David Mumford

Archive for Reprints, Notes, Talks, and Blog

Professor Emeritus
Brown and Harvard Universities

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Work on the Statistics of Images

Benoit Mandelbrot taught me two fundamental facts about the statistics of the world. The first is that univariate statistics can be expected to be normal only if they are averages of many independent variables. This is the heuristic converse to the central limit theorem. The second is that when space or time is involved, most phenomena show self-similarity (with suitable powers) under scaling. In the 90's through the lectures I was giving my students on vision, I came to realize how strongly both of his principles were realized in images: the distribution of values of the convolution of (i) random images of the world and (ii) any zero mean filter with small support has heavy tails and its histogram doesn't change if you average and down-sample the image. The background to this page, by the way, is Gaussian noise whose power spectrum is intermediate between white noise and scale-invariant noise (1/f2)

  • The Statistical Description of Visual Signals, ICIAM 95 ed. K.Kirshgassner, O.Mahrenholtz & R.Mennicken, Akademie Verlag, 1996. Scanned manuscript.
  • Learning generic prior models for visual computation (with S.C.Zhu), Proc. IEEE Conf. Comp. Vision and Pattern Rec. 1997, 463-469, Comp Sci Press.
    Digital reprint and DASH reprint .
  • Minimax Entropy Principle and its Application to Texture Modeling (with S.C.Zhu and Y.N.Wu), Neural Computation, 9, 1997, 1627-60. Scanned reprint.
  • Prior Learning and Gibbs Reaction-Diffusion (with Song Chun Zhu), IEEE Trans. Patt. Anal. and Mach. Int., 19, 1997, 1236-50.
    Digital reprint and DASH reprint.
  • FRAME: Filters, Random Field and Maximum Entropy, (with S.C.Zhu and Y.Wu), International Journal Computer Vision, 27, 1998.
    Digital reprint and DASH reprint.
  • GRADE: Gibbs reaction and diffusion equations (with Song Chun Zhu), Proceedings of the Sixth International Conference on Computer Vision 1998, IEEE Computer Society, pp. 847-854.
    Digital reprint and DASH reprint.
  • The Statistics of Natural Images and Models (with J.Huang), Proc. IEEE Conf. Comp. Vision and Pattern Rec. 1999, pp.541-547, Comp Sci Press.
    Digital reprint and DASH reprint.
  • Statistics of range images (with Jinggang Huang and Ann Lee), Proc. IEEE Conf. Comp. Vision and Pattern Rec. 2000, pp. 324-331, Comp Sci Press.
    Digital reprint and DASH reprint.
  • Stochastic Models for Generic Images (with Basilis Gidas), Quarterly Appl. Math., 59, 2001, pp.85-111. Scanned reprint.
  • Occlusion models for natural images: A statistical study of a scale-invariant dead-leaves model, (with Ann Lee and Jinggang Huang), Int. J. Computer Vision, 41, 2001, pp. 35-59.
    Digital reprint and DASH reprint.
  • The Nonlinear Statistics of High-contrast Patches in Natural Images (with Ann Lee and Kim Pedersen), Int. J. Comp. Vision, 54, 2003, pp.83-103.
    Digital reprint and DASH reprint.
  • Empirical Statistics and Stochastic Models for Visual Signals, in Brain and Systems: New Directions in Statistical Signal Processing, ed. by S.Haykin, J.Principe, T.Sejnowski, and J.McWhirter, MIT Press, 2006.
    Digital manuscript and DASH reprint.