Abstract

The human ability to discriminate structured from uniformly random binary textures has been shown to exploit third- and higher-order pixel correlations. We examine this ability in an experiment using a large number of texture families that can only be distinguished on the basis of these higher-order correlations. This study investigates statistical models based on possible explanatory variables involving spatial interactions of up to four pixels. Some of these explanatory variables have been recently associated with natural images, and others are somewhat less intuitive and are used here for the first time, to our knowledge. Our models are constructed using intraclass and cross-class feature selection by means of lasso/elastic net optimization and extensive cross-validation. We focus on a special set of locally countable image measures that seem to parsimoniously capture the observed discrimination performance. Among the measures underpinning the best models, we highlight a concept that can only exist in nine-pixel or larger image patches, but nonetheless is calculable based on the multiplicity of specific four-pixel patches in a texture. We show that this single geometric concept provides significant clues to explain texture discrimination.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. T. Maddess, Y. Nagai, J. Victor, and R. Taylor, “Multilevel isotrigon textures,” J. Opt. Soc. Am. A 24, 278–293 (2007).
    [CrossRef]
  4. J. D. Victor, “Complex visual textures as a tool for studying the VEP,” Vis. Res. 25, 1811–1827 (1985).
    [CrossRef]
  5. J. D. Victor and M. M. Conte, “Cortical interactions in texture processing: Scale and dynamics,” Vis. Neurosci. 2, 297–313 (1989).
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  6. L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
    [CrossRef]
  7. L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. E. Schneidman, S. Still, M. J. Berry, and W. Bialek, “Network information and connected correlations,” Phys. Rev. Lett. 91, 238701 (2003).
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  19. G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
    [CrossRef]
  20. L. Knüfing, H. Schollmeyer, H. Riegler, and K. Mecke, “Fractal analysis methods for solid alkane monolayer domains at SiO2/Air interfaces,” Langmuir 21, 992–1000 (2005).
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  21. R. A. Neher, K. Mecke, and H. Wagner, “Topological estimation of percolation thresholds,” J. Stat. Mechan. 2008, P01011 (2008).
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  24. W. K. Pratt, Digital Image Processing, 2nd edition (Wiley-Interscience, 1991).
  25. S. B. Gray, “Local properties of binary images in two dimensions,” IEEE Trans. Comput. C-20, 551–561 (1971).
    [CrossRef]
  26. T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vis. Res. 44, 1093–1113 (2004).
    [CrossRef]
  27. J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Software 33, 1–22 (2010).
  28. R. R. L. Taylor, T. Maddess, and Y. Nagai, “Spatial biases and computational constraints on the encoding of complex local image structure,” J. Vis. 8(7):19, 1–13 (2008).
    [CrossRef]

2012 (2)

J. D. Victor and M. M. Conte, “Local image statistics: maximum-entropy constructions and perceptual salience,” J. Opt. Soc. Am. A 29, 1313–1345 (2012).
[CrossRef]

F. Nielsen and R. Nock, “A closed-form expression for the Sharma–Mittal entropy of exponential families,” J. Phys. A 45, 032003 (2012).
[CrossRef]

2011 (1)

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

2010 (2)

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Software 33, 1–22 (2010).

G. Tkačik, J. S. Prentice, J. D. Victor, and V. Balasubramanian, “Local statistics in natural scenes predict the saliency of synthetic textures,” Proc. Nat. Acad. Sci. USA 107, 18149–18154 (2010).
[CrossRef]

2008 (2)

R. R. L. Taylor, T. Maddess, and Y. Nagai, “Spatial biases and computational constraints on the encoding of complex local image structure,” J. Vis. 8(7):19, 1–13 (2008).
[CrossRef]

R. A. Neher, K. Mecke, and H. Wagner, “Topological estimation of percolation thresholds,” J. Stat. Mechan. 2008, P01011 (2008).
[CrossRef]

2007 (1)

2005 (3)

M. O. Franz and B. Schölkopf, “Implicit wiener series for higher-order image analysis,” Adv. Neural Info. Process. Sys. 17, 465–472 (2005).

