Delaunay Triangulation

A new optimization method is proposed for solving an inverse problem concerning the shape and topology of the inductors used in the electromagnetic casting technique of the metallurgical industry. The method is based on an sparse convex quadratic programming version of a recently proposed topology optimization formulation of the inverse electromagnetic casting problem. Regular 0-1 solutions are found by adding to the original Kohn-Vogelius objective function an appropriate penalty term that preserves the quadratic programming structure of the problem, allowing the use of efficient interior-point algorithms. Results for some numerical examples are presented, showing that the technique proposed is effective and can successfully find inductors of optimal shape and topology.

CONSTRAINED DELAUNAY TRIANGULATION FOR DIAGNOSIS AND GRADING OF COLON CANCER Süleyman Tuncer Erdoğan M.S. in Computer Engineering Supervisor: Assist. Prof. Dr. Çiğdem Gündüz Demir July, 2009 In our century, the increasing rate of cancer incidents makes it inevitable to employ computerized tools that aim to help pathologists more accurately diagnose and grade cancerous tissues. These mathematical tools offer more stable and objective frameworks, which cause a reduced rate of intraand inter-observer variability. There has been a large set of studies on the subject of automated cancer diagnosis/grading, especially based on textural and/or structural tissue analysis. Although the previous structural approaches show promising results for different types of tissues, they are still unable to make use of the potential information that is provided by tissue components rather than cell nuclei. However, this additional information is one of the major information sources for the tissue types wi...

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This paper presents an efficient technique for generating adaptive triangular meshes from range images. The algorithm consists of two stages. First, a user-defined number of points is adaptively sampled from the given range image. Those points are chosen by taking into account the surface shapes represented in the range image in such a way that points tend to group in areas of high curvature and to disperse in low-variation regions. This selection process is done through a noniterative, inherently parallel algorithm in order to gain efficiency. Once the image has been subsampled, the second stage applies a two and one half-dimensional Delaunay triangulation to obtain an initial triangular mesh. To favor the preservation of surface and orientation discontinuities (jump and crease edges) present in the original range image, the aforementioned triangular mesh is iteratively modified by applying an efficient edge flipping technique. Results with real range images show accurate triangular approximations of the given range images with low processing times.

This paper * presents a GPU parallel algorithm to generate a new kind of polygonal meshes obtained from Delaunay triangulations. To generate the polygonal mesh, the algorithm first uses a classification system to label each edge of an input triangulation; second it builds polygons (simple or not) from terminal-edge regions using the label system, and third it transforms each non-simple polygon from the previous phase into simple ones, convex or not convex polygons. We show some preliminary experiments to test the scalability of the algorithm and compare it with the sequential version. We also run a very simple test to show that these meshes can be useful for the virtual element method.

This dissertation describes a new methodology for multi-modal (2-D + 3-D) face modeling and recognition. There are advantages in using each modality for face recognition. For example, the problems of pose variation and illumination condition, which cannot be resolved easily by using the 2-D data, can be handled by using the 3-D data. However, texture, which is provided by 2-D data, is an important cue that cannot be ignored. Therefore, we use both the 2-D and 3-D modalities for face recognition and fuse the results of face recognition by each modality to boost the overall performance of the system. In this dissertation, we consider two different cases for multi-modal face modeling and recognition. In the first case, the 2-D and 3-D data are registered. In this case we develop a unified graph model called Attributed Relational Graph (ARG) for face modeling and recognition. Based on the ARG model, the 2-D and 3-D data are included in a single model. The developed ARG model consists of nodes, edges, and mutual relations. The nodes of the graph correspond to the landmark points that are Behzad M. Dogahe, Dr. Hongsheng Zhang, Muhammad Rushdi, and Dr. Nuno Gracias for their collaboration during the completion of this dissertation. In addition, I want to thank Dr. Tapia, Dr. Asfour, and Mr. Ali Habashi for their support and help during my study at the University of Miami. And last but not least, I am indebted to my landlord, Mr. Carlos Fernandez, and Ms. Maria C. Fernandez for their kind and generous hospitality during the past four years that I lived with them. Specially, Maria treated me as her son when I became sick and cooked delicious Mexican food for me. v

