Abstract

Triple correlation uniqueness refers to the fact that every monochromatic image of finite size is uniquely determined up to translation by its triple correlation function. Here that fact is used to prove that every finite image composed of discrete colors is determined up to translation by its third-order statistics. Consequently, if two texture samples have identical third-order statistics, they must be physically identical images and thus visually nondiscriminable by definition. It follows that a third-order version of Julesz’s long-abandoned conjecture about spontaneous texture discrimination is necessarily true (notwithstanding such well-known counterexamples as the odd and even textures). The second-order (i.e., original) version of that conjecture is not necessarily true: physically distinct finite images with identical second-order statistics can be constructed, so counterexamples are possible. However, the counterexamples that one finds in the literature are either nonexact or difficult to reconstruct. A new principle is described that permits the easy construction of discriminable black and white texture samples that have strictly identical second-order statistics and thus provide exact counterexamples to the Julesz conjecture.

© 1993 Optical Society of America

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  1. B. Julesz, “Visual pattern discrimination,”IRE Trans. Inf. Theory IT-8, 84–92 (1962).
    [CrossRef]
  2. B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics—revisited,” Perception 2, 391–405 (1973).
    [CrossRef]
  3. B. Julesz, “Experiments in the visual perception of texture,” Sci. Am. 232, 34–43 (1975).
    [CrossRef] [PubMed]
  4. T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part 1,” Biol. Cybernet. 28, 167–175 (1978).
    [CrossRef]
  5. T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part 2,” Biol. Cybernet. 29, 201–214 (1978).
    [CrossRef]
  6. B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybernet. 31, 137–140 (1978).
    [CrossRef]
  7. P. Diaconis, D. Freedman, “On the statistics of vision: the Julesz conjecture,”J. Math. Psychol. 24, 112–138 (1981).
    [CrossRef]
  8. B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature (London) 290, 91–97 (1981).
    [CrossRef]
  9. J. R. Bergen, “Theories of visual texture perception,” in Spatial Vision, Vol. 10 of Vision and Visual Dysfunction, D. M. Regan, ed. (Macmillan, New York, 1991).
  10. H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
    [CrossRef] [PubMed]
  11. J. I. Yellott, G. J. Iverson, “Uniqueness properties of higher-order autocorrelation functions,” J. Opt. Soc. Am. A 9, 388–404 (1992).
    [CrossRef]
  12. A. Gagalowicz, “A new method for texture fields synthesis: some applications to the study of human vision,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-3, 520–533 (1981).
    [CrossRef]
  13. E. N. Gilbert, “Random colorings of a lattice of squares in the plane,” SIAM J. Algebra Discrete Meth. 1, 152–159 (1980).
    [CrossRef]
  14. J. D. Victor, M. M. Conte, “Spatial organization of nonlinear interactions in form perception,” Vision Res. 31, 1457–1488 (1991).
    [CrossRef] [PubMed]
  15. A. W. Lohmann, B. Wirnitzer, “Triple correlation,” Proc. IEEE 72, 889–901 (1984).
    [CrossRef]
  16. C. L. Nikias, J. M. Mendl, eds., Workshop on Higher-Order Spectral Analysis, Cat. No. 89TH0267-5 (Institute of Electrical and Electronics Engineers, New York, 1989).
  17. S. A. Klein, C. W. Tyler, “Phase discrimination of compound gratings: generalized autocorrelation analysis,” J. Opt. Soc. Am. A 3, 868–879 (1986). That paper was apparently written without awareness of the fact that finite-sized images are uniquely determined up to translation by their triple correlation functions. It focused on the higher-order autocorrelation functions of infinitely extended periodic images, which are not uniquely determined by their triple correlations, since they do not have bounded support: two periodic functions can have identical triple correlations without being translations of each other. (A one-dimensional example pointed out by Klein and Tyler is the pair 2 + cos x± cos 3x. Because infinitely supported periodic images have impulsive spectra, the uniqueness properties of their higher-order autocorrelations are quite different from those of finite images. Those properties are discussed in Ref. 11, and the complete theory is worked out in Ref. 18.)
    [CrossRef] [PubMed]
  18. R. Kakarala, G. J. Iverson, “Characterization of periodic functions by higher-order spectra,” (Institute for Mathematical Behavioral Sciences, University of California, Irvine, Irvine, Calif., 1991).
  19. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1966), Vol. II.
  20. J. Aczel, Lectures on Functional Equations (Academic, New York, 1966).
  21. J. D. Victor, “Complex visual textures as a tool for studying the VEP,” Vision Res. 25, 1811–1828 (1985).
    [CrossRef] [PubMed]
  22. J. D. Victor, Department of Neurology and Neuroscience, Cornell University Medical College, Ithaca, New York 10021 (personal communication).

