This paper is devoted to the understanding of regularization process in the shape optimization approach to the so-called Dirichlet inverse obstacle problem for elliptic operators. More precisely, we study two different regularization of the very classical shape optimization approach consisting in minimizing a mismatched functional. The first one is an implicit regularization when working in the class of inclusion having a uniform ε-cone property, a natural class in shape optimization. As this regularity is not trivial to guarantee numerically, we discuss the regularization by perimeter penalization. We show that this second regularization provides a stability gain in the minimization process.