Abstract

Three-dimensional (3D) cluttered scenes consist of a large number of small surfaces distributed randomly in a 3D view volume. The canonical example is the foliage of a tree or bush. 3D cluttered scenes are challenging for vision tasks such as object recognition and depth perception because most surfaces or objects are only partly visible. This paper examines the probabilities of surface visibility in 3D cluttered scenes. We model how the probabilities of visible gaps, depth discontinuities, and binocular and half-occluded points depend on scene parameters such as the size and density of the surfaces that make up the clutter, as well as on depth and inverse depth. Inverse depth is of particular interest since both binocular disparity and motion parallax depend directly on it. The probability models are verified using data from synthetic 3D cluttered scenes, which are generated using computer graphics.

© 2012 Optical Society of America

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  1. R. Rosenholtz, Y. Li, and L. Nakano, “Measuring visual clutter,” J. Vision 7, 1–22 (2007).
    [CrossRef]
  2. J. M. Wolfe, “Guided search 2.0: a revised model of visual search,” Psychon. Bull. Rev. 1, 202–238 (1994).
  3. M. A. Changizi and S. Shimojo, “X-ray vision and the evolution of forward-facing eyes,” J. Theor. Biol. 254, 756–767 (2008).
    [CrossRef]
  4. T. E. Avery and H. E. Burkhart, Forest Measurements, 5th ed. (McGraw-Hill, 2002).
  5. P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants (Springer-Verlag, 1990).
  6. P. Prusinkiewicz, “Modeling of spatial structure and development of plants: a review,” Scientia Horticulturae 74, 113–149 (1998).
    [CrossRef]
  7. T. Nilson, “A theoretical analysis of the frequency of gaps in plant stands,” Agric. Meteor. 8, 25–38 (1971).
  8. P. R. van Gardingen, G. E. Jackson, S. Hernandez-Daumas, G. Russell, and L. Sharp, “Leaf area index estimates obtained for clumped canopies using hemispherical photography,” Agric. Forest Meteor. 94, 243–257 (1999).
  9. J. M. Chen and J. Cihlar, “Plant canopy gap-size analysis theory for improving optical measurements of leaf-area index,” Appl. Opt. 34, 6211–6222 (1995).
    [CrossRef]
  10. H. Sinoquet, G. Sonohat, J. Phattaralerphong, and C. Godin, “Foliage randomness and light interception in 3-d digitized trees: an analysis from multiscale discretization of the canopy,” Plant Cell Environ. 28, 1158–1170 (2005).
    [CrossRef]
  11. D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
    [CrossRef]
  12. G. Matheron, Random Sets and Integral Geometry (Wiley, 1975).
  13. J. P. Serra, Image Analysis and Mathematical Morphology (Academic, 1982).
  14. D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
    [CrossRef]
  15. D. L. Ruderman, “Origins of scaling in natural images,” Vis. Res. 37, 3385–3398 (1997).
    [CrossRef]
  16. R. M. Balboa, C. W. Tyler, and N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vis. Res. 41, 955–964 (2001).
    [CrossRef]
  17. J. Huang, A. Lee, and D. Mumford, “Statistics of range images,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 324–331.
  18. A. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: a statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
    [CrossRef]
  19. Z. Yang and D. Purves, “Image/source statistics of surfaces in natural scenes,” Network Comput. Neural Syst. 14, 371–390 (2003).
    [CrossRef]
  20. B. Potetz and T. S. Lee, “Statistical correlations between two-dimensional images and three-dimensional structures in natural scenes,” J. Opt. Soc. Am. A 20, 1292–1303 (2003).
    [CrossRef]
  21. D. Calow and M. Lappe, “Local statistics of retinal optic flow for self-motion through natural sceneries,” Network Comput. Neural Syst. 18, 343–374 (2007).
    [CrossRef]
  22. S. Roth and M. J. Black, “On the spatial statistics of optical flow,” Int. J. Comput. Vis. 74, 33–50 (2007).
    [CrossRef]
  23. P. B. Hibbard, “A statistical model of binocular disparity,” Vis. Cogn. 15, 149–165 (2007).
  24. Y. Liu, A. Bovik, and L. Cormack, “Disparity statistics in natural scenes,” J. Vision 8(2):17, 1–22 (2008).
    [CrossRef]
  25. J. Harris and L. Wilcox, “The role of monocularly visible regions in the perception of three-dimensional scenes,” Vis. Res. 49, 2666–2685 (2009).
    [CrossRef]
  26. J. Forte, J. W. Peirce, and P. Lennie, “Binocular integration of partially occluded surfaces,” Vis. Res. 42, 1225–1235(2002).
    [CrossRef]
  27. P. Belhumeur, “A Bayesian approach to binocular stereopsis,” Int. J. Comput. Vis. 19, 237–260 (1996).
    [CrossRef]
  28. S. Kang, R. Szeliski, and J. Chai, “Handling occlusions in dense multi-view stereo,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. I:103–110.
  29. G. Egnal and R. Wildes, “Detecting binocular half-occlusions: Empirical comparisons of five approaches,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1127–1133 (2002).
    [CrossRef]
  30. V. Kolmogorov and R. Zabih, “Computing visual correspondence with occlusions via graph cuts,” in Proceedings of 8th IEEE International Conference on Computer Vision (IEEE, 2001), pp. II:508–515.
  31. Y. Wei and L. Quan, “Asymmetrical occlusion handling using graph cut for multi-view stereo,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Vol. 2 (IEEE, 2005), pp. 902–909.
  32. V. Vaish, M. Levoy, R. Szeliski, C. L. Zitnick, and S. B. Kang, “Reconstructing occluded surfaces using synthetic apertures: stereo, focus and robust measures,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2006), pp. 2331–2338.
  33. A. Mittal and L. Davis, “A general method for sensor planning in multi-sensor systems: extension to random occlusion,” Int. J. Comput. Vis. 76, 31–52 (2007).
    [CrossRef]
  34. P. Hall, Introduction to the Theory of Coverage Processes(Wiley, 1988).
  35. S. Zacks, Stochastic Visibility in Random Fields, Lecture Notes in Statistics 95 (Springer-Verlag, 1994).
  36. B. Nadler, G. Fibich, S. Lev-Yehudi, and D. Cohen-Or, “A qualitative and quantitative visibility analysis in urban scenes,” Comput. Graphics 23, 655–666 (1999).
  37. J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).
  38. M. Langer, “Surface visibility probabilities in 3d cluttered scenes,” in Proceedings of 10th European Conference on Computer Vision (Springer-Verlag, 2008), pp. I:401–412.
  39. H. Longuet-Higgins and K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. B 208, 385–397 (1980).
    [CrossRef]
  40. G. W. Larson and R. Shakespeare, Rendering with Radiance: The Art and Science of Lighting Visualization (Morgan Kaufmann, 1998).
  41. R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
    [CrossRef]
  42. “The blender greenhouse,” http://yorik.uncreated.net/greenhouse.html .
  43. “Ngplant—open source plant modeling suite,” http://ngplant.sourceforge.net/ .
  44. G. Kanizsa, Organization in Vision: Essays on Gestalt Perception (Praeger, 1979).
  45. M. Hansard, “Binocular projection of a random scene,” in Proceedings of British Machine Vision Conference (Springer-Verlag, 2012), to be published.
  46. D. C. Knill, “Surface orientation from texture: Ideal observers, generic observers and the information content of texture cues,” Vis. Res. 38, 1655–1682 (1998).
    [CrossRef]
  47. R. van Ee and B. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001).
    [CrossRef]
  48. L. Zhang and S. M. Seitz, “Estimating optimal parameters for MRF stereo from a single image pair,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 331–342 (2007).
  49. K. Engel, M. Hadwiger, J. M. Kniss, and D. W. C. Rezk-Salama, Real-Time Volume Graphics (AK Peters, 2006).