A new numerical methodology to solve the 3D Navier-Stokes equations for incompressible fluids within complex boundaries and unstructured body-fitted tetrahedral mesh is presented and validated with three literature and one real-case tests. We apply a fractional time step procedure where a predictor and a corrector problem are sequentially solved. The predictor step is solved applying the MAST (Marching in Space and Time) procedure, which explicitly handles the non-linear terms in the momentum equations, allowing numerical stability for Courant number greater than one. Correction steps are solved by a Mixed Hybrid Finite Elements discretization that assumes positive distances among tetrahedrons circumcentres. In 3D problems, non-Delaunay meshes are provided by most of the mesh generators. To maintain good matrix properties for non-Delaunay meshes, a continuity equation is integrated over each tetrahedron, but the momentum equations are integrated over clusters of tetrahedrons, such that each external face shared by two clusters belongs to two tetrahedrons whose circumcentres have positive distance. A numerical procedure is proposed to compute the velocities inside clusters with more than one tetrahedron. Model preserves mass balance at the machine error and there is no need to compute pressure at each time iteration, but only at target simulation times.

Transport problems occurring in porous media and including convection, diffusion and chemical reactions, can be well represented by systems of Partial Differential Equations. In this paper, a numerical procedure is proposed for the fast and robust solution of flow and transport problems in 2D heterogeneous saturated media. The governing equations are spatially discretized with unstructured triangular meshes that must satisfy the Delaunay condition. The solution of the flow problem is split from the solution of the transport problem and it is obtained with an approach similar to the Mixed Hybrid Finite Elements method, that always guarantees the M-property of the resulting linear system. The transport problem is solved applying a prediction/correction procedure. The prediction step analytically solves the convective/reactive components in the context of a MAST Finite Volume scheme. The correction step computes the anisotropic diffusive components in the context of a recently proposed Finite Elements scheme. Massa balance is locally and globally satisfied in all the solution steps. Convergence order and computational costs are investigated and model results are compared with literature ones.

A new methodology for the solution of the 2D diffusive shallow water equations over Delaunay unstructured triangular meshes is presented. Before developing the new algorithm, the following question is addressed: it is worth developing and using a simplified shallow water model, when well established algorithms for the solution of the complete one do exist? The governing Partial Differential Equations are discretized using a procedure similar to the linear conforming Finite Element Galerkin scheme, with a different flux formulation and a special flux treatment that requires Delaunay triangulation but entire solution monotonicity. A simple mesh adjustment is suggested, that attains the Delaunay condition for all the triangle sides without changing the original nodes location and also maintains the internal boundaries. The original governing system is solved applying a fractional time step procedure, that solves consecutively a convective prediction system and a diffusive correction system. The non linear components of the problem are concentrated in the prediction step, while the correction step leads to the solution of a linear system of the order of the number of computational cells. A semi-analytical procedure is applied for the solution of the prediction step. The discretized formulation of the governing equations allows to handle also wetting and drying processes without any additional specific treatment. Local energy dissipations, mainly the effect of vertical walls and hydraulic jumps, can be easily included in the model. Several numerical experiments have been carried out in order to test (1) the stability of the proposed model with regard to the size of the Courant number and to the mesh irregularity, (2) its computational performance, (3) the convergence order by means of mesh refinement. The model results are also compared with the results obtained by a fully dynamic model. Finally, the application to a real field case with a Venturi channel is presented.

In the present paper it is first shown that, due to their structure, the general governing equations of uncompressible real fluids can be regarded as an ''anisotropic'' potential flow problem and closed streamlines cannot occur at any time. For a discretized velocity field, a fast iterative procedure is proposed to order the computational elements at the beginning of each time level, allowing a sequential solution element by element of the advection problem. Some closed circuits could appear due to the discretization error and the elements involved in these circuits could not be ordered. We prove in the paper that the total flux of these not ordered elements goes to zero by refining the computational mesh and that it is possible to order all the remaining elements by neglecting the minimum inter-element flux inside each circuit, with a very small resulting error. The methodology is then applied to the solution of the 2D shallow water equations. The governing Partial Differential Equations are discretized over a generally unstructured triangular mesh, which attains the generalised Delaunay property. Solution is obtained applying a prediction-correction time step procedure. The prediction problem is solved applying a MArching in Space and Time (MAST) procedure, where the computational elements are required to be ordered and explicitly solved. In the correction step, a large linear well-conditioned system is solved. Model results are compared with experimental data and other numerical literature results. Computational costs have been estimated and the convergence order has been investigated according to a known exact solution.

