A shape and topology optimization method for the resolution of inverse problems

We propose a general shape optimization approach for the resolution of different inverse problems in tomography. For instance, in the case of Electrical Impedance Tomography (EIT), we reconstruct the electrical conductivity while in the case of Fluorescence Diffuse Optical Tomography (FDOT), the unknown is a fluorophore concentration. In the two cases, the underlying partial differential equation are different but the reconstruction method essentially stays the same. These problems are in general severely ill-posed, and a standard cure is to make additional assumptions on the unknowns to regularize the problem. Our approach consists in assuming that the functions to be reconstructed are piecewise constants.

Thanks to this hypothesis, we are looking for the shape of one or several inclusions in the domain and the problem essentially boils down to a shape optimization problem. The sensitivity of a certain cost functional with respect to small perturbations of the shapes of these inclusions is analysed. The algorithm consists in initializing the inclusions using the notion of topological derivative, which measures the variation of the cost functional when a small inclusion is introduced in the domain, then to reconstruct the shape of the inclusions by modifying their boundaries with the help of the so-called shape derivative.