{
  "id": "2501.01929",
  "version": "v1",
  "published": "2025-01-03T18:00:09.000Z",
  "updated": "2025-01-03T18:00:09.000Z",
  "title": "Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI",
  "authors": [
    "Giovanni S. Alberti",
    "Alessandro Felisi",
    "Matteo Santacesaria",
    "S. Ivan Trapasso"
  ],
  "comment": "47 pages",
  "categories": [
    "math.FA",
    "cs.IT",
    "math.IT",
    "math.OC"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-03T18:00:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C40",
      "94A20",
      "35R30"
    ],
    "keywords": [
      "inverse problems",
      "compressed sensing",
      "source recovery",
      "sparse radon transform",
      "deconvolution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}