Abstract

A discrete finite image I is a function assigning colors to a finite, rectangular array of discrete pixels. A dipole is a triple, ((dR, dC), α, β), where dR and dC are vertical and horizontal, integer-valued displacements and α and β are colors. For any such dipole, DI((dR, dC), α, β) gives the number of pixel pairs ((r1, c1), (r2, c2)) of I such that I[r1, c1]=α, I[r2, c2]=β and (r2, c2)-(r1, c1)=(dR, dC). The function DI is called the dipole histogram of I. The information directly encoded by the image I is purely locational, in the sense that I assigns colors to locations in space. By contrast, the information directly encoded by DI is purely relational, in the sense that DI registers only the frequencies with which pairs of intensities stand in various spatial relations. Previously we showed that any discrete, finite image I is uniquely determined by DI [Vision Res. 40, 485 (2000)]. The visual relevance of dipole histogram representations is questionable, however, for at least two reasons: (1) Even when an image viewed by the eye nominally contains only a small number of discrete color values, photon noise and the random nature of photon absorption in photoreceptors imply that the effective neural image will contain a far greater (and unknown) range of values, and (2) DI is generally of much greater cardinality than I. First we introduce “soft” dipole representations, which forgo the perfect registration of intensity implicit in the definition of DI, and show that such soft representations uniquely determine the images to which they correspond; then we demonstrate that there exists a relatively small dipole representation of any image. Specifically, we prove that for any discrete finite image I with N>1 pixels, there always exists a restriction Q of DI (with the domain of Q dependent on I) of cardinality at most N-1 sufficient to uniquely determine I, provided that one also knows N; thus there always exists a purely relational representation of I whose order of complexity is no greater than that of I itself.

© 2002 Optical Society of America

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  1. R. Desimone, C. G. Gross, “Visual areas in the temporal cortex of the macaque,” J. Neurosci. 4, 2051–2062 (1979).
  2. M. Ito, H. Tamura, I. Fujita, K. Tanaka, “Size and position invariance of neuronal responses in monkey inferotemporal cortex,” J. Neurophysiol. 73, 218–226 (1995).
    [PubMed]
  3. E. Kobatake, K. Tanaka, “Neural selectivities to complex object features in the ventral visual pathway of the macaque cerebral cortex,” J. Neurophysiol. 71, 856–867 (1994).
    [PubMed]
  4. K. Tanaka, H. Saito, Y. Fukada, M. Moriya, “Coding visual images of objects in the inferotemporal cortex of the macaque monkey,” J. Neurophysiol. 66, 170–189 (1991).
    [PubMed]
  5. D. I. Perrett, E. T. Rolls, W. Caan, “Visual neurons responsive to faces in monkey temporal cortex,” Exp. Brain Res. 47, 329–342 (1982).
    [CrossRef]
  6. E. T. Rolls, G. C. Baylis, “Size and contrast have only small effects on the responses to faces of neurons in the cortex of the superior temporal sulcus of the monkey,” Exp. Brain Res. 65, 38–48 (1986).
    [CrossRef] [PubMed]
  7. P. Diaconis, D. Freedman, “On the statistics of vision: the Julesz conjecture,” J. Math. Psychol. 24, 112–138 (1981).
    [CrossRef]
  8. 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]
  9. E. N. Gilbert, “Random colorings of a lattice on squares in the plane,” SIAM J. Alg. Disc. Meth. 1, 152–159 (1980).
    [CrossRef]
  10. B. Julesz, “Visual pattern discrimination,” IRE Trans. Inf. Theory IT-8, 84–92 (1962).
    [CrossRef]
  11. B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91–97 (1981).
    [CrossRef] [PubMed]
  12. 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,” Perception 2, 391–405 (1973).
    [CrossRef]
  13. B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third order statistics,” Biol. Cybern. 31, 137–149 (1978).
    [CrossRef] [PubMed]
  14. J. D. Victor, “Images, statistics and textures: Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture: comment,” J. Opt. Soc. Am. A 11, 1680–1684 (1994).
    [CrossRef]
  15. J. D. Victor, S. Brodie, “Discriminable textures with identical Buffon needle statistics,” Biol. Cybern. 31, 231–234 (1978).
    [CrossRef]
  16. J. D. Victor, M. Conte, K. Purpura, E. Katz, “Isodipole textures: a window on cortical mechanisms of form processing,” in Early Vision and Beyond, T. V. Papathomas, C. Chubb, A. Gorea, E. Kowler, eds. (MIT Press, Cambridge, Mass., 1995), pp. 99–107.
  17. J. I. Yellott, “Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture,” J. Opt. Soc. Am. A 10, 777–793 (1993).
    [CrossRef]
  18. J. R. Bergen, M. S. Landy, “The computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds., (MIT Press, Cambridge, Mass., 1991), pp. 253–271.
  19. C. Chubb, J. Econopouly, M. S. Landy, “Histogram contrast analysis and the visual segregation of IID textures,” J. Opt. Soc. Am. A 11, 2350–2374 (1994).
    [CrossRef]
  20. N. Graham, “Non-linearities in texture segregation,” in Higher-order Processing in the Visual System. Ciba Foundation Symposium, Vol. 184, J. A. Goode, G. R. Bock, eds. (Wiley, Chichester, UK, 1994), pp. 309–329.
  21. M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
    [CrossRef] [PubMed]
  22. J. Malik, P. Perona, “Preattentive texture discrimination with early vision mechanisms,” J. Opt. Soc. Am. A 7, 923–932 (1990).
    [CrossRef] [PubMed]
  23. C. Chubb, J. I. Yellott, “Every discrete, finite image is uniquely determined by its dipole histogram,” Vision Res. 40, 485–492 (2000).
    [CrossRef] [PubMed]
  24. J. Doner, “Dipole information complementarity in discrete 2D patterns,” J. Math. Psychol. 43, 355–393 (1999).
    [CrossRef] [PubMed]

