Information-theoretic Interpretation of Besov Spaces

Besov spaces classify signals and images through the Besov norm, which is based on a deterministic smoothness measurement. Recently, we revealed the relationship between the Besov norm and the likelihood of an independent generalized Gaussian wavelet probabilistic model. In this paper, we extend this result by providing an information-theoretic interpretation of the Besov norm as the Shannon codelength for signal compression under this probabilistic mode. This perspective unites several seemingly disparate signal/image processing methods, including denoising by Besov norm regularization, complexity regularized denoising, minimum description length (MDL) processing, and maximum smoothness interpolation. By extending the wavelet probabilistic model (to a locally adapted Gaussian model), we broaden the notion of smoothness space to more closely characterize real-world data. The locally Gaussian model leads directly to a powerful wavelet-domain Wiener ltering algorithm for denoising.