, 449–461 (1990).
[CrossRef] [PubMed]

I. Daubechies, “The wavelet transform, time frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

1989

1988

C. Morrone, D. J. Burr, “Feature detection in human vision: a phase dependent energy model,” Proc. R. Soc. London Ser. B 235, 221–245 (1988).
[CrossRef]

C. Zetzsche, B. Wegmann, “Coding properties of local amplitude and phase of two-dimensional filter outputs,” Perception 17, 396 (1988).

J. G. Daugman, “Complete discrete 2-d Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust., Speech, Signal Process. 36, 1169–1179 (1988).
[CrossRef]

1987

1986

C. Tyler, A. Gorea, “Different encoding mechanisms for phase and contrast,” Vision Res. 26, 1073–1082 (1986).
[CrossRef] [PubMed]

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple-cell receptive field properties in the cat’s visual cortex,” Proc. R. Soc. London Ser. B 228, 379–400 (1986).
[CrossRef]

1985

1984

M. J. Sabin, R. M. Gray, “Product code vector quantizers for waveform and voice coding,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 474–488 (1984).
[CrossRef]

1983

R. C. Reininger, J. D. Gibson, “Distributions of the two-dimensional DCT coefficients for images,” IEEE Trans. Commun. C-31, 835–839 (1983).
[CrossRef]

1982

D. G. Albrecht, D. B. Hamilton, “Striate cortex of monkeys and cat: contrast response functions,” J. Neurophysiol. 48, 217–237 (1982).
[PubMed]

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London Ser. B 216, 427–459 (1982).
[CrossRef]

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

G. Sclar, R. Freeman, “Orientation selectivity in the cat’s striate cortex is invariant with stimulus contrast,” Exp. Brain Res. 46, 457–461 (1982).
[CrossRef]

1981

B. Julesz, R. Schumer, “Early visual perception,” Annu. Rev. Psychol. 32, 575–627 (1981).
[CrossRef] [PubMed]

1980

1972

H. B. Barlow, “Single units and sensation: a neuron doctrine for perceptual psychology,” Perception 1, 371–394 (1972).
[CrossRef]

1968

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

1959

D. H. Hubel, T. N. Wiesel, “Receptive fields of single neurones in the cat’s striate cortex,” J. Physiol. (London) 148, 574–591 (1959).

1954

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

1949

Adelson, E. H.

Albrecht, D.

D. Albrecht, W. Geisler, “Motion selectivity and the contrast-response function of simple cells in the visual cortex,” Visual Neurosci. 7, 531–546 (1991).
[CrossRef]

Albrecht, D. G.

W. S. Geisler, D. G. Albrecht, “Bayesian analysis of identification performance in monkey visual cortex: nonlinear mechanisms and stimulus certainty,” Vision Res. 35, 2723–2730 (1995).
[CrossRef] [PubMed]

D. G. Albrecht, D. B. Hamilton, “Striate cortex of monkeys and cat: contrast response functions,” J. Neurophysiol. 48, 217–237 (1982).
[PubMed]

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

Atick, J. J.

J. J. Atick, “Could information theory provide an ecological theory of sensory processing?” Network 3, 213–251 (1992).
[CrossRef]

Attneave, F.

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Banks, M. S.

P. J. Bennett, M. S. Banks, “The effects of contrast, spatial scale, and orientation on foveal and peripheral phase discrimination,” Vision Res. 31, 1759–1786 (1991).
[CrossRef] [PubMed]

Barlow, H.

H. Barlow, “What is the computational goal of the neocortex?” in Large-Scale Neuronal Theories of the Brain, C. Koch, ed. (MIT Press, Cambridge, Mass., 1994), pp. 1–22.

Barlow, H. B.

H. B. Barlow, “Unsupervised learning,” Neural Comput. 1, 295–311 (1989).
[CrossRef]

H. B. Barlow, “Single units and sensation: a neuron doctrine for perceptual psychology,” Perception 1, 371–394 (1972).
[CrossRef]

H. B. Barlow, “Sensory mechanisms, the reduction of redundancy and intelligence,” in National Physical Laboratory Symposium No. 10 (Her Majesty’s Stationary Office, London, 1959).

Barth, E.

C. Zetzsche, E. Barth, G. Krieger, B. Wegmann, “Neural network models and the visual cortex: the missing link between cortical orientation selectivity and the natural environment,” Neurosci. Lett. 228(3), 155–158 (1997).
[CrossRef] [PubMed]

C. Zetzsche, E. Barth, B. Wegmann, “The importance of intrinsically two-dimensional image features in biological vision and picture coding,” in Digital Images and Human Vision, A. Watson, ed. (MIT Press, Cambridge, Mass., 1993), pp. 109–138.

