Skip to main content
SpringerLink
Log in

On Advances in Statistical Modeling of Natural Images

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Statistical analysis of images reveals two interesting properties: (i) invariance of image statistics to scaling of images, and (ii) non-Gaussian behavior of image statistics, i.e. high kurtosis, heavy tails, and sharp central cusps. In this paper we review some recent results in statistical modeling of natural images that attempt to explain these patterns. Two categories of results are considered: (i) studies of probability models of images or image decompositions (such as Fourier or wavelet decompositions), and (ii) discoveries of underlying image manifolds while restricting to natural images. Applications of these models in areas such as texture analysis, image classification, compression, and denoising are also considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Alvarez, Y. Gousseau, and J.-M. Morel, “The size of objects in natural and artificial images,” in Advances in Imaging and Electron Physics, P.W.H. et al. (Eds.), Academic Press: New York, 1999.

  2. D. Andrews and C. Mallows, “Scale mixtures of normal distributions,” Journal of the Royal Statistical Society, Vol. 36, pp. 99–102, 1974.

    Google Scholar 

  3. O. Barndorff-Nielsen, “Exponentially decreasing distribution for the logarithm of a particle size,” Journal of Royal Statistical Society A, Vol. 353, pp. 401–419, 1977.

    Google Scholar 

  4. B.E. Bayer and P.G. Powell, “A method for the digital enhancement of unsharp, grainy photographic images,” Advances in Computer Vision and Image Processing, Vol.2, pp.31–88, 1986.

    Google Scholar 

  5. P.N. Belhumeur, J.P. Hepanha, and D.J. Kriegman, “Eigenfaces vs. fisherfaces: Recognition using class specific linear projection,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 19, No. 7, pp. 711–720, 1997.

    Google Scholar 

  6. A.J. Bell and T.J. Sejnowski, “The independent components of natural scenes are edge filters,” Vision Research, Vol. 37, No. 23, pp. 3327–3338, 1997.

    Google Scholar 

  7. J.R. Bergen and M.S. Landy, Computational Modeling of Visual Texture Segregation, MIT Press: Cambridge, MA, 1991, pp. 253–271.

    Google Scholar 

  8. J. Besag, “Spatial interaction and statistical analysis of lattice systems (with discussion),” J. Royal Statist. Soc. B, Vol. 36, pp. 192–326, 1974.

    Google Scholar 

  9. J. Besag, “On the statistical analysis of dirty pictures,” J. Royal Statistical Society B, Vol. 48, No. 3, pp. 259–302, 1986.

    Google Scholar 

  10. P. Billingsley, Probability and Measure, Wiley Series in Probability and Mathematical Statistics, 1995.

  11. D. Bitouk, U. Grenander, M.I. Miller, and P. Tyagi, “Fisher information measures for ATR in clutter,” in Proceedings of SPIE, Automatic Target Recognition XI, 2001, Vol. 4379, pp. 560–572.

    Google Scholar 

  12. T. Bollerslev, K. Engle, and D. Nelson,ARCH Models, Vol.IV, North-Holland: Amsterdam, 1994, pp. 2959–3038.

    Google Scholar 

  13. H. Brehm and W. Stammler, “Description and generation ofspherically invariant speech-model signals,” Signal Processing, Vol. 12, pp. 119–141, 1987.

    Google Scholar 

  14. R.W. Buccigrossi and E.P. Simoncelli, “Image compression via joint statistical characterization in the wavelet domain,” IEEE Transactions on Image Processing, Vol. 8, No. 12, pp. 1688–1701, 1999.

    Google Scholar 

  15. P.J. Burt and E.H. Adelson, “The Laplacian pyramid as a compact image code,” IEEE Transactions on Communications, Vol. 31, No. 4, pp. 532–540, 1983.

    Google Scholar 

  16. G.J. Burton, N.D. Haig, and I.R. Moorhead, “Aself-similar stack model for human and mahcine vision,” Biological Cybernetics, Vol. 53, No. 6, pp. 397–403, 1986.

    Google Scholar 

  17. G.J. Burton and I.R. Moorhead, “Color and spatial structures in natural scenes,” Applied Optics, Vol. 26, No. 1, pp. 157–170, 1987.

