From restoration by topological gradient to medical image segmentation via an asymptotic expansion

The aim of this article is to present an application of the topological asymptotic expansion to the medical image segmentation problem. We first recall the classical variational of the image restoration problem, and its resolution by topological asymptotic analysis in which the identification of the diffusion coefficient can be seen as an inverse conductivity problem. The conductivity is set either to a small positive coefficient (on the edge set), or to its inverse (elsewhere). In this paper a technique based on a power series expansion of the solution to the image restoration problem with respect to this small coefficient is introduced. By considering the limit when this coefficient goes to zero, we obtain a segmented image, but some numerical issues do not allow a too small coefficient. The idea is to use the series expansion to approximate the asymptotic solution with several solutions corresponding to positive (larger than a threshold) conductivity coefficients via a quadrature formula. We illustrate this approach with some numerical results on medical images.