C. Zetzsche, E. Barth, B. Wegmann, “Nonlinear aspects of primary vision: entropy reduction beyond decorrelation,” in SID International Symposium—Digest of Technical Papers, XXIV, J. Morreale, ed. (Society for Information Display, Playa del Ray, Calif., 1993), Vol. XXIV, pp. 933–936.

Bell, A.

A. Bell, T. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

Bennett, P. J.

P. J. Bennett, M. S. Banks, “The effects of contrast, spatial scale, and orientation on foveal and peripheral phase discrimination,” Vision Res. 31, 1759–1786 (1991).
[CrossRef] [PubMed]

Bergen, J. R.

Bowne, S.

S. Bowne, “Contrast discrimination cannot explain spatial frequency, orientation or temporal frequency discrimination,” Vision Res. 30, 449–461 (1990).
[CrossRef] [PubMed]

Braun, J.

L. Itti, J. Braun, D. K. Lee, C. Koch, “A model of early visual processing,” in Advances in Neural Information Processing Systems, M. Kearns, S. Solla, eds. (MIT Press, Cambridge, Mass., 1998), Vol. 10, pp. 173–179.

Brown, W.

Buccigrossi, R. W.

R. W. Buccigrossi, E. P. Simoncelli, “Image compression via joint statistical characterization in the wavelet domain,” (General Robotics and Active Sensory Perception Laboratory, University of Pennsylvania, Philadelphia, 1997).

Burr, D. C.

D. C. Burr, “Sensitivity to spatial phase,” Vision Res. 20, 391–396 (1980).
[CrossRef] [PubMed]

Burr, D. J.

C. Morrone, D. J. Burr, “Feature detection in human vision: a phase dependent energy model,” Proc. R. Soc. London Ser. B 235, 221–245 (1988).
[CrossRef]

Campbell, F. W.

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Carandini, M.

M. Carandini, D. G. Heeger, “Summation and division by neurons in primate visual cortex,” Science 264, 1333–1336 (1994).
[CrossRef] [PubMed]

Cheng, M. I. K.

I. Fujita, M. Ito, M. I. K. Cheng, “Columns for visual features of objects in monkey inferotemporal cortex,” Nature (London) 360, 343–346 (1992).
[CrossRef]

Daubechies, I.

I. Daubechies, “The wavelet transform, time frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

Daugman, J. G.

J. G. Daugman, “Complete discrete 2-d Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust., Speech, Signal Process. 36, 1169–1179 (1988).
[CrossRef]

J. G. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters,” J. Opt. Soc. Am. A 2, 1160–1169 (1985).
[CrossRef] [PubMed]

DeValois, K. K.

R. L. DeValois, K. K. DeValois, Spatial Vision (Oxford U. Press, New York, 1988).

DeValois, R. L.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

R. L. DeValois, K. K. DeValois, Spatial Vision (Oxford U. Press, New York, 1988).

Dubs, A.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London Ser. B 216, 427–459 (1982).
[CrossRef]

Field, D.

B. Olshausen, D. Field, “Wavelet-like receptive fields emerge from a network that learns sparse codes for natural images,” Nature (London) 381, 607–609 (1996).
[CrossRef]

Field, D. J.

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
[CrossRef] [PubMed]

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple-cell receptive field properties in the cat’s visual cortex,” Proc. R. Soc. London Ser. B 228, 379–400 (1986).
[CrossRef]

Foley, J. M.

Freeman, R.

G. Sclar, R. Freeman, “Orientation selectivity in the cat’s striate cortex is invariant with stimulus contrast,” Exp. Brain Res. 46, 457–461 (1982).
[CrossRef]

Fujita, I.

I. Fujita, M. Ito, M. I. K. Cheng, “Columns for visual features of objects in monkey inferotemporal cortex,” Nature (London) 360, 343–346 (1992).
[CrossRef]

Geisler, W.

D. Albrecht, W. Geisler, “Motion selectivity and the contrast-response function of simple cells in the visual cortex,” Visual Neurosci. 7, 531–546 (1991).
[CrossRef]

Geisler, W. S.

W. S. Geisler, D. G. Albrecht, “Bayesian analysis of identification performance in monkey visual cortex: nonlinear mechanisms and stimulus certainty,” Vision Res. 35, 2723–2730 (1995).
[CrossRef] [PubMed]

Gersho, A.

A. Gersho, M. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, Mass., 1992).

Gibson, J. D.

R. C. Reininger, J. D. Gibson, “Distributions of the two-dimensional DCT coefficients for images,” IEEE Trans. Commun. C-31, 835–839 (1983).
[CrossRef]

Gorea, A.

C. Tyler, A. Gorea, “Different encoding mechanisms for phase and contrast,” Vision Res. 26, 1073–1082 (1986).
[CrossRef] [PubMed]

Gouled-Smith, B.

