Abstract

Many successful methods in various vision tasks rely on statistical analysis of visual patterns. However, we are interested in covering the gap between the underlying mathematical representation of the visual patterns and their statistics. With this general trend, in this paper a relationship between phase structure of a class of patterns and their moments after and before filtering have been considered. First, a general formula between the phase structure and moments of the images is obtained. Second, a theorem is developed that states under which conditions two visual patterns with the same frequencies but different phases have the same moments up to a certain moment. Finally, a theorem is developed that explains, given a set of filters, under which conditions two visual patterns with both different frequencies and different phases have the same subband statistics.

© 2013 Optical Society of America

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  26. J. Gluckman, “Visually distinct patterns with matching subband statistics,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 252–264 (2005).
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  28. J. H. van Hateren and A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. B 265, 359–366 (1998).
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  29. J. H. van Hateren and D. L. Ruderman, “Independent component analysis of natural image sequences yields spatiotemporal filters similar to simple cells in primary visual cortex,” Proc. R. Soc. B 265, 2315–2320 (1998).
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  32. E. P. Simoncelli, “Statistical modeling of photographic images,” in Handbook of Image and Video Processing, 2nd ed. (Academic, 2005), pp. 431–441.
  33. A. Mooijaart, “Factor analysis for non-normal variables,” Psychometrica 50, 323–342 (1985).
    [CrossRef]
  34. J. F. Cardoso, “Source separation using higher order moments” Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’89), Glasgow (IEEE, 1989), pp. 2109–2112.
  35. C. Jutten and J. Hérault, “Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture,” Signal Process. 24, 1–10 (1991).
    [CrossRef]
  36. A. J. Bell and T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vis. Res. 37, 3327–3338 (1997).
    [CrossRef]
  37. A. Hyvärinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw. 13, 411–430 (2000).
    [CrossRef]
  38. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley-Interscience, 2001).
  39. B. A. Olshausen and D. J. Field, “Sparse coding of sensory inputs,” Curr. Opin. Neurobiol. 14, 481–487 (2004).
    [CrossRef]
  40. V. Zarzoso and P. Comon, “Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size,” IEEE Trans. Neural Netw. 21, 248–261 (2010).
    [CrossRef]
  41. P. Kovesi, “Image features from phase congruency,” Videre: J. Comput. Vis. Res. 1, 1–26 (1999).
  42. Li. Zhang, Le. Zhang, D. Zhang, and Z. Guo, “Phase congruency induced local features for finger-knuckle-print recognition,” Pattern Recogn. 45, 2522–2531 (2012).
    [CrossRef]

2012 (2)

Li. Zhang, Le. Zhang, D. Zhang, and Z. Guo, “Phase congruency induced local features for finger-knuckle-print recognition,” Pattern Recogn. 45, 2522–2531 (2012).
[CrossRef]

D. J. Field and D. M. Chandler, “Method for estimating the relative contribution of phase and power spectra to the total information in natural-scene patches,” J. Opt. Soc. Am. A 29, 55–67 (2012).
[CrossRef]

2010 (2)

V. Zarzoso and P. Comon, “Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size,” IEEE Trans. Neural Netw. 21, 248–261 (2010).
[CrossRef]

D. J. Graham and C. Redies, “Statistical regularities in art: relations with visual coding and perception,” Vis. Res. 50, 1503–1509 (2010).
[CrossRef]

2007 (3)

2006 (1)

F. A. Wichmann, D. I. Braun, and K. R. Gegenfurtner, “Phase noise and the classification of natural images,” Vis. Res. 46, 1520–1529 (2006).
[CrossRef]

2005 (2)

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transforms,” Multiscale Model Simul. 5, 861–899 (2005).
[CrossRef]

J. Gluckman, “Visually distinct patterns with matching subband statistics,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 252–264 (2005).
[CrossRef]

2004 (2)

N. Guyader, A. Chauvin, C. Peyrin, J. Hérault, and C. Marendaz, “Image phase or amplitude? Rapid scene categorization is an amplitude-based process,” C. R. Biol. 327, 313–318 (2004).
[CrossRef]

