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 by D Finch, S K Patch and Rakesh (2004 SIAM J. Math. Anal. 35 1213–1240) for odd spatial dimension and by D Finch, M Haltmeier and Rakesh (2007 SIAM J. Appl. Math. 68 392–412) 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 by Y Salman (2014 J. Math. Anal. Appl. 420 612–20). For the higher dimensional case 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 by F Natterer (2012 Inverse Problems Imaging 6 315–20) for n = 3 and by V Palamodov (2012 Inverse Problems 28 065014) for arbitrary dimension. The results of Natterer have later been generalized to arbitrary dimension by M Haltmeier (2014 SIAM J. Math. Anal. 46 214–32). Note that these formulas are different from the ones derived in the present paper.