Sparsity Based Methods for Overparameterized Variational Problems
This work addresses a gap in traditional variational methods for signal and image processing, offering a novel approach to parameter recovery, though it appears incremental by building on existing sparsity and variational strategies.
The paper tackles the problem of recovering space-varying parameters in overparameterized variational problems, such as optical flow estimation and denoising, by introducing a sparsity-based framework that bridges sparse approximation with total-variation minimization, demonstrating efficiency in recovering piecewise linear and polynomial functions for 1D signals and applying it to image denoising and segmentation.
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, which may potentially help in bridging sparse approximation based methods to the traditional total-variation minimization. Based on this, we introduce a sparsity based framework for solving overparameterized 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 addressed by traditional variational methods. We first demonstrate the efficiency of the new framework for one dimensional signals in recovering a piecewise linear and polynomial function. Then, we illustrate how the new technique can be used for denoising and segmentation of images.