1992 (1)

1991 (1)

J. D. Victor, M. M. Conte, “Spatial organization of nonlinear interactions in form perception,” Vision Res. 31, 1457–1488 (1991).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

J. D. Victor, “Complex visual textures as a tool for studying the VEP,” Vision Res. 25, 1811–1828 (1985).
[CrossRef] [PubMed]

1984 (2)

1981 (3)

P. Diaconis, D. Freedman, “On the statistics of vision: the Julesz conjecture,”J. Math. Psychol. 24, 112–138 (1981).
[CrossRef]

B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature (London) 290, 91–97 (1981).
[CrossRef]

A. Gagalowicz, “A new method for texture fields synthesis: some applications to the study of human vision,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-3, 520–533 (1981).
[CrossRef]

1980 (1)

E. N. Gilbert, “Random colorings of a lattice of squares in the plane,” SIAM J. Algebra Discrete Meth. 1, 152–159 (1980).
[CrossRef]

1978 (3)

T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part 1,” Biol. Cybernet. 28, 167–175 (1978).
[CrossRef]

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part 2,” Biol. Cybernet. 29, 201–214 (1978).
[CrossRef]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybernet. 31, 137–140 (1978).
[CrossRef]

1975 (1)

B. Julesz, “Experiments in the visual perception of texture,” Sci. Am. 232, 34–43 (1975).
[CrossRef] [PubMed]

1973 (1)

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics—revisited,” Perception 2, 391–405 (1973).
[CrossRef]

1962 (1)

B. Julesz, “Visual pattern discrimination,”IRE Trans. Inf. Theory IT-8, 84–92 (1962).
[CrossRef]

Aczel, J.

J. Aczel, Lectures on Functional Equations (Academic, New York, 1966).

Bartelt, H.

Bergen, J. R.

J. R. Bergen, “Theories of visual texture perception,” in Spatial Vision, Vol. 10 of Vision and Visual Dysfunction, D. M. Regan, ed. (Macmillan, New York, 1991).

Caelli, T. M.

T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part 1,” Biol. Cybernet. 28, 167–175 (1978).
[CrossRef]

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part 2,” Biol. Cybernet. 29, 201–214 (1978).
[CrossRef]

Conte, M. M.

J. D. Victor, M. M. Conte, “Spatial organization of nonlinear interactions in form perception,” Vision Res. 31, 1457–1488 (1991).
[CrossRef] [PubMed]

Diaconis, P.

P. Diaconis, D. Freedman, “On the statistics of vision: the Julesz conjecture,”J. Math. Psychol. 24, 112–138 (1981).
[CrossRef]

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1966), Vol. II.

Freedman, D.

P. Diaconis, D. Freedman, “On the statistics of vision: the Julesz conjecture,”J. Math. Psychol. 24, 112–138 (1981).
[CrossRef]

Frisch, H. L.

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics—revisited,” Perception 2, 391–405 (1973).
[CrossRef]

Gagalowicz, A.

