Richard Kowar

Richard Kowar

In case of non-dissipative tissue the inverse problem of thermoacoustic imaging basically consists of two inverse problems. First, a function $ϕ$ depending on the \emph{electromagnetic absorption function}, is estimated from one of three types of projections (spherical, circular or planar) and secondly, the \emph{electromagnetic absorption function} is estimated from $ϕ$. In case of dissipative tissue, it is no longer possible to calculate explicitly the projection of $ϕ$ from the respective pressure data (measured by point, planar or line detectors). The goal of this paper is to derive for each of the three types of pressure data, an integral equation that allows estimating the respective projection of $ϕ$. The advantage of this approach is that all known reconstruction formulas for $ϕ$ from the respective projection can be exploited.

Richard Kowar

This article starts over the backwards diffusion problem by replacing the \emph{noncausal} diffusion equation, the direct problem, by the \emph{causal} diffusion model developed in \cite{Kow11} for the case of constant diffusion speed. For this purpose we derive an analytic representation of the Green function of causal diffusion in the wave vector-time space for arbitrary (wave vector) dimension $N$. We prove that the respective backwards diffusion problem is ill-posed, but not exponentially ill-posed, if the data acquisition time is larger than a characteristic time period $τ$ ($2\,τ$) for space dimension $N\geq 3$ (N=2). In contrast to the noncausal case, the inverse problem is well-posed for N=1. Moreover, we perform a theoretical and numerical comparison between causal and noncausal diffusion in the \emph{space-time domain} and the \emph{wave vector-time domain}. The paper is concluded with numerical simulations of the backwards diffusion problem via the Landweber method.

Richard Kowar

In this paper we discuss the problem of small frequency approximation of the causal dissipative pressure wave model proposed in \cite{KoScBo:11}. We show that for appropriate situations the Green function $G^c$ of the causal wave model can be approximated by a noncausal Green function $G_M^{pl}$ that has frequencies only in the small frequency range $[-M,M]$ ($M\leq 1/τ_0$, $τ_0$ relaxation time) and obeys a power law. For such cases, the noncausal wave $G^{pl}_M$ contains partial waves propagating arbitrarily fast but the sum of the noncausal waves is small in the $L^2-$sense.

Markus Haltmeier, Richard Kowar, Linh V. Nguyen

The development of efficient and accurate reconstruction methods is an important aspect of tomographic imaging. In this article, we address this issue for photoacoustic tomography. To this aim, we use models for acoustic wave propagation accounting for frequency dependent attenuation according to a wide class of attenuation laws that may include memory. We formulate the inverse problem of photoacoustic tomography in attenuating medium as an ill-posed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. Our approach comes with a clear convergence analysis. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. In contrast to time reversal, the employed adjoint wave equation is again damping and, thus has a stable solution. This stability property can be clearly seen in our numerical results. Moreover, the presented numerical results clearly demonstrate the Efficiency and accuracy of the derived iterative reconstruction algorithms in various situations including the limited view case.