*Article* **Full Field Inversion in Photoacoustic Tomography with Variable Sound Speed**

#### **Gerhard Zangerl 1, Markus Haltmeier 1,\*, Linh V. Nguyen 2,3 and Robert Nuster 4**


Received: 23 January 2019; Accepted: 8 April 2019; Published: 15 April 2019

**Abstract:** To accelerate photoacoustic data acquisition, in [R. Nuster, G. Zangerl, M. Haltmeier, G. Paltauf (2010). Full field detection in photoacoustic tomography. Optics express, 18(6), 6288–6299] a novel measurement and reconstruction approach has been proposed, where the measured data consist of projections of the full 3D acoustic pressure distribution at a certain time instant *T*. Existing reconstruction algorithms for this kind of setup assume a constant speed of sound. This assumption is not always met in practice and thus can lead to erroneous reconstructions. In this paper, we present a two-step reconstruction method for full field detection photoacoustic tomography that takes variable speed of sound into account. In the first step, by applying the inverse Radon transform, the pressure distribution at the measurement time is reconstructed point-wise from the projection data. In the second step, a final time wave inversion problem is solved where the initial pressure distribution is recovered from the known pressure distribution at time *T*. We derive an iterative solution approach for the final time wave inversion problem and compute the required adjoint operator. Moreover, as the main result of this paper, we derive its uniqueness and stability. Our numerical results demonstrate that the proposed reconstruction scheme is fast and stable, and that ignoring sound speed variations significantly degrades the reconstruction.

**Keywords:** photoacoustic tomography; full-field detection; wave equation; final time inversion; uniqueness; stability; iterative reconstruction
