Universal Inversion Formulas for Recovering a Function from Spherical Means

Abstract

The problem of reconstruction of a function from spherical means is at the heart of several modern imaging modalities and other applications. In this paper we derive universal back-projection-type reconstruction formulas for recovering a function in arbitrary dimension from averages over spheres centered on the boundary of an arbitrarily shaped bounded convex domain with smooth boundary. Provided that the unknown function is supported inside that domain, the derived formulas recover the unknown function up to an explicitly computed integral operator. For elliptical domains the integral operator is shown to vanish and hence we establish exact inversion formulas for recovering a function from spherical means centered on the boundary of elliptical domains in arbitrary dimension.

The universal back-projection formula for spherical means and the wave equation on certain quadric hypersurfaces

“…Recently, explicit formulas for inverting (1.2) and (1.3) on elliptical domains have been derived in [6,[20][21][22]31,38]. For the special case of spherical domains, the formulas in [20,21,31] reduce the ones earlier derived in [27,46]. According to the notion of [46] we call these formulas the universal back-projection formulas.…”

“…In the present paper, we deal with the problem of reconstructing an unknown function f ∈ C ∞ c (Ω) from the data on the boundary ∂Ω, which either consists of the spherical means of f or the solution of the standard free-space wave equation with initial data (f, 0). In particular, we investigate the universal back-projection formula (see [20,21,27,31,33,46]) on quadric hypersurfaces that can be approximated by elliptic hypersurfaces. For such type of quadrics we will show that the universal back-projection formula provides an exact reconstruction.…”

“…It should be noted that in [20,21,31,33] the use of the universal back-projection formula on general convex bounded domains Ω was analyzed. As a result, on a general convex bounded domain, the universal back-projection formula recovers the unknown function f up to an additional error term that has been explicitly computed in [20,21,31].…”

Stochastic Proximal Gradient Algorithms for Multi-Source Quantitative Photoacoustic Tomography

“…One important reconstruction problem in PAT is recovering the initial pressure distribution (see, for example, [3][4][5][6][7][8][9][10]). The initial pressure distribution only provides qualitative information about the tissue-relevant parameters, as it is the product of the optical absorption coefficient and the spatially varying optical intensity, which again indirectly depends on the tissue parameters.…”

Range description for a spherical mean transform on spaces of constant curvature

“…The first is to find closed form inversion formulas for R S . This problem has been solved when S is a sphere, cylinder, hyperplane, or some other special hypersurface; see, e.g., [FPR04,FHR07,Kun07,Ngu09,Pal12,Kun11,Nat12,Pal12,Sal14,Hal14]. The second problem is to find the non-injectivity sets S, i.e., to characterize those hypersurfaces S such that R S ( f ) ≡ 0 does not imply f ≡ 0.…”