2009

J. Harris and L. Wilcox, “The role of monocularly visible regions in the perception of three-dimensional scenes,” Vis. Res. 49, 2666–2685 (2009).
[CrossRef]

2008

Y. Liu, A. Bovik, and L. Cormack, “Disparity statistics in natural scenes,” J. Vision 8(2):17, 1–22 (2008).
[CrossRef]

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

M. A. Changizi and S. Shimojo, “X-ray vision and the evolution of forward-facing eyes,” J. Theor. Biol. 254, 756–767 (2008).
[CrossRef]

2007

L. Zhang and S. M. Seitz, “Estimating optimal parameters for MRF stereo from a single image pair,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 331–342 (2007).

R. Rosenholtz, Y. Li, and L. Nakano, “Measuring visual clutter,” J. Vision 7, 1–22 (2007).
[CrossRef]

D. Calow and M. Lappe, “Local statistics of retinal optic flow for self-motion through natural sceneries,” Network Comput. Neural Syst. 18, 343–374 (2007).
[CrossRef]

S. Roth and M. J. Black, “On the spatial statistics of optical flow,” Int. J. Comput. Vis. 74, 33–50 (2007).
[CrossRef]

P. B. Hibbard, “A statistical model of binocular disparity,” Vis. Cogn. 15, 149–165 (2007).

A. Mittal and L. Davis, “A general method for sensor planning in multi-sensor systems: extension to random occlusion,” Int. J. Comput. Vis. 76, 31–52 (2007).
[CrossRef]

2005

H. Sinoquet, G. Sonohat, J. Phattaralerphong, and C. Godin, “Foliage randomness and light interception in 3-d digitized trees: an analysis from multiscale discretization of the canopy,” Plant Cell Environ. 28, 1158–1170 (2005).
[CrossRef]

2003

Z. Yang and D. Purves, “Image/source statistics of surfaces in natural scenes,” Network Comput. Neural Syst. 14, 371–390 (2003).
[CrossRef]

B. Potetz and T. S. Lee, “Statistical correlations between two-dimensional images and three-dimensional structures in natural scenes,” J. Opt. Soc. Am. A 20, 1292–1303 (2003).
[CrossRef]

2002

J. Forte, J. W. Peirce, and P. Lennie, “Binocular integration of partially occluded surfaces,” Vis. Res. 42, 1225–1235(2002).
[CrossRef]

G. Egnal and R. Wildes, “Detecting binocular half-occlusions: Empirical comparisons of five approaches,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1127–1133 (2002).
[CrossRef]

2001

R. van Ee and B. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001).
[CrossRef]

R. M. Balboa, C. W. Tyler, and N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vis. Res. 41, 955–964 (2001).
[CrossRef]

A. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: a statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[CrossRef]

1999

P. R. van Gardingen, G. E. Jackson, S. Hernandez-Daumas, G. Russell, and L. Sharp, “Leaf area index estimates obtained for clumped canopies using hemispherical photography,” Agric. Forest Meteor. 94, 243–257 (1999).

