Linear Time Recognition Algorithms for Topological Invariants in 3D

TL;DR

This paper introduces linear time algorithms for recognizing topological invariants like genus and homology groups in 3D, facilitating advanced pattern recognition in 3D images, especially in medical imaging.

Contribution

It presents novel linear time algorithms for computing topological invariants in 3D digital images, leveraging discrete geometry and Alexander duality.

Findings

Algorithms run in linear time for 3D topological invariants.

Effective recognition of genus and homology groups in 3D images.

Applications demonstrated in 3D medical image analysis.

Abstract

In this paper, we design linear time algorithms to recognize and determine topological invariants such as the genus and homology groups in 3D. These properties can be used to identify patterns in 3D image recognition. This has tremendous amount of applications in 3D medical image analysis. Our method is based on cubical images with direct adjacency, also called (6,26)-connectivity images in discrete geometry. According to the fact that there are only six types of local surface points in 3D and a discrete version of the well-known Gauss-Bonnett Theorem in differential geometry, we first determine the genus of a closed 2D-connected component (a closed digital surface). Then, we use Alexander duality to obtain the homology groups of a 3D object in 3D space.