M. Masi, “A step beyond tsallis and rényi entropies,” Phys. Lett. A 338, 217–224 (2005).
[CrossRef]

L. Knüfing, H. Schollmeyer, H. Riegler, and K. Mecke, “Fractal analysis methods for solid alkane monolayer domains at SiO2/Air interfaces,” Langmuir 21, 992–1000 (2005).
[CrossRef]

2004 (1)

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vis. Res. 44, 1093–1113 (2004).
[CrossRef]

2003 (1)

E. Schneidman, S. Still, M. J. Berry, and W. Bialek, “Network information and connected correlations,” Phys. Rev. Lett. 91, 238701 (2003).
[CrossRef]

2001 (3)

C. Zetzsche and F. Rhrbein, “Nonlinear and extra-classical receptive field properties and the statistics of natural scenes,” Network 12, 331–350 (2001).
[CrossRef]

K. Michielsen and H. De Raedt, “Integral-geometry morphological image analysis,” Phys. Rep. 347, 461–538 (2001).
[CrossRef]

T. Maddess and Y. Nagai, “Discriminating of isotrigon textures,” Vis. Res. 41, 3837–3860 (2001).
[CrossRef]

2000 (1)

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

1998 (1)

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

1994 (2)

K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Nat. Acad. Sci. USA 91, 8482–8486 (1994).
[CrossRef]

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]

1991 (1)

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

1989 (1)

J. D. Victor and M. M. Conte, “Cortical interactions in texture processing: Scale and dynamics,” Vis. Neurosci. 2, 297–313 (1989).
[CrossRef]

1985 (1)

J. D. Victor, “Complex visual textures as a tool for studying the VEP,” Vis. Res. 25, 1811–1827 (1985).
[CrossRef]

1971 (1)

S. B. Gray, “Local properties of binary images in two dimensions,” IEEE Trans. Comput. C-20, 551–561 (1971).
[CrossRef]

Alexander, G. E.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Ankiewcz, A.

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vis. Res. 44, 1093–1113 (2004).
[CrossRef]

Armstrong, P.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

Azari, N. P.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Balasubramanian, V.

G. Tkačik, J. S. Prentice, J. D. Victor, and V. Balasubramanian, “Local statistics in natural scenes predict the saliency of synthetic textures,” Proc. Nat. Acad. Sci. USA 107, 18149–18154 (2010).
[CrossRef]

Beason-Held, L. L.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Berry, M. J.

E. Schneidman, S. Still, M. J. Berry, and W. Bialek, “Network information and connected correlations,” Phys. Rev. Lett. 91, 238701 (2003).
[CrossRef]

Bialek, W.

E. Schneidman, S. Still, M. J. Berry, and W. Bialek, “Network information and connected correlations,” Phys. Rev. Lett. 91, 238701 (2003).
[CrossRef]

Breidenbach, B.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, F. M. Schaller, B. Breidenbach, D. Hug, and K. Mecke, “Minkowski tensors of anisotropic spatial structure,” arXiv:1009.2340, 1–17 (2010).

Conte, M. M.

J. D. Victor and M. M. Conte, “Local image statistics: maximum-entropy constructions and perceptual salience,” J. Opt. Soc. Am. A 29, 1313–1345 (2012).
[CrossRef]

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

J. D. Victor and M. M. Conte, “Cortical interactions in texture processing: Scale and dynamics,” Vis. Neurosci. 2, 297–313 (1989).
[CrossRef]

Daly, E. M.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

De Hosson, J.

K. Michelsen, H. De Raedt, and J. De Hosson, “Aspects of mathematical morphology,” in Advances in Imaging and Electron Physics, B. K. Peter, W. Hawkes, and T. Mulvey, eds., Vol. 125 (Elsevier, 2003), pp. 119–194.

De Raedt, H.

K. Michielsen and H. De Raedt, “Integral-geometry morphological image analysis,” Phys. Rep. 347, 461–538 (2001).
[CrossRef]

K. Michelsen, H. De Raedt, and J. De Hosson, “Aspects of mathematical morphology,” in Advances in Imaging and Electron Physics, B. K. Peter, W. Hawkes, and T. Mulvey, eds., Vol. 125 (Elsevier, 2003), pp. 119–194.

Desmond, R. E.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

Franz, M. O.

M. O. Franz and B. Schölkopf, “Implicit wiener series for higher-order image analysis,” Adv. Neural Info. Process. Sys. 17, 465–472 (2005).

Friedman, J.

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Software 33, 1–22 (2010).

Grady, C. L.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Gray, S. B.

S. B. Gray, “Local properties of binary images in two dimensions,” IEEE Trans. Comput. C-20, 551–561 (1971).
[CrossRef]

Hastie, T.