There are a number of great attempts to develop an indoor navigation that provide the most optimal path and guidance. Finding a way in large buildings can be a challenging task. In order to represent the real situation to a maximum extent, a representation of the whole room as one single indivisible object is not enough as such representation is very abstract and this could make the navigation difficult and may result into inefficient route planning. In order to provide a smooth navigation path, the presence of humans within the indoor environment and the natural movement of individuals should be taken into consideration. In this paper a two-step indoor space subdivision for indoor navigation is described. Firstly, the indoor space is subdivided into navigable and non-navigable areas considering human perceptions of the environment and human behaviour. Secondly, the navigable space is subdivided applying a constrained Delaunay triangulation. Finally, the guidelines for generation of the navigation network and verification of the proposed model are presented.

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter-Weyl groups W(A) and W(D) are constructed. The face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell, n-simplex, n-octahedron (cross polytope), n-cube and n-hemicube and their volumes are calculated. The Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is shown that the Delone cells centred at the origin of the root lattice A are the polytopes of the fundamental weights ω, ω,…, ω and the Delone cells of the root lattice D are the polytopes obtained from the weights ω, ω and ω. A simple mechanism explains the tessellation of the root lattice by Delone cells. It is proved that the (n-1)-facet of the Voronoi cell of the root lattice A is an (n-...

The World Health Organization (WHO) has identified coronavirus disease (COVID-19), as a global pandemic due to its quick global spread to more than 183 countries. Many countries have used movement control orders (MCO) and high alert levels to halt the spread. The primary goal of this research is to provide a geofencing architecture that is specially tailored to the MCO's standard requirement for monitoring an individual's whereabouts during a lockdown. Whenever an individual tests Corona positive, Geofencing uses technology to notify an anticipated network of people who may be affected and to enable traceability for potential patients. Computational techniques such as Delaunay triangulation (inpolygon) and triangle weight characterization (inside polygon) are applied to analyze the geographical boundary in which the patient is isolated. Convex hull, on the other hand, is a better technique than computational algorithms. It is considered the best mathematical technique because it takes the least amount of time (0.014985 sec) to detect the patient within the geofence layer and has the lowest standard deviation when compared with the other computational techniques.

This paper proposes a method for the deployment of a multi-agent system of unmanned aerial vehicles (UAVs) as a shield with potential applications in the protection of infrastructures. The shield shape is modeled as a quadric surface in the 3D space. To design the desired formation (target distances between agents and interconnections), an algorithm is proposed where the input parameters are just the parametrization of the quadric and the number of agents of the system. This algorithm guarantees that the agents are almost uniformly distributed over the virtual surface and that the topology is a Delaunay triangulation. Moreover, a new method is proposed to check if the resulting triangulation meets that condition and is executed locally. Because this topology ensures that the formation is rigid, a distributed control law based on the gradient of a potential function is proposed to acquire the desired shield shape and proofs of stability are provided. Finally, simulation and experimental results illustrate the effectiveness of the proposed approach.

We give an expected-case analysis of Delaunay triangulations. To avoid edge effects we consider a unit-intensity Poisson process in Euclidean d-space, and then limit attention to the portion of the triangulation within a cube of side n 1/d . For d equal to two, we calculate the expected maximum edge length, the expected minimum and maximum angles, and the average aspect ratio of a triangle. We also show that in any fixed dimension the expected maximum vertex degree is Θ(log n/ log log n). Altogether our results provide some measure of the suitability of the Delaunay triangulation for certain applications, such as interpolation and mesh generation.