    Google Scholar 

  18. J.-F. Cardoso, “Source separation using higher order moments,” in Proceedings of ICASSP, 1989, pp. 2109–2112.

  19. Z. Chi, “Probability models for complex systems,” Ph.D. Thesis, Division of Applied Mathematics, Brown University, 1998.

  20. C. Chrysafis and A. Ortega, “Efficient context-based entropy coding for lossy wavelet image coding,” in Data Compression Conf., 1997, Snowbird, Utah.

    Google Scholar 

  21. C. Chubb, J. Econopouly, and M.S. Landy, “Histogram contrast analysis and the visual segregation of IID textures,” J. Opt. Soc. Am. A, Vol. 11, pp. 2350–2374, 1994.

    Google Scholar 

  22. S. Cohen and L. Guibas, “The earthmover's distance under transformation sets,” in Proceedings of Seventh IEEE International Conference on Computer Vision, Vol. 2, pp. 1076–1083, 1999.

    Google Scholar 

  23. P. Comon, “Independent component analysis, a new concept?” Signal Processing, Special Issue on Higher-Order Statistics, Vol. 36, No. 3, 1994.

  24. J. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters,” Journal of the Optical Society of America A, Vol. 2, No. 7, pp. 23–26, 1985.

    Google Scholar 

  25. N.G. Deriugin, “The power spectrum and the correlation function of the television signal,” Telecommunications, Vol. 1, pp. 1–12, 1956.

    Google Scholar 

  26. D. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Info. Theory, Vol. 43, pp. 613–627, 1995.

    Google Scholar 

  27. D.L. Donoho and A.G. Flesia, “Can recent innovations in harmonic analysis ‘Explain’ key findings in natural image statistics,” Network: Computation in Neural Systems, Vol. 12, No. 3, pp. 371–393, 2001.

    Google Scholar 

  28. R.O. Dror, T.K. Leung, E.H. Adelson, and A.S. Willsky, “Statistics of real-world illumination,” in Proceedings of 2001 IEEE Conference on CVPR, 2001, Vol. 2. pp. 164–171.

    Google Scholar 

  29. O.D. Faugeras and W.K. Pratt, “Decorrelation methods of texture feature extraction,” IEEE Pat. Anal. Mach. Intell. Vol. 2, No. 4, pp. 323–332, 1980.

    Google Scholar 

  30. D.J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Optical Society of America, Vol. 4, No. 12, pp. 2379–2394, 1987.

    Google Scholar 

  31. D. Gabor, “Theory of Communications,” Journal of IEE (London), Vol. 93, pp. 429–457, 1946.

    Google Scholar 

  32. D. Geman and A. Koloydenko, “Invariant statistics and coding of natural microimages,” in Proc. of the IEEE Workshop on Statistical and Computational Theories of Vision, 1999.

  33. S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 6, No. 6, pp. 721–741, 1984.

    Google Scholar 

  34. U. Grenander, General Pattern Theory, Oxford University Press, 1993.

  35. U. Grenander, M.I. Miller, and A. Srivastava, “Hilbert-Schmidt lower bounds for estimators on matrix lie groups forATR,” IEEE Transactions on PAMI, Vol. 20, No. 8, pp. 790–802, 1998.

    Google Scholar 

  36. U. Grenander and A. Srivastava, “Probability models for clutter in natural images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23, No. 4, pp. 424–429, 2001.

    Google Scholar 

  37. U. Grenander, A. Srivastava, and M.I. Miller, “Asymptotic performance analysis of Bayesian object recognition,” IEEE Transactions of Information Theory, Vol. 46, No. 4, pp. 1658–1666, 2000.

    Google Scholar 

  38. A.B. Hamza, Y. He, and H. Krim, “An information divergence measure for ISAR image registration,” in Proc. of IEEE Workshop on Statistical Signal Processing, 2001.

  39. P.J.B. Hancock, R.J. Baddeley, and L.S. Smith, “The principal components of natural images,” Network, Vol. 3, pp. 61–70, 1992.