B. Nadler, G. Fibich, S. Lev-Yehudi, and D. Cohen-Or, “A qualitative and quantitative visibility analysis in urban scenes,” Comput. Graphics 23, 655–666 (1999).

1998

D. C. Knill, “Surface orientation from texture: Ideal observers, generic observers and the information content of texture cues,” Vis. Res. 38, 1655–1682 (1998).
[CrossRef]

P. Prusinkiewicz, “Modeling of spatial structure and development of plants: a review,” Scientia Horticulturae 74, 113–149 (1998).
[CrossRef]

1997

D. L. Ruderman, “Origins of scaling in natural images,” Vis. Res. 37, 3385–3398 (1997).
[CrossRef]

1996

P. Belhumeur, “A Bayesian approach to binocular stereopsis,” Int. J. Comput. Vis. 19, 237–260 (1996).
[CrossRef]

1995

1994

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

J. M. Wolfe, “Guided search 2.0: a revised model of visual search,” Psychon. Bull. Rev. 1, 202–238 (1994).

1992

J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).

1987

1980

H. Longuet-Higgins and K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. B 208, 385–397 (1980).
[CrossRef]

1971

T. Nilson, “A theoretical analysis of the frequency of gaps in plant stands,” Agric. Meteor. 8, 25–38 (1971).

Agarwala, A.

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

Anderson, B.

R. van Ee and B. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001).
[CrossRef]

Avery, T. E.

T. E. Avery and H. E. Burkhart, Forest Measurements, 5th ed. (McGraw-Hill, 2002).

Balboa, R. M.

R. M. Balboa, C. W. Tyler, and N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vis. Res. 41, 955–964 (2001).
[CrossRef]

Belhumeur, P.

P. Belhumeur, “A Bayesian approach to binocular stereopsis,” Int. J. Comput. Vis. 19, 237–260 (1996).
[CrossRef]

Bialek, W.

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

Black, M. J.

S. Roth and M. J. Black, “On the spatial statistics of optical flow,” Int. J. Comput. Vis. 74, 33–50 (2007).
[CrossRef]

Bovik, A.

Y. Liu, A. Bovik, and L. Cormack, “Disparity statistics in natural scenes,” J. Vision 8(2):17, 1–22 (2008).
[CrossRef]

Burkhart, H. E.

T. E. Avery and H. E. Burkhart, Forest Measurements, 5th ed. (McGraw-Hill, 2002).

Calow, D.

D. Calow and M. Lappe, “Local statistics of retinal optic flow for self-motion through natural sceneries,” Network Comput. Neural Syst. 18, 343–374 (2007).
[CrossRef]

Chai, J.

S. Kang, R. Szeliski, and J. Chai, “Handling occlusions in dense multi-view stereo,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. I:103–110.

Changizi, M. A.

M. A. Changizi and S. Shimojo, “X-ray vision and the evolution of forward-facing eyes,” J. Theor. Biol. 254, 756–767 (2008).
[CrossRef]

Chen, J. M.

Cihlar, J.

Cohen-Or, D.

B. Nadler, G. Fibich, S. Lev-Yehudi, and D. Cohen-Or, “A qualitative and quantitative visibility analysis in urban scenes,” Comput. Graphics 23, 655–666 (1999).

Cormack, L.

Y. Liu, A. Bovik, and L. Cormack, “Disparity statistics in natural scenes,” J. Vision 8(2):17, 1–22 (2008).
[CrossRef]

Davis, L.

A. Mittal and L. Davis, “A general method for sensor planning in multi-sensor systems: extension to random occlusion,” Int. J. Comput. Vis. 76, 31–52 (2007).
[CrossRef]

Egnal, G.

G. Egnal and R. Wildes, “Detecting binocular half-occlusions: Empirical comparisons of five approaches,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1127–1133 (2002).
[CrossRef]

Engel, K.

K. Engel, M. Hadwiger, J. M. Kniss, and D. W. C. Rezk-Salama, Real-Time Volume Graphics (AK Peters, 2006).

Fibich, G.

B. Nadler, G. Fibich, S. Lev-Yehudi, and D. Cohen-Or, “A qualitative and quantitative visibility analysis in urban scenes,” Comput. Graphics 23, 655–666 (1999).

Field, D. J.

Forte, J.

J. Forte, J. W. Peirce, and P. Lennie, “Binocular integration of partially occluded surfaces,” Vis. Res. 42, 1225–1235(2002).
[CrossRef]

Godin, C.

H. Sinoquet, G. Sonohat, J. Phattaralerphong, and C. Godin, “Foliage randomness and light interception in 3-d digitized trees: an analysis from multiscale discretization of the canopy,” Plant Cell Environ. 28, 1158–1170 (2005).
[CrossRef]

Grzywacz, N. M.

