Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI
Giovanni S. Alberti, Alessandro Felisi, Matteo Santacesaria, S. Ivan Trapasso
TL;DR
This work extends infinite-dimensional compressed sensing to ill-posed inverse problems by embedding forward models into an abstract framework whose guarantees rely on quasi-diagonalization, multiscale coherence, and a balancing property, with sampling designed via a density f_ν that matches the coherence structure. It unifies three practical problems—sparse deconvolution, sparse inverse source recovery for elliptic PDEs, and ill-posed Fourier sampling relevant to MRI—under a single theory, providing explicit recovery bounds and sample complexities that scale with the ill-posedness parameter b and the truncation level j0. Key contributions include a rigorous abstract recovery theorem, optimized sampling strategies (including nonuniform density choices), and concrete error bounds for cartoon-like images and MRI-like settings, all framed in a wavelet dictionary context. The results offer a principled, theoretically grounded foundation for applying compressed sensing to diverse inverse problems, with direct implications for improved sampling design and reconstruction guarantees in practice.
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.