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Reconstruction of convex bodies from surface tensors

By Astrid Kousholt and Markus Kiderlen
CSGB Research Reports
No. 10, June 2015

We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank $s$ for some natural number $s$. The second algorithm uses harmonic intrinsic volumes which are certain values of the surface tensors and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based on measurements subject to noise is established under certain assumptions on the noise variables.

Keywords: Convex body, shape, reconstruction algorithm, surface tensor, harmonic intrinsic volume, generalized Wirtinger's inequality

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