Surface meshing plays a fundamental role in graphics and visualization. Many geometric processing tasks involve solving geometric PDEs on meshes. The numerical stability, convergence rates and approximation errors are largely determined by the mesh qualities. In practice, Delaunay refinement algorithms offer satisfactory solutions to high quality mesh generations. The theoretical proofs for volume based and surface based Delaunay refinement algorithms have been established, but those for conformal parameterization based ones remain wide open. This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms. Given a metric surface, the proposed approach triangulates its conformal uniformization domain by the planar Delaunay refinement algorithms, and produces a high quality mesh. We give explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh. In contrast to the conventional results based on volumetric Delaunay refinement, our stronger estimates are independent of the mesh structure and directly guarantee the convergence of curvature measures. Meanwhile, our result on Gaussian curvature measure is intrinsic to the Riemannian metric and independent of the embedding. In practice, our meshing algorithm is much easier to implement and much more efficient. The experimental results verified our theoretical results and demonstrated the efficiency of the meshing algorithm.

Surface meshing plays a fundamental role in graphics and visualization. Many geometric processing tasks involve solving geometric PDEs on meshes. The numerical stability, convergence rates and approximation errors are largely determined by the mesh qualities. In practice, Delaunay refinement algorithms offer satisfactory solutions to high quality mesh generations. The theoretical proofs for volume based and surface based Delaunay refinement algorithms have been established, but those for conformal parameterization based ones remain wide open. This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms. Given a metric surface, the proposed approach triangulates its conformal uniformization domain by the planar Delaunay refinement algorithms, and produces a high quality mesh. We give explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh. In contrast to the conventional results based on volumetric Delaunay refinement, our stronger estimates are independent of the mesh structure and directly guarantee the convergence of curvature measures. Meanwhile, our result on Gaussian curvature measure is intrinsic to the Riemannian metric and independent of the embedding. In practice, our meshing algorithm is much easier to implement and much more efficient. The experimental results verified our theoretical results and demonstrated the efficiency of the meshing algorithm.

In this paper, we present the first nontrivial theoretical bound on the quality of the 3D solids generated by any contour interpolation method. Given two arbitrary parallel contour slices with n vertices in 3D, let α be the smallest angle in the constrained Delaunay triangulation of the corresponding 2D contour overlay, we present a contour interpolation method which reconstructs a 3D solid with the minimum dihedral angle of at least α/8. Our algorithm runs in O(nlogn) time where n is the size of the contour overlay.We also present a heuristic algorithm that optimizes the dihedral angles of a mesh representing a surface in 3D.

This paper studies the point location problem in Delaunay triangulations without preprocessing and additional storage. The proposed procedure finds the query point simply by "walking through" the triangulation, after selecting a "good starting point" by random sampling. The analysis generalizes and extends a recent result for d = 2 dimensions by proving this procedure to take expected time close to O(nl/(d+l)) for point location in Delaunay triangulations of n random points in d = 3 dimensions. Empirical results in both two and three dimensions show that this procedure is efficient in practice.

Purpose -The purpose of this paper is to investigate the extension of Delaunay refinement algorithms to work directly with a curved geometry in arbitrary dimensional spaces, which is also able to refine geometry pieces of different dimensions altogether. Design/methodology/approach -The extension of Delaunay refinement is based on ideas of the Bowyer-Watson algorithm and Ruppert algorithm. Findings -The attempt to extend the fundamental ideas of Delaunay refinement to cope with curved geometries led to an algorithm whose performance in practice, regarding speed and mesh quality, is comparable to classical Delaunay refinement for flat geometries. Unfortunately, there are only theoretical guarantees that the refinement itself works under some conditions. No theoretical mesh quality bounds are provided. Research limitations/implications -A mesh refinement algorithm that deals with curved geometries is a key feature for adaptive mesh generators, so that points are inserted properly in the curved pieces instead of in linear approximations of them. For instance, it is well known that sharp edges are singular points of finite element formulations. This singularity fulfills in practice as mesh is refined around them. Those corners can be rounded up to avoid singularities. Furthermore, with this kind of tool, for instance, a user could start to mesh a disc from a single triangle representing it. Points would be efficiently inserted in the circle as needed during refinement. Originality/value -This paper introduces the concept of manifold complex and also an extension of Delaunay refinement algorithm to deal with curved geometries.

