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Probabilistic Marching Cubes
"... In this paper we revisit the computation and visualization of equivalents to isocontours in uncertain scalar fields. We model uncertainty by discrete random fields and, in contrast to previous methods, also take arbitrary spatial correlations into account. Starting with joint distributions of the ra ..."
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Cited by 23 (3 self)
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In this paper we revisit the computation and visualization of equivalents to isocontours in uncertain scalar fields. We model uncertainty by discrete random fields and, in contrast to previous methods, also take arbitrary spatial correlations into account. Starting with joint distributions of the random variables associated to the sample locations, we compute level crossing probabilities for cells of the sample grid. This corresponds to computing the probabilities that the wellknown symmetryreduced marching cubes cases occur in random field realizations. For Gaussian random fields, only marginal density functions that correspond to the vertices of the considered cell need to be integrated. We compute the integrals for each cell in the sample grid using a Monte Carlo method. The probabilistic ansatz does not suffer from degenerate cases that usually require case distinctions and solutions of illconditioned problems. Applications in 2D and 3D, both to synthetic and real data from ensemble simulations in climate research, illustrate the influence of spatial correlations on the spatial distribution of uncertain isocontours. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Picture/Image
C.: Revisiting histograms and isosurface statistics
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract—Recent results have shown a link between geometric properties of isosurfaces and statistical properties of the underlying sampled data. However, this has two defects: not all of the properties described converge to the same solution, and the statistics computed are not always invariant unde ..."
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Cited by 19 (3 self)
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Abstract—Recent results have shown a link between geometric properties of isosurfaces and statistical properties of the underlying sampled data. However, this has two defects: not all of the properties described converge to the same solution, and the statistics computed are not always invariant under isosurfacepreserving transformations. We apply Federer’s Coarea Formula from geometric measure theory to explain these discrepancies. We describe an improved substitute for histograms based on weighting with the inverse gradient magnitude, develop a statistical model that is invariant under isosurfacepreserving transformations, and argue that this provides a consistent method for algorithm evaluation across multiple datasets based on histogram equalization. We use our corrected formulation to reevaluate recent results on average isosurface complexity, and show evidence that noise is one cause of the discrepancy between the expected figure and the observed one. 1
Computing Robustness and Persistence for Images
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"... We are interested in 3dimensional images given as arrays of voxels with intensity values. Extending these values to a continuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbation needed to destroy these classes. The structur ..."
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Cited by 15 (5 self)
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We are interested in 3dimensional images given as arrays of voxels with intensity values. Extending these values to a continuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbation needed to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can be visualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchical algorithm using the dual complexes of octtree approximations of the function. In addition, we show that for balanced octtrees, the dual complexes are geometrically realized in R³ and can thus be used to construct level and interlevel sets. We apply these tools to study 3dimensional images of plant root systems.
studies
, 1992
"... Comparison of the safety and efficacy of fixeddose combination of arterolane maleate and piperaquine phosphate with chloroquine in acute, uncomplicated Plasmodium vivax malaria: a phase III, multicentric, openlabel ..."
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Cited by 13 (1 self)
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Comparison of the safety and efficacy of fixeddose combination of arterolane maleate and piperaquine phosphate with chloroquine in acute, uncomplicated Plasmodium vivax malaria: a phase III, multicentric, openlabel
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 11 (5 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
From multiple views to textured 3D meshed: a GPUpowered approach
 ECCV 2010 Workshop on Computer Vision on GPUs
, 2010
"... Abstract. We present work on exploiting modern graphics hardware towards the realtime production of a textured 3D mesh representation of a scene observed by a multicamera system. The employed computational infrastructure consists of a network of four PC workstations each of which is connected to a ..."
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Cited by 7 (3 self)
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Abstract. We present work on exploiting modern graphics hardware towards the realtime production of a textured 3D mesh representation of a scene observed by a multicamera system. The employed computational infrastructure consists of a network of four PC workstations each of which is connected to a pair of cameras. One of the PCs is equipped with a GPU that is used for parallel computations. The result of the processing is a list of texture mapped triangles representing the reconstructed surfaces. In contrast to previous works, the entire processing pipeline (foreground segmentation, 3D reconstruction, 3D mesh computation, 3D mesh smoothing and texture mapping) has been implemented on the GPU. Experimental results demonstrate that an accurate, high resolution, texturemapped 3D reconstruction of a scene observed by eight cameras is achievable in real time. 1
On the Fractal Dimension of Isosurfaces
"... Fig. 1: Visible male data set (www.stereofx.org): Fractal box span dimension, number of isosurface components and sample isosurfaces. Noisy isosurface at isovalue 68 corresponds to high fractal dimension and large topological noise. Topological noise is measured by the number of connected components ..."
