Overcomplete wavelet representations have become increasingly popular for their ability to provide highly sparse and robust descriptions of natural signals. We describe a method for incorporating an overcomplete wavelet representation as part of a statistical model of images which includes a sparse prior distribution over the wavelet coefficients. The wavelet basis functions are parameterized by a small set of 2-D functions. These functions are adapted to maximize the average log-likelihood of the model for a large database of natural images. When adapted to natural images, these functions become selective to different spatial orientations, and they achieve a superior degree of sparsity on natural images as compared with traditional wavelet bases. The learned basis is similar to the Steerable Pyramid basis, and yields slightly higher SNR for the same number of active coefficients. Inference with the learned model is demonstrated for applications such as denoising, with results that compare favorably with other methods.
We show how a wavelet basis may be adapted to best represent natural images in terms of sparse coefficients. The wavelet basis, which may be either complete or overcomplete, is specified by a small number of spatial functions which are repeated across space and combined in a recursive fashion so as to be self-similar across scale. These functions are adapted to minimize the estimated code length under a model that assumes images are composed as a linear superposition of sparse, independent components. When adapted to natural images, the wavelet bases become selective to different spatial orientations, and they achieve a superior degree of sparsity on natural images as compared with traditional wavelet bases.
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