signalrecovery

Signal recovery is the process of reconstructing a signal of interest from observed data that are incomplete, corrupted by noise, or distorted by the measurement process. It is typically modeled as an inverse problem: y = Φx + n, where x is the unknown signal, y is the observed data, Φ is a measurement operator, and n is noise. Because Φ can be ill-conditioned or undersampled, direct inversion is unstable or impossible, and recovery relies on regularization or prior information to constrain the solution.

Common approaches include optimization-based methods that minimize a data fidelity term together with a regularization penalty, such as L2, L1, or total variation. Sparse representations play a central role, enabling compressed sensing where x is assumed sparse in some dictionary. Bayesian methods treat x probabilistically. Practical algorithms include ISTA/FISTA, greedy pursuit, and expectation-maximization, often iterating between signal estimation and model adaptation. Applications include deconvolution, denoising, and super-resolution.

Signal recovery is essential in many fields. In communications, it improves symbol detection; in medical imaging and astronomy, it enables higher-quality reconstructions from limited measurements; in geophysics and radar, it recovers subsurface or scene information. The quality of recovery depends on the design of Φ, the level and structure of noise, and the availability of accurate priors or models.

Theoretical guarantees in signal recovery often rely on properties like sparsity, incoherence, and the restricted isometry property, which yield conditions for exact or stable recovery under noise. The field grew notably with the advent of compressed sensing in the mid-2000s, building on decades of inverse problem theory and sampling.