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Data adaptive multi-scale representations for image analysis

Data adaptive tight frame methods have been proven a powerful sparse approximation tool in a variety of settings. We introduce a model of a data adaptive representation that also provides a multi-scale structure. Our idea is to design a multi-scale frame representation for a given data set, with scaling properties similar to the ones of a wavelet basis, but without the necessary self-similar structure. The adaptivity provides better sparsity properties, using Besov-like norm structure both induces sparsity and helps in identifying important features. We focus on investigating the efficiency of a weighted l1 constraint in the context of sparse recovery from noisy data and compare it to the weighted l0 model alongside. Numerical experiments confirm that the recovered frame vectors assigned lower weights correspond to image elements of larger scale and lower local variation, thus indicating that weighted sparsity in natural images leads to a natural scale separation.