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Software 33, 1–22 (2010).

Hoffmann, M. J. F.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

Horwitz, B.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Hug, D.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, F. M. Schaller, B. Breidenbach, D. Hug, and K. Mecke, “Minkowski tensors of anisotropic spatial structure,” arXiv:1009.2340, 1–17 (2010).

Inayat, A.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

James, A. C.

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vis. Res. 44, 1093–1113 (2004).
[CrossRef]

Kapfer, S. C.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, F. M. Schaller, B. Breidenbach, D. Hug, and K. Mecke, “Minkowski tensors of anisotropic spatial structure,” arXiv:1009.2340, 1–17 (2010).

Katz, E.

K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Nat. Acad. Sci. USA 91, 8482–8486 (1994).
[CrossRef]

Klatt, M. A.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

Kleppmann, N.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

Knüfing, L.

L. Knüfing, H. Schollmeyer, H. Riegler, and K. Mecke, “Fractal analysis methods for solid alkane monolayer domains at SiO2/Air interfaces,” Langmuir 21, 992–1000 (2005).
[CrossRef]

Krasuski, J. S.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

Maddess, T.

R. R. L. Taylor, T. Maddess, and Y. Nagai, “Spatial biases and computational constraints on the encoding of complex local image structure,” J. Vis. 8(7):19, 1–13 (2008).
[CrossRef]

T. Maddess, Y. Nagai, J. Victor, and R. Taylor, “Multilevel isotrigon textures,” J. Opt. Soc. Am. A 24, 278–293 (2007).
[CrossRef]

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vis. Res. 44, 1093–1113 (2004).
[CrossRef]

T. Maddess and Y. Nagai, “Discriminating of isotrigon textures,” Vis. Res. 41, 3837–3860 (2001).
[CrossRef]

Mangot, D. J.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Masi, M.

M. Masi, “A step beyond tsallis and rényi entropies,” Phys. Lett. A 338, 217–224 (2005).
[CrossRef]

Mecke, K.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

R. A. Neher, K. Mecke, and H. Wagner, “Topological estimation of percolation thresholds,” J. Stat. Mechan. 2008, P01011 (2008).
[CrossRef]

L. Knüfing, H. Schollmeyer, H. Riegler, and K. Mecke, “Fractal analysis methods for solid alkane monolayer domains at SiO2/Air interfaces,” Langmuir 21, 992–1000 (2005).
[CrossRef]

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, F. M. Schaller, B. Breidenbach, D. Hug, and K. Mecke, “Minkowski tensors of anisotropic spatial structure,” arXiv:1009.2340, 1–17 (2010).

Mentis, M. J.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Michelsen, K.

K. Michelsen, H. De Raedt, and J. De Hosson, “Aspects of mathematical morphology,” in Advances in Imaging and Electron Physics, B. K. Peter, W. Hawkes, and T. Mulvey, eds., Vol. 125 (Elsevier, 2003), pp. 119–194.

Michielsen, K.

K. Michielsen and H. De Raedt, “Integral-geometry morphological image analysis,” Phys. Rep. 347, 461–538 (2001).
[CrossRef]

Mickel, W.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, F. M. Schaller, B. Breidenbach, D. Hug, and K. Mecke, “Minkowski tensors of anisotropic spatial structure,” arXiv:1009.2340, 1–17 (2010).

Nagai, Y.

R. R. L. Taylor, T. Maddess, and Y. Nagai, “Spatial biases and computational constraints on the encoding of complex local image structure,” J. Vis. 8(7):19, 1–13 (2008).
[CrossRef]

T. Maddess, Y. Nagai, J. Victor, and R. Taylor, “Multilevel isotrigon textures,” J. Opt. Soc. Am. A 24, 278–293 (2007).
[CrossRef]

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vis. Res. 44, 1093–1113 (2004).
[CrossRef]

T. Maddess and Y. Nagai, “Discriminating of isotrigon textures,” Vis. Res. 41, 3837–3860 (2001).
[CrossRef]

Neher, R. A.

R. A. Neher, K. Mecke, and H. Wagner, “Topological estimation of percolation thresholds,” J. Stat. Mechan. 2008, P01011 (2008).
[CrossRef]

Nielsen, F.

F. Nielsen and R. Nock, “A closed-form expression for the Sharma–Mittal entropy of exponential families,” J. Phys. A 45, 032003 (2012).
[CrossRef]

Nock, R.

