On local region models and a statistical interpretation of the piecewise smooth Mumford-Shah functional

The Mumford-Shah functional is a general and quite popular variational model for image segmentation. In particular, it provides the possibility to represent regions by smooth approximations. In this paper, we derive a statistical interpretation of the full (piecewise smooth) Mumford-Shah functional by relating it to recent works on local region statistics. Moreover, we show that this statistical interpretation comes along with several implications. Firstly, one can derive extended versions of the Mumford-Shah functional including more general distribution models. Secondly, it leads to faster implementations. Finally, thanks to the analytical expression of the smooth approximation via Gaussian convolution, the coordinate descent can be replaced by a true gradient descent.