Graham, N.

N. Graham, Visual Pattern Analyzers (Oxford U. Press, New York, 1989).

Gray, M.

A. Gersho, M. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, Mass., 1992).

Gray, R. M.

M. J. Sabin, R. M. Gray, “Product code vector quantizers for waveform and voice coding,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 474–488 (1984).
[CrossRef]

Hamilton, D. B.

D. G. Albrecht, D. B. Hamilton, “Striate cortex of monkeys and cat: contrast response functions,” J. Neurophysiol. 48, 217–237 (1982).
[PubMed]

Heeger, D.

D. Heeger, “Nonlinear model of neural responses in cat visual cortex,” in Computational Models of Visual Processing, M. Landy, J. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 119–133.

Heeger, D. G.

M. Carandini, D. G. Heeger, “Summation and division by neurons in primate visual cortex,” Science 264, 1333–1336 (1994).
[CrossRef] [PubMed]

Hubel, D. H.

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Figures (11)

Fig. 1
Fig. 1

Different transformations of the probability-density function (pdf) that can be achieved with linear and with nonlinear methods. Principal-components analysis, PCA in the figure, can linearly transform the pdf into a new feature space with rotated Cartesian coordinates. The new components are decorrelated, which amounts to statistical independence in the case of Gaussian statistics. Linear independent-components analysis, (L)ICA, can realize an affine transform into a feature space with sheared Cartesian coordinates. This permits the separation of a linear mixture of independent sources. The separation of nonlinear mixtures requires a deformation of the intrinsic structure of the coordinate system, i.e., a feature space with curved coordinates. This is possible only with an appropriate nonlinear transformation (nonlinear independent-components analysis, NICA).

Fig. 2
Fig. 2

Even- and odd-symmetric linear filters with standard parameters (radial octave bandwidth, ±15-deg orientation bandwidth) were applied to the set of natural images shown on the left, and the individual filter statistics p(e), p(o), and the joint pdf p(e, o) were evaluated to determine whether p(e, o)=pˆ(e, o)=p(e)p(o), i.e., whether the linear Cartesian components are statistically independent. The predicted pdf pˆ(e, o) and the actual joint pdf p(e, o) are shown in two versions, as surface plots (upper row) and as contour plots (bottom row) on the left and on the right side, respectively. Obviously, a Cartesian encoding cannot even approach the desired statistical independence, since the shape of the true pdf p(e, o) is far from the predicted starlike shape of x˙(e, o) and exhibits instead a strong circular symmetry. Further tests showed that this basic result is independent of the special type of imagery and calibration.

Fig. 3
Fig. 3

Schematic examples of stimuli in the increment-threshold measurements. The form of a Gabor patch changes systematically in dependence of the variation of the parameters in the two-dimensional feature space. This is illustrated by the upper set of images, which correspond to signals located in the upper right quadrant of the feature space. These images could be equivalently indexed by the two Cartesian variables e, o, or by the two polar variables r, ϕ, i.e., by a local amplitude (contrast) and a local phase value. For example, the images that are directly located on the e or the o axis (bottom row and left column) are pure even-symmetric or odd-symmetric patterns, respectively, while the images in between have mixed symmetry. Equivalently, the contrast of the images increases in proportion to their distance r from the origin, and their phase φ varies continuously with the angle.

Fig. 4
Fig. 4

Incremental threshold curves predicted by a Cartesian representation (left) and by a polar representation (right). The jnd figures will reflect the intrinsic coordinate system of the representation used in early vision. When the relevant representation is organized in a Cartesian fashion, the jnd figures should be squarelike or circlelike, whereas they should be similar to a sector of a circular ring or kidneylike if the representation is polar.

Fig. 5
Fig. 5

Psychophysically measured jnd curve for Gabor patches. The axes and the test patterns are schematically indicated in Fig. 3. The jnd curve that results in this experiment is kidneylike and is thus inseparable in Cartesian coordinates but separable in polar coordinates. This is a clear indication for a polar representation in early vision. The task of the subjects was to decide at which point they can perceive a difference between a reference Gabor patch and a modified patch that is varied with respect to its local phase, its amplitude, or with respect to a combination of both parameters. The position of the reference patch in the parameter space is indicated by the square in the center of the jnd figure. The circles show the measured incremental thresholds (jnd’s) that result if the parameters of the test patch are varied in direction from the reference toward the respective circle. For example, a pure contrast change leads to the circles located on the diagonal. Error bars represent the standard deviation (rms error) pooled over all trials and subjects and separated into an amplitude and a phase component. A detailed description of the experiment can be found in Appendix B. A control experiment in which the increments were varied in Cartesian coordinates yielded a jnd figure with basically identical shape.