In this paper, the accuracy of two mesh-free approximation approaches, the Gravity model and Radial Basis Function, are compared. The two schemes’ convergence behaviors prove that RBF is faster and more accurate than the Gravity model. As a case study, the interpolation of temperature at different locations in Tennessee, USA, are compared. Delaunay mesh generation is used to create random points inside and on the border, which data can be incorporated in these locations. 49 MERRA weather stations as used as data sources to provide the temperature at a specific day and hour. The contours of interpolated temperatures provided in the result section assert RBF is a more accurate method than the Gravity model by showing a smoother and broader range of interpolated data. INTRODUCTION In current data driven world, maintaining the quality of data and its derivatives is of most importance. Complete data sets are necessary to properly understand, interpret and use data for 1 Ghasemi, February...

In this paper, the accuracy of two mesh-free approximation approaches, the Gravity model and Radial Basis Function, are compared. The two schemes' convergence behaviors prove that RBF is faster and more accurate than the Gravity model. As a case study, the interpolation of temperature at different locations in Tennesse, USA, are compared. Delaunay mesh generation is used to create random points inside and on the border, which data can be incorporated in these locations. 49 MERRA weather stations as used as data sources to provide the temperature at a specific day and hour. The contours of interpolated temperatures provided in the result section assert RBF is a more accurate method than the Gravity model by showing a smoother and broader range of interpolated data.

Artificial neural networks are a set of tools that are widely used for the information classification. Its expansion within artificial intelligence has been due to its use in the Machine Learning area. A fundamental part of the artificial neural networks algorithm is the so-called activation function, the above because it is the part that triggers the process as a whole and due to its result the neuron/perceptron sends its outputs. Back-propagation activation function of an artificial neural network is also described; this is artificial neural network with a simpler functioning whose adaptation has made it especially attractive to pattern recognition; also, a different algorithm such as Swish is introduced. As part of the pattern recognition study, three wind classifications present on the Mexican Republic Atlantic coast are formed, each group is made up of graphic files referring to meteorological maps with wind indicators in order to feed the network and as new maps are generated,...

The main objective of the research is to suggest the use of unstructured meshes in the SWAN wave model because structured meshes require more adjustments in geographically small areas such as rivers, beaches or even bays. Third-party tools and data such as bathymetry, Meteorological Research Forecast (WRF), WaveWatch III (WWIII) and Simulating Waves Nearshore (SWAN) models were used to carry out the research. The aforementioned software and its own developments were used in Acapulco Bay, Guerrero, Mexico for the elaboration of maps that would serve as input in the generation of Delaunay diagrams, which in turn were fundamental parts in the configuration of the experiments and, with this, perform the analysis on the images of the simulations. With the results obtained, it is expected to confirm the validity of the wave data for the bay; therefore, it is considered a useful tool in the process of analysing the natural phenomena present in the port.

A domain must be decomposed in sufficiently small elements to attain the desired accuracy in a numerical solution of partial differential equations that simulate a physical problem. In this work an unstructured mesh of triangular three nodes elements is generated automatically by a computer code that uses the Delaunay algorithm. To generate the first mesh, a set of points is distributed in the domain internal region according to user specified local mesh density parameter. To improve the iterative solution convergence rate of the algebraic equations system, a Delaunay algorithm is used to found the set of three vertices (mesh nodes) of each triangle maximizing the smallest internal angle of all mesh elements. In the triangular mesh obtained with the Delaunay algorithm, the circle whose circumference contains the triangle vertices is void, that is, no another mesh node is inside this circle. When the application of the mesh refinement error estimator over obtained results is higher than an user specified value, additional points are inserted in the more critical regions. The meshes obtained by this procedure are regular with nearly six triangle connected to each node and improve the overall numerical solution process.