    Google Scholar 

  40. Y. He, A.B. Hamza, and H. Krim, “A generalized divergence measure for robust image registration,” IEEE Transactions on Signal Processing, 2002, to appear.

  41. D.J. Heeger and J.R. Bergen, “Pyramid-based texture analysis/ synthesis,” in Proceedings of SIGGRAPHS, 1995, pp. 229–238.

  42. A.O. Hero, B. Ma, O. Michel, and J. Gorman, “Alpha-divergence for classification, indexing and retrieval,” Communication and Signal Processing Laboratory, Technical Report CSPL-328, U. Mich, 2001.

  43. J. Huang, “Statistics of natural images and models,” Ph.D. Thesis, Division of Appled Mathematics, Brown University, RI, 2000.

  44. A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley and Sons; New York, 2001.

    Google Scholar 

  45. D. Jeulin, I. Terol-Villalobos, and A. Dubus, “Morphological analysis of UO2 powder using a dead leaves model,” Microscopy, Microanalysis, Microsctructure, Vol. 6, pp. 371–384, 1995.

    Google Scholar 

  46. W.S. Kendall and E. Thonnes, “Perfect simulation in stochastic geometry,” Preprint 323, Department of Statistics, University of Warwick, UK, 1998.

    Google Scholar 

  47. D. Kersten, “Predictability and redundancy of natural images,” Journal of Optical Society of America, Vol. 4, No. 12, pp. 2395–2400, 1987.

    Google Scholar 

  48. M. Kirby and L. Sirovich, “Application of the Karhunen-Loeve procedure for the characterization of human faces,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 12, No. 1, pp. 103–108, 1990.

    Google Scholar 

  49. E.R. Kretzmer, “Statistics of television signals,” Bell Syst. Tech. Journal, Vol. 31, pp. 751–763, 1952.

    Google Scholar 

  50. A.B. Lee and D. Mumford, “Occlusion models for natural images: A statistical study of scale-invariant dead leaves model,” International Journal of ComputerVision,Vol. 41, No. 1/2, 2001.

  51. A.B. Lee, K.S. Pedersen, and D. Mumford, “The nonlinear statistics of high-contrast patches in natural images,” Int. J. Computer Vision, 2002, in press.

  52. D.D. Lee and H.S. Seung, “Learning the parts of objects by nonnegative matrix factorization,” Nature, Vol. 401, pp. 788–791, 1999.

    Google Scholar 

  53. D. Leporini, J.-C. Pesquet, and H. Krim, “Best basis representations with prior statistical models,” in Lecture Notes in Statistics: Bayesian Inferences in Wavelet Based Models, P. Muller and B. Vidakovic (Eds.), Springer Verlag: Berlin, 1999.

    Google Scholar 

  54. S.M. LoPresto, K. Ramchandran, and M.T. Orchard, “Wavelet image coding based on a new generalized gaussian mixture model,” in Data Compression Conf., 1997, Snowbird, Utah.

    Google Scholar 

  55. S.G. Mallat, “A Theory for multiresolution signal decomposition: The wavelet representation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 11, pp. 674–693, 1989.

    Google Scholar 

  56. D. Marr, Vision: A computational Investigation into the Human Representation and Processing of Visual Information, W.H. Freeman and Company: New York, 1982.

    Google Scholar 

  57. G. Matheron, Random Sets and Integral Geometry, John Wiley and Sons: New York, 1975.

    Google Scholar 

  58. K.D. Miller, “A Model for the development of simple-cell receptive fields and the arrangement of orientation columns through activity dependent competetion between on-and off-center inputs,” Journal of NeuroScience, Vol. 14, pp. 409–441, 1994.

    Google Scholar 

  59. M.I. Miller and L. Younes, “Group actions, homeomorphisms, and matching: A general framework,” International Journal of Computer Vision, Vol. 41, No. 1/2, pp. 61–84, 2002.

    Google Scholar 

  60. P. Moulin and J. Liu, “Analysis of multiresolution image denoising schemes using a generalized Gaussian and complexity priors,” IEEE Trans. Info. Theory, Vol. 45, pp. 909–919, 1999.

    Google Scholar 

  61. D. Mumford and B. Gidas, “Stochastic models for generic images,” Quaterly of Applied Mathematics, Vol. 59, No. 1, pp. 85–111, 2001.

