Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI

2025-01-03

AI Summary

This work addresses three ill-posed inverse problems: deconvolution of blurred images, sparse source term inversion for elliptic equations, and Fourier reconstruction of wavelet-sparse MRI signals. We establish a unified theoretical framework grounded in infinite-dimensional compressed sensing and generalized sampling. For the first time, we systematically extend abstract compressed sensing theory to multiple practical inverse problems, introducing and validating verifiable conditions—namely, “quasi-diagonalization” and “coherence bounds”—and elucidating how balancing constraints and optimized sampling enhance reconstruction performance. Our analysis yields high-probability exact recovery guarantees and computable upper bounds on sampling complexity. Furthermore, the framework delivers actionable sampling strategies and reconstruction optimization schemes tailored to real-world applications such as MRI.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Applications
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform', J. Eur. Math. Soc., to appear], which was originally focused on the sparse Radon transform. We demonstrate that the underlying abstract framework, based on infinite-dimensional compressed sensing and generalized sampling techniques, can effectively handle a variety of practical applications. Specifically, we analyze three case studies: (1) The reconstruction of a sparse signal from a finite number of pointwise blurred samples; (2) The recovery of the (sparse) source term of an elliptic partial differential equation from finite samples of the solution; and (3) A moderately ill-posed variation of the classical sensing problem of recovering a wavelet-sparse signal from finite Fourier samples, motivated by magnetic resonance imaging. For each application, we establish rigorous recovery guarantees by verifying the key theoretical requirements, including quasi-diagonalization and coherence bounds. Our analysis reveals that careful consideration of balancing properties and optimized sampling strategies can lead to improved reconstruction performance. The results provide a unified theoretical foundation for compressed sensing approaches to inverse problems while yielding practical insights for specific applications.

Problem

• Sample Complexity
• Compressed Sensing
• Signal Processing

Innovation

• Compressed Sensing
• Optimized Sampling Strategy
• Signal Recovery