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"Wenger, Rephael"
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Isosurfaces : geometry, topology, and algorithms
\"Ever since Lorensen and Cline published their paper on the marching cubes algorithm, isosurfaces have been a standard technique for the visualization of 3D volumetric data. Yet there is no book exclusively devoted to isosurfaces. This book presents the basic algorithms for isosurface construction and gives a rigorous mathematical perspective to some of the algorithms and results. It offers a solid introduction to research in this area as well as an organized overview of the various algorithms associated with isosurfaces\"-- Provided by publisher.
Book
Isosurfaces
This work represents the first book to focus on basic algorithms for isosurface construction. It also gives a rigorous mathematical perspective on some of the algorithms and results. In color throughout, the book covers the Marching Cubes algorithm and variants, dual contouring algorithms, multilinear interpolation, multiresolution isosurface extraction, isosurfaces in four dimensions, interval volumes, and contour trees. It also describes data structures for faster isosurface extraction as well as methods for selecting significant isovalues.
eBook
Filtering of Small Components for Isosurface Generation
Let \\(f: R^3 R\\) be a scalar field. An isosurface is a piecewise linear approximation of a level set \\(f^-1()\\) for some \\( ın R\\) built from some regular grid sampling of \\(f\\). Isosurfaces constructed from scanned data such as CT scans or MRIs often contain extremely small components that distract from the visualization and do not form part of any geometric model produced from the data. Simple prefiltering of the data can remove such small components while having no effect on the large components that form the body of the visualization. We present experimental results on such filtering.
Paper
Stability of Critical Points with Interval Persistence
Scalar functions defined on a topological space [Omega] are at the core of many applications such as shape matching, visualization and physical simulations. Topological persistence is an approach to characterizing these functions. It measures how long topological structures in the sub-level sets [chi] [belongs to] [Omega]: f([chi]) [less than or equal to] c persist as c changes. Recently it was shown that the critical values defining a topological structure with relatively large persistence remain almost unaffected by small perturbations. This result suggests that topological persistence is a good measure for matching and comparing scalar functions. We extend these results to critical points in the domain by redefining persistence and critical points and replacing sub-level sets [chi] [belongs to] [Omega]: f([chi]) [less than or equal to] c with interval sets [chi] [belongs to] [Omega]: a [less than or equal to] f([chi]) < b. With these modifications we establish a stability result for critical points. This result is strengthened for maxima that can be used for matching two scalar functions. [PUBLICATION ABSTRACT]
Journal Article
Marching Cubes and Variants
In the introduction, we mentioned four different approaches to isosurface construction. In this chapter, we describe one of those approaches to isosurface
construction, the widely used Marching Cubes algorithm by Lorensen and
Cline [Lorensen and Cline, 1987a].
Book Chapter
Interval Volumes
The region between two level sets is called an interval volume. Formally, the
interval volume for a function φ : R3 → R is defined asIφ(σ0, σ1) = {x ∈ R3 : σ0 ≤ φ(x) ≤ σ1},
where σ0, σ1 ∈ R and σ0 < σ1.
Book Chapter
Multiresolution Polyhedral Meshes
In the previous chapter, we presented an algorithm for constructing an isosurface
using a multiresolution tetrahedral mesh. This algorithm requires the entire
mesh to be broken into tetrahedra, even if most of the mesh is uniform and only
a small portion is at high resolution.
Book Chapter