Homological spanning forest framework for 2D image analysis

A 2D topology-based digital image processing framework is presented here. This framework consists of the computation of a flexible geometric graph-based structure, starting from a raster representation of a digital image I. This structure is called Homological Spanning Forest (HSF for short), and it is built on a cell complex associated to I. The HSF framework allows an efficient and accurate topological analysis of regions of interest (ROIs) by using a four-level architecture. By topological analysis, we mean not only the computation of Euler characteristic, genus or Betti numbers, but also advanced computational algebraic topological information derived from homological classification of cycles. An initial HSF representation can be modified to obtain a different one, in which ROIs are almost isolated and ready to be topologically analyzed. The HSF framework is susceptible of being parallelized and generalized to higher dimensions

Homological Spanning Forests for Discrete Objects

Computing and representing topological information form an important part in many applications such as image representation and compression, classification, pattern recognition, geometric modelling, etc. The homology of digital objects is an algebraic notion that provides a concise description of their topology in terms of connected components, tunnels and cavities. The purpose of this work is to develop a theoretical and practical frame- work for efficiently extracting and exploiting useful homological information in the context of nD digital images. To achieve this goal, we intend to combine known techniques in algebraic topology, and image processing. The main notion created for this purpose consists of a combinatorial representation called Homological Spanning Forest (or HSF, for short) of a digital object or a digital image. This new model is composed of a set of directed forests, which can be constructed under an underlying cell complex format of the image. HSF’s are based on the algebraic concept of chain homotopies and they can be considered as a suitable generalization to higher dimensional cell complexes of the topological meaning of a spanning tree of a geometric graph. Based on the HSF representation, we present here a 2D homology-based framework for sequential and parallel digital image processing.Premio Extraordinario de Doctorado U

Topological Signals of Singularities in Ricci Flow

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.Comment: 24 pages, 14 figure

Efficient topology-aware simplification of large triangulated terrains

A common first step in the terrain processing pipeline of large Triangulated Irregular Networks (TINs) is simplifying the TIN to make it manageable for further processing. The major problem with TIN simplification algorithms is that they create or remove critical points in an uncontrolled way. Topology-aware operators have been defined to solve this issue by coarsening a TIN without affecting the topology of its underlying terrain, i.e., without modifying critical simplices describing pits, saddles, peaks, and their connectivity. While effective, existing algorithms are sequential in nature and are not scalable enough to perform well with large terrains on multicore systems. Here, we consider the problem of topology-aware simplification of very large meshes. We define a topology-aware simplification algorithm on a compact and distributed data structure for triangle meshes, namely the Terrain trees. Terrain trees reduce both the memory and time requirements of the simplification procedure by adopting a batched processing strategy of the mesh elements. Furthermore, we define a new parallel topology-aware simplification algorithm that takes advantage of the spatial domain decomposition at the basis of Terrain trees. Scalability and efficiency are experimentally demonstrated on real-world TINs originated from topographic and bathymetric LiDAR data. Our experiments show that topology-aware simplification on Terrain trees uses 40% less memory and half the time than the same approach implemented on the most compact and efficient connectivity-based data structure for TINs. Beyond that, our parallel algorithm on the Terrain trees reaches a 12x speedup when using 20 threads