2000 (1)

C. Chubb, J. I. Yellott, “Every discrete, finite image is uniquely determined by its dipole histogram,” Vision Res. 40, 485–492 (2000).
[CrossRef] [PubMed]

1999 (1)

J. Doner, “Dipole information complementarity in discrete 2D patterns,” J. Math. Psychol. 43, 355–393 (1999).
[CrossRef] [PubMed]

1995 (1)

M. Ito, H. Tamura, I. Fujita, K. Tanaka, “Size and position invariance of neuronal responses in monkey inferotemporal cortex,” J. Neurophysiol. 73, 218–226 (1995).
[PubMed]

1994 (3)

1993 (1)

J. I. Yellott, “Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture,” J. Opt. Soc. Am. A 10, 777–793 (1993).
[CrossRef]

1991 (2)

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

K. Tanaka, H. Saito, Y. Fukada, M. Moriya, “Coding visual images of objects in the inferotemporal cortex of the macaque monkey,” J. Neurophysiol. 66, 170–189 (1991).
[PubMed]

1990 (1)

1986 (1)

E. T. Rolls, G. C. Baylis, “Size and contrast have only small effects on the responses to faces of neurons in the cortex of the superior temporal sulcus of the monkey,” Exp. Brain Res. 65, 38–48 (1986).
[CrossRef] [PubMed]

1982 (1)

D. I. Perrett, E. T. Rolls, W. Caan, “Visual neurons responsive to faces in monkey temporal cortex,” Exp. Brain Res. 47, 329–342 (1982).
[CrossRef]

1981 (3)

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

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

1980 (1)

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

1979 (1)

R. Desimone, C. G. Gross, “Visual areas in the temporal cortex of the macaque,” J. Neurosci. 4, 2051–2062 (1979).

1978 (2)

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third order statistics,” Biol. Cybern. 31, 137–149 (1978).
[CrossRef] [PubMed]

J. D. Victor, S. Brodie, “Discriminable textures with identical Buffon needle statistics,” Biol. Cybern. 31, 231–234 (1978).
[CrossRef]

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,” Perception 2, 391–405 (1973).
[CrossRef]

1962 (1)

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

Baylis, G. C.

