1,304 research outputs found

    Error-Bounded and Feature Preserving Surface Remeshing with Minimal Angle Improvement

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    The typical goal of surface remeshing consists in finding a mesh that is (1) geometrically faithful to the original geometry, (2) as coarse as possible to obtain a low-complexity representation and (3) free of bad elements that would hamper the desired application. In this paper, we design an algorithm to address all three optimization goals simultaneously. The user specifies desired bounds on approximation error {\delta}, minimal interior angle {\theta} and maximum mesh complexity N (number of vertices). Since such a desired mesh might not even exist, our optimization framework treats only the approximation error bound {\delta} as a hard constraint and the other two criteria as optimization goals. More specifically, we iteratively perform carefully prioritized local operators, whenever they do not violate the approximation error bound and improve the mesh otherwise. In this way our optimization framework greedily searches for the coarsest mesh with minimal interior angle above {\theta} and approximation error bounded by {\delta}. Fast runtime is enabled by a local approximation error estimation, while implicit feature preservation is obtained by specifically designed vertex relocation operators. Experiments show that our approach delivers high-quality meshes with implicitly preserved features and better balances between geometric fidelity, mesh complexity and element quality than the state-of-the-art.Comment: 14 pages, 20 figures. Submitted to IEEE Transactions on Visualization and Computer Graphic

    High-Quality Simplification and Repair of Polygonal Models

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    Because of the rapid evolution of 3D acquisition and modelling methods, highly complex and detailed polygonal models with constantly increasing polygon count are used as three-dimensional geometric representations of objects in computer graphics and engineering applications. The fact that this particular representation is arguably the most widespread one is due to its simplicity, flexibility and rendering support by 3D graphics hardware. Polygonal models are used for rendering of objects in a broad range of disciplines like medical imaging, scientific visualization, computer aided design, film industry, etc. The handling of huge scenes composed of these high-resolution models rapidly approaches the computational capabilities of any graphics accelerator. In order to be able to cope with the complexity and to build level-of-detail representations, concentrated efforts were dedicated in the recent years to the development of new mesh simplification methods that produce high-quality approximations of complex models by reducing the number of polygons used in the surface while keeping the overall shape, volume and boundaries preserved as much as possible. Many well-established methods and applications require "well-behaved" models as input. Degenerate or incorectly oriented faces, T-joints, cracks and holes are just a few of the possible degenaracies that are often disallowed by various algorithms. Unfortunately, it is all too common to find polygonal models that contain, due to incorrect modelling or acquisition, such artefacts. Applications that may require "clean" models include finite element analysis, surface smoothing, model simplification, stereo lithography. Mesh repair is the task of removing artefacts from a polygonal model in order to produce an output model that is suitable for further processing by methods and applications that have certain quality requirements on their input. This thesis introduces a set of new algorithms that address several particular aspects of mesh repair and mesh simplification. One of the two mesh repair methods is dealing with the inconsistency of normal orientation, while another one, removes the inconsistency of vertex connectivity. Of the three mesh simplification approaches presented here, the first one attempts to simplify polygonal models with the highest possible quality, the second, applies the developed technique to out-of-core simplification, and the third, prevents self-intersections of the model surface that can occur during mesh simplification

    A case study in hexahedral mesh generation: Simulation of the human mandible

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    We provide a case study for the generation of pure hexahedral meshes for the numerical simulation of physiological stress scenarios of the human mandible. Due to its complex and very detailed free-form geometry, the mandible model is very demanding. This test case is used as a running example to demonstrate the applicability of a combinatorial approach for the generation of hexahedral meshes by means of successive dual cycle eliminations, which has been proposed by the second author in previous work. We report on the progress and recent advances of the cycle elimination scheme. The given input data, a surface triangulation obtained from computed tomography data, requires a substantial mesh reduction and a suitable conversion into a quadrilateral surface mesh as a first step, for which we use mesh clustering and b-matching techniques. Several strategies for improved cycle elimination orders are proposed. They lead to a significant reduction in the mesh size and a better structural quality. Based on the resulting combinatorial meshes, gradient-based optimized smoothing with the condition number of the Jacobian matrix as objective together with mesh untangling techniques yielded embeddings of a satisfactory quality. To test our hexahedral meshes for the mandible model within an FEM simulation we used the scenario of a bite on a ‘hard nut.’ Our simulation results are in good agreement with observations from biomechanical experiments

    Convergence of the restricted Nelder-Mead algorithm in two dimensions

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    The Nelder-Mead algorithm, a longstanding direct search method for unconstrained optimization published in 1965, is designed to minimize a scalar-valued function f of n real variables using only function values, without any derivative information. Each Nelder-Mead iteration is associated with a nondegenerate simplex defined by n+1 vertices and their function values; a typical iteration produces a new simplex by replacing the worst vertex by a new point. Despite the method's widespread use, theoretical results have been limited: for strictly convex objective functions of one variable with bounded level sets, the algorithm always converges to the minimizer; for such functions of two variables, the diameter of the simplex converges to zero, but examples constructed by McKinnon show that the algorithm may converge to a nonminimizing point. This paper considers the restricted Nelder-Mead algorithm, a variant that does not allow expansion steps. In two dimensions we show that, for any nondegenerate starting simplex and any twice-continuously differentiable function with positive definite Hessian and bounded level sets, the algorithm always converges to the minimizer. The proof is based on treating the method as a discrete dynamical system, and relies on several techniques that are non-standard in convergence proofs for unconstrained optimization.Comment: 27 page