R. M. Balboa, C. W. Tyler, and N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vis. Res. 41, 955–964 (2001).
[CrossRef]

Hadwiger, M.

K. Engel, M. Hadwiger, J. M. Kniss, and D. W. C. Rezk-Salama, Real-Time Volume Graphics (AK Peters, 2006).

Hall, P.

P. Hall, Introduction to the Theory of Coverage Processes(Wiley, 1988).

Hansard, M.

M. Hansard, “Binocular projection of a random scene,” in Proceedings of British Machine Vision Conference (Springer-Verlag, 2012), to be published.

Harris, J.

J. Harris and L. Wilcox, “The role of monocularly visible regions in the perception of three-dimensional scenes,” Vis. Res. 49, 2666–2685 (2009).
[CrossRef]

Hernandez-Daumas, S.

P. R. van Gardingen, G. E. Jackson, S. Hernandez-Daumas, G. Russell, and L. Sharp, “Leaf area index estimates obtained for clumped canopies using hemispherical photography,” Agric. Forest Meteor. 94, 243–257 (1999).

Hibbard, P. B.

P. B. Hibbard, “A statistical model of binocular disparity,” Vis. Cogn. 15, 149–165 (2007).

Huang, J.

A. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: a statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[CrossRef]

J. Huang, A. Lee, and D. Mumford, “Statistics of range images,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 324–331.

Jackson, G. E.

P. R. van Gardingen, G. E. Jackson, S. Hernandez-Daumas, G. Russell, and L. Sharp, “Leaf area index estimates obtained for clumped canopies using hemispherical photography,” Agric. Forest Meteor. 94, 243–257 (1999).

Kang, S.

S. Kang, R. Szeliski, and J. Chai, “Handling occlusions in dense multi-view stereo,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. I:103–110.

Kang, S. B.

V. Vaish, M. Levoy, R. Szeliski, C. L. Zitnick, and S. B. Kang, “Reconstructing occluded surfaces using synthetic apertures: stereo, focus and robust measures,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2006), pp. 2331–2338.

Kanizsa, G.

G. Kanizsa, Organization in Vision: Essays on Gestalt Perception (Praeger, 1979).

Knill, D. C.

D. C. Knill, “Surface orientation from texture: Ideal observers, generic observers and the information content of texture cues,” Vis. Res. 38, 1655–1682 (1998).
[CrossRef]

Kniss, J. M.

K. Engel, M. Hadwiger, J. M. Kniss, and D. W. C. Rezk-Salama, Real-Time Volume Graphics (AK Peters, 2006).

Kolmogorov, V.

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

V. Kolmogorov and R. Zabih, “Computing visual correspondence with occlusions via graph cuts,” in Proceedings of 8th IEEE International Conference on Computer Vision (IEEE, 2001), pp. II:508–515.

Langer, M.

M. Langer, “Surface visibility probabilities in 3d cluttered scenes,” in Proceedings of 10th European Conference on Computer Vision (Springer-Verlag, 2008), pp. I:401–412.

Lappe, M.

D. Calow and M. Lappe, “Local statistics of retinal optic flow for self-motion through natural sceneries,” Network Comput. Neural Syst. 18, 343–374 (2007).
[CrossRef]

Larson, G. W.

G. W. Larson and R. Shakespeare, Rendering with Radiance: The Art and Science of Lighting Visualization (Morgan Kaufmann, 1998).

Lee, A.

A. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: a statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[CrossRef]

J. Huang, A. Lee, and D. Mumford, “Statistics of range images,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 324–331.

Lee, T. S.

Lennie, P.

J. Forte, J. W. Peirce, and P. Lennie, “Binocular integration of partially occluded surfaces,” Vis. Res. 42, 1225–1235(2002).
[CrossRef]

Levoy, M.

V. Vaish, M. Levoy, R. Szeliski, C. L. Zitnick, and S. B. Kang, “Reconstructing occluded surfaces using synthetic apertures: stereo, focus and robust measures,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2006), pp. 2331–2338.

Lev-Yehudi, S.

B. Nadler, G. Fibich, S. Lev-Yehudi, and D. Cohen-Or, “A qualitative and quantitative visibility analysis in urban scenes,” Comput. Graphics 23, 655–666 (1999).

Li, Y.

R. Rosenholtz, Y. Li, and L. Nakano, “Measuring visual clutter,” J. Vision 7, 1–22 (2007).
[CrossRef]

Lindenmayer, A.

P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants (Springer-Verlag, 1990).

Liu, Y.

Y. Liu, A. Bovik, and L. Cormack, “Disparity statistics in natural scenes,” J. Vision 8(2):17, 1–22 (2008).
[CrossRef]

Longuet-Higgins, H.

H. Longuet-Higgins and K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. B 208, 385–397 (1980).
[CrossRef]

Matheron, G.

G. Matheron, Random Sets and Integral Geometry (Wiley, 1975).

Mittal, A.

A. Mittal and L. Davis, “A general method for sensor planning in multi-sensor systems: extension to random occlusion,” Int. J. Comput. Vis. 76, 31–52 (2007).
[CrossRef]

Mumford, D.

A. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: a statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[CrossRef]

J. Huang, A. Lee, and D. Mumford, “Statistics of range images,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 324–331.

Nadler, B.

B. Nadler, G. Fibich, S. Lev-Yehudi, and D. Cohen-Or, “A qualitative and quantitative visibility analysis in urban scenes,” Comput. Graphics 23, 655–666 (1999).

Nakano, L.

R. Rosenholtz, Y. Li, and L. Nakano, “Measuring visual clutter,” J. Vision 7, 1–22 (2007).
[CrossRef]

Nilson, T.

T. Nilson, “A theoretical analysis of the frequency of gaps in plant stands,” Agric. Meteor. 8, 25–38 (1971).

Peirce, J. W.

J. Forte, J. W. Peirce, and P. Lennie, “Binocular integration of partially occluded surfaces,” Vis. Res. 42, 1225–1235(2002).
[CrossRef]

Phattaralerphong, J.

H. Sinoquet, G. Sonohat, J. Phattaralerphong, and C. Godin, “Foliage randomness and light interception in 3-d digitized trees: an analysis from multiscale discretization of the canopy,” Plant Cell Environ. 28, 1158–1170 (2005).
[CrossRef]

Potetz, B.

Prazdny, K.

H. Longuet-Higgins and K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. B 208, 385–397 (1980).
[CrossRef]

Prusinkiewicz, P.

P. Prusinkiewicz, “Modeling of spatial structure and development of plants: a review,” Scientia Horticulturae 74, 113–149 (1998).
[CrossRef]

P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants (Springer-Verlag, 1990).

Purves, D.

Z. Yang and D. Purves, “Image/source statistics of surfaces in natural scenes,” Network Comput. Neural Syst. 14, 371–390 (2003).
[CrossRef]

Quan, L.

Y. Wei and L. Quan, “Asymmetrical occlusion handling using graph cut for multi-view stereo,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Vol. 2 (IEEE, 2005), pp. 902–909.

Rezk-Salama, D. W. C.

K. Engel, M. Hadwiger, J. M. Kniss, and D. W. C. Rezk-Salama, Real-Time Volume Graphics (AK Peters, 2006).

Rosenholtz, R.

R. Rosenholtz, Y. Li, and L. Nakano, “Measuring visual clutter,” J. Vision 7, 1–22 (2007).
[CrossRef]

Roth, S.

S. Roth and M. J. Black, “On the spatial statistics of optical flow,” Int. J. Comput. Vis. 74, 33–50 (2007).
[CrossRef]

Rother, C.

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

Ruderman, D. L.

D. L. Ruderman, “Origins of scaling in natural images,” Vis. Res. 37, 3385–3398 (1997).
[CrossRef]

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

Russell, G.

P. R. van Gardingen, G. E. Jackson, S. Hernandez-Daumas, G. Russell, and L. Sharp, “Leaf area index estimates obtained for clumped canopies using hemispherical photography,” Agric. Forest Meteor. 94, 243–257 (1999).

Scharstein, D.

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

Seitz, S. M.

L. Zhang and S. M. Seitz, “Estimating optimal parameters for MRF stereo from a single image pair,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 331–342 (2007).

Serra, J. P.

J. P. Serra, Image Analysis and Mathematical Morphology (Academic, 1982).

Shakespeare, R.

G. W. Larson and R. Shakespeare, Rendering with Radiance: The Art and Science of Lighting Visualization (Morgan Kaufmann, 1998).

Sharp, L.

P. R. van Gardingen, G. E. Jackson, S. Hernandez-Daumas, G. Russell, and L. Sharp, “Leaf area index estimates obtained for clumped canopies using hemispherical photography,” Agric. Forest Meteor. 94, 243–257 (1999).

Shimojo, S.

M. A. Changizi and S. Shimojo, “X-ray vision and the evolution of forward-facing eyes,” J. Theor. Biol. 254, 756–767 (2008).
[CrossRef]

Sinoquet, H.

H. Sinoquet, G. Sonohat, J. Phattaralerphong, and C. Godin, “Foliage randomness and light interception in 3-d digitized trees: an analysis from multiscale discretization of the canopy,” Plant Cell Environ. 28, 1158–1170 (2005).
[CrossRef]

Sonohat, G.

H. Sinoquet, G. Sonohat, J. Phattaralerphong, and C. Godin, “Foliage randomness and light interception in 3-d digitized trees: an analysis from multiscale discretization of the canopy,” Plant Cell Environ. 28, 1158–1170 (2005).
[CrossRef]

Szeliski, R.

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

V. Vaish, M. Levoy, R. Szeliski, C. L. Zitnick, and S. B. Kang, “Reconstructing occluded surfaces using synthetic apertures: stereo, focus and robust measures,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2006), pp. 2331–2338.

S. Kang, R. Szeliski, and J. Chai, “Handling occlusions in dense multi-view stereo,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. I:103–110.

Tappen, M.

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

Tyler, C. W.

R. M. Balboa, C. W. Tyler, and N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vis. Res. 41, 955–964 (2001).
[CrossRef]

Vaish, V.

V. Vaish, M. Levoy, R. Szeliski, C. L. Zitnick, and S. B. Kang, “Reconstructing occluded surfaces using synthetic apertures: stereo, focus and robust measures,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2006), pp. 2331–2338.

van Ee, R.

