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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms
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by Gabriele Bianchi, Richard J. Gardner and Markus Kiderlen PDF
J. Amer. Math. Soc. 24 (2011), 293-343 Request permission

Abstract:

We propose strongly consistent algorithms for reconstructing the characteristic function $1_K$ of an unknown convex body $K$ in $\mathbb {R}^n$ from possibly noisy measurements of the modulus of its Fourier transform $\widehat {1_K}$. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing $K$ from noisy measurements of its covariogram, the function giving the volume of the intersection of $K$ with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use $O(k^n)$ noisy covariogram measurements to construct a convex polytope $P_k$ that approximates $K$ or its reflection $-K$ in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all three-dimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between $P_k$ and $\pm K$ tends to zero as $k$ tends to infinity.
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Additional Information
  • Gabriele Bianchi
  • Affiliation: Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/A, Firenze, Italy I-50134
  • MR Author ID: 239050
  • Email: gabriele.bianchi@unifi.it
  • Richard J. Gardner
  • Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063
  • MR Author ID: 195745
  • Email: Richard.Gardner@wwu.edu
  • Markus Kiderlen
  • Affiliation: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK–8000 Aarhus C, Denmark
  • Email: kiderlen@imf.au.dk
  • Received by editor(s): July 7, 2009
  • Received by editor(s) in revised form: June 30, 2010, and July 23, 2010
  • Published electronically: October 5, 2010
  • Additional Notes: The second author was supported in part by U.S. National Science Foundation grant DMS-0603307.
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 24 (2011), 293-343
  • MSC (2010): Primary 42--04, 42B10, 52--04, 52A20; Secondary 52B11, 62H35
  • DOI: https://doi.org/10.1090/S0894-0347-2010-00683-2
  • MathSciNet review: 2748395