Morphological Semigroups and Scale-Spaces on Ultrametric Spaces

Abstract

Abstract. Ultrametric spaces are the natural mathematical structure to deal with data embedded into a hierarchical representation. This kind of representations is ubiquitous in morphological image processing, from pyramids of nested partitions to more abstract dendograms from minimum spanning trees. This paper is a formal study of morphological operators for functions dened on ultrametric spaces. First, the notion of ultrametric structuring function is introduced. Then, using as basic ingredient the convolution in (max,min)-algebra, the multi-scale ultrametric dilation and erosion are dened and their semigroup properties are stated. It is proved in particular that they are idempotent operators and consequently they are algebraically ultrametric closing and opening too. Some preliminary examples illustrate the behavior and practical interest of ultrametric dilations/erosions.

Eigenfunctions of Ultrametric Morphological Openings and Closings

“…The property d * (x, y) • min,max d * (x, y) = d * (x, y) is the basic ingredient in [2] to prove the existency of the supremal ultrametric morphological semigroups and the fact that ultrametric erosion and dilation are idempotent operators. Let us also notice that using expression (13), one has…”

“…In this work, we are interested on relating the notion of spectral analysis in maxmin algebra to ultrametric morphological operators [2]. Indeed, morphological semigroups in ultrametric spaces are essentially based on that algebra.…”

“…Before going further, let us recall basic facts on ultrametric morphological openings and closings. One can refer to [2] for details.…”

Some open questions on morphological operators and representations in the deep learning era

“…When an image, or any other element in a dataset embedded into an edge-weighted graph, is represented by a dendrogram, the use of deep learning techniques and in particular of ultrametric convolutional neural networks requires to be have a specific definition of the typical layers for dendrograms: their structure is very different from metric graphs. Classical image/data processing opeartors and transforms, including Gaussian and Laplacian operators, convolution, morphological semigroups, etc., have been formulated on ultrametric spaces [4,5], and the basic ingredients are the ultrametric distance and the distribution of diameters of the ultrametric balls.…”

Hierarchical Laplacian and Its Spectrum in Ultrametric Image Processing

“…Once an image has been endowed with a hierarchical structure (i.e., the image domain is an ultrametric space), the image can be not only segmented, but also ltered out, enhanced and so on, according to such representation. In this context, we have introduced the corresponding ultrametric morphological semigroups and scalespaces [1]. This paper is a step forwards in the program of revisiting classical image/data processing on ultrametric representations.…”