    Approximation Algorithms for Geometric Networks

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    The main contribution of this thesis is approximation algorithms for several computational geometry problems. The underlying structure for most of the problems studied is a geometric network. A geometric network is, in its abstract form, a set of vertices, pairwise connected with an edge, such that the weight of this connecting edge is the Euclidean distance between the pair of points connected. Such a network may be used to represent a multitude of real-life structures, such as, for example, a set of cities connected with roads. Considering the case that a specific network is given, we study three separate problems. In the first problem we consider the case of interconnected `islands' of well-connected networks, in which shortest paths are computed. In the second problem the input network is a triangulation. We efficiently simplify this triangulation using edge contractions. Finally, we consider individual movement trajectories representing, for example, wild animals where we compute leadership individuals. Next, we consider the case that only a set of vertices is given, and the aim is to actually construct a network. We consider two such problems. In the first one we compute a partition of the vertices into several subsets where, considering the minimum spanning tree (MST) for each subset, we aim to minimize the largest MST. The other problem is to construct a tt-spanner of low weight fast and simple. We do this by first extending the so-called gap theorem. In addition to the above geometric network problems we also study a problem where we aim to place a set of different sized rectangles, such that the area of their corresponding bounding box is minimized, and such that a grid may be placed over the rectangles. The grid should not intersect any rectangle, and each cell of the grid should contain at most one rectangle. All studied problems are such that they do not easily allow computation of optimal solutions in a feasible time. Instead we consider approximation algorithms, where near-optimal solutions are produced in polynomial time. In addition to the above geometric network problems we also study a problem where we aim to place a set of different sized rectangles, such that the area of their corresponding bounding box is minimized, and such that a grid may be placed over the rectangles. The grid should not intersect any rectangle, and each cell of the grid should contain at most one rectangle. All studied problems are such that they do not easily allow computation of optimal solutions in a feasible time. Instead we consider approximation algorithms, where near-optimal solutions are produced in polynomial time

    Appearance Preserving Rendering of Out-of-Core Polygon and NURBS Models

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    In Computer Aided Design (CAD) trimmed NURBS surfaces are widely used due to their flexibility. For rendering and simulation however, piecewise linear representations of these objects are required. A relatively new field in CAD is the analysis of long-term strain tests. After such a test the object is scanned with a 3d laser scanner for further processing on a PC. In all these areas of CAD the number of primitives as well as their complexity has grown constantly in the recent years. This growth is exceeding the increase of processor speed and memory size by far and posing the need for fast out-of-core algorithms. This thesis describes a processing pipeline from the input data in the form of triangular or trimmed NURBS models until the interactive rendering of these models at high visual quality. After discussing the motivation for this work and introducing basic concepts on complex polygon and NURBS models, the second part of this thesis starts with a review of existing simplification and tessellation algorithms. Additionally, an improved stitching algorithm to generate a consistent model after tessellation of a trimmed NURBS model is presented. Since surfaces need to be modified interactively during the design phase, a novel trimmed NURBS rendering algorithm is presented. This algorithm removes the bottleneck of generating and transmitting a new tessellation to the graphics card after each modification of a surface by evaluating and trimming the surface on the GPU. To achieve high visual quality, the appearance of a surface can be preserved using texture mapping. Therefore, a texture mapping algorithm for trimmed NURBS surfaces is presented. To reduce the memory requirements for the textures, the algorithm is modified to generate compressed normal maps to preserve the shading of the original surface. Since texturing is only possible, when a parametric mapping of the surface - requiring additional memory - is available, a new simplification and tessellation error measure is introduced that preserves the appearance of the original surface by controlling the deviation of normal vectors. The preservation of normals and possibly other surface attributes allows interactive visualization for quality control applications (e.g. isophotes and reflection lines). In the last part out-of-core techniques for processing and rendering of gigabyte-sized polygonal and trimmed NURBS models are presented. Then the modifications necessary to support streaming of simplified geometry from a central server are discussed and finally and LOD selection algorithm to support interactive rendering of hard and soft shadows is described

    Multi-Resolution Meshes for Feature-Aware Hardware Tessellation

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    International audienceHardware tessellation is de facto the preferred mechanism to adaptively control mesh resolution with maximal performances. However, owing to its fixed and uniform pattern, leveraging tessellation for feature-aware LOD rendering remains a challenging problem. We relax this fundamental constraint by introducing a new spatial and temporal blending mechanism of tessellation levels, which is built on top of a novel hierarchical representation of multi-resolution meshes. This mechanism allows to finely control topological changes so that vertices can be removed or added at the most appropriate location to preserve geometric features in a continuous and artifact-free manner. We then show how to extend edge-collapse based decimation methods to build feature-aware multi-resolution meshes that match the tessellation patterns. Our approach is fully compatible with current hardware tessellators and only adds a small overhead on memory consumption and tessellation cost
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