Abstract: Urban reconstruction from a video captured by a surveying vehicle constitutes a core module of automated mapping. When computational power represents a limited resource and, a detailed map is not the primary goal, the reconstruction can be performed incrementally, from a monocular video, carving a 3D Delaunay triangulation of sparse points; this allows online incremental mapping for tasks such as traversability analysis or obstacle avoidance. To exploit the sharp edges of urban landscape, we propose to use a Delaunay triangulation of Edge-Points, which are the 3D points corresponding to image edges. These points constrain the edges of the 3D Delaunay triangulation to real-world edges. Besides the use of the Edge-Points, a second contribution of this paper is the Inverse Cone Heuristic that preemptively avoids the creation of artifacts in the reconstructed manifold surface. We force the reconstruction of a manifold surface since it makes it possible to apply computer graphics or photometric refinement algorithms to the output mesh. We evaluated our approach on four real sequences of the public available KITTI dataset by comparing the incremental reconstruction against Velodyne measurements.

Abstract: Voronoi diagrams are useful for spatial reasoning, and the robust and efficient construction of the ordinary Voronoi diagram of points is well known. However, its counterpart for circular disks in R2 and spherical balls in R3 remains a challenge. In this article, we propose a topology-oriented incremental algorithm which robustly and efficiently computes a Voronoi diagram by incrementing a new disk generator to an existing one. The key idea is to enforce the convexity of the Voronoi cell corresponding to the incrementing disk so that a simple variation of the algorithm for points proposed by Sugihara in 1992 can be applied. A benchmark using both random and degenerate disks shows that the proposed algorithm is superior to CGAL in both computational efficiency and algorithmic robustness.

Abstract: In contrast to network information flow proposed by Ahlswede et al., Space Information Flow (SIF) studies network coding in a geometric space such as a Euclidean space, in which additional relay nodes are allowed for reducing the communication cost. This work focuses on the problem of mincost multicast network coding in 2-D Euclidean space. We prove several properties of an optimal solution to the problem, and propose a new polynomial-time heuristic algorithm, combining techniques of Delaunay triangulation and non-uniform partitioning. The introduction of Delaunay triangulation aims to adapt the new algorithm to any density distribution of relay and terminal nodes, while non-uniform partitioning can handle any density distribution among terminal nodes. The two complementary techniques work in concert to eliminate an approaching-infinity problem that recent algorithms are known to be prone of, and consequently make the new algorithm fast-converging. Theoretic analysis and simulation results verify the effectiveness of the new algorithm.

Abstract: This article proposed an image segmentation algorithm based on Voronoi Diagram (VD). VD is a well-known technique in computational geometry, which generates clusters of intensity values using information from the vertices of the external boundary of Delaunay triangulation (DT). The image domain is partitioned into a group of sub-regions by Voronoi tessellation, each of which is a component of homogeneous regions. In this way, it is possible to produce segmented image regions. Voronoi-based image segmentation can be extended to RGB images without converting them into grayscale. Its mathematical formulation and practical implementations are also discussed and given. We test the method on and also compare it with K-means algorithms using segmentation examples; the results demonstrate excellent performance and competence of the proposed method.

Abstract: We present a practical reconstruction algorithm that generates a surface triangulation from an input pointset. In the result, the input points appear as vertices of the generated triangulation. The algorithm has several interesting properties: it is very simple to implement, it is time and memory efficient, and it is trivially parallelized. On a standard hardware (core i7, 16Gb) it takes less than 10 seconds to reconstruct a surface from 1 million points, and scales-up to 36 million points (then it takes 350 seconds). On a telephone (ARMV7 Neon, quad core), it takes 55 seconds to reconstruct a surface from 900K points. The algorithm computes the Delaunay triangulation of the input pointset restricted to a "thickening" of the pointset (similarly to several existing methods, like alpha-shapes, crust and co-cone). By considering the problem from the Voronoi point of view (rather than Delaunay), we use a simple observation (radius of security) that makes the problem simpler. The Delaunay triangulation data structure and associated algorithms are replaced by simpler ones (KD-Tree and convex clipping) while the same set of triangles is provably obtained. The restricted Delaunay triangulation can thus be computed by an algorithm that is not longer than 200 lines of code, memory efficient and parallel. The so-computed restricted Delaunay triangulation is finally post-processed to remove the non-manifold triangles that appear in regions where the sampling was not regular/dense enough. Sensitivity to outliers and noise is not addressed here. Noisy inputs need to be pre-processed with a pointset filtering method. In the presented experimental results, we are using two iterations of projection onto the best approximating plane of the 30 nearest neighbours (more sophisticated ones may be used if the input pointset has many outliers).

Abstract: We present a data-parallel algorithm for the construction of Delaunay triangulations on the sphere. Our method combines a variant of the classical Bowyer-Watson point insertion algorithm [2, 14] with the recently published parallelization technique by Jacobsen et al. [7]. It resolves a breakdown situation of the latter approach and is suitable for practical implementation due to its compact formulation. Some complementary aspects are discussed such as the parallel workload, floating-point arithmetics and an application to interpolation of scattered data.

