A family of inversion formulas in thermoacoustic tomography

Abstract

We present a family of closed form inversion formulas in thermoacoustic tomography in the case of a constant sound speed. The formulas are presented in both time-domain and frequency-domain versions. As special cases, they imply most of the previously known filtered backprojection type formulas.1991 Mathematics Subject Classification. Primary: 92C55, 35L05, 35R30; Secondary: 44A99, 45Q05.

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

“…In [15] it has been shown that for a spherical observation surface in two and three spatial dimensions, the universal back-projection formula can be derived from the inversion formulas of [16,17]. In [32] a class of inversion formulas has been obtained that includes the formulas of [16,17] as well as the universal back-projection formulas.…”

“…Such formulas are currently only known for special boundaries ∂Ω. For example, explicit inversion formulas have been derived for hyperplanes [5,7,10,13,30,46], spheres [16,17,27,32,46] and cylinders [36]. Reconstruction formulas for some polygons and polyhedra in two and three spatial dimensions have been obtained in [29].…”

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.…”

Well-Posedness for Photoacoustic Tomography with Fabry--Perot Sensors

“…For qualitative photoacoustic tomography, the goal is to form an image of the initial state of the pressure field using boundary measurements of the transient pressure waves. Most of the reconstruction methods assume that the actual pressure field (Dirichlet data) can be measured at the boundary [2,3,4,5,11,22,32,35,36,37,38,42,44,51,52,55,59,60,61,63]. In reality, ultrasound sensors are not able to measure the pressure field directly.…”