Recovering linear images of sparse signals from indirect observations

2026-09-05

AI Summary

本文开发并分析了从间接噪声观测中恢复稀疏信号线性图像的技术，采用ℓ1-最小化方法而不对感知矩阵作特殊假设。

Abstract

In this paper, we develop and analyze techniques for recovering a linear image $Bx$ of an unknown signal $x$ from indirect noisy observation $\omega=Ax+\xi$. It is {\em a priori} known that $x\in \cX$, a given convex compact set, and that $x$ is $s$-sparse---has at most $s$ nonvanishing entries. The proposed estimates belong to a large family of recovery routines by $\ell_1$-minimization. However, unlike the classical result describing performance of such estimates, we do not make any special (and hard to check) assumptions about the sensing matrix $A$ such as nullspace or Restricted Isometry condition and the like. As a consequence, parameters of the estimates and the upper bounds on their risks are not available in a closed analytic form, but are delivered instead by efficient computation as solutions to explicit convex optimization problems.

Problem

• sparse signal
• linear image recovery
• noisy observation

Innovation

• s-sparse signal
• indirect noisy observation
• convex optimization
• $\ell_1$-minimization