R. van Ee and B. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001).
[CrossRef]

van Gardingen, P. R.

P. R. van Gardingen, G. E. Jackson, S. Hernandez-Daumas, G. Russell, and L. Sharp, “Leaf area index estimates obtained for clumped canopies using hemispherical photography,” Agric. Forest Meteor. 94, 243–257 (1999).

van Hateren, J. H.

J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).

Veksler, O.

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

Wei, Y.

Y. Wei and L. Quan, “Asymmetrical occlusion handling using graph cut for multi-view stereo,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Vol. 2 (IEEE, 2005), pp. 902–909.

Wilcox, L.

J. Harris and L. Wilcox, “The role of monocularly visible regions in the perception of three-dimensional scenes,” Vis. Res. 49, 2666–2685 (2009).
[CrossRef]

Wildes, R.

G. Egnal and R. Wildes, “Detecting binocular half-occlusions: Empirical comparisons of five approaches,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1127–1133 (2002).
[CrossRef]

Wolfe, J. M.

J. M. Wolfe, “Guided search 2.0: a revised model of visual search,” Psychon. Bull. Rev. 1, 202–238 (1994).

Yang, Z.

Z. Yang and D. Purves, “Image/source statistics of surfaces in natural scenes,” Network Comput. Neural Syst. 14, 371–390 (2003).
[CrossRef]

Zabih, R.

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

V. Kolmogorov and R. Zabih, “Computing visual correspondence with occlusions via graph cuts,” in Proceedings of 8th IEEE International Conference on Computer Vision (IEEE, 2001), pp. II:508–515.

Zacks, S.

S. Zacks, Stochastic Visibility in Random Fields, Lecture Notes in Statistics 95 (Springer-Verlag, 1994).

Zhang, L.

L. Zhang and S. M. Seitz, “Estimating optimal parameters for MRF stereo from a single image pair,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 331–342 (2007).

Zitnick, C. L.

V. Vaish, M. Levoy, R. Szeliski, C. L. Zitnick, and S. B. Kang, “Reconstructing occluded surfaces using synthetic apertures: stereo, focus and robust measures,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2006), pp. 2331–2338.

Agric. Forest Meteor.

P. R. van Gardingen, G. E. Jackson, S. Hernandez-Daumas, G. Russell, and L. Sharp, “Leaf area index estimates obtained for clumped canopies using hemispherical photography,” Agric. Forest Meteor. 94, 243–257 (1999).

Agric. Meteor.

T. Nilson, “A theoretical analysis of the frequency of gaps in plant stands,” Agric. Meteor. 8, 25–38 (1971).

Appl. Opt.

Comput. Graphics

B. Nadler, G. Fibich, S. Lev-Yehudi, and D. Cohen-Or, “A qualitative and quantitative visibility analysis in urban scenes,” Comput. Graphics 23, 655–666 (1999).

IEEE Trans. Pattern Anal. Mach. Intell.

G. Egnal and R. Wildes, “Detecting binocular half-occlusions: Empirical comparisons of five approaches,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1127–1133 (2002).
[CrossRef]

L. Zhang and S. M. Seitz, “Estimating optimal parameters for MRF stereo from a single image pair,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 331–342 (2007).

IEEE Trans. Pattern Anal. Machine Intell.

R. Szeliski, R. Zabih, D. Scharstein, O. Veksler, V. Kolmogorov, A. Agarwala, M. Tappen, and C. Rother, “A comparative study of energy minimization methods for Markov random fields with smoothness-based priors,” IEEE Trans. Pattern Anal. Machine Intell. 30, 1068–1080 (2008).
[CrossRef]

Int. J. Comput. Vis.

A. Mittal and L. Davis, “A general method for sensor planning in multi-sensor systems: extension to random occlusion,” Int. J. Comput. Vis. 76, 31–52 (2007).
[CrossRef]

P. Belhumeur, “A Bayesian approach to binocular stereopsis,” Int. J. Comput. Vis. 19, 237–260 (1996).
[CrossRef]

A. Lee, D. Mumford, and J. Huang, “Occlusion models for natural images: a statistical study of a scale-invariant dead leaves model,” Int. J. Comput. Vis. 41, 35–59 (2001).
[CrossRef]

S. Roth and M. J. Black, “On the spatial statistics of optical flow,” Int. J. Comput. Vis. 74, 33–50 (2007).
[CrossRef]

J. Comp. Physiol. A

J. H. van Hateren, “Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation,” J. Comp. Physiol. A 171, 157–170 (1992).

J. Opt. Soc. Am. A

J. Theor. Biol.

M. A. Changizi and S. Shimojo, “X-ray vision and the evolution of forward-facing eyes,” J. Theor. Biol. 254, 756–767 (2008).
[CrossRef]

J. Vision

R. Rosenholtz, Y. Li, and L. Nakano, “Measuring visual clutter,” J. Vision 7, 1–22 (2007).
[CrossRef]

Y. Liu, A. Bovik, and L. Cormack, “Disparity statistics in natural scenes,” J. Vision 8(2):17, 1–22 (2008).
[CrossRef]

Nature

R. van Ee and B. Anderson, “Motion direction, speed, and orientation in binocular matching,” Nature 410, 690–694 (2001).
[CrossRef]

Network Comput. Neural Syst.