E. T. Rolls, G. C. Baylis, “Size and contrast have only small effects on the responses to faces of neurons in the cortex of the superior temporal sulcus of the monkey,” Exp. Brain Res. 65, 38–48 (1986).
[CrossRef] [PubMed]

Bergen, J. R.

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

J. R. Bergen, M. S. Landy, “The computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds., (MIT Press, Cambridge, Mass., 1991), pp. 253–271.

Brodie, S.

J. D. Victor, S. Brodie, “Discriminable textures with identical Buffon needle statistics,” Biol. Cybern. 31, 231–234 (1978).
[CrossRef]

Caan, W.

D. I. Perrett, E. T. Rolls, W. Caan, “Visual neurons responsive to faces in monkey temporal cortex,” Exp. Brain Res. 47, 329–342 (1982).
[CrossRef]

Chubb, C.

C. Chubb, J. I. Yellott, “Every discrete, finite image is uniquely determined by its dipole histogram,” Vision Res. 40, 485–492 (2000).
[CrossRef] [PubMed]

C. Chubb, J. Econopouly, M. S. Landy, “Histogram contrast analysis and the visual segregation of IID textures,” J. Opt. Soc. Am. A 11, 2350–2374 (1994).
[CrossRef]

Conte, M.

J. D. Victor, M. Conte, K. Purpura, E. Katz, “Isodipole textures: a window on cortical mechanisms of form processing,” in Early Vision and Beyond, T. V. Papathomas, C. Chubb, A. Gorea, E. Kowler, eds. (MIT Press, Cambridge, Mass., 1995), pp. 99–107.

Desimone, R.

R. Desimone, C. G. Gross, “Visual areas in the temporal cortex of the macaque,” J. Neurosci. 4, 2051–2062 (1979).

Diaconis, P.

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

Doner, J.

J. Doner, “Dipole information complementarity in discrete 2D patterns,” J. Math. Psychol. 43, 355–393 (1999).
[CrossRef] [PubMed]

Econopouly, J.

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,” Perception 2, 391–405 (1973).
[CrossRef]

Fujita, I.

M. Ito, H. Tamura, I. Fujita, K. Tanaka, “Size and position invariance of neuronal responses in monkey inferotemporal cortex,” J. Neurophysiol. 73, 218–226 (1995).
[PubMed]

Fukada, Y.

K. Tanaka, H. Saito, Y. Fukada, M. Moriya, “Coding visual images of objects in the inferotemporal cortex of the macaque monkey,” J. Neurophysiol. 66, 170–189 (1991).
[PubMed]

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 on squares in the plane,” SIAM J. Alg. Disc. Meth. 1, 152–159 (1980).
[CrossRef]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third order statistics,” Biol. Cybern. 31, 137–149 (1978).
[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,” Perception 2, 391–405 (1973).
[CrossRef]

Graham, N.

N. Graham, “Non-linearities in texture segregation,” in Higher-order Processing in the Visual System. Ciba Foundation Symposium, Vol. 184, J. A. Goode, G. R. Bock, eds. (Wiley, Chichester, UK, 1994), pp. 309–329.

Gross, C. G.

R. Desimone, C. G. Gross, “Visual areas in the temporal cortex of the macaque,” J. Neurosci. 4, 2051–2062 (1979).

Ito, M.

M. Ito, H. Tamura, I. Fujita, K. Tanaka, “Size and position invariance of neuronal responses in monkey inferotemporal cortex,” J. Neurophysiol. 73, 218–226 (1995).
[PubMed]

Julesz, B.

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

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third order statistics,” Biol. Cybern. 31, 137–149 (1978).
[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,” Perception 2, 391–405 (1973).
[CrossRef]

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

Katz, E.

J. D. Victor, M. Conte, K. Purpura, E. Katz, “Isodipole textures: a window on cortical mechanisms of form processing,” in Early Vision and Beyond, T. V. Papathomas, C. Chubb, A. Gorea, E. Kowler, eds. (MIT Press, Cambridge, Mass., 1995), pp. 99–107.

