  In this paper we introduce a new phase field approximation of the
Mumford-Shah functional similar to the well-known one from Ambrosio and
Tortorelli. However, in our setting the phase field is allowed to be a function
of bounded variation, instead of an $H^1$-function. In the context of image
segmentation, we also show how this new approximation can be used for numerical
computations, which contains a total variation minimization of the phase field
variable, as it appears in many problems of image processing. A comparison to
the classical Ambrosio-Tortorelli approximation, where the phase field is an
$H^1$-function, shows that the new model leads to sharper phase fields.