Abstract: Polygons can conveniently represent real world objects. In automatic character recognition, shapes of individual letters are represented by polygons. In robotics, obstacles are represented using polygons. In computer graphics programming, solid objects are represented using polygons on the two dimensional screen. The polygons can be easily manipulated using known mathematical operations. That is the reason for representing real world objects using polygons. However, polygons can be in complicated shapes. Therefore, it is better if there is a way to partition a polygon into smaller pieces. Triangulation is a particular way of doing this from which polygons are partitioned into triangles. The basic triangulation algorithm is widely used in applications where 100% accuracy is necessary. Algorithms with better asymptotic order than the basic triangulation algorithm exist. However they are not 100% accurate and use advanced data structures causing higher memory consumption. This paper proposes a simple, efficient and 100% accurate algorithm which uses lowest amount of memory. The proposed algorithm is more suitable for embedded systems which do not possess large amount of memory. The proposed algorithm was experimentally compared with the basic triangulation algorithm. The experimental results prove that the proposed algorithm is faster than the basic triangulation algorithm.

Abstract: Delaunay triangulation has been widely used in terrain expression,surface reconstruction,digital city and other GIS areasIn Delaunay triangulation network construction algorithm,the commonly seen incremental insertion algorithm is simple and flexible,but its efficiency is bounded by searching for triangleA method based on grid division for building Delaunay triangulation was proposed,this algorithm was optimized especially in point location and local optimization procedure of traditional incremental insertion methodFirst,the discrete point data were sorted by grid division,and then algorithms of the coordinate of triangle area and walking straight were merged,the process of point positioning could significantly shorten the search path and quickly navigate to the insertion point of the target triangleExperimental results showed that the performance of the improved algorithm in both time and space;its executing efficiency was remarkable,and location path was the best

Abstract: When Delaunay triangulation is performed in an incremental fashion, different steps are involved in the process. Within those steps “reconstruction” is the most important stage when a new point is randomly inserted. Although there are several techniques to perform this reconstruction, one of the most relevant is a validation technique called “empty circle”, described by Boris Delone. In this paper, we focus on the use of the Conformal Geometric Algebra (CGA) to perform such validation. In addition, the proposal includes a mathematical environment change to show the advantages of using CGA’s geometric entities and use them inside a module for validating the triangulation.

Abstract: The Delaunay triangulation and the Voronoi diagram are two classic geometric structures in the field of computational geometry Their success can perhaps be attributed to two main reasons: Firstly, there exist practical, efficient algorithms to construct them; and secondly, they have an enormous number of useful applications ranging from meshing and 3D-reconstruction to interpolation

Abstract: Voronoi diagrams are fundamental data structures in computational geometry, with applications in areas such as physics-based simulations For non-Euclidean distances, the Voronoi diagram must be performed over a grid-graph, where the edges encode the required distance information The major bottleneck in this case is a shortest path algorithm that must be computed multiple times during the simulation The authors present a GPU algorithm for solving the shortest path problem from multiple sources using a generalized distance function Their algorithm was designed to leverage the grid-based nature of the underlying graph that represents the deformable objects Experimental results report speed-ups up to 65x over a current reference sequential method

Abstract: A novel mesh deformation technique is developed based on the Delaunay graph mapping method and the inverse distance weighting (IDW) interpolation. The algorithm maintains the advantages of the efficiency of Delaunay-graph-mapping mesh deformation while possess the ability for better controlling the near surface mesh quality. The Delaunay graph is used to divide the mesh domain into a number of sub-domains. On each of the sub-domains, the inverse distance weighting interpolation is applied to build a much smaller sized translation matrix between the original mesh and the deformed mesh, resulting a similar efficiency for the mesh deformation as compared to the fast Delaunay graph mapping method. The paper will show how the near-wall mesh quality is controlled and improved by the new method while the computational time is compared with the original Delaunay graph mapping method.

Abstract: Voronoi diagrams tessellate the space where each cell corresponds to an associated generator under an a priori defined distance and have been extensively used to solve geometric problems of various disciplines. Additively-weighted Voronoi diagrams, also called the Voronoi diagram of disks and spheres, have many critical applications and a few algorithms are known. However, algorithmic robustness remains a major hurdle to use these Voronoi diagrams in practice. There are two important yet different approaches to design robust algorithms: the exact-computation and topology-oriented approaches. The former uses high-precision arithmetic and guarantees the correctness mathematically with the cost of a significant use of computational resources. The latter focuses on topological properties to keep consistency using logical computation rather than numerical computation. In this paper, we present a robust and efficient algorithm for computing the Voronoi diagram of disks using a topology-oriented incremental method. The algorithm is rather simple as it primarily checks topological changes only during each disk is incrementally inserted into a previously constructed Voronoi diagram of some other disks.

Abstract: To solve the problem that the K-means clustering algorithm is over dependent on the K value and the clustering center, we proposed an improved K-means clustering algorithm, VK-means algorithm in this paper. In the initial stage, the Voronoi diagram was adapted in the K-means algorithm to get a better K value and clustering center. By means of weighted average of K-means algorithm, the results of the criterion function is improved. This method could fast convergence and improve performance of the algorithm. The superiority of the improved algorithm was verified by experiments on Weka platform.

