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45
Mesh optimization
, 1993
"... We present a method for solving the following problem: Given a set of data points scattered in three dimensions and an initial triangular mesh M0, produce a mesh M, of the same topological type as M0, that fits the data well and has a small number of vertices. Our approach is to minimize an energy f ..."
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Cited by 352 (9 self)
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We present a method for solving the following problem: Given a set of data points scattered in three dimensions and an initial triangular mesh M0, produce a mesh M, of the same topological type as M0, that fits the data well and has a small number of vertices. Our approach is to minimize an energy function that explicitly models the competing desires of conciseness of representation and fidelity to the data. We show that mesh optimization can be effectively used in at least two applications: surface reconstruction from unorganized points, and mesh simplification (the reduction of the number of vertices in an initially dense mesh of triangles).
Octrees for faster isosurface generation
 IEEE TRANSACTIONS ON MEDICAL IMAGING
, 2000
"... The large size of many volume data sets often prevents visualization algorithms from providing interactive rendering. The use of hierarchical data structures can ameliorate this problem by storing summary information to prevent useless exploration of regions of little or no current interest within ..."
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Cited by 274 (3 self)
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The large size of many volume data sets often prevents visualization algorithms from providing interactive rendering. The use of hierarchical data structures can ameliorate this problem by storing summary information to prevent useless exploration of regions of little or no current interest within the volume. This paper discusses research into the use of the octree hierarchical data structure when the regions of current interest can vary during the application, and are not known a priori. Octrees are well suited to the sixsided cell structure of many volumes. A new spaceefficient design is introduced for octree representations of volumes whose resolutions are not conveniently a power of two; octrees following this design are called branchonneed octrees (BONOs). Also, a caching method is described that essentially passes information between octree neighbors whose visitation times may be quite different, then discards it when its useful life is over. Using the application of octrees to isosurface generation as a focus, space and time comparisons for octreebased versus more traditional "marching" methods are presented.
HardwareAccelerated Volume and Isosurface Rendering Based on CellProjection
, 2000
"... We present two beneficial rendering extensions to the Projected Tetrahedra (PT) algorithm by Shirley and Tuchman. These extensions are compatible with any cell sorting technique, for example the BSPXMPVO sorting algorithm for unstructured meshes. ..."
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Cited by 85 (13 self)
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We present two beneficial rendering extensions to the Projected Tetrahedra (PT) algorithm by Shirley and Tuchman. These extensions are compatible with any cell sorting technique, for example the BSPXMPVO sorting algorithm for unstructured meshes.
The Marching Lines Algorithm: New results and proofs
, 1993
"... This research report is a compilation of two articles describing new results concerning the 3D Marching Lines algorithm. The Marching Lines extracts, with subvoxel accuracy, characteristic 3D lines out of 3D images, such as the Crest Lines. Those feature lines can then be used to perform higher lev ..."
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Cited by 30 (6 self)
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This research report is a compilation of two articles describing new results concerning the 3D Marching Lines algorithm. The Marching Lines extracts, with subvoxel accuracy, characteristic 3D lines out of 3D images, such as the Crest Lines. Those feature lines can then be used to perform higher level Image Processing tasks, such as 3D image registration, or automatic labeling of anatomical structures, when Medical Images are processed. The first paper concentrates on the computation of the differential characteristics of isointensity surfaces, and shows how to characterize Crest Lines points directly from the differentials of the 3D Image. The second paper brings the proof of the good topological properties of the reconstructed surfaces and 3D curves obtained with the Marching Lines algorithm. New experiments on real and synthetic data are also presented, showing the high precision and stability of the extracted feature lines.
Kernel Functions in Convolution Surfaces: A Comparative Analysis
 The Visual Computer
, 1999
"... A comprehensive analysis of various convolution kernels is presented. Computational complexity and compatibility between the kernels and a number of modeling primitives are examined. A number of practical suggestions are given how to choose the proper kernel function, with a special attention to pol ..."
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Cited by 24 (0 self)
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A comprehensive analysis of various convolution kernels is presented. Computational complexity and compatibility between the kernels and a number of modeling primitives are examined. A number of practical suggestions are given how to choose the proper kernel function, with a special attention to polynomial kernels. Mathematical formulations for convolved line segments are given. Key words: Geometric modeling  Isosurfaces  Polynomial line segments  Implicit modeling primitives 1 Introduction A convolution surface is the set of points (x; y; z) that satisfy f(x; y; z) = T (1) where T is some scalar value and the field function f(x; y; z) is obtained via a 3D convolution of a kernel function h(p) and a skeleton function g(p): f(p) = Z S g(r)h(p \Gamma r) dr; (2) integrating for all points r that belong to the skeleton S. Skeleton elements may be points, line segments, curves, polygons, and other geometrical modeling primitives. Kernels may be represented by a number of funct...
