Sparsity Based Methods for Overparametrized Variational Problems

Two complementary approaches have been extensively used in signal and image processing leading to novel results, the sparse representation methodology and the variational strategy. Recently, a new sparsity based model has been proposed, the cosparse analysis framework, that has established a very interesting connection between the two, highlighting shown how the traditional total variation minimization problem can be viewed as a sparse approximation problem. Based on this work we introduce a sparsity based framework for solving overparametrized variational problems. The latter has been used to improve the estimation of optical flow and also for general denoising of signals and images. However, the recovery of the space varying parameters involved was not adequately solved by the traditional variational methods. We first demonstrate the efficiency of the new framework for one dimensional signals in recovering a piecewise linear and polynomial function. We present also performance guarantees for recovery of picewise polynomial functions from a few number of measurements. Then for images we illustrate how the new technique can be used for denoising, geometrical inpainting and segmentation.