Fig. 6
Fig. 6

jnd curves of Fig. 5 replotted in polar coordinates.

Fig. 7
Fig. 7

Amplitude/phase response surface of a simple cell predicted by the Cartesian standard model (upper half) and by the nonlinear cortical gain-control model of Heeger46,47 (lower half). The response surfaces are shown in Cartesian coordinates (left column) and in polar coordinates (right column). The response surface of the linear model will separate the signal properties with respect to Cartesian coordinates while confounding them with respect to polar coordinates. The response surface of the gain-controlled unit can provide the desired separation with respect to the polar coordinates. Gain-controlled units are hence most appropriately interpreted as performing an encoding in terms of a polar feature space.

Fig. 8
Fig. 8

Amplitude/phase response surface of a simple cell. The figure is derived from the actual measurements by Albrecht and Geisler of responses of a cortical simple cell to counterphase flickering gratings presented at different relative positions and different contrast levels.45 Since the contrast measurements have not been continued beyond the contrast saturation level, we have extrapolated the responses to the 100% contrast level, as indicated by the dashed lines. It is obvious that cortical neurons are best described as operating in polar feature space. Units like the one shown here, which reach their saturation at low contrasts, act effectively as phase-encoding devices over most of the contrast range.

Fig. 9
Fig. 9

Schematic comparison of shape-gain vector quantization in communication engineering (left) with the distributed encoding by a set of neural gain-controlled units (right). Shown is the simple case of the processing of a one-dimensional signal by a three-unit encoding scheme. The two schemes have equivalent overall signal-encoding properties; i.e., they yield an identical partitioning of the axis, although their implementation differs in detail. In both schemes a signal can be distinguished from another one if there is a change of the global system state, i.e., if at least one of the units changes its activity from ON to OFF or vice versa (e.g., a transition 010001 in the vector quantizer corresponds to a transition 110111 in the biological scheme). Differences are as follows: (i) Biological units can produce graded responses in the transition regions; hence the discrimination sensitivity will be higher than with simple binary units. (ii) The biological scheme is more robust against a loss of units. A unit of the technical scheme will be switched into OFF state once its amplitude range is exceeded, whereas a cortical unit remains in ON state once it is driven into saturation (neural responses have been reported to decrease again for higher contrasts in some cases, however (e.g., Ref. 44). Whereas loss of a unit in vector quantization will cause the complete ignorance of certain signals, the error-protection redundancy of the biological scheme will increase with contrast such that the risk for ignoring high-contrast features is minimized.

Fig. 10
Fig. 10

In the case of the two-dimensional feature space considered here, shape-gain vector quantization and the distributed encoding by neural gain-controlled units will both produce the same kind of polar partitioning of the feature space. The shaded area exemplifies one region of the two-dimensional feature space corresponding to one of the possible system states. All signals falling into this region appear to the system to be basically equivalent. Note the structural similarity of this figure to the jnd figure measured psychophysically (Fig. 5). Only if two signals are sufficiently dissimilar to cause distinct states of the system can they be distinguished from each other.

Fig. 11
Fig. 11

Exploitation of a radially separable pdf by distributed encoding with gain-controlled units. Shown is a schematized view of the multidimensional pdf of a local image patch, where bright and dark areas indicate regions of high and low probability, respectively. Since the radial component pr(r) can be separated from the multivariate pdf, the characteristic angular probability pattern within concentric (hyper)spherical shells remains essentially constant, up to a scaling for the different radii by pr(r). (Only for the special case of a spherically symmetric pdf will this probability pattern be a constant). The statistical redundancies given by this specific type of pdf can be exploited by a partitioning of the feature space into segments of hypercones, the extensions of which will depend inversely on the magnitude of the angular probability-density function. This corresponds to a distributed encoding by gain-controlled units with various gains and tuning functions, where the combination of gain control and subsequent expansive or sigmoid nonlinearities can address much more specific regions of the hyperspace than can any linear decomposition.

Equations (12)

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g(x, y; e, o)
=exp[-πa(x2+y2)]{e cos[2π(u0x+ν0y)]
+o sin[2π(u0x+ν0y)]},
g(x, y; r, φ)
=exp[-πa(x2+y2)]r{cos φ cos[2π(u0x+ν0y)]
+sin φ sin[2π(u0x+ν0y)]}.
px(x1,, xN)=pr(r)pΘ(Θ1,, ΘN-1).
c(x)=si(x)*sj(-x),
C(u)=Si(u)Sj(-u)=L(u)Hi(u)L(-u)Hj(-u),
|L(u)|2Hi(u)Hj*(u)du=!0.
Hi(u)Hj(u)=0, u,
Hij(-u)=-Hij(u).

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