A Weighted $\ell_1$-Minimization Approach For Wavelet Reconstruction of Signals and Images

TL;DR

This paper introduces a convex weighted -minimization method for wavelet-based signal and image reconstruction, effectively exploiting sparsity and structure to improve recovery from limited or noisy data.

Contribution

It proposes a novel weighted -regularization approach with specific weights and an adaptive reweighting scheme for enhanced wavelet signal reconstruction.

Findings

Numerical experiments show efficient recovery from subsampled or noisy data.

The method exploits structured sparsity common in signals and images.

Effective with both orthonormal wavelets and wavelet frames.

Abstract

In this effort, we propose a convex optimization approach based on weighted $\ell_1$-regularization for reconstructing objects of interest, such as signals or images, that are sparse or compressible in a wavelet basis. We recover the wavelet coefficients associated to the functional representation of the object of interest by solving our proposed optimization problem. We give a specific choice of weights and show numerically that the chosen weights admit efficient recovery of objects of interest from either a set of sub-samples or a noisy version. Our method not only exploits sparsity but also helps promote a particular kind of structured sparsity often exhibited by many signals and images. Furthermore, we illustrate the effectiveness of the proposed convex optimization problem by providing numerical examples using both orthonormal wavelets and a frame of wavelets. We also provide an adaptive choice of weights which is a modification of the iteratively reweighted $\ell_1$-minimization method.