B. A. Olshausen and D. J. Field, “Sparse coding of sensory inputs,” Curr. Opin. Neurobiol. 14, 481–487 (2004).
[CrossRef]

2002 (1)

2001 (1)

B. Willmore and D. J. Tolhurst, “Characterizing the sparseness of neural codes,” Network 12, 255–270 (2001).
[CrossRef]

2000 (3)

M. G. A. Thomson, D. H. Foster, and R. J. Summers, “Human sensitivity to phase perturbations in natural images: a statistical framework,” Perception 29, 1057–1069 (2000).
[CrossRef]

A. Hyvärinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw. 13, 411–430 (2000).
[CrossRef]

J. Portilla and E. P. Simoncelli, “A parametric texture model based on joint statistics of complex wavelet coefficients,” Int. J. Comput. Vis. 40, 49–70 (2000).
[CrossRef]

1999 (1)

P. Kovesi, “Image features from phase congruency,” Videre: J. Comput. Vis. Res. 1, 1–26 (1999).

1998 (3)

S. C. Zhu, Y. Wu, and D. Mumford, “Filters, random-fields and maximum-entropy frame: towards a unified theory for texture modeling,” Int. J. Comput. Vis. 27, 107–126 (1998).
[CrossRef]

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

J. H. van Hateren and D. L. Ruderman, “Independent component analysis of natural image sequences yields spatiotemporal filters similar to simple cells in primary visual cortex,” Proc. R. Soc. B 265, 2315–2320 (1998).
[CrossRef]

1997 (1)

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

1996 (2)

B. A. Olshausen and D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature 381, 607–609 (1996).
[CrossRef]

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vis. Res. 36, 2759–2770 (1996).
[CrossRef]

1995 (1)

D. J. Heeger and J. R. Bergen, “Pyramid-based texture analysis/synthesis,” Comput. Graph. 22, 229–238 (1995).

1994 (2)

D. J. Field, “What is the goal of sensory coding?” Neural Comput. 6, 559–601 (1994).
[CrossRef]

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef]

1992 (2)

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol Opt. 12, 229–232 (1992).

P. J. B. Hancock, R. J. Baddeley, and L. S. Smith, “The principal components of natural images,” Network 3, 61–70 (1992).
[CrossRef]

1991 (1)

C. Jutten and J. Hérault, “Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture,” Signal Process. 24, 1–10 (1991).
[CrossRef]

1989 (1)

T. Sanger, “Optimal unsupervised learning in a single-layered linear feedforward network,” Neural Netw. 2, 459–473 (1989).
[CrossRef]

1988 (1)

R. Linsker, “Self-organization in a perceptual network,” Computer 21, 105–117 (1988).
[CrossRef]

1987 (2)

1985 (1)

A. Mooijaart, “Factor analysis for non-normal variables,” Psychometrica 50, 323–342 (1985).
[CrossRef]

1982 (1)

L. N. Piotrowski and F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef]

1981 (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

1962 (1)

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

Baddeley, R. J.

P. J. B. Hancock, R. J. Baddeley, and L. S. Smith, “The principal components of natural images,” Network 3, 61–70 (1992).
[CrossRef]

Bell, A. J.

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

Bergen, J. R.

D. J. Heeger and J. R. Bergen, “Pyramid-based texture analysis/synthesis,” Comput. Graph. 22, 229–238 (1995).

Bex, P. J.

Bialek, W.

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef]

Braun, D. I.

F. A. Wichmann, D. I. Braun, and K. R. Gegenfurtner, “Phase noise and the classification of natural images,” Vis. Res. 46, 1520–1529 (2006).
[CrossRef]

Burton, G. J.

Campbell, F. W.

L. N. Piotrowski and F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef]

Candès, E. J.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transforms,” Multiscale Model Simul. 5, 861–899 (2005).
[CrossRef]

Cardoso, J. F.

J. F. Cardoso, “Source separation using higher order moments” Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’89), Glasgow (IEEE, 1989), pp. 2109–2112.

Chandler, D. M.

Chao, T.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol Opt. 12, 229–232 (1992).

