Chromatic Alpha Complexes

TL;DR

This work develops a multicolor topological framework for analyzing spatial interactions among colored point sets by extending alpha complexes to the chromatic setting via the chromatic Delaunay mosaic and a chromatic radius function. It introduces a $6$-pack of persistence diagrams to capture the interaction between color classes, proves that the chromatic radius function is a generalized discrete Morse function under chromatic genericity, and provides a linear-time algorithm (in fixed dimension and colors) to compute the chromatic alpha complexes. A lifting construction ties the chromatic theory to ordinary Delaunay complexes, enabling efficient computation and interpretation via color-augmented embeddings. The paper also presents a comprehensive persistent-homology framework for chromatic complexes, explores relations across diagrams and $6$-packs, and demonstrates a tri-chromatic case study with concrete mingling patterns, offering practical tools for applications in spatial biology and related fields.

Abstract

Motivated by applications in the medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.

Figures (15)

• Figure 1: Mingling patterns distinguished by the number of colors forming a cycle and the number of additional colors filling this cycle. The drawings are caricatures of similar patterns for cycles different from circles and fillings different from disks. The patterns are but a first attempt to differentiate types of interactions, and they are by no means precise or exhaustive. For example, two additional colors can fill a cycle in at least two different ways (see the pattern of type 1+2): in a collaboration as suggested in the drawing, or each individually, like two different patterns of type 1+1.
• Figure 2: On the left: a set, $A \subseteq {\mathbb R}{\hbox{${\mathbb R}$}}^2$, together with its Voronoi tessellation, ${\rm Vor}_{}{({A})}{\hbox{${\rm Vor}_{}{({A})}$}}$, and one Voronoi ball highlighted. On the right: the union of disks, $\bigcup_{a\in A} {B}_{r}{({a})}{\hbox{${B}_{r}{({a})}$}}$, with the alpha complex, ${\rm Alf}_{r}{({A})}{\hbox{${\rm Alf}_{r}{({A})}$}}$, superimposed.
• Figure 3: Two empty stacks in ${\mathbb R}{\hbox{${\mathbb R}$}}^2$ that pass through one blue point, two green points, and one orange point forming a simplex $\nu\in{\rm Del}_{}{({\chi})}{\hbox{${\rm Del}_{}{({\chi})}$}}$. (In fact, the stack on the right passes through two orange points, and it also passes through the one orange point that the left orange circle passes through.) The set of centers of all empty stacks that pass through these four points is the intersection of three Voronoi cells: a blue $2$-cell, a green $1$-cell, and an orange $2$-cell. The right panel shows the smallest empty stack in this collection: its center lies on the boundary of the intersection of Voronoi cells, which is the reason why one of its circles passes through an extra point.
• Figure 4: An obtuse triangle with two blue points and an orange point at the obtuse angle. The smallest empty sphere that passes through the three points on the left has strictly larger radius than the smallest empty stack that passes through the three points on the right. Therefore, the triangle belongs to both, the Delaunay complex and the chromatic Delaunay complex, but it has a different value under the two radius functions.
• Figure 5: On the left: a chromatic set together with the Voronoi tessellations of the blue and orange points overlaid, and one chromatic Voronoi ball highlighted. On the right: the union of blue and the union of orange disks.

Theorems & Definitions (39)

• Definition 2.1: Conventional Genericity
• Lemma 2.2
• proof
• Definition 2.3
• Proposition 2.4: BaEd17
• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.3
• proof