# Shape reconstruction from moments

CSGB seminar

Thursday, 10 October, 2013, at 13:15-14:00, in Koll. G (1532-214)

Julia Hörrmann (Karlsruhe Institute of Technology)

Abstract:

In the last two decades the reconstruction of a shape from its moments has received considerable attention. Milanfar

*et al.*developed in [1] an inversion algorithm for 2 -dimensional polygons. Restricting to convex polygons they proved that every*m*- gon is uniquely determined by its complex moments up to order 2*m*-3 . Recently Gravin*et al.*showed in [2] that an*n*-dimensional convex polytope*P*with*m*vertices is uniquely determined by its moments up to order 2*m*-1 . Though there exist strong inversion results for the polytopes, a dense subset of the convex bodies, it is not known how much information can be retrieved from finitely many moments of an arbitrary convex body. We will answer this question partially showing that the symmetric difference of two-dimensional convex bodies*K*and*L*with identical moments up to order*m*is smaller than $c\,(1/m)$ , where*c*>0 is a constant independent of*K*,*L*and*m*. The approach is based on orthogonal polynomials. Furthermore we will discuss some open questions in this context.**References**

[1] P. Milanfar , G. Verghese , W. Karl, and A. Willsky , Reconstructing polygons from moments with connections to array processing, * IEEE Transactions * * on Signal Processing*, no. 2, 1995.

[2] N. Gravin , J. Lasserre , Pasechnik , Dmitrii V., and S. Robins, The inverse moment problem for convex polytopes, 2011. [ Online ]. Available: http:// arxiv . org /pdf/1106. 5723v2 .pdf