Grayscale and Color Image Segmentation using Computational Topology

In this paper, we present image segmentation algorithms based on tools from computational algebraic topology and Morse theory. We build our implementations on a very general clustering algorithm [4], developed by Chazal et al., which has been adapted for both grayscale and color image segmentation. By building up a simplicial complex incrementally filtering its simplices by values of a scalar function we can assign a quantity called persistence to its topological features, measuring their ”lifetime” in the construction. Combined with concepts from Morse theory, this allows us to construct and simplify a watershedtype segmentation of the complex using a Union-Find algorithm, guided by a single intuitive scalar parameter and supported by theoretical guarantees for topological consistency and robustness. In the case of grayscale images, the complex is an 8-connected pixel adjacency graph which is filtered by pixel values or the absolute value of the image gradient. For color images a point cloud in an appropriate color space is clustered by filtering a Vietoris-Rips neigbourhood graph via Gaussian density estimation and taking spatial proximity into account.