Properties of Brownian Image Models in Scale-Space
In this paper it is argued that the Brownian image model is the least committed, scale invariant, statistical image model which describes the second order statistics of natural images. Various properties of three different types of Gaussian image models (white noise, Brownian and fractional Brownian images) will be discussed in relation to linear scale-space theory, and it will be shown empirically that the second order statistics of natural images mapped into jet space may, within some scale interval, be modeled by the Brownian image model. This is consistent with the 1/f 2 power spectrum law that apparently governs natural images. Furthermore, the distribution of Brownian images mapped into jet space is Gaussian and an analytical expression can be derived for the covariance matrix of Brownian images in jet space. This matrix is also a good approximation of the covariance matrix of natural images in jet space. The consequence of these results is that the Brownian image model can be used as a least committed model of the covariance structure of the distribution of natural images.
KeywordsWhite Noise Gaussian White Noise Natural Image Stochastic Function Spatial Covariance Function
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- 3.D. J. Field. Relations between the statistics of natural images and the response properties of cortical cells. J. Optic. Soc. of Am., 4(12):2379–2394, 1987.Google Scholar
- 4.L. Florack. Image Structure. Kluwer Academic Publishers, 1997.Google Scholar
- 6.G. Friedlander and M. Joshi. Introduction to The Theory of Distributions. Cambridge University Press, 2nd edition, 1998.Google Scholar
- 8.J. Huang and D. Mumford. Statistics of natural images and models. In Proc. of IEEE Conf. on Computer Vision and Pattern Recognition, 1999.Google Scholar
- 12.M. A. Lifshits. Gaussian Random Functions. Kluwer Academic Publishers, 1995.Google Scholar
- 14.P. Majer. Self-similarity of noise in scale-space. In Proc. of Scale-Space’99, LNCS 1682, pages 423–428. Springer Verlag, 1999.Google Scholar
- 17.D. Mumford. The statistical description of visual signals. In ICIAM’95, 1996.Google Scholar
- 19.M. Nielsen and M. Lillholm. What do features tell about images? In Proc. of Scale-Space’01, LNCS 2106, pages 39–50. Springer, 2001.Google Scholar
- 20.B. Øksendal. Stochastic Differential Equations. Springer, 5 edition, 2000.Google Scholar
- 21.K. S. Pedersen and A. B. Lee. Toward a full probability model of edges in natural images. In Proc. of 7th ECCV, LNCS 2350, pages 328–342. Springer Verlag, 2002.Google Scholar
- 25.A. Srivastava, X. Liu, and U. Grenander. Universal analytical forms for modeling image probabilities. IEEE Trans. on PAMI, 24(9):1200–1214, September 2002.Google Scholar