Kobatake, E.

E. Kobatake, K. Tanaka, “Neural selectivities to complex object features in the ventral visual pathway of the macaque cerebral cortex,” J. Neurophysiol. 71, 856–867 (1994).
[PubMed]

Landy, M. S.

C. Chubb, J. Econopouly, M. S. Landy, “Histogram contrast analysis and the visual segregation of IID textures,” J. Opt. Soc. Am. A 11, 2350–2374 (1994).
[CrossRef]

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

J. R. Bergen, M. S. Landy, “The computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds., (MIT Press, Cambridge, Mass., 1991), pp. 253–271.

Malik, J.

Moriya, M.

K. Tanaka, H. Saito, Y. Fukada, M. Moriya, “Coding visual images of objects in the inferotemporal cortex of the macaque monkey,” J. Neurophysiol. 66, 170–189 (1991).
[PubMed]

Perona, P.

Perrett, D. I.

D. I. Perrett, E. T. Rolls, W. Caan, “Visual neurons responsive to faces in monkey temporal cortex,” Exp. Brain Res. 47, 329–342 (1982).
[CrossRef]

Purpura, K.

J. D. Victor, M. Conte, K. Purpura, E. Katz, “Isodipole textures: a window on cortical mechanisms of form processing,” in Early Vision and Beyond, T. V. Papathomas, C. Chubb, A. Gorea, E. Kowler, eds. (MIT Press, Cambridge, Mass., 1995), pp. 99–107.

Rolls, E. T.

E. T. Rolls, G. C. Baylis, “Size and contrast have only small effects on the responses to faces of neurons in the cortex of the superior temporal sulcus of the monkey,” Exp. Brain Res. 65, 38–48 (1986).
[CrossRef] [PubMed]

D. I. Perrett, E. T. Rolls, W. Caan, “Visual neurons responsive to faces in monkey temporal cortex,” Exp. Brain Res. 47, 329–342 (1982).
[CrossRef]

Saito, H.

K. Tanaka, H. Saito, Y. Fukada, M. Moriya, “Coding visual images of objects in the inferotemporal cortex of the macaque monkey,” J. Neurophysiol. 66, 170–189 (1991).
[PubMed]

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,” Perception 2, 391–405 (1973).
[CrossRef]

Tamura, H.

M. Ito, H. Tamura, I. Fujita, K. Tanaka, “Size and position invariance of neuronal responses in monkey inferotemporal cortex,” J. Neurophysiol. 73, 218–226 (1995).
[PubMed]

Tanaka, K.

M. Ito, H. Tamura, I. Fujita, K. Tanaka, “Size and position invariance of neuronal responses in monkey inferotemporal cortex,” J. Neurophysiol. 73, 218–226 (1995).
[PubMed]

E. Kobatake, K. Tanaka, “Neural selectivities to complex object features in the ventral visual pathway of the macaque cerebral cortex,” J. Neurophysiol. 71, 856–867 (1994).
[PubMed]

K. Tanaka, H. Saito, Y. Fukada, M. Moriya, “Coding visual images of objects in the inferotemporal cortex of the macaque monkey,” J. Neurophysiol. 66, 170–189 (1991).
[PubMed]

Victor, J. D.

J. D. Victor, “Images, statistics and textures: Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture: comment,” J. Opt. Soc. Am. A 11, 1680–1684 (1994).
[CrossRef]

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third order statistics,” Biol. Cybern. 31, 137–149 (1978).
[CrossRef] [PubMed]

J. D. Victor, S. Brodie, “Discriminable textures with identical Buffon needle statistics,” Biol. Cybern. 31, 231–234 (1978).
[CrossRef]

J. D. Victor, M. Conte, K. Purpura, E. Katz, “Isodipole textures: a window on cortical mechanisms of form processing,” in Early Vision and Beyond, T. V. Papathomas, C. Chubb, A. Gorea, E. Kowler, eds. (MIT Press, Cambridge, Mass., 1995), pp. 99–107.

