  The Ambrosio-Tortorelli functional is a phase-field approximation of the
Mumford-Shah functional that has been widely used for image segmentation. The
approximation has the advantages of being easy to implement, maintaining the
segmentation ability, and $\Gamma$-converging to the Mumford-Shah functional.
However, it has been observed in actual computation that the segmentation
ability of the Ambrosio-Tortorelli functional varies significantly with
different values of the parameter and it even fails to $\Gamma$-converge to the
original functional for some cases. In this paper we present an asymptotic
analysis on the gradient flow equation of the Ambrosio-Tortorelli functional
and show that the functional can have different segmentation behavior for small
but finite values of the regularization parameter and eventually loses its
segmentation ability as the parameter goes to zero when the input image is
treated as a continuous function. This is consistent with the existing
observation as well as the numerical examples presented in this work. A
selection strategy for the regularization parameter and a scaling procedure for
the solution are devised based on the analysis. Numerical results show that
they lead to good segmentation of the Ambrosio-Tortorelli functional for real
images.
