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Shape Estimation from Support and Diameter Functions

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Abstract

We address the problem of reconstructing a planar shape from a finite number of noisy measurements of its support function or its diameter function. New linear and non-linear algorithms are proposed, based on the parametrization of the shape by its Extended Gaussian Image. This parametrization facilitates a systematic statistical analysis of the problem via the Cramér-Rao lower bound (CRLB), which provides a fundamental lower bound on the performance of estimation algorithms. Using CRLB, we also generate confidence regions which conveniently display the effect of parameters like eccentricity, scale, noise, and measurement direction set, on the quality of the estimated shapes, as well as allow a performance analysis of the algorithms.

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Authors and Affiliations

  1. Department of Computer Engineering, University of California, Santa Cruz, CA, 95064, USA

    Amyn Poonawala

  2. Department of Electrical Engineering, University of California, Santa Cruz, CA, 95064, USA

    Peyman Milanfar

  3. Department of Mathematics, Western Washington University, Bellingham, WA, 98825, USA

    Richard J. Gardner

Authors
  1. Amyn Poonawala
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  2. Peyman Milanfar
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  3. Richard J. Gardner
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Additional information

Supported in part by U.S. National Science Foundation grants CCR-9984246 and DMS-0203527.

Amyn Poonawala received the B.E. degree from the University of Mumbai, India, in 2001, and the M.S. degree from the University of California, Santa Cruz (UCSC), in 2004, both in computer engineering. He is currently pursuing the Ph.D. degree in computer engineering at UCSC. His technical interests include statistical signal and image processing and inverse problems in microlithography.

Peyman Milanfar received the B.S. degree in electrical engineering/mathematics from the University of California, Berkeley, in 1988, and the S.M., E.E., and Ph.D. degrees in electrical engineering from the Massachusetts Institute of Technology, in 1990, 1992, and 1993, respectively. Until 1999, he was a Senior Research Engineer at SRI International, Menlo Park, CA. He is currently Associate Professor of Electrical Engineering at the University of California, Santa Cruz. He was a Consulting Assistant Professor of computer science at Stanford University from 1998-2000, and a visiting Associate Professor there in 2002. His technical interests are in statistical signal and image processing, and inverse problems. He won a National Science Foundation CAREER award in 2000, was associate editor for the IEEE Signal Processing Letters from 1998 to 2001, and is a Senior member of the IEEE.

Richard Gardner holds B.Sc. and Ph.D. degrees in mathematics from University College London and was awarded a D.Sc. degree from the University of London in 1988 for contributions to measure theory and convex geometry. He has held positions at universities and research institutions in several countries and has been Professor of Mathematics at Western Washington University since 1991. He founded geometric tomography, an area of geometric inverse problems involving data concerning sections by and projections on lines or planes, and published a book on the subject in 1995.

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Poonawala, A., Milanfar, P. & Gardner, R.J. Shape Estimation from Support and Diameter Functions. J Math Imaging Vis 24, 229–244 (2006). https://doi.org/10.1007/s10851-005-3625-z

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