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

Abstract

In the mathematical analysis of photoacoustic imaging, it is usually assumed that the acoustic pressure (Dirichlet data) is measured on a detection surface. However, actual ultrasound detectors gather data of a different type. In this paper, we propose a more realistic mathematical model of ultrasound measurements acquired by the Fabry-Perot sensor. This modeling incorporates directional response of such sensors. We study the solvability of the resulting photoacoustic tomography problem, concluding that the problem is well-posed under certain assumptions. Numerical reconstructions are implemented using the Landweber iterations, after discretization of the governing equations using the finite element method.

Solvability for Photoacoustic Imaging with Idealized Piezoelectric Sensors

“…This assumption is rigorously justified in the next section. The model ( 13) for the piezoelectric sensor is qualitatively similar to the model for the Fabry-Perot transducer proposed in [3] in spite of the completely different physical principles from which they are derived.…”

“…As in [3], we make some geometric assumptions about the domain Ω p occupied by the piezoelectric film. We let Ω p = {y ∈ Ω c : 0 < dist(y, Γ) < }.…”

“…Similar models for photoacoustics are studied in [4,5,3,2] As an example for the response of the piezoelectric sensor design, we can analyze its behavior for plane waves and for a flat boundary Γ. Both the boundary value problem ( 22)-( 23) and the model for the measurements ( 13) play an important role in this analysis.…”

“…The most common mechanisms for ultrasound applications are based on the piezoelectric effect [44,40,41], on Fabry-Perot interferometry [9,14,50,22] or on fiber-optic refractometry [43,42,45]. In [3] we formulated a model specifically tailored to the Fabry-Perot sensor design. In this paper, we derive a similar model for piezoelectric sensors and analyze the well-posedness of photoacoustic imaging with such measurements.…”

Recovering the Initial Data of the Wave Equation from Neumann Traces

“…However, as pointed out in [11,35,8], PAT measurements are often more accurately modeled by a linear combination of the Dirichlet trace and the normal derivative u a,b (x, t) = au(x, t) + b∂ ν u(x, t) for (x, t) ∈ ∂Ω × (0, ∞) , (1.2) which we refer to as the mixed trace on ∂Ω; here a, b ≥ 0 are constants. See also [2,1,7,29] for modeling and analysis of detectors characteristics in PAT. Measurements of the form (1.2) with a = 0 corresponds to the Neumann trace and with b = 0 to the Dirichlet trace on ∂Ω.…”