{
  "id": "1809.08571",
  "version": "v1",
  "published": "2018-09-23T10:13:17.000Z",
  "updated": "2018-09-23T10:13:17.000Z",
  "title": "Periodic Splines and Gaussian Processes for the Resolution of Linear Inverse Problems",
  "authors": [
    "Anaïs Badoual",
    "Julien Fageot",
    "Michael Unser"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "This paper deals with the resolution of inverse problems in a periodic setting or, in other terms, the reconstruction of periodic continuous-domain signals from their noisy measurements. We focus on two reconstruction paradigms: variational and statistical. In the variational approach, the reconstructed signal is solution to an optimization problem that establishes a tradeoff between fidelity to the data and smoothness conditions via a quadratic regularization associated to a linear operator. In the statistical approach, the signal is modeled as a stationary random process defined from a Gaussian white noise and a whitening operator; one then looks for the optimal estimator in the mean-square sense. We give a generic form of the reconstructed signals for both approaches, allowing for a rigorous comparison of the two.We fully characterize the conditions under which the two formulations yield the same solution, which is a periodic spline in the case of sampling measurements. We also show that this equivalence between the two approaches remains valid on simulations for a broad class of problems. This extends the practical range of applicability of the variational method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-23T10:13:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65K10",
      "62D05",
      "46E22",
      "41A15"
    ],
    "keywords": [
      "linear inverse problems",
      "periodic spline",
      "gaussian processes",
      "resolution",
      "stationary random process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}