[Paper Review] Chromatic Alpha Complexes

Sebastiano Cultrera di Montesano, Ondřej Draganov+2 more|arXiv (Cornell University)|Dec 6, 2022

Topological and Geometric Data Analysis4 citations

TL;DR

This paper introduces chromatic alpha complexes, a topological construction that extends the classical alpha complex by incorporating color-coded point sets. By analyzing gradient vectors derived from point configurations and applying linear algebra techniques to ensure linear independence, the authors prove the uniqueness of dual solutions in a generalized Voronoi-type decomposition, enabling robust topological inference in multi-colored point clouds.

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.

Motivation & Objective

• To develop a topological framework that incorporates color information in point cloud data for enhanced geometric and topological inference.
• To generalize the classical alpha complex by introducing color-based partitioning of point sets and associated gradient vector systems.
• To prove the linear independence of gradient vectors derived from multi-colored point configurations, ensuring uniqueness of dual solutions in the complex.
• To establish a robust mathematical foundation for constructing chromatic alpha complexes using vector space techniques and coordinate transformations.
• To enable stable and generic topological constructions in computational topology through color-aware decomposition of point sets.

Proposed method

• Construct a matrix of gradient vectors by combining point differences: for colors 0 to k, subtract a common center point c₀; for colors k+1 to s, subtract a fixed representative point cⱼ from each color class.
• Apply coordinate transformations: replace ∇gₓ with ∇gₓ + ∇hⱼ − ∇h₀ for x ∈ ηⱼ and 1 ≤ j ≤ k, shifting the pivot position in the matrix.
• Replace ∇gₓ with ∇gₓ − ∇g_c₀ for all x ≠ c₀ in η₀, and similarly for other colors j ≥ k+1, using fixed reference points cⱼ to normalize vectors.
• Reorganize the matrix to isolate columns with topmost d coordinates corresponding to vectors proven linearly independent in Lemma LABEL:lem:equivalent_genericity_conditions (b).
• Use the resulting pivot structure to demonstrate linear independence across all columns, ensuring the dual solution is unique.
• Leverage the structure of block matrices and vector space properties to show that the chromatic alpha complex admits a well-defined, unique dual cell decomposition.

Experimental results

Research questions

• RQ1 Can a topological complex be constructed from colored point sets that preserves geometric and topological structure across color classes?
• RQ2 How can gradient vectors from multi-colored point configurations be transformed to ensure linear independence in a matrix representation?
• RQ3 What conditions guarantee the uniqueness of the dual solution in a chromatic alpha complex construction?
• RQ4 How does the choice of reference points cⱼ affect the linear independence of the resulting gradient vectors?
• RQ5 To what extent can coordinate transformations preserve the topological integrity of the alpha complex under color-based decomposition?

Key findings

• The gradient vectors derived from color classes 0 to k, after subtracting a common center point c₀, form a set of linearly independent vectors in the top d coordinates.
• For colors j ≥ k+1, the vectors obtained by subtracting a fixed representative point cⱼ from each point in the color class are linearly independent when combined with the transformed vectors from earlier blocks.
• The matrix of gradient vectors, after appropriate coordinate and pivot transformations, exhibits full column rank, proving linear independence of all columns.
• The pivot structure of the matrix ensures that each column has a unique pivot position, confirming the absence of linear dependencies.
• The construction guarantees a unique dual solution in the chromatic alpha complex, which is essential for stable topological inference.
• The method ensures genericity of the complex by transforming vectors in a way that preserves linear independence under color-based partitioning.