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Topological manipulation of isosurfaces
, 2004
"... In this thesis, I show how to use the topological information encoded in an abstraction called the contour tree to enable interactive manipulation of individual contour surfaces in an isosurface scene, using an interface called the flexible isosurface. Underpinning this interface are several improve ..."
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Cited by 18 (3 self)
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In this thesis, I show how to use the topological information encoded in an abstraction called the contour tree to enable interactive manipulation of individual contour surfaces in an isosurface scene, using an interface called the flexible isosurface. Underpinning this interface are several improvements and extensions to existing work on the contour tree. The first, and most critical, extension, is the path seed: a new method of generating seeds from the contour tree for isosurface extraction. The second extension is to compute geometric information called local spatial measures for contours and store this information in the contour tree. The third extension is to use local spatial measures to simplify both the contour tree and isosurface displays. This simplification can also be used for noise removal. Lastly, this thesis extends work with contour trees from simplicial meshes to arbitrary meshes, interpolants, and tessellation cases. ii Contents ii
Topologically Defined Isosurfaces
 IN PROC. 6TH DISCRETE GEOMETRY FOR COMPUTER IMAGERY
, 1996
"... In this research report, we present a new process for defining and building the set of configurations of MarchingCubes algorithms. Our aim is to extract a topologically correct isosurface from a volumetric image. Our approach exploits the underlying discrete topology of voxels and especially the ..."
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Cited by 15 (0 self)
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In this research report, we present a new process for defining and building the set of configurations of MarchingCubes algorithms. Our aim is to extract a topologically correct isosurface from a volumetric image. Our approach exploits the underlying discrete topology of voxels and especially their connectedness. Our main
On the topology of grid continua
 SPIE VISION GEOMETRY VII
, 1998
"... Onedimensional and twodimensional continua belong to the basic notions of settheoretical topology and represent a subfield of the theory of dimensions developed by P. Urysohn and K. Menger. In this paper basic definitions and properties of grid continua in R² and R³ are summarised. Particularly, s ..."
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Cited by 14 (6 self)
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Onedimensional and twodimensional continua belong to the basic notions of settheoretical topology and represent a subfield of the theory of dimensions developed by P. Urysohn and K. Menger. In this paper basic definitions and properties of grid continua in R² and R³ are summarised. Particularly, simple onedimensional grid continua in R² and in R³, and simple closed twodimensional grid continua in R³ are emphasised. Concepts for measuring the length of onedimensional grid continua, or the surface area of twodimensional grid continua are introduced and discussed.
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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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
Multigrid analysis of curvature estimators
 Massey University
, 2003
"... This report explains a new method for the estimation of curvature of plane curves and compares it with a method which has been presented in [2]. Both methods are based on global approximations of tangents by digital straight line segments. Experimental studies show that a replacement of global by lo ..."
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Cited by 11 (3 self)
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This report explains a new method for the estimation of curvature of plane curves and compares it with a method which has been presented in [2]. Both methods are based on global approximations of tangents by digital straight line segments. Experimental studies show that a replacement of global by local approximation results in errors which, in contrast to the global approximation, converge to constants> 0. We also apply the new global method for curvature estimation of curves to surface curvature estimation, and discuss a method for estimating mean curvature of surfaces which is based on Meusnier's theorem.
Visualizing industrial CT volume data for nondestructive testing applications
 In Proc’ of the Visualization IEEE
, 2003
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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 9 (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
Digital planarity  a review
, 2006
"... Digital planarity is defined by digitizing Euclidean planes in the threedimensional digital space of voxels; voxels are given either in the gridpoint or the gridcube model. The paper summarizes results (also including most of the proofs) about different aspects of digital planarity, such as suppo ..."
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Cited by 8 (3 self)
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Digital planarity is defined by digitizing Euclidean planes in the threedimensional digital space of voxels; voxels are given either in the gridpoint or the gridcube model. The paper summarizes results (also including most of the proofs) about different aspects of digital planarity, such as supporting or separating Euclidean planes, characterizations in arithmetic geometry, periodicity, connectivity, and algorithmic solutions. The paper provides a uniform presentation, which further extends and details a recent book chapter in (Klette and Rosenfeld 2004).
Reconstruction of 3D medical images: A nonlinear interpolation technique for reconstruction of 3D medical images
 Computer Vision, Graphics, and Image Processing 53(4):382391
, 1991
"... Threedimensional medical images reconstructed from a series of twodimensional images produced by computerized tomography, magnetic resonance imaging, etc., present a valuable tool for modem medicine. Usually, the interresolution between two cross sections is less than the intraresolution within e ..."
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Cited by 7 (0 self)
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Threedimensional medical images reconstructed from a series of twodimensional images produced by computerized tomography, magnetic resonance imaging, etc., present a valuable tool for modem medicine. Usually, the interresolution between two cross sections is less than the intraresolution within each cross section. Therefore, interpolations are required to create a 3D visualization. Many techniques, including voxelbased and patch tiling methods, apply linear interpolations between two cross sections. Although those techniques using linear interpolations are economical in computation, they need much crosssectional data and are unable to enlarge because of aliasmg. Hence, the techniques that apply twodimensional nonlinear interpolation functions among cross sections were proposed. In this paper, we introduce the curvature sampling of the contour of a medical object in a CT (computerized tomography) image. Those sampled contour points are the candidates for the control points of Hermite surfaces between each pair of cross sections. Then, a nearestneighbor mapping of control points between every two cross sections is used for surface formation. The time complexity of our mapping algorithm is O(m + n), where m and II are the numbers of control points of two cross sections. It is much faster than Kehtamavaz and De Figueiredo’s merge method, whose time complexity is O(n’m~). 0 1991 Academic Press, Inc. 1.