Efficient algorithms for robust recovery of images from compressed data

TL;DR

This paper introduces efficient algorithms for robust compressed sensing that directly solve the problem, significantly improving computational speed and extending applicability to various complex imaging scenarios.

Contribution

The paper develops direct, computationally efficient algorithms for robust compressed sensing and extends the formulation to multiple settings, surpassing previous iterative methods.

Findings

New algorithms outperform previous methods in speed and robustness.

Extensions enable robust CS in more complex imaging tasks.

Algorithms effectively handle additional constraints and regularizations.

Abstract

Compressed sensing (CS) is an important theory for sub-Nyquist sampling and recovery of compressible data. Recently, it has been extended by Pham and Venkatesh to cope with the case where corruption to the CS data is modeled as impulsive noise. The new formulation, termed as robust CS, combines robust statistics and CS into a single framework to suppress outliers in the CS recovery. To solve the newly formulated robust CS problem, Pham and Venkatesh suggested a scheme that iteratively solves a number of CS problems, the solutions from which converge to the true robust compressed sensing solution. However, this scheme is rather inefficient as it has to use existing CS solvers as a proxy. To overcome limitation with the original robust CS algorithm, we propose to solve the robust CS problem directly in this paper and drive more computationally efficient algorithms by following latest advances in large-scale convex optimization for non-smooth regularization. Furthermore, we also extend the robust CS formulation to various settings, including additional affine constraints, $\ell_1$-norm loss function, mixed-norm regularization, and multi-tasking, so as to further improve robust CS. We also derive simple but effective algorithms to solve these extensions. We demonstrate that the new algorithms provide much better computational advantage over the original robust CS formulation, and effectively solve more sophisticated extensions where the original methods simply cannot. We demonstrate the usefulness of the extensions on several CS imaging tasks.