Exact Reconstruction Formula for the Spherical Mean Radon Transform on Ellipsoids

Abstract

Many modern imaging and remote sensing applications require reconstructing a function from spherical averages (mean values). Examples include photoacoustic tomography, ultrasound imaging or SONAR. Several formulas of the back-projection type for recovering a function in $n$ spatial dimensions from mean values over spheres centered on a sphere have been derived in [D. Finch, S. K. Patch, and Rakesh, SIAM J. Math. Anal. 35(5), pp. 1213--1240, 2004] for odd spatial dimension and in [D. Finch, M. Haltmeier, and Rakesh, SIAM J. Appl. Math. 68(2), pp. 392--412, 2007] for even spatial dimension. In this paper we generalize some of these formulas to the case where the centers of integration lie on the boundary of an arbitrary ellipsoid. For the special cases $n=2$ and $n=3$ our results have recently been established in [Y. Salman, J. Math. Anal. Appl., 2014, in press]. For the higher dimensional case $n>3$ we establish proof techniques extending the ones in the above references. Back-projection type inversion formulas for recovering a function from spherical means with centers on an ellipsoid have first been derived in [F. Natterer, Inverse Probl. Imaging 6(2), pp. 315--320, 2012] for $n=3$ and in [V. Palamodov, Inverse Probl. 28(6), 065014, 2012] for arbitrary dimension. The results of Natterer have later been generalized to arbitrary dimension in [M. Haltmeier, SIAM J. Math. Anal. 46(1), pp. 214--232, 2014]. Note that these formulas are different from the ones derived in the present paper.

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