Photoacoustic tomography with direction dependent data: an exact series reconstruction approach

Abstract

Photoacoustic image reconstruction often assumes that the restriction of the acoustic pressure on the detection surface is given. However, commonly used detectors often have a certain directivity and frequency dependence, in which case the measured data are more accurately described as a linear combination of the acoustic pressure and its normal derivative on the detection surface. In this paper, we consider the inverse source problem for data that are a combination of an acoustic pressure of the wave equation and its normal derivative For the special case of a spherical detection geometry we derive exact frequency domain reconstruction formulas. We present numerical results showing the robustness and validity of the derived formulas. Moreover, we compare several different combinations of the pressure and its normal derivative showing that used measurement model significantly affects the recovered initial pressure.

Recovering the Initial Data of the Wave Equation from Neumann Traces

“…However, few theoretical results for Neumann as well as for mixed traces are known. To the best of our knowledge, the only results for this inverse problem are presented in [38], where a series inversion formula for spheres in arbitrary dimension has been established, and [8], where an exact reconstruction formula of a so-called back-projection type has been provided in the case of two spatial variables.…”

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

“…Instead, they measure certain combinations of the field and its derivatives. This challenge has been noted in [77] and investigated by Finch [31] and by Zangerl, Moon and Haltmeier [79].…”

Photoacoustic inversion formulas using mixed data on finite time intervals

“…In [11,14], an inversion formula for a smooth function g from the normal gradient of u in (1.1) with initial data (0, g) on spheres in 3D is presented. Recently, in [38] a series formula for spheres and in [7,8] an exact inversion formula of back-projection type for ellipsoids in arbitrary dimensions has been derived.…”

“…Accounting for realistic measurements mentioned above, a more general measurement model, which includes both the Dirichlet and Neumann case, is given by the trace of u a,b (x, t) := au(x, t) + b∂ ν u(x, t), (x, t) ∈ ∂Ω × (0, ∞) on the measurement surface ∂Ω × (0, ∞) for some weights a, b ∈ R. Throughout this article, we refer to these measurements as mixed data or mixed trace. Recovering the absorption coefficient from mixed data has previously been studied in [38] for spheres in arbitrary dimensions and in [7] for circles in two dimensions. Another mathematical model for piezoelectric sensors is proposed, for example, in [1].…”