F. Nielsen and R. Nock, “A closed-form expression for the Sharma–Mittal entropy of exponential families,” J. Phys. A 45, 032003 (2012).
[CrossRef]

Optican, L. M.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Peukert, W.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 2nd edition (Wiley-Interscience, 1991).

Prentice, J. S.

G. Tkačik, J. S. Prentice, J. D. Victor, and V. Balasubramanian, “Local statistics in natural scenes predict the saliency of synthetic textures,” Proc. Nat. Acad. Sci. USA 107, 18149–18154 (2010).
[CrossRef]

Purpura, K. P.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Nat. Acad. Sci. USA 91, 8482–8486 (1994).
[CrossRef]

Rapoport, S. I.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Reichelsdorfer, M.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

Rhrbein, F.

C. Zetzsche and F. Rhrbein, “Nonlinear and extra-classical receptive field properties and the statistics of natural scenes,” Network 12, 331–350 (2001).
[CrossRef]

Riegler, H.

L. Knüfing, H. Schollmeyer, H. Riegler, and K. Mecke, “Fractal analysis methods for solid alkane monolayer domains at SiO2/Air interfaces,” Langmuir 21, 992–1000 (2005).
[CrossRef]

Schaller, F. M.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, F. M. Schaller, B. Breidenbach, D. Hug, and K. Mecke, “Minkowski tensors of anisotropic spatial structure,” arXiv:1009.2340, 1–17 (2010).

Schapiro, M. B.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Schneider, R.

R. Schneider and W. Weil, Stochastic and Integral Geometry (Springer, 2008).

Schneidman, E.

E. Schneidman, S. Still, M. J. Berry, and W. Bialek, “Network information and connected correlations,” Phys. Rev. Lett. 91, 238701 (2003).
[CrossRef]

Schölkopf, B.

M. O. Franz and B. Schölkopf, “Implicit wiener series for higher-order image analysis,” Adv. Neural Info. Process. Sys. 17, 465–472 (2005).

Schollmeyer, H.

L. Knüfing, H. Schollmeyer, H. Riegler, and K. Mecke, “Fractal analysis methods for solid alkane monolayer domains at SiO2/Air interfaces,” Langmuir 21, 992–1000 (2005).
[CrossRef]

Schröder-Turk, G. E.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, F. M. Schaller, B. Breidenbach, D. Hug, and K. Mecke, “Minkowski tensors of anisotropic spatial structure,” arXiv:1009.2340, 1–17 (2010).

Schwieger, W.

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

Still, S.

E. Schneidman, S. Still, M. J. Berry, and W. Bialek, “Network information and connected correlations,” Phys. Rev. Lett. 91, 238701 (2003).
[CrossRef]

Taylor, R.

Taylor, R. R. L.

R. R. L. Taylor, T. Maddess, and Y. Nagai, “Spatial biases and computational constraints on the encoding of complex local image structure,” J. Vis. 8(7):19, 1–13 (2008).
[CrossRef]

Tibshirani, R.

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Software 33, 1–22 (2010).

Tkacik, G.

G. Tkačik, J. S. Prentice, J. D. Victor, and V. Balasubramanian, “Local statistics in natural scenes predict the saliency of synthetic textures,” Proc. Nat. Acad. Sci. USA 107, 18149–18154 (2010).
[CrossRef]

Van Meter, J. W.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

VanMeter, J. W.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

Victor, J.

Victor, J. D.

J. D. Victor and M. M. Conte, “Local image statistics: maximum-entropy constructions and perceptual salience,” J. Opt. Soc. Am. A 29, 1313–1345 (2012).
[CrossRef]

G. Tkačik, J. S. Prentice, J. D. Victor, and V. Balasubramanian, “Local statistics in natural scenes predict the saliency of synthetic textures,” Proc. Nat. Acad. Sci. USA 107, 18149–18154 (2010).
[CrossRef]

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]

K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Nat. Acad. Sci. USA 91, 8482–8486 (1994).
[CrossRef]

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

J. D. Victor and M. M. Conte, “Cortical interactions in texture processing: Scale and dynamics,” Vis. Neurosci. 2, 297–313 (1989).
[CrossRef]

J. D. Victor, “Complex visual textures as a tool for studying the VEP,” Vis. Res. 25, 1811–1827 (1985).
[CrossRef]

Wagner, H.

