Effective Discretization of Direct Reconstruction Schemes For Photoacoustic Imaging In Spherical Geometries

where Sd−1 = { u ∈ Rd : |u| = 1 } denotes the unit sphere and σ its surface measure. In particular is σ(S1) = 2π and σ(S2) = 4π. The variable t ≥ 0 is called measurement time and the variable ξ ∈ Rd detector position or center point. In all practical applications these center points are located on a curve or surface and we consider the classical case ξ ∈ Sd−1 here. Moreover, we restrict ourselves to functions f : Rd → R with support suppf ⊂ B, where B = { x ∈ Rd : |x| < 1 } denotes the open unit ball, such that the spherical mean values Rf(ξ, t) vanish for t ≥ 2. As outlined in [3, 1], the function f can be approximated from the spherical mean values Rf(ξ, t), ξ ∈ Sd−1, t ∈ [0, 2], using specific summability kernels. These kernels are families of