Complete lattice learning for multivariate mathematical morphology

Complete lattice learning for multivariate mathematical morphology

The generalization of mathematical morphology to multivariate vector spaces is addressed in this paper. The proposed approach is fully unsupervised and consists in learning a complete lattice from an image as a nonlinear bijective mapping, interpreted in the form of a learned rank transformation together with an ordering of vectors. This unsupervised ordering of vectors relies on three steps: d...

Learning complete lattices for manifold mathematical morphology

Mathematical Morphology (MM) is a nonlinear approach to image processing that relies on a fundamental structure, the complete lattice L [7] (a nonempty set equipped with an ordering relation). With the complete lattice theory, it is possible to define morphological operators for any type of data once a proper ordering is established [1]. If Mathematical Morphology is well defined for binary and...

A comparative study on multivariate mathematical morphology

The successful application of univariate morphological operators on several domains, along with the increasing need for processing the plethora of available multivalued images, have been the main motives behind the efforts concentrated on extending the mathematical morphology framework to multivariate data. The few theoretical requirements of this extension, consisting primarily of a ranking sc...

On lexicographical ordering in multivariate mathematical morphology

Since mathematical morphology is based on complete lattice theory, a vector ordering method becomes indispensable for its extension to multivariate images. Among the several approaches developed with this purpose, lexicographical orderings are by far the most frequent, as they possess certain desirable theoretical properties. However, their main drawback consists of the excessive priority attri...

Mathematical Morphology