Chauvin, A.

N. Guyader, A. Chauvin, C. Peyrin, J. Hérault, and C. Marendaz, “Image phase or amplitude? Rapid scene categorization is an amplitude-based process,” C. R. Biol. 327, 313–318 (2004).
[CrossRef]

Comon, P.

V. Zarzoso and P. Comon, “Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size,” IEEE Trans. Neural Netw. 21, 248–261 (2010).
[CrossRef]

Demanet, L.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transforms,” Multiscale Model Simul. 5, 861–899 (2005).
[CrossRef]

Donoho, D. L.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transforms,” Multiscale Model Simul. 5, 861–899 (2005).
[CrossRef]

Field, D. J.

Foster, D. H.

M. G. A. Thomson, D. H. Foster, and R. J. Summers, “Human sensitivity to phase perturbations in natural images: a statistical framework,” Perception 29, 1057–1069 (2000).
[CrossRef]

Gegenfurtner, K. R.

F. A. Wichmann, D. I. Braun, and K. R. Gegenfurtner, “Phase noise and the classification of natural images,” Vis. Res. 46, 1520–1529 (2006).
[CrossRef]

Gluckman, J.

J. Gluckman, “Visually distinct patterns with matching subband statistics,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 252–264 (2005).
[CrossRef]

Graham, D. J.

D. J. Graham and C. Redies, “Statistical regularities in art: relations with visual coding and perception,” Vis. Res. 50, 1503–1509 (2010).
[CrossRef]

Guo, Z.

Li. Zhang, Le. Zhang, D. Zhang, and Z. Guo, “Phase congruency induced local features for finger-knuckle-print recognition,” Pattern Recogn. 45, 2522–2531 (2012).
[CrossRef]

Guyader, N.

N. Guyader, A. Chauvin, C. Peyrin, J. Hérault, and C. Marendaz, “Image phase or amplitude? Rapid scene categorization is an amplitude-based process,” C. R. Biol. 327, 313–318 (2004).
[CrossRef]

Hancock, P. J. B.

P. J. B. Hancock, R. J. Baddeley, and L. S. Smith, “The principal components of natural images,” Network 3, 61–70 (1992).
[CrossRef]

Hansen, B. C.

Heeger, D. J.

D. J. Heeger and J. R. Bergen, “Pyramid-based texture analysis/synthesis,” Comput. Graph. 22, 229–238 (1995).

Hérault, J.

N. Guyader, A. Chauvin, C. Peyrin, J. Hérault, and C. Marendaz, “Image phase or amplitude? Rapid scene categorization is an amplitude-based process,” C. R. Biol. 327, 313–318 (2004).
[CrossRef]

C. Jutten and J. Hérault, “Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture,” Signal Process. 24, 1–10 (1991).
[CrossRef]

Hess, R. F.

Hoyer, P. O.

A. Hyvärinen, J. Hurri, and P. O. Hoyer, Natural Image Statistics (Springer, 2009).

Hup, X.

X. Ni and X. Hup, “Statistical interpretation of the importance of phase information in signal and image reconstruction,” Statist. Probability Lett. 77, 447–454 (2007).

Hurri, J.

A. Hyvärinen, J. Hurri, and P. O. Hoyer, Natural Image Statistics (Springer, 2009).

Hyvärinen, A.

A. Hyvärinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw. 13, 411–430 (2000).
[CrossRef]

A. Hyvärinen, J. Hurri, and P. O. Hoyer, Natural Image Statistics (Springer, 2009).

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley-Interscience, 2001).

Julesz, B.

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

Jutten, C.

C. Jutten and J. Hérault, “Blind separation of sources, part I: an adaptive algorithm based on neuromimetic architecture,” Signal Process. 24, 1–10 (1991).
[CrossRef]

Karhunen, J.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley-Interscience, 2001).

Kovesi, P.

P. Kovesi, “Image features from phase congruency,” Videre: J. Comput. Vis. Res. 1, 1–26 (1999).

Lim, J. S.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Linsker, R.

