217 research outputs found
A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations
Computing Delaunay triangulations in involves evaluating the
so-called in\_sphere predicate that determines if a point lies inside, on
or outside the sphere circumscribing points . This
predicate reduces to evaluating the sign of a multivariate polynomial of degree
in the coordinates of the points . Despite
much progress on exact geometric computing, the fact that the degree of the
polynomial increases with makes the evaluation of the sign of such a
polynomial problematic except in very low dimensions. In this paper, we propose
a new approach that is based on the witness complex, a weak form of the
Delaunay complex introduced by Carlsson and de Silva. The witness complex
is defined from two sets and in some metric space
: a finite set of points on which the complex is built, and a set of
witnesses that serves as an approximation of . A fundamental result of de
Silva states that if .
In this paper, we give conditions on that ensure that the witness complex
and the Delaunay triangulation coincide when is a finite set, and we
introduce a new perturbation scheme to compute a perturbed set close to
such that . Our perturbation
algorithm is a geometric application of the Moser-Tardos constructive proof of
the Lov\'asz local lemma. The only numerical operations we use are (squared)
distance comparisons (i.e., predicates of degree 2). The time-complexity of the
algorithm is sublinear in . Interestingly, although the algorithm does not
compute any measure of simplex quality, a lower bound on the thickness of the
output simplices can be guaranteed.Comment: 24 page
A Probabilistic Approach to Reducing Algebraic Complexity of Delaunay Triangulations
International audienceWe propose algorithms to compute the Delaunay triangulation of a point set L using only (squared) distance comparisons (i.e., predicates of degree 2). Our approach is based on the witness complex, a weak form of the Delaunay complex introduced by Carlsson and de Silva. We give conditions that ensure that the witness complex and the Delaunay triangulation coincide and we introduce a new perturbation scheme to compute a perturbed set LâČ close to L such that the Delaunay triangulation and the witness complex coincide. Our perturbation algorithm is a geometric application of the Moser-Tardos constructive proof of the LovĂĄsz local lemma
Computational Geometric and Algebraic Topology
Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity.
At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field
Only distances are required to reconstruct submanifolds
In this paper, we give the first algorithm that outputs a faithful
reconstruction of a submanifold of Euclidean space without maintaining or even
constructing complicated data structures such as Voronoi diagrams or Delaunay
complexes. Our algorithm uses the witness complex and relies on the stability
of power protection, a notion introduced in this paper. The complexity of the
algorithm depends exponentially on the intrinsic dimension of the manifold,
rather than the dimension of ambient space, and linearly on the dimension of
the ambient space. Another interesting feature of this work is that no explicit
coordinates of the points in the point sample is needed. The algorithm only
needs the distance matrix as input, i.e., only distance between points in the
point sample as input.Comment: Major revision, 16 figures, 47 page
Active Exploration Using Gaussian Random Fields and Gaussian Process Implicit Surfaces
In this work we study the problem of exploring surfaces and building compact
3D representations of the environment surrounding a robot through active
perception. We propose an online probabilistic framework that merges visual and
tactile measurements using Gaussian Random Field and Gaussian Process Implicit
Surfaces. The system investigates incomplete point clouds in order to find a
small set of regions of interest which are then physically explored with a
robotic arm equipped with tactile sensors. We show experimental results
obtained using a PrimeSense camera, a Kinova Jaco2 robotic arm and Optoforce
sensors on different scenarios. We then demonstrate how to use the online
framework for object detection and terrain classification.Comment: 8 pages, 6 figures, external contents (https://youtu.be/0-UlFRQT0JI
Distributed Graph Isomorphism using Quantum Walks
Graph isomorphism being an NP problem, most of the systems that solves the graph isomorphism are constrained with some classes of the graph, and do not work for all types of graphs in polynomial time. We exploited the two particle quantum walks on different classes of graphs including strongly regular graphs which are co-spectral in nature. We simulated two particle quantum walks on graph using distributed algorithm. To show the effectiveness of the technique, we applied it to the large graphs derived from images using Delauney triangulation. The results show a remarkable speedup for large data. The two-particle quantum walks is implemented in map-reduce programming technique which scales the computation as the cluster get scaled to account Big data. We checked the isomorphism of the graphs with upto 100 vertices in polynomial time. The system is scalable to accept big inputs from any other domain in graph format.
DOI: 10.17762/ijritcc2321-8169.15021
Purposive three-dimensional reconstruction by means of a controlled environment
Retrieving 3D data using imaging devices is a relevant task for many applications in medical imaging, surveillance, industrial quality control, and others. As soon as we gain procedural control over parameters of the imaging device, we encounter the necessity of well-defined reconstruction goals and we need methods to achieve them. Hence, we enter next-best-view planning. In this work, we present a formalization of the abstract view planning problem and deal with different planning aspects, whereat we focus on using an intensity camera without active illumination. As one aspect of view planning, employing a controlled environment also provides the planning and reconstruction methods with additional information. We incorporate the additional knowledge of camera parameters into the Kanade-Lucas-Tomasi method used for feature tracking. The resulting Guided KLT tracking method benefits from a constrained optimization space and yields improved accuracy while regarding the uncertainty of the additional input. Serving other planning tasks dealing with known objects, we propose a method for coarse registration of 3D surface triangulations. By the means of exact surface moments of surface triangulations we establish invariant surface descriptors based on moment invariants. These descriptors allow to tackle tasks of surface registration, classification, retrieval, and clustering, which are also relevant to view planning. In the main part of this work, we present a modular, online approach to view planning for 3D reconstruction. Based on the outcome of the Guided KLT tracking, we design a planning module for accuracy optimization with respect to an extended E-criterion. Further planning modules endow non-discrete surface estimation and visibility analysis. The modular nature of the proposed planning system allows to address a wide range of specific instances of view planning. The theoretical findings in this work are underlined by experiments evaluating the relevant terms
Geometric and Topological Combinatorics
The 2007 Oberwolfach meeting âGeometric and Topological Combinatoricsâ presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions
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