There are so many applications of ellipse equation, such that graphical representation and finding problem of any elliptical shapes object, computer graphics, space science, engineering design and so on. In this paper state about general equation of ellipse. Generally, there are various equations for drawing an ellipse in 2d plane but not exactly the equations are general, must have some condition and rule. So, here observed those things and found a problem to create a general equation of ellipse. And tried to invent a polynomial equation of curve that can represent an ellipse and focus of ellipse act as variable of equation (and a starting point of major axis (vertex) also act as a variable).

A method of encryption of the 3D plane in Geographic Information Systems (GIS) is presented. The method is derived using Voronoi-Delaunay triangulation and properties of Catalan numbers. The Voronoi-Delaunay incremental algorithm is presented as one of the most commonly used triangulation techniques for random point selection. In accordance with the multiple applications of Catalan numbers in solving combinatorial problems and their "bit-balanced" characteristic, the process of encrypting and decrypting the coordinates of points using the Lattice Path method (walk on the integer lattice) or LIFO model is given. The triangulation of the plane started using decimal coordinates of a set of given planar points. Afterward, the resulting decimal values of the coordinates are converted to corresponding binary records and the encryption process starts by a random selection of the Catalan key according to the LIFO model. These binary coordinates are again converted into their origi...

This paper presents a new method of steganography based on a combination of Catalan objects and Voronoi–Delaunay triangulation. Two segments are described within the proposed method. The first segment describes the process of embedding data and generating a complex stego key. The second segment explains the extraction of a hidden message. The main goal of this paper is to transfer a message via the Internet (or some other medium) using an image so that the image remains absolutely unchanged. In this way, we prevented the potential attacker from noticing some secret message hidden in that picture. Additionally, the complex stego key consists of three completely different parts (the image, the encrypted Delaunay triangulation, and the array Rk in Base64 code), which are very difficult to relate with each other. Finally, a few security analyses of the proposed method are conducted, as well as the corresponding steganalysis.

In this paper we describe the architecture of a simple evacuation model which is based on a graph representation of the scene. Such graphs are typically constructed using Medial Axis Transform (MAT) or Straight Medial Axis Transform (S-MAT) transformations, the former being a part of the Voronoi diagram (Dirichlet tessellation) of the floor plan. In our work we construct such graphs for floor plans using Voronoi diagram along with the dual Delaunay triangulation of a set of points approximating the scene. Information supplied by Delaunay triangulation complements the graph in two ways: it determines capacities of some paths associated with edges, and provides a bijection between graph vertices and a set of regions forming a partition of the floor plan. We call the representation granular for this reason. We provide an exposition of the representation of fire scene that aids egress time calculations, discuss the algorithm of construction of this representation and briefly discuss the applicability of network flow models (e.g. the Ford-Fulkerson method or the push-relabel method) in our setting.

In this paper we describe an evacuation modeling framework based on a graph representation of the scene which is derived from its geometric description. Typically such graphs (geometric networks) are constructed using Medial Axis Transform (MAT) or Straight Medial Axis Transform (S-MAT). In our work we use Voronoi tessellation of a set of points approximating the scene (a single floor plan) along with the dual graph -Delaunay triangulation. Using these two graphs we extract not only the information about paths in the building, but also information about path widths and areas assigned to vertices. Typically only path lengths from MAT or S-MAT based geometric networks are used in evacuation modeling. Our approach enables us to include flow analysis and e.g. locate bottlenecks. We discuss a typical density-based evacuation model coupled with a partial behavioral evacuation model within proposed framework.

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This paper describes a two level classification algorithm to discriminate the handwritten elements from the printed text in a printed document. The proposed technique is independent of size, slant, orientation, translation and other variations in handwritten text. At the first level of classification, we use two classifiers and present a comparison between the nearest neighbour classifier and Support Vector Machines(SVM) classifier to localize the handwritten text. The features that are extracted from the document are seven invariant central moments and based on these features, we classify the text as hand-written. At the second level, we use Delaunay triangulation to reclassify the misclassified elements. When Delaunay triangulation is imposed on the centroid points of the connected components, we extract features based on the triangles and reclassify the misclassified elements. We remove the noise components in the document as part of the pre-processing step.

