Mumford-Shah Regularizer with Spatial Coherence

As recently discussed by Bar, Kiryati, and Sochen in [3], the Ambrosio-Tortorelli approximation of the Mumford-Shah functional defines an extended line process regularization where the regularizer has an additional constraint introduced by the term ρ|∇v|. This term mildly forces some spatial organization by demanding that the edges are smooth. However, it does not force spatial coherence such as edge direction compatibility or edge connectivity, as in the traditional edge detectors such as Canny. Using the connection between regularization and diffusion filters, we incorporate further spatial structure into the regularization process of the Mumford-Shah model. The new model combines smoothing, edge detection and edge linking steps of the traditional approach to boundary detection. Importance of spatial coherence is best observed if the image noise is salt and pepper like. Proposed approach is able to deal with difficult noise cases without using non-smooth cost functions such as L in the data fidelity or regularizer.