R. A. Neher, K. Mecke, and H. Wagner, “Topological estimation of percolation thresholds,” J. Stat. Mechan. 2008, P01011 (2008).
[CrossRef]

Weil, W.

R. Schneider and W. Weil, Stochastic and Integral Geometry (Springer, 2008).

Zetzsche, C.

C. Zetzsche and F. Rhrbein, “Nonlinear and extra-classical receptive field properties and the statistics of natural scenes,” Network 12, 331–350 (2001).
[CrossRef]

Adv. Mater. (1)

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, M. A. Klatt, F. M. Schaller, M. J. F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, and K. Mecke, “Minkowski tensor shape analysis of cellular, granular and porous structures,” Adv. Mater. 23, 2535–2553 (2011).
[CrossRef]

Adv. Neural Info. Process. Sys. (1)

M. O. Franz and B. Schölkopf, “Implicit wiener series for higher-order image analysis,” Adv. Neural Info. Process. Sys. 17, 465–472 (2005).

Exp. Brain Res. (1)

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, “Striate cortex in humans demonstrates the relationship between activation and variations in visual form,” Exp. Brain Res. 130, 221–226 (2000).
[CrossRef]

IEEE Trans. Comput. (1)

S. B. Gray, “Local properties of binary images in two dimensions,” IEEE Trans. Comput. C-20, 551–561 (1971).
[CrossRef]

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

J. Phys. A (1)

F. Nielsen and R. Nock, “A closed-form expression for the Sharma–Mittal entropy of exponential families,” J. Phys. A 45, 032003 (2012).
[CrossRef]

J. Stat. Mechan. (1)

R. A. Neher, K. Mecke, and H. Wagner, “Topological estimation of percolation thresholds,” J. Stat. Mechan. 2008, P01011 (2008).
[CrossRef]

J. Stat. Software (1)

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Software 33, 1–22 (2010).

J. Vis. (1)

R. R. L. Taylor, T. Maddess, and Y. Nagai, “Spatial biases and computational constraints on the encoding of complex local image structure,” J. Vis. 8(7):19, 1–13 (2008).
[CrossRef]

Langmuir (1)

L. Knüfing, H. Schollmeyer, H. Riegler, and K. Mecke, “Fractal analysis methods for solid alkane monolayer domains at SiO2/Air interfaces,” Langmuir 21, 992–1000 (2005).
[CrossRef]

Network (1)

C. Zetzsche and F. Rhrbein, “Nonlinear and extra-classical receptive field properties and the statistics of natural scenes,” Network 12, 331–350 (2001).
[CrossRef]

Phys. Lett. A (1)

M. Masi, “A step beyond tsallis and rényi entropies,” Phys. Lett. A 338, 217–224 (2005).
[CrossRef]

Phys. Rep. (1)

K. Michielsen and H. De Raedt, “Integral-geometry morphological image analysis,” Phys. Rep. 347, 461–538 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

E. Schneidman, S. Still, M. J. Berry, and W. Bialek, “Network information and connected correlations,” Phys. Rev. Lett. 91, 238701 (2003).
[CrossRef]

Proc. Nat. Acad. Sci. USA (2)

G. Tkačik, J. S. Prentice, J. D. Victor, and V. Balasubramanian, “Local statistics in natural scenes predict the saliency of synthetic textures,” Proc. Nat. Acad. Sci. USA 107, 18149–18154 (2010).
[CrossRef]

K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Nat. Acad. Sci. USA 91, 8482–8486 (1994).
[CrossRef]

Vis. Neurosci. (2)

J. D. Victor and M. M. Conte, “Cortical interactions in texture processing: Scale and dynamics,” Vis. Neurosci. 2, 297–313 (1989).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, “PET reveals occipitotemporal pathway activation during elementary form perception in humans,” Vis. Neurosci. 15, 503–510 (1998).
[CrossRef]

Vis. Res. (4)

J. D. Victor, “Complex visual textures as a tool for studying the VEP,” Vis. Res. 25, 1811–1827 (1985).
[CrossRef]

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

T. Maddess and Y. Nagai, “Discriminating of isotrigon textures,” Vis. Res. 41, 3837–3860 (2001).
[CrossRef]

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vis. Res. 44, 1093–1113 (2004).
[CrossRef]

Other (4)

R. Schneider and W. Weil, Stochastic and Integral Geometry (Springer, 2008).

K. Michelsen, H. De Raedt, and J. De Hosson, “Aspects of mathematical morphology,” in Advances in Imaging and Electron Physics, B. K. Peter, W. Hawkes, and T. Mulvey, eds., Vol. 125 (Elsevier, 2003), pp. 119–194.

