Related papers: Algorithms for Computing Topological Invariants in…
This paper deals with computing topological invariants such as connected components, boundary surface genus, and homology groups. For each input data set, we have designed or implemented algorithms to calculate connected components,…
Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in unorganized clouds in the plane. The holes in a given cloud are quantified by the topological persistence of their boundary…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
The number of holes in a connected component in 2D images is a basic invariant. In this note, a simple formula was proven using our previous results in digital topology (Chen 2004, Chen and Rong (2010). The new is: $h =1+ (|C_4|-|C_2|)/4$,…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
Quantum algorithms provide a potential strategy for solving computational problems that are intractable by classical means. Computing the topological invariants of topological matter is one central problem in research on quantum materials,…
Topological insulators are exotic material that possess conducting surface states protected by the topology of the system. They can be classified in terms of their properties under discrete symmetries and are characterized by topological…
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…
In this note we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures we show that the tile counting group associated to a set $T$ of tiles and a…
We describe algorithms based on invariant theory to solve problems on the geometry of curves, mainly those of genus 2, 3 and 4. New theoretical results building on the first author's PhD thesis are also included.
We propose three fast algorithms for solving the inverse problem of the thermoacoustic tomography corresponding to certain acquisition geometries. Two of these methods are designed to process the measurements done with point-like detectors…
In this letter, we introduce several algorithms for geometry inpainting of 3D point clouds with large holes. The algorithms are examplar-based: hole filling is performed iteratively using templates near the hole boundary to find the best…
In the field of data-driven 3D shape analysis and generation, the estimation of global topological features from localized representations such as point clouds, voxels, and neural implicit fields is a longstanding challenge. This paper…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part we discus the main structures…
This paper presents a new algorithm for filling holes in Level of Detail 2 (LoD2) building mesh models, addressing the challenges posed by geometric inaccuracies and topological errors. Unlike traditional methods that often alter the…
We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains…
Topological quantum matter represents a flexible playground to engineer unconventional excitations. While non-interacting topological single-particle systems have been studied in detail, topology in quantum many-body systems remains an open…
Topological invariants allow to characterize Hamiltonians, predicting the existence of topologically protected in-gap modes. Those invariants can be computed by tracing the evolution of the occupied wavefunctions under twisted boundary…
Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical…
We reexamine fundamental problems from computational geometry in the word RAM model, where input coordinates are integers that fit in a machine word. We develop a new algorithm for offline point location, a two-dimensional analog of sorting…