Morphological Operator Design from Training Data

Mathematical morphology offers a set of powerful tools for image processing and analysis. From a practical perspective, the expected results of many morphological operators can be intuitively explained in terms of geometrical and topological characteristics of the images. From a formal perspective, mathematical morphology is based on complete lattices, which provides a solid theoretical framework for the study of algebraic properties of the operators. Despite of these nice characteristics, designing morphological operators is not a trivial task; it requires knowledge and experience. In this chapter, a self-contained exposition on the design of translation-invariant morphological operators from training data is presented. The described training procedure relies on the canonical sup-decomposition theorem of morphological operators, which in the context of binary images states that any translation-invariant operator can be expressed uniquely in terms of two elementary operators, erosions and dilations, plus set operations. An important issue considered in this exposition is how the bias-variance tradeoff manifests within the training context and how its understanding can lead to approaches that generate better results. Several application examples that illustrate the usefulness of the described design procedure are also presented.