A. Gagalowicz, “A new method for texture fields synthesis: some applications to the study of human vision,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-3, 520–533 (1981).
[CrossRef]

Gilbert, E. N.

E. N. Gilbert, “Random colorings of a lattice of squares in the plane,” SIAM J. Algebra Discrete Meth. 1, 152–159 (1980).
[CrossRef]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybernet. 31, 137–140 (1978).
[CrossRef]

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part 2,” Biol. Cybernet. 29, 201–214 (1978).
[CrossRef]

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics—revisited,” Perception 2, 391–405 (1973).
[CrossRef]

Iverson, G. J.

J. I. Yellott, G. J. Iverson, “Uniqueness properties of higher-order autocorrelation functions,” J. Opt. Soc. Am. A 9, 388–404 (1992).
[CrossRef]

R. Kakarala, G. J. Iverson, “Characterization of periodic functions by higher-order spectra,” (Institute for Mathematical Behavioral Sciences, University of California, Irvine, Irvine, Calif., 1991).

Julesz, B.

B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature (London) 290, 91–97 (1981).
[CrossRef]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybernet. 31, 137–140 (1978).
[CrossRef]

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part 2,” Biol. Cybernet. 29, 201–214 (1978).
[CrossRef]

T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part 1,” Biol. Cybernet. 28, 167–175 (1978).
[CrossRef]

B. Julesz, “Experiments in the visual perception of texture,” Sci. Am. 232, 34–43 (1975).
[CrossRef] [PubMed]

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics—revisited,” Perception 2, 391–405 (1973).
[CrossRef]

B. Julesz, “Visual pattern discrimination,”IRE Trans. Inf. Theory IT-8, 84–92 (1962).
[CrossRef]

Kakarala, R.

R. Kakarala, G. J. Iverson, “Characterization of periodic functions by higher-order spectra,” (Institute for Mathematical Behavioral Sciences, University of California, Irvine, Irvine, Calif., 1991).

Klein, S. A.

Lohmann, A. W.

Shepp, L. A.

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics—revisited,” Perception 2, 391–405 (1973).
[CrossRef]

Tyler, C. W.

Victor, J. D.

J. D. Victor, M. M. Conte, “Spatial organization of nonlinear interactions in form perception,” Vision Res. 31, 1457–1488 (1991).
[CrossRef] [PubMed]

J. D. Victor, “Complex visual textures as a tool for studying the VEP,” Vision Res. 25, 1811–1828 (1985).
[CrossRef] [PubMed]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybernet. 31, 137–140 (1978).
[CrossRef]

J. D. Victor, Department of Neurology and Neuroscience, Cornell University Medical College, Ithaca, New York 10021 (personal communication).

Wirnitzer, B.

Yellott, J. I.

Appl. Opt. (1)

Biol. Cybernet. (3)

T. M. Caelli, B. Julesz, “On perceptual analyzers underlying visual texture discrimination: Part 1,” Biol. Cybernet. 28, 167–175 (1978).
[CrossRef]

T. M. Caelli, B. Julesz, E. N. Gilbert, “On perceptual analyzers underlying visual texture discrimination: Part 2,” Biol. Cybernet. 29, 201–214 (1978).
[CrossRef]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybernet. 31, 137–140 (1978).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

A. Gagalowicz, “A new method for texture fields synthesis: some applications to the study of human vision,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-3, 520–533 (1981).
[CrossRef]

IRE Trans. Inf. Theory (1)

B. Julesz, “Visual pattern discrimination,”IRE Trans. Inf. Theory IT-8, 84–92 (1962).
[CrossRef]

J. Math. Psychol. (1)

P. Diaconis, D. Freedman, “On the statistics of vision: the Julesz conjecture,”J. Math. Psychol. 24, 112–138 (1981).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nature (London) (1)

B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature (London) 290, 91–97 (1981).
[CrossRef]

Perception (1)