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Cited by 7 (2 self)
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Fig. 1: Visible male data set (www.stereofx.org): Fractal box span dimension, number of isosurface components and sample isosurfaces. Noisy isosurface at isovalue 68 corresponds to high fractal dimension and large topological noise. Topological noise is measured by the number of connected components in the isosurface. (Visible male data set provided by the National Library of Medicine, USA.) Abstract—A (3D) scalar grid is a regular n1 × n2 × n3 grid of vertices where each vertex v is associated with some scalar value sv. Applying trilinear interpolation, the scalar grid determines a scalar function g where g(v) = sv for each grid vertex v. An isosurface with isovalue σ is a triangular mesh which approximates the level set g −1 (σ). The fractal dimension of an isosurface represents the growth in the isosurface as the number of grid cubes increases. We define and discuss the fractal isosurface dimension. Plotting the fractal dimension as a function of the isovalues in a data set provides information about the isosurfaces determined by the data set. We present statistics on the average fractal dimension of 60 publicly available benchmark data sets. We also show the fractal dimension is highly correlated with topological noise in the benchmark data sets, measuring the topological noise by the number of connected components in the isosurface. Lastly, we present a formula predicting the fractal dimension as a function of noise and validate the formula with experimental results. Index Terms—Isosurfaces, scalar data, fractal dimension. 1
Computational topology
 Algorithms and Theory of Computation Handbook
, 2010
"... According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in t ..."
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Cited by 6 (3 self)
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According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in turn, undertakes the challenge of studying topology using a computer.
Direct Interval Volume Visualization
, 2010
"... We extend direct volume rendering with a unified model for generalized isosurfaces, also called interval volumes, allowing a wider spectrum of visual classification. We generalize the concept of scaleinvariant opacity—typical for isosurface rendering— to semitransparent interval volumes. Scalein ..."
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Cited by 6 (2 self)
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We extend direct volume rendering with a unified model for generalized isosurfaces, also called interval volumes, allowing a wider spectrum of visual classification. We generalize the concept of scaleinvariant opacity—typical for isosurface rendering— to semitransparent interval volumes. Scaleinvariant rendering is independent of physical space dimensions and therefore directly facilitates the analysis of data characteristics. Our model represents sharp isosurfaces as limits of interval volumes and combines them with features of direct volume rendering. Our objective is accurate rendering, guaranteeing that all isosurfaces and interval volumes are visualized in a crackfree way with correct spatial ordering. We achieve simultaneous direct and interval volume rendering by extending preintegration and explicit peak finding with datadriven splitting of ray integration and hybrid computation in physical and data domains. Our algorithm is suitable for efficient parallel processing for interactive applications as demonstrated by our CUDA implementation.
Edge groups: An approach to understanding the mesh quality of marching methods
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract — Marching Cubes is the most popular isosurface extraction algorithm due to its simplicity, efficiency and robustness. It has been widely studied, improved, and extended. While much early work was concerned with efficiency and correctness issues, lately there has been a push to improve the ..."
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Cited by 4 (2 self)
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Abstract — Marching Cubes is the most popular isosurface extraction algorithm due to its simplicity, efficiency and robustness. It has been widely studied, improved, and extended. While much early work was concerned with efficiency and correctness issues, lately there has been a push to improve the quality of Marching Cubes meshes so that they can be used in computational codes. In this work we present a new classification of MC cases that we call Edge Groups, which helps elucidate the issues that impact the triangle quality of the meshes that the method generates. This formulation allows a more systematic way to bound the triangle quality, and is general enough to extend to other polyhedral cell shapes used in other polygonization algorithms. Using this analysis, we also discuss ways to improve the quality of the resulting triangle mesh, including some that require only minor modifications of the original algorithm. Index Terms—Isosurface extraction, Marching Cubes.