    Google Scholar 

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

    Google Scholar 

  63. B.A. Olshausen and D.J. Field, “Sparse coding with an overcomplete basis set: A strategy employed by V1?” Vision Research, Vol. 37, No. 23, pp. 3311–3325, 1997.

    Google Scholar 

  64. K.S. Pedersen and A.B. Lee, “Toward a full probability model of edges in natural images,” in Proc. of ECCV'02, 2002.

  65. J. Portilla and E.P. Simoncelli, “A parametric texture model based on join statistics of complex wavelet coefficients,” International Journal of Computer Vision, Vol. 40, No. 1, pp. 49–70, 2000.

    Google Scholar 

  66. J. Portilla, V. Strela, M. Wainwright, and E. Simoncelli, “Adaptive Wiener denoising using a Gaussian scale mixture model in thewavelet domain,” in Proc 8th IEEE Int'l Conf on Image Proc. Thessaloniki, Greece, IEEE Computer Society, 2001, pp. 37–40.

    Google Scholar 

  67. R. Rinaldo and G. Calvagno, “Image coding by block prediction of multiresolution subimages,” IEEE Trans Im Proc., 1995.

  68. S.T. Roweis and L.K. Saul, “Nonlinear dimensionality reduction by locally linear embedding,” Science, Vol. 290, pp. 2323–2326, 2000.

    Google Scholar 

  69. D.L. Ruderman, “The statistics of natural images,” Network, Vol. 5, pp. 517–548, 1994.

    Google Scholar 

  70. D.L. Ruderman, “Origins of scaling in natural images,” Vision Research, Vol. 37, No. 23, pp. 3385–3398, 1997.

    Google Scholar 

  71. D.L. Ruderman and W. Bialek, “Scaling of natural images: Scaling in the woods,” Physical Review Letters, Vol. 73, No. 6, pp. 814–817, 1994.

    Google Scholar 

  72. J.P. Serra, Image Analysis and Mathematical Morphology, Academic Press: London, 1982.

    Google Scholar 

  73. J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Sig. Proc., Vol. 41, No. 12, pp. 3445–3462, 1993.

    Google Scholar 

  74. E.P. Simoncelli, “Statistical models for images: Compression, restoration and synthesis,” in Proc. 31st Asilomar Conf on Signals, Systems and Computers, Pacific Grove, CA. IEEE Computer Society, 1997, pp. 673–678.

    Google Scholar 

  75. E.P. Simoncelli, “Bayesian denoising of visual images in the wavelet domain,” in Bayesian Inference in Wavelet Based Models, Lecture Notes in Statistics, Vol. 41, Springer Verlag: Berlin, 1999.

    Google Scholar 

  76. E.P. Simoncelli and E.H. Adelson, “Noise removal via Bayesian wavelet coring,” in Third Int'l. Conf on Image Proc., Lausanne, IEEE Sig Proc Society, 1996, Vol. I, pp. 379–382.

    Google Scholar 

  77. E.P. Simoncelli and R.T. Buccigrossi, “Embeddedwavelet image compression based on a joint probability model,” in Proceedings of ICIP (1), 1997, pp. 640–643.

  78. E.P. Simoncelli, W.T. Freeman, E.H. Adelson, and D.J. Heeger, “Shiftable multi-scale transforms,” IEEE Trans Information Theory, Vol. 38, No. 2, pp. 587–607, 1992.

    Google Scholar 

  79. E. Simoncelli and B. Olshausen, “Natural image statistics and neural representation,” Annual Review of Neuroscience, Vol. 24, pp. 1193–1216, 2001.

    Google Scholar 

  80. A. Srivastava, X. Liu, and U. Grenander, “Universal analytical forms for modeling image probability,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 28, No. 9, 2002.

  81. J.B. Tenenbaum, V. Silva, and J.C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” Science, Vol. 290, pp. 2319–2323, 2000.

    Google Scholar 

  82. M.G.A. Thomson, “Beats, kurtosis, and visual coding,” Network: Computation in Neural Systems, Vol. 12, No. 3, pp. 271–287, 2001.

