Related papers: The Topological Complexity of Spaces of Digital Jo…
Manifolds occur naturally as configuration spaces of robotic systems. They provide global descriptions of local coordinate systems that are common tools in expressing positions of robots. The purpose of this survey is threefold. Firstly, we…
Knowledge of the road network topology is crucial for autonomous planning and navigation. Yet, recovering such topology from a single image has only been explored in part. Furthermore, it needs to refer to the ground plane, where also the…
Numerical computation of shortest paths or geodesics on curved domains, as well as the associated geodesic distance, arises in a broad range of applications across digital geometry processing, scientific computing, computer graphics, and…
This book provides an introduction to the theory of digital (molecular) spaces (TDS). Digital spaces are combinatorial models of continuous spaces. TDS is one of alternative branches of digital topology that studies constructing and…
The matching problem for a given Jordan curve in the complex plane asks to find two nonconstant functions, one analytic in the bounded complementary component of the curve and the other analytic in the unbounded complementary component of…
This paper introduces the structure and axioms for a temporal digital topology (TDT) with the focus on digital connectedness, continuity and proximities in TDT spaces. Results are given for temporal digital adjacencies, connectedness and…
Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between…
This paper gives a concise proof of the Jordan curve theorem on discrete surfaces. We also embed the discrete surface in the 2D plane to prove the original version of the Jordan curve theorem. This paper is a simple version of L. Chen, Note…
We study the topology of the complex points of the algebraic loop space of a smooth curve.
When one considers the collection $\mathcal{H}(\mathbb{R}^n)$ of all compact subsets of $\mathbb{R}^n$ and equip it with a topology, many questions can be asked about the topological space one ends up with. This is an example of a…
We study an elementary problem of topological robotics: rotation of a line, which is fixed by a revolving joint at a base point: one wants to bring the line from its initial position to a final position by a continuous motion in the space.…
We study the topological construction called Mapper in the context of simply connected domains, in particular on images. The Mapper construction can be considered as a generalization for contour, split, and joint trees on simply connected…
Topological learning is a wide research area aiming at uncovering the mutual spatial relationships between the elements of a set. Some of the most common and oldest approaches involve the use of unsupervised competitive neural networks.…
In this survey, we explore recent literature on finding the cores of higher graphs using geometric and topological means. We study graphs, hypergraphs, and simplicial complexes, all of which are models of higher graphs. We study the notion…
Jordan geometries are defined as spaces equipped with point reflections depending on triples of points, exchanging two of the points and fixing the third. In a similar way, symmetric spaces have been defined by Loos (Symmetric Spaces I,…
The analysis of digital images using homological procedures is an outstanding topic in the area of Computational Algebraic Topology. In this paper, we describe a certified reduction strategy to deal with digital images, but preserving their…
Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a…
The topology of digital images has been studied much in recent years, but no attempt has been made to exhaustively catalog the structure of binary images of small numbers of points. We produce enumerations of several classes of digital…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
We design a motion planning algorithm to coordinate the movements of two robots along a figure eight track, in such a way that no collisions occur. We use a topological approach to robot motion planning that relates instabilities in motion…