Sparsity Based Methods for Overparameterized Variational Problems

TL;DR

This paper introduces a sparsity-based framework leveraging cosparse analysis to improve overparameterized variational problems, enhancing optical flow estimation, denoising, and segmentation in signals and images.

Contribution

It presents a novel sparsity-based approach that bridges sparse approximation and total-variation methods for better parameter recovery in variational problems.

Findings

Effective recovery of piecewise linear and polynomial signals.

Improved denoising and segmentation of images.

Enhanced optical flow estimation.

Abstract

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.