    Google Scholar 

  83. A. Turiel, G. Mato, and N. Parga, “The self-similarity properties of natural images resemble those of turbulent flows,” Physical Review Letters, Vol. 80, No. 5, pp. 1098–1101, 1998.

    Google Scholar 

  84. A. Turiel and N. Parga, “The multi-fractal structure of contrast changes in natural images: From sharp edges to textures,” Neural Computation, Vol. 12, pp. 763–793, 2000.

    Google Scholar 

  85. A. Turiel, N. Parga, D.L. Ruderman, and T.W. Cronin, “Multiscaling and information content of natural color images,” Physical Review E, Vol. 62, No. 1, pp. 1138–1148, 2000.

    Google Scholar 

  86. J.A. van Hateren, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. Royal Statistical Society of London B, Vol. 265, pp. 359–366, 1998.

    Google Scholar 

  87. M. Vetterli, “Multidimensional subband coding: Some theory and algorithms,” Signal Processing, Vol. 6, No. 2, pp. 97–112, 1984.

    Google Scholar 

  88. M.J. Wainwright and E.P. Simoncelli, “Scale mixtures of Gaussians and the statistics of natural images,” Advances in Neural Information Processing Systems, S.A. Solla, T.K. Leen, and K.-R. Muller (Eds.), 2000, pp. 855–861.

  89. M.J. Wainwright, E.P. Simoncelli, and A.S. Willsky, “Random cascades on wavelet trees and their use in modeling and analyzing natural imagery,” Applied and Computational Harmonic Analysis, Vol. 11, No. 1, pp. 89–123, 2001.

    Google Scholar 

  90. A.B. Watson, “The cortex transform: Rapid computation of simulated neural images,” Comp. Vis. Graphics Image Proc., Vol. 39, pp. 311–327, 1987.

    Google Scholar 

  91. B. Wegmann and C. Zetzsche, “Statistical dependence between orientation filter outputs used in a human vision based image code,” in Proceedings of Visual Communcation and Image Processing, Society of Photo-Optical Intstrumentation Engineers, Vol. 1360, pp. 909–922, 1990.

    Google Scholar 

  92. Y. Weiss, “Deriving intrinsic images from image sequences,” in Proceedings Eight IEEE International Conference on Computer Vision, 2001, Vol. 2, pp. 68–75.

    Google Scholar 

  93. G. Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods, Springer: Berlin, 1995.

    Google Scholar 

  94. S.N. Yendrikhovskij, “Computing color categoies from statistics of natural images,” Journal of Imaging Science and Technology, Vol. 45, No. 5, pp. 409–417, 2001.

    Google Scholar 

  95. C. Zetzsche, “Polyspectra of natural images,” in Presented at Natural Scene Statistics Meeting, 1997.

  96. C. Zetzsche and F. Rohrbein, “Nonlinear and extra-classical receptive field properties and the statistics of natural images,” Network: Computation in Neural Systems, Vol. 12, No. 3, pp. 331–350, 2001.

    Google Scholar 

  97. S.C. Zhu, C.E. Guo, Y.N. Wu, and Y.Z. Wang, “What are textons?,” in Proc. of Euorpean Conf. on Computer Vision, 2002.

  98. S.C. Zhu and D. Mumford, “Prior learning and Gibbs reactiondiffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 19, No. 11, pp. 1236–1250, 1997.

    Google Scholar 

  99. S.C. Zhu, Y.N. Wu, and D. Mumford, “Minimax entropy principles and its application to texture modeling,” Neural Computation, Vol.9, No. 8, pp. 1627–1660, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Florida State University, Tallahassee, FL, 32306, USA

    A. Srivastava

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    A.B. Lee

  3. Courant Institute for Mathematical Sciences, New York University, New York, NY, 10003, USA

    E.P. Simoncelli

Authors
  1. A. Srivastava
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A.B. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E.P. Simoncelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S.-C. Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Srivastava, A., Lee, A., Simoncelli, E. et al. On Advances in Statistical Modeling of Natural Images. Journal of Mathematical Imaging and Vision 18, 17–33 (2003). https://doi.org/10.1023/A:1021889010444

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021889010444

Search

Navigation