R. Linsker, “Self-organization in a perceptual network,” Computer 21, 105–117 (1988).
[CrossRef]

Makous, W.

Marendaz, C.

N. Guyader, A. Chauvin, C. Peyrin, J. Hérault, and C. Marendaz, “Image phase or amplitude? Rapid scene categorization is an amplitude-based process,” C. R. Biol. 327, 313–318 (2004).
[CrossRef]

Mooijaart, A.

A. Mooijaart, “Factor analysis for non-normal variables,” Psychometrica 50, 323–342 (1985).
[CrossRef]

Moorehead, T. R.

Mumford, D.

S. C. Zhu, Y. Wu, and D. Mumford, “Filters, random-fields and maximum-entropy frame: towards a unified theory for texture modeling,” Int. J. Comput. Vis. 27, 107–126 (1998).
[CrossRef]

Ni, X.

X. Ni and X. Hup, “Statistical interpretation of the importance of phase information in signal and image reconstruction,” Statist. Probability Lett. 77, 447–454 (2007).

Oja, E.

A. Hyvärinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw. 13, 411–430 (2000).
[CrossRef]

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley-Interscience, 2001).

Olshausen, B. A.

B. A. Olshausen and D. J. Field, “Sparse coding of sensory inputs,” Curr. Opin. Neurobiol. 14, 481–487 (2004).
[CrossRef]

B. A. Olshausen and D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature 381, 607–609 (1996).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Peyrin, C.

N. Guyader, A. Chauvin, C. Peyrin, J. Hérault, and C. Marendaz, “Image phase or amplitude? Rapid scene categorization is an amplitude-based process,” C. R. Biol. 327, 313–318 (2004).
[CrossRef]

Piotrowski, L. N.

L. N. Piotrowski and F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef]

Portilla, J.

J. Portilla and E. P. Simoncelli, “A parametric texture model based on joint statistics of complex wavelet coefficients,” Int. J. Comput. Vis. 40, 49–70 (2000).
[CrossRef]

Redies, C.

D. J. Graham and C. Redies, “Statistical regularities in art: relations with visual coding and perception,” Vis. Res. 50, 1503–1509 (2010).
[CrossRef]

Ruderman, D. L.

J. H. van Hateren and D. L. Ruderman, “Independent component analysis of natural image sequences yields spatiotemporal filters similar to simple cells in primary visual cortex,” Proc. R. Soc. B 265, 2315–2320 (1998).
[CrossRef]

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef]

Sanger, T.

T. Sanger, “Optimal unsupervised learning in a single-layered linear feedforward network,” Neural Netw. 2, 459–473 (1989).
[CrossRef]

Sejnowski, T. J.

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

Simoncelli, E. P.

J. Portilla and E. P. Simoncelli, “A parametric texture model based on joint statistics of complex wavelet coefficients,” Int. J. Comput. Vis. 40, 49–70 (2000).
[CrossRef]

E. P. Simoncelli, “Statistical modeling of photographic images,” in Handbook of Image and Video Processing, 2nd ed. (Academic, 2005), pp. 431–441.

Smith, L. S.

P. J. B. Hancock, R. J. Baddeley, and L. S. Smith, “The principal components of natural images,” Network 3, 61–70 (1992).
[CrossRef]

Summers, R. J.

M. G. A. Thomson, D. H. Foster, and R. J. Summers, “Human sensitivity to phase perturbations in natural images: a statistical framework,” Perception 29, 1057–1069 (2000).
[CrossRef]

Tadmor, Y.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol Opt. 12, 229–232 (1992).

Thomson, M. G. A.

M. G. A. Thomson, D. H. Foster, and R. J. Summers, “Human sensitivity to phase perturbations in natural images: a statistical framework,” Perception 29, 1057–1069 (2000).
[CrossRef]

Tolhurst, D. J.

B. Willmore and D. J. Tolhurst, “Characterizing the sparseness of neural codes,” Network 12, 255–270 (2001).
[CrossRef]

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol Opt. 12, 229–232 (1992).

van der Schaaf, A.

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

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vis. Res. 36, 2759–2770 (1996).
[CrossRef]

van Hateren, J. H.

