Mumford–Shah functional

The Mumford–Shah functional is a functional that is used to establish an optimality criterion for segmenting an image into sub-regions. An image is modeled as a piecewise-smooth function. The functional penalizes the distance between the model and the input image, the lack of smoothness of the model within the sub-regions, and the length of the boundaries of the sub-regions. By minimizing the functional one may compute the best image segmentation. The functional was proposed by mathematicians David Mumford and Jayant Shah in 1989. The Mumford–Shah functional is a functional that is used to establish an optimality criterion for segmenting an image into sub-regions. An image is modeled as a piecewise-smooth function. The functional penalizes the distance between the model and the input image, the lack of smoothness of the model within the sub-regions, and the length of the boundaries of the sub-regions. By minimizing the functional one may compute the best image segmentation. The functional was proposed by mathematicians David Mumford and Jayant Shah in 1989. Consider an image I with a domain of definition D, call J the image's model, and call B the boundaries that are associated with the model: the Mumford–Shah functional E is defined as Optimization of the functional may be achieved by approximating it with another functional, as proposed by Ambrosio and Tortorelli. Ambrosio and Tortorelli showed that Mumford–Shah functional E can be obtained as the limit of a family of energy functionals E where the boundary B is replaced by continuous function z whose magnitude indicates the presence of a boundary. Their analysis show that the Mumford–Shah functional has a well-defined minimum. It also yields an algorithm for estimating the minimum.