On an inverse problem in photoacoustics (Q2422504)

The paper is devoted to the standard wave equation for a pressure wave $u(x,t)$ \[ \begin{cases} u_{tt}-\Delta u=0,\quad x\in\mathbb{R}^3,\quad t>0,\\ u|_{t=0}=a(x),\quad u_t|_{t=0}=b(x), \end{cases} \] which appears in the photoacoustic imaging process. Here the initial data $a(x)$ and $b(x)$ describe spatial distribution of the absorption of laser radiation energy and its temporal change. By assuming that a transducer at time $t$ located at point $x$ measures the value of $u( x,t)$, the function $F:=u|_{S\times [0,T]}$ is known for some $T>0$, where $S$ is the unit sphere and $\text{supp}\,a,\,b$ are located inside $S$. The authors investigate the inverse problem which consists in finding the initial values $a(x)$ and $b(x)$ using the data $F(x,t)$. They propose a new approach by combining the harmonic decomposition of the measured data and equating the Fourier coefficients of the solutions, with the filtered back projection type formulas. Two cases, namely the $3$rd case and the $2$d case are investigated, and the comparison between the method of solving of the inverse problem used in this paper with other methods is finally presented.