We present a novel algorithm to perform continuous collision detection for articulated models. Given two discrete configurations of the links of an articulated model, we use an "arbitrary in-between motion" to interpolate its motion between two successive time steps and check the resulting trajectory for collisions. Our approach uses a three-stage pipeline: (1) dynamic bounding-volume hierarchy (D-BVH) culling based on interval arithmetic; (2) culling refinement using the swept volume of line swept sphere (LSS) and graphics hardware accelerated queries; (3) exact contact computation using OBB-trees and continuous collision detection between triangular primitives. The overall algorithm computes the time of collision, contact locations and prevents any interpenetration between the articulated model with the environment. We have implemented the algorithm and tested its performance on a 2.4 GHz Pentium PC with 1 Gbyte of RAM and a NVIDIA GeForce FX 5800 graphics card. In practice, our algorithm is able to perform accurate and continuous collision detection between articulated models and complex environments at nearly interactive rates.

This paper proposes and proves a formula for calculating the maximum number of integrated polygons of a given type that can be formed within a larger polygon in a single combination without overlapping. The formula is expressed as Θ(α,n)=⌊α−2/n−2⌋, where n is the number of sides of the initial polygon, and α is the number of sides of the integrated polygon. The formula is derived from the properties of diagonal partitioning in polygons and provides a general approach for solving such problems. Examples are provided to illustrate the practical application of the formula.

The insufficiency of labeled training data for representing the distribution of the entire dataset is a major obstacle in automatic semantic annotation of large-scale video database. Semi-supervised learning algorithms along with adaptive techniques, attempt to learn from both labeled and unlabeled data, are promising to solve this problem. In this paper, retrieving videos using key words requires obtaining the semantic features of the videos. Most work reported in the literature focuses on different key frame extraction techniques and shot detection methods along with annotating a video shot with a fixed number of key words, no matter how much information is contained in the video shot.Two level keyframe extraction is performed for removal of redundant keyframes.The aim is to find minimal of Keyframes.

This dissertation places intersection homology and local homology within the framework of persistence, which was originally developed for ordinary homology by Edelsbrunner, Letscher, and Zomorodian. The eventual goal, begun but not completed here, is to provide analytical tools for the study of embedded stratified spaces, as well as for high-dimensional and possibly noisy datasets for which the number of degrees of freedom may vary across the parameter space. Specifically, we create a theory of persistent intersection homology for a filtered stratified space and prove several structural theorems about the pair groups associated to such a filtration. We prove the correctness of a cubic algorithm which computes these pair groups in a simplicial setting. We also define a series of intersection homology elevation functions for an embedded stratified space and characterize their local maxima in dimension one. In addition, we develop a theory of persistence for a multi-scale analogue of the local homology groups of a stratified space at a point. This takes the form of a series of local homology vineyards which allow one to assess the homological structure within a one-parameter family of neighborhoods of the point. Under the assumption of dense sampling, we prove the correctness of this assessment at a variety of radius scales.

Boundary representation models reverse engineered from 3D range data suffer from various inaccuracies caused by noise in the measured data and the model building software. Beautification aims to improve such models in a post-processing step solely working with the boundary representation model. The improved model should exhibit topological and geometric regularities representing the original, ideal design intent. This paper gives an overview of algorithms for a complete beautification system suitable for improving the topology and the geometry of low to medium complexity reverse engineered models.

The greedy triangulation of a finite planar point set is obtained by repeatedly inserting a shortest diagonal that does not cross those already in the plane. The Delaunay triangulation, which is the straight-line dual of the Voronoi diagram, can be produced in O(n log n) worst-case time, and often even faster, by several practical algorithms. In this paper we show that for any planar point set S, if the Delaunay triangulation of S is given, then the greedy triangulation of S can be computed in linear worst-case time (and linear space).