B. Julesz, E. N. Gilbert, L. A. Shepp, H. L. Frisch, “Inability of humans to discriminate between visual textures that agree in second-order statistics—revisited,” Perception 2, 391–405 (1973).
[CrossRef]

Proc. IEEE (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlation,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Sci. Am. (1)

B. Julesz, “Experiments in the visual perception of texture,” Sci. Am. 232, 34–43 (1975).
[CrossRef] [PubMed]

SIAM J. Algebra Discrete Meth. (1)

E. N. Gilbert, “Random colorings of a lattice of squares in the plane,” SIAM J. Algebra Discrete Meth. 1, 152–159 (1980).
[CrossRef]

Vision Res. (2)

J. D. Victor, M. M. Conte, “Spatial organization of nonlinear interactions in form perception,” Vision Res. 31, 1457–1488 (1991).
[CrossRef] [PubMed]

J. D. Victor, “Complex visual textures as a tool for studying the VEP,” Vision Res. 25, 1811–1828 (1985).
[CrossRef] [PubMed]

Other (6)

J. D. Victor, Department of Neurology and Neuroscience, Cornell University Medical College, Ithaca, New York 10021 (personal communication).

C. L. Nikias, J. M. Mendl, eds., Workshop on Higher-Order Spectral Analysis, Cat. No. 89TH0267-5 (Institute of Electrical and Electronics Engineers, New York, 1989).

R. Kakarala, G. J. Iverson, “Characterization of periodic functions by higher-order spectra,” (Institute for Mathematical Behavioral Sciences, University of California, Irvine, Irvine, Calif., 1991).

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1966), Vol. II.

J. Aczel, Lectures on Functional Equations (Academic, New York, 1966).

J. R. Bergen, “Theories of visual texture perception,” in Spatial Vision, Vol. 10 of Vision and Visual Dysfunction, D. M. Regan, ed. (Macmillan, New York, 1991).

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Figures (12)

Fig. 1
Fig. 1

(a) 50 × 50 pixel samples of odd (right side) and even (left side) texture. Tick marks indicate the boundary between the two. (These are matched samples, as explained in Section 4.) (b) 50 × 50 samples of odd (right side) versus coin-toss (left side) texture. (c) 50 × 50 samples of even (left side) versus coin-toss (right side) texture.

Fig. 2
Fig. 2

(a) Scatterplot that compares all the third-order statistics [Eq. (9′)] of the odd and even texture samples shown in Fig. 1. The x axis corresponds to even texture statistic values, and the y axis corresponds to odd texture values. Both axes run from 0 to 0.6, with tick marks at 0.25 intervals. If the odd and even samples had identical third-order statistics, all the points would lie on the diagonal line. If all the statistics equaled their expected values (i.e., if the sample statistics equaled the ensemble statistics), all the points would fall at either (1/2, 1/2), (1/4, 1/4), or (1/8, 1/8). The two other scatterplots compare the third-order statistics of the odd versus coin-toss texture samples (b) and the even versus coin-toss samples (c) from Fig. 1. In (b) and (c) the y axis corresponds to coin-toss statistic values.

Fig. 3
Fig. 3

(a) Scatterplots that compare all the second-order statistics [Eq. (8′)] of the texture samples from Fig. 1. The axes are the same as those in Fig. 2. (a) Odd versus even, (b) odd versus coin-toss, (c) even versus coin-toss.

Fig. 4
Fig. 4

Exact one-dimensional counterexample to the Julesz conjecture, based on the construction principle explained in Section 5. The texture samples in (a) and (b) have identical second-order statistics. Sample (a) is composed of four replicas of the micropattern shown below in Fig. 12(a); sample (b) is composed of four replicas of the micropattern shown in Fig. 12(b). In (c) the two texture samples are displayed side by side.

Fig. 5
Fig. 5

Two-dimensional counterexample to the Julesz conjecture, based on the construction principle described in Section 5. The texture samples on the left and the right have identical second-order statistics.

