Publications
Preprints
Alberti, Giovanni S.; Felisi, Alessandro; Santacesaria, Matteo; Trapasso, S. Ivan
Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI Preprint
2025.
Abstract | Links | BibTeX | Tags: compressed sensing, harmonic analysis, inverse problems, mathematical imaging, sparse recovery
@workingpaper{AFST_CSIP2_25,
title = {Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI},
author = {Giovanni S. Alberti and Alessandro Felisi and Matteo Santacesaria and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2501.01929, arXiv},
year = {2025},
date = {2025-01-03},
urldate = {2025-01-03},
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.},
keywords = {compressed sensing, harmonic analysis, inverse problems, mathematical imaging, sparse recovery},
pubstate = {published},
tppubtype = {workingpaper}
}
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.
Journal Articles
Alberti, Giovanni S.; Felisi, Alessandro; Santacesaria, Matteo; Trapasso, S. Ivan
Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform Journal Article
In: J. Eur. Math. Soc. (JEMS), iss. to appear, 2025.
Abstract | Links | BibTeX | Tags: compressed sensing, harmonic analysis, inverse problems, mathematical imaging, Radon transform, sparse recovery
@article{AFST_JEMS_24,
title = {Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform},
author = {Giovanni S. Alberti and Alessandro Felisi and Matteo Santacesaria and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2302.03577, arXiv},
doi = {10.4171/jems/1617},
year = {2025},
date = {2025-04-01},
urldate = {2025-04-01},
journal = {J. Eur. Math. Soc. (JEMS)},
issue = {to appear},
abstract = {Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In this work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map.
As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles (theta_1,dots,theta_m)), which models computed tomography, in both the parallel-beam and the fan-beam settings. In the case when the unknown signal is (s)-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable recovery under the condition [ mgtrsimß, ] up to logarithmic factors.},
keywords = {compressed sensing, harmonic analysis, inverse problems, mathematical imaging, Radon transform, sparse recovery},
pubstate = {published},
tppubtype = {article}
}
Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In this work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map.
As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles \(\theta_1,\dots,\theta_m\)), which models computed tomography, in both the parallel-beam and the fan-beam settings. In the case when the unknown signal is \(s\)-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable recovery under the condition \[ m\gtrsimß, \] up to logarithmic factors.
As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles \(\theta_1,\dots,\theta_m\)), which models computed tomography, in both the parallel-beam and the fan-beam settings. In the case when the unknown signal is \(s\)-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable recovery under the condition \[ m\gtrsimß, \] up to logarithmic factors.