W. K. Pratt, Digital Image Processing, 2nd edition (Wiley-Interscience, 1991).

G. E. Schröder-Turk, W. Mickel, S. C. Kapfer, F. M. Schaller, B. Breidenbach, D. Hug, and K. Mecke, “Minkowski tensors of anisotropic spatial structure,” arXiv:1009.2340, 1–17 (2010).

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Figures (6)

Fig. 1.
Fig. 1.

Binary textures: Samples of the isotrigons (top and bottom) and isodipoles (middle) used in this study, displayed according to their group (V3L2, V2L2, VNL2). Also shown for each sample are the gliders (greatly magnified), rules, and identification labels.

Fig. 2.
Fig. 2.

Ideal observer: (a) Human performance in discriminating the texture families of Fig. 1 from uniformly random point patterns. In the middle and bottom inserts are the Sharma–Mittal divergences, measuring departure from uniformity, encompassing both (b) Tsallis and (c) Rényi divergences. The Kullback–Leibler divergence, the ideal observer response, is highlighted by the thicker lines; it is recovered in (b) when ( r q and q 1 ) and in (c) when ( r 1 and q 1 ).

Fig. 3.
Fig. 3.

Image decomposition: Simple example illustrating the decomposition of an image (left) into its disjoint (open) building blocks (right): points, lines and open bodies, respectively, vertices, sides, and interior of a black square.

Fig. 4.
Fig. 4.

Models and psychophysical data: Fit of a cross-validated, minimum deviance, linear model based on low order moments of the additive Minkowski functionals (A, P, χ ) (a), of the multi-information measures α G (b), and of both (c). Models based on moments of the quads distribution are in (d) and based on a full predictor set in (e).

Fig. 5.
Fig. 5.

Values of selected predictor variables: Measures on the vertical axis are selected by the lasso procedure for the model with all considered predictors. Their value for each of the texture families, on the horizontal axis, is displayed using the gray scale on the right.

Fig. 6.
Fig. 6.

Contribution to the rsmd error, on the whole data set, for each of the coefficients of the minimum variance model using all features.

Tables (4)

Tables Icon

Table 2. Model Properties

Tables Icon

Table 3. Values of the Minkowski Functionals for 2D Open Bodies

Tables Icon

Table 4. Coefficients of Predictor Variables, Including the Intercept I 0 , for Both Full ( m 0 ) and Sparse ( m 1 ) Models and All Five Predictor Classesa

Equations (27)