Yellott, J. I.

C. Chubb, J. I. Yellott, “Every discrete, finite image is uniquely determined by its dipole histogram,” Vision Res. 40, 485–492 (2000).
[CrossRef] [PubMed]

J. I. Yellott, “Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture,” J. Opt. Soc. Am. A 10, 777–793 (1993).
[CrossRef]

Biol. Cybern. (2)

B. Julesz, E. N. Gilbert, J. D. Victor, “Visual discrimination of textures with identical third order statistics,” Biol. Cybern. 31, 137–149 (1978).
[CrossRef] [PubMed]

J. D. Victor, S. Brodie, “Discriminable textures with identical Buffon needle statistics,” Biol. Cybern. 31, 231–234 (1978).
[CrossRef]

Exp. Brain Res. (2)

D. I. Perrett, E. T. Rolls, W. Caan, “Visual neurons responsive to faces in monkey temporal cortex,” Exp. Brain Res. 47, 329–342 (1982).
[CrossRef]

E. T. Rolls, G. C. Baylis, “Size and contrast have only small effects on the responses to faces of neurons in the cortex of the superior temporal sulcus of the monkey,” Exp. Brain Res. 65, 38–48 (1986).
[CrossRef] [PubMed]

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. Neurophysiol. (1)

E. Kobatake, K. Tanaka, “Neural selectivities to complex object features in the ventral visual pathway of the macaque cerebral cortex,” J. Neurophysiol. 71, 856–867 (1994).
[PubMed]

J. Math. Psychol. (2)

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

J. Doner, “Dipole information complementarity in discrete 2D patterns,” J. Math. Psychol. 43, 355–393 (1999).
[CrossRef] [PubMed]

J. Neurophysiol. (2)

K. Tanaka, H. Saito, Y. Fukada, M. Moriya, “Coding visual images of objects in the inferotemporal cortex of the macaque monkey,” J. Neurophysiol. 66, 170–189 (1991).
[PubMed]

M. Ito, H. Tamura, I. Fujita, K. Tanaka, “Size and position invariance of neuronal responses in monkey inferotemporal cortex,” J. Neurophysiol. 73, 218–226 (1995).
[PubMed]

J. Neurosci. (1)

R. Desimone, C. G. Gross, “Visual areas in the temporal cortex of the macaque,” J. Neurosci. 4, 2051–2062 (1979).

J. Opt. Soc. Am. A (1)

J. I. Yellott, “Implications of triple correlation uniqueness for texture statistics and the Julesz conjecture,” J. Opt. Soc. Am. A 10, 777–793 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nature (1)

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

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,” Perception 2, 391–405 (1973).
[CrossRef]

SIAM J. Alg. Disc. Meth. (1)

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

Vision Res. (2)

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

C. Chubb, J. I. Yellott, “Every discrete, finite image is uniquely determined by its dipole histogram,” Vision Res. 40, 485–492 (2000).
[CrossRef] [PubMed]

Other (3)

J. R. Bergen, M. S. Landy, “The computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds., (MIT Press, Cambridge, Mass., 1991), pp. 253–271.

N. Graham, “Non-linearities in texture segregation,” in Higher-order Processing in the Visual System. Ciba Foundation Symposium, Vol. 184, J. A. Goode, G. R. Bock, eds. (Wiley, Chichester, UK, 1994), pp. 309–329.

J. D. Victor, M. Conte, K. Purpura, E. Katz, “Isodipole textures: a window on cortical mechanisms of form processing,” in Early Vision and Beyond, T. V. Papathomas, C. Chubb, A. Gorea, E. Kowler, eds. (MIT Press, Cambridge, Mass., 1995), pp. 99–107.

