Wavelet shrinkage using adaptive structured sparsity constraints

Structured sparsity approaches have recently received much attention in the statistics, machine learning, and signal processing communities. A common strategy is to exploit or assume prior information about structural dependencies inherent in the data; the solution is encouraged to behave as such by the inclusion of an appropriate regularization term which enforces structured sparsity constraints over sub-groups of data. An important variant of this idea considers the tree-like dependency structures often apparent in wavelet decompositions. However, both the constituent groups and their associated weights in the regularization term are typically defined a priori. We here introduce an adaptive wavelet denoising framework whereby a sparsity-inducing regularizer is modified based on information extracted from the signal itself. In particular, we use the same wavelet decomposition to detect the location of salient features in the signal, such as jumps or sharp bumps. Given these locations, the weights in the regularizer associated to the groups of coefficients that cover these time locations are modified in order to favour retention of those coefficients. Denoising experiments show that, not only does the adaptive method preserve the salient features better than the non-adaptive constraints, but it also delivers significantly better shrinkage over the signal as a whole.