Recent Developments in Inverse Problems - Abstract

Quellmalz, Michael

Optimal mollifiers for reconstructing spherical images from circular means

Michael Quellmalz, Ralf Hielscher We take a look at the reconstruction of spherical images from mean values along great or small circles on the two-dimensional sphere $mathbbS^2$. The mean operator $mathcal M f(boldsymbol eta,t)$ computes the mean values of a function $f colon mathbbS^2 to mathbbC$ along all circles beginequation* mathcal C(boldsymbol eta,t) = boldsymbol xi in mathbbS^2 mid boldsymbol xi cdot boldsymbol eta = t , quad (boldsymbol eta,t) in mathbbS^2 times [-1,1]. endequation* We consider the reconstruction of $f$ given only finitely many samples of $mathcal Mf$, that are corrupted by white noise. Our approach combines the mollifier method, cf. citeLoRiSpSp11, with a quadrature rule to derive an estimator $f_psi$ of $f$. In this talk, we will give optimal rates of the minimax error beginequation* inf_psi sup_leftVertfrightVert_sleq S mathbb E leftVertf - f_psirightVert_2^2 endequation* for functions $f$ with bounded Sobolev norm $leftVert f rightVert_s$, together with asymptotically optimal mollifiers $psi$ as the number of sampling points goes to infitnity. Finally, we illustrate our findings by numerical experiments and discuss fast algorithms that make use of the fast spherical Fourier transform. beginthebibliography1 bibitemHiQu15 Ralf Hielscher and Michael Quellmalz. newblock Optimal mollifiers for spherical deconvolution. newblock em Preprint 2015-04, Faculty of Mathematics, Technische Universität Chemnitz, 2015. bibitemLoRiSpSp11 Alfred Karl Louis, Martin Riplinger, Malte Spiess, and Evgeny Spodarev. newblock Inversion algorithms for the spherical Radon and cosine transform. newblock em Inverse Problems, 27(3):035015, March 2011. endthebibliography