{
  "id": "1302.6193",
  "version": "v2",
  "published": "2013-02-25T18:56:03.000Z",
  "updated": "2013-06-12T16:28:06.000Z",
  "title": "The Broken Ray Transform On The Square",
  "authors": [
    "Mark Hubenthal"
  ],
  "comment": "Submitted to Journal of Fourier Analysis and Applications 26 pages, 4 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for $C_{0}^{2}$ perturbations of the constant unit weight. Given an open subset $E$ of the boundary, we measure the attenuation of all broken rays starting and ending at $E$ with the standard optical reflection rule. Using the analytic microlocal approach of Frigyik, Stefanov, and Uhlmann for the X-ray transform on generic families of curves, we show injectivity via a path unfolding argument under suitable conditions on the available broken rays. Then we show that with a suitable decomposition of the measurement operator via smooth cutoff functions, the associated normal operator is a classical pseudo differential operator of order -1 plus a smoothing term with $C_{0}^{\\infty}$ Schwartz kernel, which leads to the desired result.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-12T16:28:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "45Q05",
      "47G30",
      "53C65"
    ],
    "keywords": [
      "broken ray transform",
      "euclidean unit square",
      "classical pseudo differential operator",
      "smooth cutoff functions",
      "analytic microlocal approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.6193H"
    }
  }
}