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

D KL = i n log ( p i / q i ) = H ( Q ) H ( P ) ,
S r q = 1 1 r [ ( 1 n p i q ) 1 r 1 q 1 ] .
T q = p i q 1 1 q , R q = log ( p i q ) 1 q .
A = L 2 2 + 1 2 i , j = 1 L σ i , j , P = L 2 1 2 i , j = 1 L σ i , j ( σ i + 1 , j + σ i , j + 1 ) ,
χ { 4 , 8 } = L 2 16 1 16 i , j = 1 L { 4 σ i , j ± 2 σ i , j ( σ i + 1 , j + σ i , j + 1 ) σ i , j ( σ i + 1 , j + 1 + σ i + 1 , j 1 ) + σ i , j ( σ i + 1 , j + σ i , j + 1 ) σ i + 1 , j + 1 + σ i + 1 , j ( σ i , j + σ i + 1 , j + 1 ) σ i , j + 1 σ i , j σ i + 1 , j σ i + 1 , j + 1 σ i , j + 1 } ,
A = 1 4 k = 1 4 ( k n { Q k } + 2 n { Q D } ) , P = k = 1 3 n { Q k } + 2 n { Q D } , χ 4 = 1 4 [ n { Q 1 } n { Q 3 } + 2 n { Q D } ] χ 8 = 1 4 [ n { Q 1 } n { Q 3 } 2 n { Q D } ] ,
P G ( 1 ) = exp { i h i 1 σ i } Z 1 , P G ( 2 ) = exp { i h i 1 σ i + ( 1 / 2 ) i j h i j 2 σ i σ j } Z 2 ,
P G ( 3 ) = exp { i h i 1 σ i + ( 1 / 2 ) i j h i j 2 σ i σ j + ( 1 / 6 ) h i j k 3 σ i σ j σ k } Z 3 ,
log Z 3 h i 1 σ i , log Z 3 h i j 3 σ i σ j , log Z 3 h i j k 3 σ i σ j σ k .
I G ( ν ) = S ( P G ( ν 1 ) ) S ( P G ( ν ) ) .
P G ( 4 ) ( σ⃗ ) = P G ( σ⃗ ) = P G ( 3 ) ( σ⃗ ) + 1 16 α G ( 4 ) Π G ( σ⃗ ) ,
α G ( 4 ) = σ Π G ( σ⃗ ) P G ( σ⃗ ) σ Π G ( σ⃗ ) P G ( 3 ) ( σ⃗ ) .
min ( β 0 , β ) R p + 1 R λ ( β 0 , β ) ,
R λ ( β 0 , β ) = 1 N D ( y i | β 0 , β ) + λ G a ( β ) ,
G a ( β ) = ( 1 a ) 1 2 β l 2 2 + a β l 1 .
D ( y i | β 0 , β ) = 2 [ y i log ( y i ( μ ^ i ) ) + ( n y i ) log ( n y i n μ ^ i ) ] ,
W ν ( d ) ( P ) = W ν ( d ) ( i = 1 l K i ) = i W ν ( d ) ( i m ) ,
i < j W ν ( d ) ( i j ) + ( 1 ) l + 1 W ν ( d ) ( 1 l ) = 0 .
W ν ( d ) ( P ) = m W ν ( d ) ( m ) n m ( P ) .
A = i , j = 1 L b 1 , P = i , j = 1 L ( 2 b 1 + b 2 + b 3 2 b 1 b 2 2 b 1 b 3 )
χ 4 = i , j = 1 L ( 3 b 2 / 8 3 b 1 / 8 + 3 b 3 / 8 + 5 b 4 / 8 b 1 b 2 / 4 b 1 b 3 / 4 b 1 b 4 b 2 b 3 3 b 2 b 4 / 4 3 b 3 b 4 / 4 + b 1 b 2 b 3 + b 1 b 2 b 4 + b 1 b 3 b 4 + b 2 b 3 b 4 b 1 b 2 b 3 b 4 ) .
P = i , j = 1 L { 4 b 1 2 b 1 b 2 2 b 1 b 3 } .
m 1 = b 1 m 2 = ( b 1 b 2 + b 1 b 3 ) m 3 = ( b 1 b 4 + b 2 b 3 ) m 4 = + ( b 1 b 2 b 3 + b 1 b 2 b 4 + b 1 b 3 b 4 + b 2 b 3 b 4 ) m 5 = b 1 b 2 b 3 b 4 ,
χ 4 = i , j = 1 L k = 1 5 m k , χ 8 = i , j = 1 L m 1 + m 2 m 5 .
b 1 n ( ( 1 0 0 0 ) ) + n ( ( 1 0 1 0 ) ) + n ( ( 1 1 0 0 ) ) + n ( ( 1 0 0 1 ) ) + n ( ( 1 0 1 1 ) ) + n ( ( 1 1 1 0 ) ) + n ( ( 1 1 0 1 ) ) + n ( ( 1 1 1 1 ) ) , b 1 b 2 n ( ( 1 0 1 0 ) ) + n ( ( 1 1 1 0 ) ) + n ( ( 1 0 1 1 ) ) + n ( ( 1 1 1 1 ) ) , b 1 b 2 b 3 n ( ( 1 0 1 1 ) ) + n ( ( 1 1 1 1 ) ) , b 1 b 2 b 3 b 4 n ( ( 1 1 1 1 ) ) .
b i b j b l a 0 , 1 δ 1 i δ 1 j δ 1 l n ( a 1 a 3 a 2 a 4 ) .
s 1 = 1 / 4 [ n { Q 1 } + 2 n { Q 2 } + 3 n { Q 3 } + 4 n { Q 4 } + 2 n { Q D } ] s 2 = n { Q 2 } 2 + n { Q 3 } + 2 n { Q 4 } s 3 = ( n { Q D } + n { Q 3 } + 2 n { Q 4 } ) s 4 = n { Q 3 } + 4 n { Q 4 } s 5 = n { Q 4 } ,

Metrics