Fig. 6
Fig. 6

(a) Scatterplot that compares all third-order statistics of small (10 × 10) samples of odd-versus-coin-toss texture (shown in the inset). (b) Scatterplot that compares the same third-order statistics for 50 × 50 samples of odd-versus-coin-toss texture. The axes are the same as those in Fig. 2, with coin-toss statistic values on the y axis.

Fig. 7
Fig. 7

(a) Scatterplot that compares all third-order statistics for two independent 10 × 10 samples of coin-toss texture. The axes are the same as those in Fig. 2. (b) Comparison of the same statistics for independent 50 × 50 coin-toss samples. (c) Comparison of all third-order statistics for the same 50 × 50 samples.

Fig. 8
Fig. 8

Solid black even image and its odd match. (b) Non-matched 50 × 50 samples of odd (right side) and even (left side) texture. The two images are created with different initial colorings for the zeroth row and column. (The odd and even images in Fig. 1 are a matched pair; the same initial coloring generated both images.)

Fig. 9
Fig. 9

(a) Scatterplot that compares all third-order statistics of 10 × 10 matched samples of odd and even texture (shown in the inset). The axes are the same as those in Fig. 2. (b) Comparison of the same third-order statistics (0 ≤ ni, mi, ≤ 9) for a matched pair of 50 × 50 odd and even samples (the samples in Fig. 1). (c) Comparison of third-order statistics in the same range as that in (b) but only for statistics whose arguments are all even numbers. For this subset of statistics, matched odd and even samples are strongly correlated.

Fig. 10
Fig. 10

Same comparisons as those in Fig. 9 but for the non-matched odd and even samples shown in Fig. 8.

Fig. 11
Fig. 11

Comparison of small-argument (0 ≤ ni, mi ≤ 9) third-order statistics of the 50 × 50 even and coin-toss texture samples fom Fig. 1. Coin-toss values are on the y axis.

Fig. 12
Fig. 12

Pairs of bar-code micropatterns with identical second-order statistics. The numbers are the positions of the bars. Pair (a) and (b) are p(x) * q(x) and p(x) * q(−x), respectively, where p and q are defined by Eqs. (22) and (23), respectively. Pair (c) and (d) result from a change in the upper limits of the sums from 4 to 3. (The functions have been translated here so that the first bars of all the patterns fall at position 0.)

