The Mumford–Shah variational model for image segmentation: An overview of the theory, implementation and use

Image segmentation is a hot topic of research given its applicability as a pre-processing technique in many image understanding applications. This paper describes the Mumford–Shah variational model for image segmentation. The mathematical framework and the main features of the model are sketched along with the procedure leading from the analytical expression of the model to its practical implementation. The Mumford–Shah functional consists of three weighted terms, the interaction of which assures that the three conditions of adherence to the data, smoothing, and discontinuity detection are met at once. The solution of the Mumford–Shah variational problem is twofold. On one side, a smooth approximation of the data is built so that the data discontinuities are explicitly preserved from being smoothed. On the other side, the model directly produces an image of the detected discontinuities. An open source software has been developed and used to perform a set of tests on synthetic and real images to demonstrate the feasibility and the effectiveness of the implementation and to give practical evidence of some theoretically foreseen properties of the model. The effect of varying the values of the weight parameters appearing in the Mumford–Shah model has been investigated. In this work, a maximum-likelihood based classifier has been concatenated to the Mumford–Shah model for the processing of a high-resolution orthophoto. The classified image has been compared against the output of the same classifier applied directly to the original orthophoto. Results clearly shows the quality and the practical convenience of variational segmentation. Some promising and interesting extensions of the Mumford–Shah model are also introduced in a dedicated section.