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Interactive ray tracing of arbitrary implicits with simd interval arithmetic
 In Proceedings of the 2nd IEEE/EG Symposium on Interactive Ray Tracing
, 2007
"... We present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any progra ..."
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Cited by 21 (7 self)
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We present a practical and efficient algorithm for interactively ray tracing arbitrary implicit surfaces. We use interval arithmetic (IA) both for robust root computation and guaranteed detection of topological features. In conjunction with ray tracing, this allows for rendering literally any programmable implicit function simply from its definition. Our method requires neither special hardware, nor preprocessing or storage of any data structure. Efficiency is achieved through SIMD optimization of both the interval arithmetic computation and coherent ray traversal algorithm, delivering interactive results even for complex implicit functions.
Simulation of Bubbles in Foam With The Volume Control Method
"... Figure 1: When the level set is advected by the BFECC [Dupont and Liu 2003] method, the simulation of a rising bubble produces volume loss (top). When the proposed volume control method is used, the volume of bubble is preserved regardless of the length of the simulation (bottom). From left to right ..."
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Cited by 12 (0 self)
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Figure 1: When the level set is advected by the BFECC [Dupont and Liu 2003] method, the simulation of a rising bubble produces volume loss (top). When the proposed volume control method is used, the volume of bubble is preserved regardless of the length of the simulation (bottom). From left to right, each column shows the bubble at t = 0, 0.0625, 0.125, 0.25, 0.5, and 10.0 second. The image on the far right shows a foam structure obtained after raising more than 400 bubbles. Liquid and gas interactions often produce bubbles that stay for a long time without bursting on the surface, making a dry foam structure. Such long lasting bubbles simulated by the level set method can suffer from a small but steady volume error that accumulates to a visible amount of volume change. We propose to address this problem by using the volume control method. We track the volume change of each connected region, and apply a carefully computed divergence that compensates undesired volume changes. To compute the divergence, we construct a mathematical model of the volume change, choose control strategies that regulate the modeled volume error, and establish methods to compute the control gains that provide robust and fast reduction of the volume error, and (if desired) the control of how the volume changes over time. 1
Edge transformations for improving mesh quality of marching cubes
 IEEE TVCG
"... Abstract—Marching Cubes is a popular choice for isosurface extraction from regular grids due to its simplicity, robustness, and efficiency. One of the key shortcomings of this approach is the quality of the resulting meshes, which tend to have many poorly shaped and degenerate triangles. This issue ..."
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Cited by 9 (5 self)
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Abstract—Marching Cubes is a popular choice for isosurface extraction from regular grids due to its simplicity, robustness, and efficiency. One of the key shortcomings of this approach is the quality of the resulting meshes, which tend to have many poorly shaped and degenerate triangles. This issue is often addressed through postprocessing operations such as smoothing. As we demonstrate in experiments with several data sets, while these improve the mesh, they do not remove all degeneracies and incur an increased and unbounded error between the resulting mesh and the original isosurface. Rather than modifying the resulting mesh, we propose a method to modify the grid on which Marching Cubes operates. This modification greatly increases the quality of the extracted mesh. In our experiments, our method did not create a single degenerate triangle, unlike any other method we experimented with. Our method incurs minimal computational overhead, requiring at most twice the execution time of the original Marching Cubes algorithm in our experiments. Most importantly, it can be readily integrated in existing Marching Cubes implementations and is orthogonal to many Marching Cubes enhancements (particularly, performance enhancements such as outofcore and acceleration structures). Index Terms—Meshing, marching cubes. Ç 1
Topology, accuracy, and quality of isosurface meshes using dynamic particles
 IEEE Transactions on Visualization and Computer Graphics
, 2007
"... This paper describes a method for constructing isosurface triangulations of sampled, volumetric, threedimensional scalar fields. The resulting meshes consist of triangles that are of consistently high quality, making them well suited for accurate interpolation of scalar and vectorvalued quantities ..."
