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Abstract and Applied Analysis
Volume 2014, Article ID 341910, 14 pages
http://dx.doi.org/10.1155/2014/341910
Research Article

Nonasymptotic Densities for Shape Reconstruction

1Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA
2Department of Mathematics, University of Tennessee Knoxville, Knoxville, TN 37996-1320, USA

Received 17 August 2013; Revised 25 February 2014; Accepted 25 February 2014; Published 20 May 2014

Academic Editor: Victor Kovtunenko

Copyright © 2014 Sharif Ibrahim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this work, we study the problem of reconstructing shapes from simple nonasymptotic densities measured only along shape boundaries. The particular density we study is also known as the integral area invariant and corresponds to the area of a disk centered on the boundary that is also inside the shape. It is easy to show uniqueness when these densities are known for all radii in a neighborhood of , but much less straightforward when we assume that we only know the area invariant and its derivatives for only one . We present variations of uniqueness results for reconstruction (modulo translation and rotation) of polygons and (a dense set of) smooth curves under certain regularity conditions.