Recovery of Sparsely Corrupted Signals

Abstract: We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, such as image inpainting, super-resolution, signal separation, and recovery of signals that are impaired by, e.g., clipping, impulse noise, or narrowband interference. We present deterministic recovery guarantees based on a novel uncertainty relation for pairs of general dictionaries and we provide corresponding practicable recovery algorithms. The recovery guarantees we find depend on the signal and noise sparsity levels, on the coherence parameters of the involved dictionaries, and on the amount of prior knowledge about the signal and noise support sets.

• The paper establishes uncertainty relations that provide deterministic recovery guarantees for signals affected by sparse noise.
• It introduces practical algorithms like ℓ1-norm minimization and Orthogonal Matching Pursuit to accurately isolate and recover corrupted signals.
• It extends classical recovery frameworks by incorporating structured noise models, enhancing reconstruction in applications such as image inpainting and communications.

Recovery of Sparsely Corrupted Signals: A Technical Overview

The paper "Recovery of Sparsely Corrupted Signals" by Christoph Studer et al. addresses the problem of recovering signals from measurements corrupted by sparse noise. This topic is relevant in numerous applications such as image inpainting, signal separation, and dealing with interference in communication systems. The authors explore a model where the measured signal is expressed as a linear combination of two components: one corresponding to the signal of interest and the other to sparse corruptions. These components are represented using arbitrary, possibly redundant, and incomplete dictionaries.

Key Contributions

1. Uncertainty Relations and Recovery Guarantees: The authors derive deterministic recovery guarantees based on an uncertainty relation tailored for pairs of general dictionaries. These results provide conditions under which the original signal can be perfectly recovered from corrupted measurements, considering the sparsity levels of both the signal and noise, as well as the coherence properties of the dictionaries involved.
2. Algorithms and Their Efficacy: The paper proposes practical algorithms for signal recovery, specifically leveraging $\ell_1$-norm minimization and greedy algorithms such as Orthogonal Matching Pursuit (OMP). The authors establish conditions under which these algorithms can successfully recover the original signal, given varying amounts of prior information about the support of the signal and noise vectors.
3. Structured Noise Consideration: The study extends the framework to account for sparsely corrupted signals, providing more robust recovery guarantees than traditional unstructured noise frameworks. This extension is particularly useful in scenarios where prior knowledge about the noise structure is available.
4. Comparison with Classical Results: The results are compared with classical bounds, such as those based on Restricted Isometry Properties (RIP) and coherence, highlighting the strengths and limitations of the proposed approach. The paper demonstrates that the proposed coherence-based guarantees, while less restrictive in some aspects, still abide by the inherent limitations like the square-root bottleneck observed in conventional sparse recovery settings.

Implications and Future Directions

The implications of this work are significant for both theoretical and practical pursuits in signal processing. Practically, the proposed methods enable precise recovery in applications plagued by sparse corruptions, enhancing the reliability and quality of signal reconstruction. Theoretically, the novel uncertainty relations for general dictionary pairs provide a broader framework for analyzing and designing recovery algorithms under diverse settings.

For future research, one avenue is to extend probabilistic recovery guarantees that might offer insightful trade-offs between recovery performance and computational complexity. Additionally, adapting these methods to operate efficiently in real-time applications or resource-constrained environments could significantly broaden their applicability. Furthermore, exploring the integration of machine learning methods with these sparse recovery algorithms could yield adaptive techniques capable of handling more complex and dynamic corruption patterns.

In closing, this paper advances the understanding and capabilities of sparse-signal recovery in the presence of sparse noise, providing both rigorous theoretical insights and applicable recovery strategies.