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Cited by 9 (2 self)
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This paper describes a method for constructing isosurface triangulations of sampled, volumetric, threedimensional scalar fields. The resulting meshes consist of triangles that are of consistently high quality, making them well suited for accurate interpolation of scalar and vectorvalued quantities, as required for numerous applications in visualization and numerical simulation. The proposed method does not rely on a local construction or adjustment of triangles as is done, for instance, in advancing wavefront or adaptive refinement methods. Instead, a system of dynamic particles optimally samples an implicit function such that the particles ’ relative positions can produce a topologically correct Delaunay triangulation. Thus, the proposed method relies on a global placement of triangle vertices. The main contributions of the paper are the integration of dynamic particles systems with surface sampling theory and PDEbased methods for controlling the local variability of particle densities, as well as detailing a practical method that accommodates Delaunay sampling requirements to generate sparse sets of points for the production of highquality tessellations. Index Terms—Isosurface extraction, particle systems, Delaunay triangulation.
Quality isosurface mesh generation using an extended marching cubes lookup table
 Comput. Graph. Forum
"... The Marching Cubes Algorithm may return degenerate, zero area isosurface triangles, and often returns isosurface triangles with small areas, edges or angles. We show how to avoid both problems using an extended Marching Cubes lookup table. As opposed to the conventional Marching Cubes lookup table, ..."
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Cited by 8 (0 self)
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The Marching Cubes Algorithm may return degenerate, zero area isosurface triangles, and often returns isosurface triangles with small areas, edges or angles. We show how to avoid both problems using an extended Marching Cubes lookup table. As opposed to the conventional Marching Cubes lookup table, the extended lookup table differentiates scalar values equal to the isovalue from scalar values greater than the isovalue. The lookup table has 3 8 = 6561 entries, based on three possible labels, ’− ’ or ’= ’ or ’+’, of each cube vertex. We present an algorithm based on this lookup table which returns an isosurface close to the Marching Cubes isosurface, but without any degenerate triangles or any small areas, edges or angles. 1.
Particlebased Sampling and Meshing of Surfaces in Multimaterial Volumes
"... Abstract — Methods that faithfully and robustly capture the geometry of complex material interfaces in labeled volume data are important for generating realistic and accurate visualizations and simulations of realworld objects. The generation of such multimaterial models from measured data poses tw ..."
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Cited by 7 (1 self)
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Abstract — Methods that faithfully and robustly capture the geometry of complex material interfaces in labeled volume data are important for generating realistic and accurate visualizations and simulations of realworld objects. The generation of such multimaterial models from measured data poses two unique challenges: first, the surfaces must be wellsampled with regular, efficient tessellations that are consistent across material boundaries; and second, the resulting meshes must respect the nonmanifold geometry of the multimaterial interfaces. This paper proposes a strategy for sampling and meshing multimaterial volumes using dynamic particle systems, including a novel, differentiable representation of the material junctions that allows the particle system to explicitly sample corners, edges, and surfaces of material intersections. The distributions of particles are controlled by fundamental sampling constraints, allowing Delaunaybased meshing algorithms to reliably extract watertight meshes of consistently highquality. Index Terms—Sampling, meshing, visualizations. 1
Verifiable visualization for isosurface extraction
 IEEE Transactions on Visualization and Computer Graphics
"... Fig. 1. Through the verification methodology presented on this paper we were able to uncover a convergence problem within a publicly available marchingbased isosurfacing code (top left) and fix it (top right). The problem causes the mesh normals to disagree with the known gradient field when refini ..."