J. H. van Hateren and D. L. Ruderman, “Independent component analysis of natural image sequences yields spatiotemporal filters similar to simple cells in primary visual cortex,” Proc. R. Soc. B 265, 2315–2320 (1998).
[CrossRef]

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

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vis. Res. 36, 2759–2770 (1996).
[CrossRef]

Wichmann, F. A.

F. A. Wichmann, D. I. Braun, and K. R. Gegenfurtner, “Phase noise and the classification of natural images,” Vis. Res. 46, 1520–1529 (2006).
[CrossRef]

Willmore, B.

B. Willmore and D. J. Tolhurst, “Characterizing the sparseness of neural codes,” Network 12, 255–270 (2001).
[CrossRef]

Wu, Y.

S. C. Zhu, Y. Wu, and D. Mumford, “Filters, random-fields and maximum-entropy frame: towards a unified theory for texture modeling,” Int. J. Comput. Vis. 27, 107–126 (1998).
[CrossRef]

Ying, L.

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying, “Fast discrete curvelet transforms,” Multiscale Model Simul. 5, 861–899 (2005).
[CrossRef]

Zarzoso, V.

V. Zarzoso and P. Comon, “Robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size,” IEEE Trans. Neural Netw. 21, 248–261 (2010).
[CrossRef]

Zhang, D.

Li. Zhang, Le. Zhang, D. Zhang, and Z. Guo, “Phase congruency induced local features for finger-knuckle-print recognition,” Pattern Recogn. 45, 2522–2531 (2012).
[CrossRef]

Zhang, Le.

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Figures (11)

Fig. 1.
Fig. 1.

Importance of phase and its relation to statistics.

Fig. 2.
Fig. 2.

Effect of filtering on phase and statistics.

Fig. 3.
Fig. 3.

Symmetric natural image with only 0 or π phases.

Fig. 4.
Fig. 4.

Two patterns with same frequency set but different phase set. This is an example of an extreme case in the sense that all of their moments are identical. (a) and (b) are the frequency content of patterns (c) and (d), respectively. (e) and (f) are the histograms of patterns (c) and (d), respectively.

Fig. 5.
Fig. 5.

Same description as Fig. 4 except that their moments are identical up to moment k=12.

Fig. 6.
Fig. 6.

Visually distinct patterns generated by the second technique (Section 5) with same marginal statistics up to certain moments. Patterns on the same row are compared with each other. Each histogram belongs to the pattern on its top.

Fig. 7.
Fig. 7.

Two patterns (a) and (b) with different frequencies and phases. (c) and (d) frequency content of (a) and (b), respectively. (e) and (f) are histogram of (a) and (b), respectively. (g) and (h) are the histogram of Gaussian filtered of (a) and (b), respectively. (h) and (j) are the histogram of the Laplacian filtered of (a) and (b), respectively.

Fig. 8.
Fig. 8.

Same description as Fig. 7. Identical moments up to k=11.

Fig. 9.
Fig. 9.

Same description as Fig. 7, except that filter type is derivative of Gaussian. Identical moments up to k=8.

Fig. 10.
Fig. 10.

Same description as Fig. 9. Identical moments up to k=12.

Fig. 11.
Fig. 11.

Same description as Fig. 9. Identical moments up to k=12.

Tables (3)

Tables Icon

Table 1. Identical Moments for Patterns in Each Figure

Tables Icon

Table 2. Moments before and after Filtering with Gaussian and Laplacian of Gaussian filters

Tables Icon

Table 3. Moments before and after Filtering with Derivative of Gaussian Filters

Equations (42)

