Arbitrary Shrinkage Rules for Approximation Schemes with Sparsity Constraints

Finding a sparse representation of a possibly noisy signal can be modeled as a variational minimization with `q-sparsity constraints for q less than one. Especially for real-time and on-line applications, one requires fast computations of these minimizers. However, there are no sufficiently fast algorithms, and to circumvent this limitation, we consider minimization up to a constant factor. We verify that q-dependent modifications of shrinkage rules provide closed formulas for such minimizers, and we introduce a new shrinkage rule which is adapted to q. To support the concept of shrinkage rules and minimizers up to a constant factor, we finally apply different shrinkage rules to wavelet-based variational image denoising. We verify in our numerical experiments that the H-curve criterion which is a parameter selection method already being successfully applied to soft-thresholding can yield better results if we replace softby other shrinkage rules.