Acceleration structures are mandatory for ray-tracing applications, allowing to cast a large number of rays per second. In 2008, Lagae and Dutré have proposed to use Constrained Delaunay Tetrahedralization (CDT) as an acceleration structure for ray tracing. Our experiments show that their traversal algorithm is not suitable for GPU applications, mainly due to arithmetic errors. This article proposes a new CDT traversal algorithm. This new algorithm is more efficient than the previous ones: it uses less arithmetic operations; it does not add extra thread divergence since it uses a fixed number of operation; at last, it is robust with 32-bits floats, contrary to the previous traversal algorithms. Hence, it is the first method usable both on CPU and GPU. This article is organized as follows: Section 2 recapitulates previous CDT works. Section 3 presents our new CDT traversal. Section 4 discusses our experiments. Finally, Section 5 concludes this paper. 2 PREVIOUS WORKS ON CDT This section first describes CDT, then it presents its construction from a geometric model, before focusing on former ray traversal methods. 2.1 CDT Description A Delaunay tetrahedralization of a set of points X ∈ E 3 is a set of tetrahedra occupying the whole

1. One of the central challenges in ecology and animal behaviour is to generate animal home range estimations. Kernel density estimate for home range has been one of the most widely used estimates these last few decades, despite its limitations. More recently a network-based kernel density (NKDE) approach has been proposed, which uses Delaunay triangulation. 2. Here we show that NKDE has a discontinuous kernel density. We then develop a new network-based method that emphasises entirely on the edges, instead of the nodes in the network. We call this Edge-Focused Network-based Kernel Density Estimation (EFNKDE). In this method, a unit weight is distributed uniformly along each edge of the network and Euclidean distance is used to compute the contribution of each segment of the edge to the kernel density at a given point. 3. This is a network based method that leads to a continuous and differentiable kernel density. An analytical expression for the same is obtained for the Gaussian kernel. By taking concrete examples and studying different methods and through different perspectives across a range of bandwidths, we show that EFNKDE has many advantages over other methods. We present a model that provides a theoretical basis to the effectiveness of EFNKDE in linear regimes. 4. EFNKDE is easy to apply, can work with minimal data, provides a smooth kernel density that highlights the network and does not overemphasise the data. This method is most suitable for estimating home ranges with narrow corridors and forbidden regions.

Nous présentons une nouvelle représentation hybride métrique/topologique/sémantique appelé MTS-Maps conçue pour la conduite automatique de véhicule autonome dans des environnements de grande taille d'intérieur ou d'extérieur. Basée sur une représentation générique, la sphère RGB-DL, elle permet de s'affranchir de la technologie du capteur utilisé. Ces différentes couches permettent de traiter le problème de localisation allant de la requête contextuelle jusqu'au calcul de l'erreur de pose utilisée dans le feedback du contrôleur. Les résultats présentés montrent ses performancesà la fois au niveau métrique par une expérimentation de conduite automatique dans un environnement urbain et au niveau sémantique par une comparaison avec une implémentation récente d'une méthode Bag-of-Words.

In this paper we propose a new polygonization method based on the classic Marching Triangle algorithm. It is an improved and efficient version of the basic algorithm which produces a complete mesh without any cracks. Our method is useful in the surface reconstruction process of scanned objects. It works over the scalar field distance transform of the object to produce the resulting triangle mesh. First we improve the original algorithm in finding new potential vertices in the mesh growing process. Second we modify the Delaunay sphere test on the new triangles. Third we consider new triangles configuration to obtain a more complete mesh. Finally we introduce an edge processing sequence to improve the overall Marching Triangle algorithm. We use a relevant error metric tool to compare results and show our new method is more accurate than Marching Cube which is the most widely used triangulation algorithm in the surface reconstruction process of scanned objects.

In this paper we propose a new polygonization method based on the classic Marching Triangle algorithm. It is an improved and efficient version of the basic algorithm which produces a complete mesh without any cracks. Our method is useful in the surface reconstruction process of scanned objects. It works over the scalar field distance transform of the object to produce the resulting triangle mesh. First we improve the original algorithm in finding new potential vertices in the mesh growing process. Second we modify the Delaunay sphere test on the new triangles. Third we consider new triangles configuration to obtain a more complete mesh. Finally we introduce an edge processing sequence to improve the overall Marching Triangle algorithm. We use a relevant error metric tool to compare results and show our new method is more accurate than Marching Cube which is the most widely used triangulation algorithm in the surface reconstruction process of scanned objects.