Source and metric estimation in the eikonal equation using optimization on a manifold

Abstract

<p style='text-indent:20px;'>We address the estimation of the source(s) location in the eikonal equation on a Riemann surface, as well as the determination of the metric when it depends on a few parameters. The available observations are the arrival times or are obtained indirectly from the arrival times by an observation operator, this frame is intended to describe electro-cardiographic imaging. The sensitivity of the arrival times is computed from <inline-formula><tex-math id="M1">\begin{document}$ {{{\rm{Log}}}}_x $\end{document}</tex-math></inline-formula> the log map wrt to the source <inline-formula><tex-math id="M2">\begin{document}$ x $\end{document}</tex-math></inline-formula> on the surface. The <inline-formula><tex-math id="M3">\begin{document}$ {{{\rm{Log}}}}_x $\end{document}</tex-math></inline-formula> map is approximated by solving an elliptic vectorial equation, using the Vector Heat Method. The <inline-formula><tex-math id="M4">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-error function between the model predictions and the observations is minimized using Gauss-Newton optimization on the Riemann surface. This allows to obtain fast convergence. We present numerical results, where coefficients describing the metric are also recovered like anisotropy and global orientation.</p>