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

Mk(I)=1|Ω|ΩI(x,y)kdΩ.
i=1nwicos(uix+viy+φi)Img.
Mk(I)=Ω[i=1n(wicos(zi+φi))]kdΩ,
(x1+x2++xn)k=k1+k2++kn=k(kk1,k2,,kn)1tnxtkt.
Mk(I)=Ωk1,k2,,knk!k1!k2!kn!i=1nwikicoski(zi+φi)dΩ,
Mk(I)=ΩκKF(κ,w)i=1ncoski(zi+φi)dΩ,
F(κ,w)=k!k1!k2!kn!(w1k1w2k2wnkn).
cosp(z)=i=0pC(i,p)cos(iz),pZ.
C(i,p)0iff parity(i)=parity(p).
Mk(I)=ΩκKF(κ,w)i=1n(ri=0kiC(ri,ki)cos(rizi+riφi))dΩ.
Mk(I)=ΩκKF(κ,w)rRki=1nC(ri,ki)cos(rizi+riφi)dΩ,
Mk(I)=κKF(κ,w)rRkC(r,κ)Ωi=1ncos(rizi+riφi)dΩ,
C(r,k)0iffiparity(ri)=parity(ki).
cos(a)cos(b)=12(cos(a+b)+cos(ab)),
Mk(I)=κKF(κ,w)rRκC(r,κ)Ω12n1d2,d3,,dncos(i=1ndi(rizi+riφi))dΩ,
Ωcos(r1(z1+φ1)±r2(z2+φ2)±±rn(zn+φn))dΩ.
r1u1+d2r2u2+dnrnun=0andr1v1+d2r2v2+dnrnvn=0,
cos(r1d1φ1+r2d2φ2++rndnφn),
Ωcos(i=1ndi(zi+φi))dΩ={4π2cos(d1r1φ1+d2r2φ2++dnrnφn)ifindirizi=00otherwise.
Mk(I)=κKF(κ,w)rRκC(r,κ)12n1d1,d2,,dnT(r,d)cos(d1r1φ1+d2r2φ2++dnrnφn),
T(r,d)={0ifi=1nridizi01otherwise.
Mk(I)=ΩκKF(κ,w)(s1k1s2k2snkn)i=1ncoski(zi)dΩ,
Mk(I)=κKF(κ,w)(s1k1s2k2snkn)rRκC(r,κ)Ω12n1d2,d3,,dncos(i=1ndirizi)dΩ.
Ωcos(i=1ndirizi)dΩ={4π2ifindirizi=00otherwise.
Mk(I)=κKF(κ,w)(s1k1s2k2snkn)rRκC(r,κ)12n1t(u,v,r),
r1u1+d2r2u2+dnrnun=0andr1v1+d2r2v2+dnrnvn=0,
c1u1+c2u2++cnun=0c1v1+c2v2++cnvn=0.
c1u1+c2u2++cnun=0c1v1+c2v2++cnvn=0,
s1k1s2k2snkn(s1)k1(s2)k2(sn)kn,
f1=f*If2=f*I.
s1k1s2k2snkn=(s1)k1(s2)k2(sn)kn
ifiF(ui,vi)=F(ui,vi),(ui,vi)is a frequency component ofI(ui,vi)is a frequency component ofI
(ui,vi)ubthere exist a(ui,vj)ubsuch thatF(ui,vi)=F(ui,vi).
DoG(x,y,m,n,σ)=m+nxmymexp((x+y)2/2σ2).
F(u,v)=(2πiu)m(2πiv)nexp((u+v)2/2σ2),
f*I=f*Ia(u,v)+f*Ib(u,v)f*I=f*Ia(u,v)f*Ib(u,v),
f1=f*I=i=1nF(ui,vi)wisicos(uix+viy)f2=f*I=i=1nF(ui,vi)wisicos(uix+viy),
f1=f*I=i=1nF(ui,vi)wisisin(uix+viy)f2=f*I=i=1nF(ui,vi)wisisin(uix+viy).
s1k1s2k2snkn=(s1)k1(s2)k2(sn)kn,
(c1u1+c2u2++cnun=0)(c1v1+c2v2++cnvn=0)(c1u1+c2u2++cnun0)(c1v1+c2v2++cnvn0).
f1=f*I=i=1nF(ui,vi)wisicos(uix+viy)f2=f*I=i=1nF(ui,vi)wisicos(uix+viy).
f1=f*I=i=1nF(ui,vi)wisisin(uix+viy)f2=f*I=i=1nF(ui,vi)wisisin(uix+viy).

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