  In this paper, we present a method for the numerical minimization of the
Mumford-Shah functional that is based on the idea of topological asymptotic
expansions. The basic idea is to cover the expected edge set with balls of
radius \epsilon > 0 and use the number of balls, multiplied with 2\epsilon, as
an estimate for the length of the edge set. We introduce a functional based on
this idea and prove that it converges in the sense of \Gamma-limits to the
Mumford-Shah functional. Moreover, we show that ideas from topological
asymptotic analysis can be used for determining where to position the balls
covering the edge set. The results of the proposed method are presented by
means of two numerical examples and compared with the results of the classical
approximation due to Ambrosio and Tortorelli.
