Lewicki MS, Olshausen BA (1999). Probabilistic framework for the adaptation and comparison of image codes.  Journal of the Optical Society of America, 16(7): 1587-1601.

We apply a Bayesian method for inferring an optimal basis to the problem of finding efficient image codes for natural scenes. The bases learned by the algorithm are oriented and localized in both space and spatial-frequency, bearing a resemblance to two-dimensional Gabor functions, and increasing the number of basis functions results in a greater sampling density in position, orientation, and scale. These properties also resemble the spatial receptive fields of neurons in the primary visual cortex of mammals, suggesting that the receptive-field structure of these neurons can be accounted for by a general efficient coding principle. The probabilistic framework provides a method for comparing the coding efficiency of different bases objectively by calculating their probability given the observed data or by measuring the entropy of the basis function coefficients. The learned bases are shown 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 results obtained by applying the learned bases to these problems are improved over those obtained with traditional techniques.