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Cited by 5 (3 self)
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Fig. 1. Through the verification methodology presented on this paper we were able to uncover a convergence problem within a publicly available marchingbased isosurfacing code (top left) and fix it (top right). The problem causes the mesh normals to disagree with the known gradient field when refining the voxel size h (bottom row). The two graphs show the convergence of the normals before and after fixing the code. Abstract—Visual representations of isosurfaces are ubiquitous in the scientific and engineering literature. In this paper, we present techniques to assess the behavior of isosurface extraction codes. Where applicable, these techniques allow us to distinguish whether anomalies in isosurface features can be attributed to the underlying physical process or to artifacts from the extraction process. Such scientific scrutiny is at the heart of verifiable visualization – subjecting visualization algorithms to the same verification process that is used in other components of the scientific pipeline. More concretely, we derive formulas for the expected order of accuracy (or convergence rate) of several isosurface features, and compare them to experimentally observed results in the selected codes. This technique is practical: in two cases, it exposed actual problems in implementations. We provide the reader with the range of responses they can expect to encounter with isosurface techniques, both under “normal operating conditions ” and also under adverse conditions. Armed with this information – the results of the verification process – practitioners can judiciously select the isosurface extraction technique appropriate for their problem of interest, and have confidence in its behavior. Index Terms—Verification, V&V, Isosurface Extraction, Marching Cubes. 1
Topology Verification for Isosurface Extraction
, 2010
"... The importance of properly implemented isosurface extraction for verifiable visualization led to a previously published paper on the general Method of Manufactured Solutions (MMS), inclusive of a supportive software infrastructure. This work builds upon that foundation, while significantly extending ..."
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Cited by 4 (2 self)
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The importance of properly implemented isosurface extraction for verifiable visualization led to a previously published paper on the general Method of Manufactured Solutions (MMS), inclusive of a supportive software infrastructure. This work builds upon that foundation, while significantly extending it. Specifically, we extend previous work on verification of geometrical properties to ensuring correctness of considerably more subtle topological characteristics that are crucial for the extracted surfaces. We first show a new theoretical synthesis of results from stratified Morse theory and digital topology for algorithms created to verify topological invariants and then we demonstrate how the MMS approach can be extended to embrace topology, consistent with the design intent for MMS. The transition to topological verification motivated these considerable theoretical advances and algorithmic development, consistent with general MMS principles. The methodology reported reveals unexpected behavior and even coding mistakes in publicly available popular isosurface codes, as presented in a case study for visualization tools that documents the
Hermite Interpolation of Implicit Surfaces with Radial Basis Functions
"... Abstract—We present the Hermite radial basis function (HRBF) implicits method to compute a global implicit function which interpolates scattered multivariate Hermite data (unstructured points and their corresponding normals). Differently from previous radial basis functions (RBF) approaches, HRBF im ..."
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Cited by 2 (1 self)
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Abstract—We present the Hermite radial basis function (HRBF) implicits method to compute a global implicit function which interpolates scattered multivariate Hermite data (unstructured points and their corresponding normals). Differently from previous radial basis functions (RBF) approaches, HRBF implicits do not depend on offset points to ensure existence and uniqueness of its interpolant. Intrinsic properties of this method allow the computation of implicit surfaces rich in details, with irregularly spaced points even in the presence of close sheets. Comparisons to previous works show the effectiveness of our approach. Further, the theoretical background of HRBF implicits relies on results from generalized interpolation theory with RBFs, making possible powerful new variants of this method and establishing connections with previous efforts based on statistical learning theory. KeywordsImplicit surfaces, Hermite data, radial basis functions, HermiteBirkhoff interpolation, scattered data approximation, geometric modeling, surface reconstruction I.
Marching Cubes without Skinny Triangles
"... Most computational codes that use irregular grids depend on the triangle quality of the single worst triangle in the grid: skinny triangles can lead to bad performance and numerical instabilities. Marching Cubes is the standard isosurface grid generation algorithm, and while most triangles it genera ..."
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Cited by 1 (0 self)
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Most computational codes that use irregular grids depend on the triangle quality of the single worst triangle in the grid: skinny triangles can lead to bad performance and numerical instabilities. Marching Cubes is the standard isosurface grid generation algorithm, and while most triangles it generates are good, it almost always generates some bad triangles. Here we show how simple changes to Marching Cubes can lead to a drastically reduced number of degenerate triangles, making it a more practical choice for isosurface grid generation, reducing or eliminating the need and costs of postprocessing. 1.