Z. Yang and D. Purves, “Image/source statistics of surfaces in natural scenes,” Network Comput. Neural Syst. 14, 371–390 (2003).
[CrossRef]

D. Calow and M. Lappe, “Local statistics of retinal optic flow for self-motion through natural sceneries,” Network Comput. Neural Syst. 18, 343–374 (2007).
[CrossRef]

Phys. Rev. Lett.

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

Plant Cell Environ.

H. Sinoquet, G. Sonohat, J. Phattaralerphong, and C. Godin, “Foliage randomness and light interception in 3-d digitized trees: an analysis from multiscale discretization of the canopy,” Plant Cell Environ. 28, 1158–1170 (2005).
[CrossRef]

Proc. R. Soc. B

H. Longuet-Higgins and K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. B 208, 385–397 (1980).
[CrossRef]

Psychon. Bull. Rev.

J. M. Wolfe, “Guided search 2.0: a revised model of visual search,” Psychon. Bull. Rev. 1, 202–238 (1994).

Scientia Horticulturae

P. Prusinkiewicz, “Modeling of spatial structure and development of plants: a review,” Scientia Horticulturae 74, 113–149 (1998).
[CrossRef]

Vis. Cogn.

P. B. Hibbard, “A statistical model of binocular disparity,” Vis. Cogn. 15, 149–165 (2007).

Vis. Res.

D. L. Ruderman, “Origins of scaling in natural images,” Vis. Res. 37, 3385–3398 (1997).
[CrossRef]

R. M. Balboa, C. W. Tyler, and N. M. Grzywacz, “Occlusions contribute to scaling in natural images,” Vis. Res. 41, 955–964 (2001).
[CrossRef]

J. Harris and L. Wilcox, “The role of monocularly visible regions in the perception of three-dimensional scenes,” Vis. Res. 49, 2666–2685 (2009).
[CrossRef]

J. Forte, J. W. Peirce, and P. Lennie, “Binocular integration of partially occluded surfaces,” Vis. Res. 42, 1225–1235(2002).
[CrossRef]

D. C. Knill, “Surface orientation from texture: Ideal observers, generic observers and the information content of texture cues,” Vis. Res. 38, 1655–1682 (1998).
[CrossRef]

Other

G. W. Larson and R. Shakespeare, Rendering with Radiance: The Art and Science of Lighting Visualization (Morgan Kaufmann, 1998).

K. Engel, M. Hadwiger, J. M. Kniss, and D. W. C. Rezk-Salama, Real-Time Volume Graphics (AK Peters, 2006).

S. Kang, R. Szeliski, and J. Chai, “Handling occlusions in dense multi-view stereo,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. I:103–110.

V. Kolmogorov and R. Zabih, “Computing visual correspondence with occlusions via graph cuts,” in Proceedings of 8th IEEE International Conference on Computer Vision (IEEE, 2001), pp. II:508–515.

Y. Wei and L. Quan, “Asymmetrical occlusion handling using graph cut for multi-view stereo,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Vol. 2 (IEEE, 2005), pp. 902–909.

V. Vaish, M. Levoy, R. Szeliski, C. L. Zitnick, and S. B. Kang, “Reconstructing occluded surfaces using synthetic apertures: stereo, focus and robust measures,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2006), pp. 2331–2338.

M. Langer, “Surface visibility probabilities in 3d cluttered scenes,” in Proceedings of 10th European Conference on Computer Vision (Springer-Verlag, 2008), pp. I:401–412.

P. Hall, Introduction to the Theory of Coverage Processes(Wiley, 1988).

S. Zacks, Stochastic Visibility in Random Fields, Lecture Notes in Statistics 95 (Springer-Verlag, 1994).

“The blender greenhouse,” http://yorik.uncreated.net/greenhouse.html .

“Ngplant—open source plant modeling suite,” http://ngplant.sourceforge.net/ .

G. Kanizsa, Organization in Vision: Essays on Gestalt Perception (Praeger, 1979).

M. Hansard, “Binocular projection of a random scene,” in Proceedings of British Machine Vision Conference (Springer-Verlag, 2012), to be published.

J. Huang, A. Lee, and D. Mumford, “Statistics of range images,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2000), pp. 324–331.

G. Matheron, Random Sets and Integral Geometry (Wiley, 1975).

J. P. Serra, Image Analysis and Mathematical Morphology (Academic, 1982).

T. E. Avery and H. E. Burkhart, Forest Measurements, 5th ed. (McGraw-Hill, 2002).

P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants (Springer-Verlag, 1990).

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Figures (12)

Fig. 1.
Fig. 1.

A surface point (x,z) is visible if the cylinder centered at the line of sight to that point does not contain any disk centers.

Fig. 2.
Fig. 2.

(Top and middle) Examples of large and small square scenes. (Bottom) The black sphere shows the position of the eye. The dashed lines show the pyramid shaped view volume. The small thick black square on the front face of the rectangular volume shows the “window” on the z=z0 plane through which the viewer sees the scene. See top row of Figs. 3 and 4.