Abstract: The stereo vision problem gives rise to two sub-problems: the correspondence and the stereo calibration. Once the sub-problems have been solved, depth is estimated by triangulation. Depth estimation, using stereo images, is based on disparity – the relative displacement between corresponding points – and the camera geometry. Small errors in disparity may produce large errors in depth estimates due to the ill-posedness nature of the stereo vision problem. Moreover, if the camera geometry is unknown, it is estimated using corresponding points. A solution of stereo vision problem may be improved by identifying corresponding points which are inaccurately estimated. In this paper, the classification of a set of estimated corresponding points uses Delaunay triangulation by restricting it to a given subset of estimated corresponding points. Delaunay edges among estimated corresponding points are used to build undirected graphs. Classification criteria based on adjacencies are defined in order to decide whether or not a corresponding point is correctly estimated. Experimental evaluation, using ground truth image sets for quantitative analysis, shown values of specificity around 70% while sensitivity up to 96%.

Abstract: Surface mesh sensitivity in the discrete adjoint framework is efficiently computed using the Delaunay graph method. The method results in an explicit algebraic map between the volume and surface mesh for which the solution of the linear mesh-adjoint system, generally associated with any implicit mesh movement, is not required. This method is used for the computation of the metric term sensitivity as well as a mesh movement. This requires a careful Delaunay decomposition of the computation domain. A new formulation of the Delaunay mesh movement is presented in order to improve the decomposition of the computational domain.

Abstract: A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semi-Riemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles.

Abstract: With a given set of points, Voronoi diagram is a technique of partitioning a plane into set of points having the same closest sites. A number of studies are made on the generalization of Voronoi diagram. These studies are mainly conducted on different kind of distance metric. Some studies are also made upon Voronoi diagram of sites those are not necessarily points. The current researchers present a new generalization of the construction of Voronoi diagram in the plane with axes parallel rectangles. Here, the interest has been shifted from points to simple convex orthogonal blocks i.e. rectangles. The current researchers have computed the partition of the set of the rectangles in the plane into sets of lines having the same closest sites (where sites are rectangles in the plane. This paper represents a method to find the Voronoi diagram, which considered rectangles as sites. Prototype implementation has been done and the results are quite encouraging. The algorithm is easy to implement and finds application in VLSI layouts.

Abstract: We trace movements of certain points in space-time along their corresponding continuous path. We partition the space at every moment of time using alpha-Complexes, Voronoi medusa is then the collection or union of restricted Voronoi cells at every moment in time. We can imagine them as a four dimensional structure formed when three dimensional restricted Voronoi cells sweeps continuously through the extra dimension of time. Similarly Delaunay medusa is the collection of the corresponding Delaunay triangulations at each moment in time. In this article we prove that these two structures are homotopic.

Abstract: In this paper, we present a completely automatic method for multi-view multi-exposure image fusion. The technique adopts the normalized cross-correlation NCC as the measurement of the similarity of interest points. With the matched feature points, we divide images into a set of triangles by Delaunay triangulation. Then we apply affine transformation to each matched triangle pairs respectively to get the registration of multi-view images. After images aligned, we partition the image domain into uniformed regions and select the images that provides the most information with certain blocks. The selected images are fused together under monotonically blending functions.

Abstract: Conventional skeleton extraction method usually limits its applicability due to the close curve or complete boundary constraint. Unfortunately in cluttered image or natural image, the close curve or complete boundary constraint happens all the time. Therefore a skeleton extraction algorithm with an incomplete boundary condition is proposed. The proposed method is based on the Delaunay triangulation of some sampled points from the input image. By using the proposed method, the incomplete boundary condition can be relaxed to some extends.

Abstract: The restricted Delaunay triangulation can be conceived as an operator that takes as input a k-manifold (typically smooth) embedded in R^d and a set of points sampled with sufficient density on that manifold, and produces as output a k-dimensional triangulation of the manifold, the input points serving as its vertices. What happens if we feed that triangulation back into the operator, replacing the original manifold, while retaining the same set of input points? If k = 2 and the sample points are sufficiently dense, we obtain another triangulation of the manifold. Iterating this process, we soon reach an iteration for which the input and output triangulations are the same. We call this triangulation a fixed point of the restricted Delaunay triangulation operator. With this observation, and a new test for distinguishing "critical points" near the manifold from those near its medial axis, we develop a provably good surface reconstruction algorithm for R^3 with unusually modest sampling requirements. We develop a similar algorithm for constructing a simplicial complex that models a 2-manifold embedded in a high-dimensional space R^d, also with modest sampling requirements (especially compared to algorithms that depend on sliver exudation). The latter algorithm builds a non-manifold representation similar to the flow complex, but made solely of Delaunay simplices. The algorithm avoids the curse of dimensionality: its running time is polynomial, not exponential, in d.

Abstract: This paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessellations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams, a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries.