# Full Field Inversion in Photoacoustic Tomography with Variable Sound Speed
## 1 Introduction  
Photoacoustic tomography (PAT) is a hybrid imaging modality that combines high spatial resolution of ultrasound and high contrast of optical tomography [1; 2; 3; 4; 5]. In PAT, a semitransparent sample is illuminated by a short laser pulse. As a result, parts of the optical energy are absorbed inside the sample. This causes an initial pressure distribution and a subsequent acoustic pressure wave. The pressure wave is detected outside the investigated object and used to recover an image of the interior. In standard PAT, the induced pressure is measured on a detection surface as a function of time. In the case of constant sound speed and when the observation surface exhibits a special geometry (planar, cylindrical, spherical), the initial pressure distribution can be recovered by closed-form inversion formulas; see [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20] and references therein. While some algorithms take the size and other properties of the detectors into account [21; 22; 23; 24], most algorithms assume that the acoustic pressure is known point-wise on a detection surface for all times in the measurement interval. Due to the finite width of the commonly used piezoelectric elements this assumption is only approximately satisfied. One approach to address this issue uses the concept of integrating detectors [25]. Integrating detectors measure integrals of the acoustic pressure over planes, lines or circles and several inversion methods have been derived in the literature [26; 27; 28; 29]. Inspired by the concept of integrating line detectors, a full field detection and reconstruction method that uses projections of the full acoustic field surrounding the sample has been developed in [30; 31]. The pressure field within the depth of field is imaged by the use of an optical phase contrast imaging system with a CCD camera. In this way, one obtains 2D projections of the pressure field at a time instant \(T\). Reconstructing 2D projections of the initial pressure has been investigated in [32; 33]. In [30; 31] we go one step further and show that projection data from different directions allow for a full 3D reconstruction of the initial pressure by Radon or Fourier transform techniques. We refer to this approach as full field detection photoacoustic tomography (FFD-PAT). A photograph of the FFD-PAT setup developed in Graz is shown in Figure 1. For practical aspects and a detailed description of the working principle of the FFD-PAT detection system we refer to the original works [30; 31]. Existing image reconstruction methods for FFD-PAT data assume a constant speed of sound. However, there are relevant cases when the assumption of constant speed of sound is inaccurate [34; 35]. For example, it is known that acoustic properties vary within female human breasts. Consequently, for accurate image reconstruction, variable speed of sound has to be incorporated in the wave propagation model. Iterative methods are capable to deal with this assumption. In the case of standard PAT, such methods have been studied in [36; 37; 38; 39; 40; 41] for spatially variable speed of sound, that is smooth and bounded from below. Moreover, it is assumed to satisfy the so-called non-trapping condition, which means that the supremum of the lengths of all geodesics connecting any two points inside the volume enclosed by the measurement surface \(S\) is finite. Under these assumptions, it is known that the initial pressure can be stably reconstructed from pressure data given on \(S\times[0,T]\). In this paper, for the first time, we study image reconstruction in FFD-PAT with a spatially variable speed of sound. We will give a precise mathematical formulation of FFD-PAT and describe the inverse problem we are dealing with (see Section 2). For its solution, we propose a two-step process. In the first step, the acoustic pressure at time \(T\) is reconstructed pointwise from the FFD data. In the second step, we recover the desired initial pressure distribution from the known valued of the pressure at time \(T\).