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Equations (30)

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

S I ( d ,   α ,   β ) = r = 0 N - 1 - d G ( I [ r ] - α ) G ( I [ r + d ] - β ) .
G ( v ) = 1 if - 1 2 < v < 1 2 0 otherwise .
G ( v ) = 1 2 π σ exp - ( v 2 / 2 σ 2 ) .
- - S I ( d ,   α ,   β ) α d α d β
= r = 0 N - 1 - d - G ( I [ r ] - α ) α d α - G ( I [ r + d ] - β ) d β = r = 0 N - 1 - d I [ r ] ,
- - [ S I ( N - 1 - d ,   α ,   β ) - S I ( N - d ,   α ,   β ) ] α
× d α d β = r = 0 d I [ r ] - r = 0 d - 1 I [ r ] = I [ d ] .
S I ( ( d r ,   d c ) ,   α ,   β ) = r = 0 N - 1 - d r c = 0 M - 1 - d c G ( I [ r ,   c ] - α ) × G ( I [ r + d r ,   c + d c ] - β ) .
J [ rM + c ] = I [ r ,   c ] ,
r = 0 ,   1 , ,   N - 1 ,
c = 0 ,   1 , ,   M - 1 .
S J ( d ,   α ,   β ) = S I ( ( d r M ,   d c ) ,   α ,   β ) ,
S J ( d ,   α ,   β ) = S I ( ( d r M ,   d c ) ,   α ,   β ) + S I ( ( ( d r + 1 ) M ,   d c - M ) ,   α ,   β ) .
N = N M N M - 1 N 0
λ [ x ] = x M i = 0 M - 1   N i + x M - 1 i = 0 M - 2 N i + + x 1 N 0 + x 0
( a ) If x + y X , then λ [ x + y ] = λ [ x ] + λ [ y ] .
( b ) If y - x X , then λ [ y - x ] = λ [ y ] - λ [ x ] .
π [ n ] - π [ 0 ] = π [ n ] = π [ N - 1 ] - π [ N - 1 - n ] .
π [ N - 1 ] - π [ N - 1 - n ]
= π [ λ [ π [ N - 1 ] - π [ N - 1 - n ] ] ] = π [ λ [ π [ N - 1 ] ] - λ [ π [ N - 1 - n ] ] ] = π [ N - 1 - [ N - 1 - n ] ] = π [ n ] .
( a ) If λ [ x ] + λ [ y ] > N - 1 , then x + y X .
( b ) If λ [ y ] - λ [ x ] < 0 , then y - x X .
H I = ( π [ N - 1 ] ,   I [ π [ 0 ] ] ,   I [ π [ N - 1 ] ] ) , ( π [ N - 2 ] ,   I [ π [ 0 ] ] ,   I [ π [ N - 2 ] ] ) , ( π [ N - 2 ] ,   I [ π [ 1 ] ] ,   I [ π [ N - 1 ] ] ) , ( π [ N - 3 ] ,   I [ π [ 0 ] ] ,   I [ π [ N - 3 ] ] ) , ( π [ N - 3 ] ,   I [ π [ 2 ] ] ,   I [ π [ N - 1 ] ] ) , ( π [ K ] ,   I [ π [ 0 ] ] ,   I [ π [ K ] ] ) , ( π [ K ] ,   I [ π [ K - 1 ] ] ,   I [ π [ N - 1 ] ] ) .
( i ) λ [ x ] j and ( ii ) λ [ y ] N - 1 - j .
C 11 [ d ] = α , β D I [ d ,   α ,   β ] α β = r = 0 N - 1 - d I [ r ] I [ r + d ]
C 10 [ d ] = α , β D I [ d ,   α ,   β ] α ( 1 - β ) = r = 0 N - 1 - d I [ r ] ( 1 - I [ r + d ] ) .
C 11 [ N - 1 - k ] + C 10 [ N - 1 - k ]
= r = 0 k I [ r ] I [ r + N - k ] + r = 0 k I [ r ] ( 1 - I [ r + N - k ] ) = r = 0 k I [ r ] .
I [ 0 ] = C 11 [ N - 1 ] + C 10 [ N - 1 ] ,
I [ k ] = C 11 [ N - 1 - k ] + C 10 [ N - 1 - k ] - r = 0 k - 1 I [ r ] .

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