217 research outputs found

    A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations

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    Computing Delaunay triangulations in Rd\mathbb{R}^d involves evaluating the so-called in\_sphere predicate that determines if a point xx lies inside, on or outside the sphere circumscribing d+1d+1 points p0,
,pdp_0,\ldots ,p_d. This predicate reduces to evaluating the sign of a multivariate polynomial of degree d+2d+2 in the coordinates of the points x, p0, 
, pdx, \, p_0,\, \ldots,\, p_d. Despite much progress on exact geometric computing, the fact that the degree of the polynomial increases with dd 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 Wit(L,W)\mathrm{Wit} (L,W) is defined from two sets LL and WW in some metric space XX: a finite set of points LL on which the complex is built, and a set WW of witnesses that serves as an approximation of XX. A fundamental result of de Silva states that Wit(L,W)=Del(L)\mathrm{Wit}(L,W)=\mathrm{Del} (L) if W=X=RdW=X=\mathbb{R}^d. In this paper, we give conditions on LL that ensure that the witness complex and the Delaunay triangulation coincide when WW is a finite set, and we introduce a new perturbation scheme to compute a perturbed set Lâ€ČL' close to LL such that Del(Lâ€Č)=wit(Lâ€Č,W)\mathrm{Del} (L')= \mathrm{wit} (L', W). 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 ∣W∣|W|. 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

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    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 Lâ€Č 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

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    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

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    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

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    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

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    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

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    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

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    The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” 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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