Binary Partition Trees-Based Spectral-Spatial Permutation Ordering

Abstract

Mathematical Morphology (MM) is founded on the mathematical branch of Lattice Theory. Morphological operations can be described as mappings between complete lattices, and complete lattices are a type of partially-ordered sets (poset). Thus, the most elementary requirement to define morphological operators on a data domain is to establish an ordering of the data. MM has been very successful defining image operators and filters for binary and gray-scale images, where it can take advantage of the natural ordering of the sets $\left\lbrace 0,1 \right\rbrace$ and ℝ. For multivariate data, i.e. RGB or hyperspectral images, there is no natural ordering. Thus, other orderings such as reduced orderings (R-orderings) have been proposed. Anyway, all these orderings are based solely on sorting the spectral set of values. Here, we propose to define an ordering based on both, the spectral and the spatial information, by means of a binary partition tree (BPT) representation of images. The proposed ordering aims to find a permutation of the pixel indexes, that is, a sorting of the pixels arrangement in the data matrix. Morphological operations using the proposed ordering are able to enlarge (shrink) spatial structures independently of their spectral values, as far as the spatial structures are encoded in the BPT representation. We provide examples of potential use of the proposed ordering using binary and RGB images.