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

s 1 , f = 1 A R 2 f ( x , y ) d x d y .
s 2 , f ( h , v ) = 1 A R 2 f ( x , y ) f ( x + h , y + v ) d x d y .
s 3 , f ( h 1 , v 1 , h 2 , v 2 ) = 1 A R 2 f ( x , y ) f ( x + h 1 , y + v 1 ) f ( x + h 2 , y + v 2 ) d x d y .
1 A R 2 f ( x , y ) [ 1 - f ( x + h , y + v ) ] d x d y = s 1 , f - s 2 , f ( h , v ) .
s 1 , i , f = 1 A R 2 f i ( x , y ) d x d y ,
s 2 , i , j , f ( h , v ) = 1 A R 2 f i ( x , y ) f j ( x + h , y + v ) d x d y ,
s 3 , i , j , k , f ( h 1 , v 1 , h 2 , v 2 ) = 1 A R 2 f i ( x , y ) f j ( x + h 1 , y + v 1 ) f k ( x + h 2 , y + v 2 ) d x d y ,
s 1 , p = 1 C R c = 0 C - 1 r = 0 R - 1 p ( c , r ) ,
s 2 , p ( n , m ) = 1 C R c = 0 C - 1 r = 0 R - 1 p ( c , r ) p ( c + n , r + m ) ,
s 3 , p ( n 1 , m 1 , n 2 , m 2 ) = 1 C R c = 0 C - 1 r = 0 R - 1 p ( c , r ) p ( c + n 1 , r + m 1 ) × p ( c + n 2 , r + m 2 ) ,
S 2 , p ( n , m ) = [ C R / ( C - n ) ( R - m ) ] s 2 , p ( n , m ) ,
S 3 , p ( n 1 , m 1 , n 2 , m 2 ) = { C R / [ ( C - max ( n 1 , n 2 ) ] × [ R - max ( m 1 , m 2 ) ] } × s 3 , p ( n 1 , m 1 , n 2 , m 2 ) ,
p ( c , r ) f ( x , y ) = d ( x , y ) * c = 0 C - 1 r = 0 R - 1 p ( c , r ) δ ( x - c ) δ ( y - r ) ,
s 2 , p ( n , m ) s 2 , f ( h , v ) = n m s 2 , p ( n , m ) δ ( h - n ) δ ( v - m ) , s 3 , p ( n 1 , m 1 , n 2 , m 2 ) s 3 , f ( h 1 , v 1 , h 2 , v 2 ) = n 1 m 1 n 2 m 2 s 3 , p ( n 1 , m 1 , n 2 , m 2 ) × δ ( h 1 - n 1 ) δ ( v 1 - m 1 ) × δ ( h 2 - n 2 ) δ ( v 2 - m 2 ) ,
a f ( h , v ) = R 2 f ( x , y ) f ( x + h , y + v ) d x d y .
t f ( h 1 , v 1 , h 2 , v 2 ) = R 2 f ( x , y ) f ( x + h 1 , y + v 1 ) f ( x + h 2 , y + v 2 ) d x d y .
T f ( α 1 , β 1 , α 2 , β 2 ) = F ( α 1 , β 1 ) F ( α 2 , β 2 ) F ( - α 1 - α 2 , - β 1 - β 2 ) ,
F ( α ) F ( β ) F ( - α - β ) = G ( α ) G ( β ) G ( - α - β ) .
G ( α ) G ( β ) / F ( α ) F ( β ) = G ( α + β ) / F ( α + β ) ,
H ( α ) H ( β ) = H ( α + β ) ,
R 2 g i ( x , y ) g i ( x + h 1 , y + v 1 ) g 1 ( x + h 2 , y + v 2 ) d x d y = R 2 f i ( x , y ) f i ( x + h 1 , y + v 1 ) f 1 ( x + h 2 , y + v 2 ) d x d y
G i ( α 1 , β 1 ) G i ( α 2 , β 2 ) G 1 ( - α 1 - α 1 , - β 1 - β 2 ) = F i ( α 1 , β 1 ) F i ( α 2 , β 2 ) F 1 ( - α 1 - α 2 , - β 1 - β 2 ) ,
exp { i 2 π [ ( a i - a 1 ) ( α 1 + α 2 ) + ( b i - b 1 ) ( β 1 + β 2 ) ] } = 1
lim R C S 3 , p ( n 1 , m 1 , n 2 , m 2 ) = E [ S 3 , p ( n 1 , m 1 , n 2 , m 2 ) ] .
p ( c , r ) = p ( c , 0 ) + p ( 0 , r ) + p ( 0 , 0 )             ( mod 2 ) ,
p ( c , r ) = p ( c , 0 ) + p ( 0 , r ) + p ( 0 , 0 ) + c r             ( mod 2 ) .
c = 0 C - 1 r = 0 R - 1 o ( c , r ) o ( c + 2 n , r + 2 m ) = c = 0 C - 1 r = 0 R - 1 ( - 1 ) 2 c r + 2 c m + 2 r n e ( c , r ) e ( c + 2 n , r + 2 m )
f ( x , y ) = ρ ( x , y ) * ϕ ( x , y ) = i = 1 N ϕ ( x - x i , y - y i ) .
p ( x ) = n = 0 4 δ ( x - 4 n 2 ) ,
q ( x ) = n = 0 4 δ [ x - ( 4 n 2 + n + 1 ) ] .

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