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.
Haptic Sculpting of Volumetric Implicit Functions
, 2001
"... Implicit functions characterized by the zeroset of polynomialbased algebraic equations and other commonlyused analytic equations are extremely powerful in graphics, geometric design, and visualization. But the potential of implicit functions is yet to be fully realized due to the lack of flexible ..."
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Cited by 18 (8 self)
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Implicit functions characterized by the zeroset of polynomialbased algebraic equations and other commonlyused analytic equations are extremely powerful in graphics, geometric design, and visualization. But the potential of implicit functions is yet to be fully realized due to the lack of flexible and interactive design techniques. This paper presents a haptic sculpting system founded upon scalar trivariate Bspline functions. All the solids sculpted in our environment are semialgebraic sets of volumetric implicit functions. We develop a large variety of sculpting toolkits equipped with an intuitive haptic interface to facilitate the direct manipulation of implicit functions in realtime. To facilitate multiresolution editing and different levels of details, we employ three techniques: hierarchical Bsplines, CSGbased functional composition, and knot insertion. Our experiments demonstrate that our algorithms and hapticsbased techniques can greatly overcome the modeling difficulties associated with implicit functions. The novel modeling techniques and their hapticsbased design principle are extensible to the design of arbitrary implicit functions.
Visualization, Analysis and Shape Reconstruction of Unorganized Data Sets
"... In this chapter mathematical models and efficient algorithms are developed for the visualization, analysis and shape reconstruction for an arbitrary data set that can include unorganized points or continuous manifolds of any codimension, such as pieces of curves and surface patches. The distance fun ..."
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Cited by 9 (0 self)
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In this chapter mathematical models and efficient algorithms are developed for the visualization, analysis and shape reconstruction for an arbitrary data set that can include unorganized points or continuous manifolds of any codimension, such as pieces of curves and surface patches. The distance function to the data set and its contours are used for fast visualization and analysis of the data set. A minimal surface and a convection model are used for shape reconstruction from the data set. All formulations and numerical algorithms are based on implicit representations on simple rectangular grids which extend to any number of dimensions and which also can easily be combined with the level set method for dynamic shape deformation and other manipulations.
Fast Ray Tracing of Arbitrary Implicit Surfaces with Interval and Affine Arithmetic
"... Existing techniques for rendering arbitraryform implicit surfaces are limited, either in performance, correctness or flexibility. Ray tracing algorithms employing interval arithmetic (IA) or affine arithmetic (AA) for rootfinding are robust and general in the class of surfaces they support, but tr ..."
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Cited by 9 (4 self)
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Existing techniques for rendering arbitraryform implicit surfaces are limited, either in performance, correctness or flexibility. Ray tracing algorithms employing interval arithmetic (IA) or affine arithmetic (AA) for rootfinding are robust and general in the class of surfaces they support, but traditionally slow. Nonetheless, implemented efficiently using a stackdriven iterative algorithm and SIMD vector instructions, these methods can achieve interactive performance for common algebraic surfaces on the CPU. A similar algorithm can also be implemented stacklessly, allowing for efficient ray tracing on the GPU. This paper presents these algorithms, as well as an inclusionpreserving reduced affine arithmetic (RAA) for faster raysurface intersection. Shader metaprogramming allows for immediate and automatic generation of symbolic expressions and their interval or affine extensions. Moreover, we are able to render even complex forms robustly, in realtime at high resolution.
Seed sets and search structures for optimal isocontour extraction
 Texas Institute of Computational and Applied Mathematics
, 1999
"... The search for intersected cells in isocontouring can be accelerated using suitable range query data structures, such as the interval tree or segment tree. The storage overhead of such search structures can be significantly reduced by searching over a subset of the cells Ë, called a seed set, which ..."
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Cited by 7 (1 self)
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The search for intersected cells in isocontouring can be accelerated using suitable range query data structures, such as the interval tree or segment tree. The storage overhead of such search structures can be significantly reduced by searching over a subset of the cells Ë, called a seed set, which contains at least one cell per connected component of every isocontour. We present three algorithms for generating seed sets and compare their time complexity and performance in terms of the number of seed cells generated. The first two algorithms are applicable to both regular and irregular grids of arbitrary dimension, while the third is a specialization for regular grids. The first algorithm produces a nearly optimal seed set, minimizing the storage overhead for the search structure. While the second and third algorithms may produce a larger seed set, they are extremely fast, have the advantage of being suitable for extremely large datasets that cannot be kept in main memory (outofcore computation), and are amenable to parallelization. In each case the resulting seed sets are orders of magnitude smaller than the total number of cells, while the computational complexity remains optimal. We compare the results of the two new algorithms with previous results and recent new work. 2