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Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform

Giovanni S. Alberti, Alessandro Felisi, Matteo Santacesaria, S. Ivan Trapasso

TL;DR

This work develops a unified, infinite-dimensional compressed sensing framework for ill-posed inverse problems with a forward operator $F$, introducing a generalized restricted isometry property (g-RIP) and a quasi-diagonalization mechanism to obtain finite-sample recovery guarantees for sparse representations. By coupling the abstract theory with a truncation-based reduction and wavelet dictionaries, it provides rigorous recovery bounds for ill-posed forward maps and, notably, rigorous sample-complexity results for the sparse Radon transform in both parallel-beam and fan-beam tomography. The key contributions include establishing how the number of measurements $m$ scales with sparsity $s$ and the forward-map conditioning via $G$ and its inverse, and deriving explicit error bounds that interpolate between noise-dominated and approximation-dominated regimes. The practical impact lies in enabling provable, sample-efficient recovery for tomography and other inverse problems using sparsity-promoting reconstructions in multiscale dictionaries, with explicit bounds that inform experimental design and algorithmic choice.

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 $θ_1,\dots,θ_m$), 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 \[ m\gtrsim s, \] up to logarithmic factors.

Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform

TL;DR

This work develops a unified, infinite-dimensional compressed sensing framework for ill-posed inverse problems with a forward operator , introducing a generalized restricted isometry property (g-RIP) and a quasi-diagonalization mechanism to obtain finite-sample recovery guarantees for sparse representations. By coupling the abstract theory with a truncation-based reduction and wavelet dictionaries, it provides rigorous recovery bounds for ill-posed forward maps and, notably, rigorous sample-complexity results for the sparse Radon transform in both parallel-beam and fan-beam tomography. The key contributions include establishing how the number of measurements scales with sparsity and the forward-map conditioning via and its inverse, and deriving explicit error bounds that interpolate between noise-dominated and approximation-dominated regimes. The practical impact lies in enabling provable, sample-efficient recovery for tomography and other inverse problems using sparsity-promoting reconstructions in multiscale dictionaries, with explicit bounds that inform experimental design and algorithmic choice.

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 ), which models computed tomography, in both the parallel-beam and the fan-beam settings. In the case when the unknown signal is -sparse with respect to an orthonormal basis of compactly supported wavelets, we prove stable recovery under the condition up to logarithmic factors.
Paper Structure (28 sections, 29 theorems, 243 equations)

This paper contains 28 sections, 29 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. State of the art
  3. Main contributions
  4. Summary
  5. Sample complexity of sparse ill-posed IP
  6. The sparse Radon transform
  7. Related research
  8. Discussion and open questions
  9. Structure of the paper
  10. Setup
  11. Weighted setup and notation
  12. Dictionaries, measurements and the minimization problem
  13. A general result on sample complexity
  14. Main result
  15. Quantitative bounds on |G-1| and quasi-diagonalization
  16. ...and 13 more sections

Key Result

Theorem 1

Under the assumptions outlined so far, there exist universal constants $C_0,C_1,C_2>0$ such that, for any integer $2\le s \le |\Lambda|$, if $\widehat{x}\in\ell^2(\Lambda)$ is a solution of the minimization problem eq-minpbm-intro and the number of samples satisfies up to logarithmic factors, then the following recovery estimate holds with overwhelming probability: the term $\sigma_s(x^\dagger)_

Theorems & Definitions (71)

  • Theorem
  • Theorem
  • Remark 2.1
  • Example 2.2: Reconstruction of wavelet coefficients from Fourier samples
  • Example 2.3: Approximation with polynomials
  • Remark 3.2
  • Remark 3.4
  • Theorem 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 61 more