Classification of small structures in piecewise-constant Mumford–Shah model

Abstract Mumford–Shah model is a very popular variational model in image restoration and image classification. As a simplification, piecewise constant Mumford–Shah model is very useful and has been extensively studied in recent two decades. An interesting topic on Mumford–Shah model is how to choose the weight parameters for implementation. This paper aims at discussing and analyzing the relation between choosing weight parameters and removing/preserving small structures, including noise, for piecewise-constant Mumford–Shah model. The main contributions are: (1) provided a necessary condition on the weight parameter of regularity term for removing a small structure from background. It is proved that whether or not a small structure could be removed from the background in the piecewise-constant Mumford–Shah model depends on two aspects: the ratio of the area to the perimeter for the smaller structure and the intensities of other classes; (2) provided a decision-making strategy on the class that a small structure will be classified to if it does not belong to the background; (3) developed a balanced Mumford–Shah model with which the scale measurements (weights for fidelity terms) can be chosen based on prior knowledge or users’ purposes.