Fig. 3.
Fig. 3.

Top row shows rendered images for scenes with (left) large and (right) small occluders. Here the image intensity is inversely related to depth to better illustrate the depth ordering of elements. Middle row shows p(z). Bottom row shows p(v). The depth and inverse depth intervals have been partitioned into nbin=20 bins. Error bars show standard error of the mean normalized histograms from 100 scenes.

Fig. 4.
Fig. 4.

Same as Fig. 3 except now the scene has randomly oriented squares.

Fig. 5.
Fig. 5.

Gap size probabilities Pgap(rR,z) for the scenes with (a) large and (b) small randomly oriented squares. The four curves in each plot from top to bottom are for ratios rR=0, 12, 1, and 2.

Fig. 6.
Fig. 6.

Theory plots for visibility at neighboring pixels, as derived in Appendix A. Top row are logP(z,z) values. Bottom row are logP(v,v) values. Left and right columns correspond to the large and small disk scenes, respectively.

Fig. 7.
Fig. 7.

Probabilities estimated from joint histograms over 100 scenes. (Top row) logP(z,z). (Bottom row) logP(v,v). Left and right columns are for large and small square scenes, respectively. The data show means over 100 scenes. The log probabilities within each scene are considerably more variable than illustrated here.

Fig. 8.
Fig. 8.

A point at depth z is visible to both eyes if there is no surface center laying in the cylinders whose axes join the point to the two eyes. (Top) When surfaces are small relative to IOD, the two cylinders overlap only a little. (Bottom) When surfaces are large relative to IOD, the cylinders overlap almost entirely.

Fig. 9.
Fig. 9.

Conditional probabilities that a surface at depth z is visible to left eye, given it is visible to right eye, for (left) large and (right) small square scenes. The vertical dotted lines mark the near and far boundaries of the clutter. The z values beyond the right dotted line are in the empty space behind the clutter.

Fig. 10.
Fig. 10.

(a) Range map where brighter means closer. (b)–(d) are p(z), p(v), and P(left|right;z). Each plot shows three curves corresponding to three different image regions. The image is partitioned into six vertical rectangles, each one-sixth of the image width. The central two rectangles define center, the two regions next to the center define periphery-center, and the outer two regions define periphery.

Fig. 11.
Fig. 11.

Joint probabilities for a tree scene. (Top) Plots of logP(z,z). (Bottom) Plots of logP(v,v). (Left) Central third of the image. (Right) Left and right peripheral regions of the image. White regions correspond to points with 0 probability, i.e., infinite log probability.

Fig. 12.
Fig. 12.

(Left) For any point that lies on a disk at depth z and is visible at pixel x, a neighboring pixel x may or may not see that same disk. Three possible cases are shown. See text. (Right) Given that the surface at pixel x has depth z, the probability that the neighboring scene point (x,z) is occluded is determined by the volume of Cx\Cx according to the Poisson model.

Equations (32)

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

V=πR2(zz0).
P(no disk centers inV)=exp{ηV}.
exp{ηV}1ηV.
P(visible disk[z,Δz])=ηπR2exp{ηπR2(zz0)}Δz
=λexp{λ(zz0)}Δz.
p(z)=λexp{λ(zz0)}.
λ=ηπR2.
v=1z,
dz=1v2(dv),
p(v)=ηπR2v2exp{ηπR2(1v1v0)}.
Vgap(r,R,z)=πz3r((R+r)3(R+z0zr)3).
Pgap(r,R,z)=exp{ηVgap(z,r,R)}.
Pgap(r,R,z)=exp{λ3(Rr)((1+rR)3(1+z0zrR)3)z}.
limr0Pgap(r,R,z)=exp{λ(zz0)}.
P(z,z)={pdiff(z,z)(Δz)2,ifzzpsame(z,z)(Δz),ifz=z
P(v,v)={pdiff(v,v)(Δv)2,ifvvpsame(v,v)(Δv),ifv=v
Ex,x(v,v)=min(|vv|k,τ),
V(CrCl)=V(Cr)+V(Cl\Cr).
Vlarge(Cl\Cr)=RTx(zz0)2z.
Vsmall(Cl\Cr)=V(Cl)=πR2(zz0).
P(left|right;z)=P(left and right;z)P(right;z)=eηV(ClCr)eηV(Cr)=eηV(Cl\Cr).
power(z)P(left and right;z)P(right;z).
P(left or right;z)=P(left;z)+P(right;z)P(left and right;z).
P(left;z)=P(right;z),
power(z)=2P(left|right;z),
P(right;z)=P(left or right;z)+P(left and right;z)2.
pdiff(z,z)=2η2πR3(zz0)|xx|exp{η(πR2(zz0)+R(z2z02)|xx|)}.
pdiff(v,v)=2η2πR3|xx|v2v3exp{η(πR2(1v1v0)+R(1v21v02)|xx|)}.
πR22R(xx)zπR2,
exp{η(πR2(zz0)+R(z2z02)|xx|)}.
psame(z,z)=η(πR22R|xx|z)exp{η(πR2(zz0)+R(z2z02)|xx|)}.
psame(v,v)=ηv2(πR22Rv|xx|)exp{η(πR2(1v1v0)+R(1v21v02)|xx|)}.

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