Probabilistic framework for the adaptation and comparison of image codes

We apply a Bayesian method for inferring an optimal basis to the problem of finding efficient image codes for natural scenes. The basis functions lear ned by the algorithm are oriented and localized in both space and frequency , bearing a resemblance to two-dimensional Gabor functions, and increasing the number of basis functions results in a greater sampling density in posi tion, orientation, and scale. These properties also resemble the spatial re ceptive fields of neurons :in the primary visual cortex of mammals, suggest ing that the receptive-field structure of these neurons can be accounted fo r by a general efficient coding principle. The probabilistic framework prov ides a method for comparing the coding efficiency of different bases object ively by calculating their probability given the observed data or by measur ing the entropy of the basis function coefficients. The learned bases are s hown to have better coding efficiency than traditional Fourier and wavelet bases. This framework also provides a Bayesian solution to the problems of image denoising and filling in of missing pixels. We demonstrate that the r esults obtained by applying the learned bases to these problems are improve d over those obtained with traditional techniques. (C) 1999 Optical Society of America [S0740-3232(99)03107-5].