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Quadric-based simplification in any dimension

Authors:

Michael Garland,

Yuan ZhouAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 24, Issue 2

Pages 209 - 239

https://doi.org/10.1145/1061347.1061350

Published: 01 April 2005 Publication History

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Abstract

We present a novel generalization of the quadric error metric used in surface simplification that can be used for simplifying simplicial complexes of any type embedded in Euclidean spaces of any dimension. We demonstrate that our generalized simplification system can produce high quality approximations of plane and space curves, triangulated surfaces, tetrahedralized volume data, and simplicial complexes of mixed type. Our method is both efficient and easy to implement. It is capable of processing complexes of arbitrary topology, including nonmanifolds, and can preserve intricate boundaries.

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Index Terms

Quadric-based simplification in any dimension
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models

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Reviews

Reviewer: Joseph J. O'Rourke

The original Garland-Heckbert surface simplification algorithm [1] was both novel and successful. The basic structure of the algorithm is to repeatedly contract edges of the surface, where, in the context of this paper, the contraction of edge (vi,vj) replaces all occurrences of vertex vj with vertex vi. Which edge to contract at each step is determined by a cost function. This cost function measures a sum of squared distances and is expressed as a quadratic form whose level surfaces are quadrics. In this new work, Garland and Zhou generalize the quadric error measure so that it still measures a squared distance error, but is expressed in terms of an orthonormal basis of manifold tangent vectors, rather than on surface normal vectors. This seemingly small variation in defining the "fundamental quadric" has a significant consequence: the computation is independent of the dimension in which the manifold is embedded. This permits them to use the same definition to simplify planar curves, 3-space curves, triangulated surfaces in 3D, and tetrahedralized volumes. Even triangulated surfaces with k extra attributes (for example, RGB colors) associated with each point succumb, viewed as manifolds embedded in a space of (3+k) dimensions. The initialization phase of the algorithm computes the fundamental quadric for each vertex v, which can be viewed as a machine to measure error of any point from v. These quadrics are in turn based on quadrics for each incident simplex. The cost of contracting edge (vi,vj) then reflects the movement errors caused by the contraction, based on the quadrics of its endpoints vi and vj. Boundary edges that should be preserved have their error quadrics boosted by a user-specified penalty factor. Contractions that invert simplices of the manifold are disallowed, as would be those that alter the topology (again, if the user so desires). After contraction, the altered error quadrics are recomputed, and the process is repeated. An impressive set of examples simplified by their GSlim C++ implementation is presented. One surprise is that GSlim outperforms the popular Douglas-Peucker line-simplification algorithm in both speed and quality of output. Even simplifying 2.7 million tetrahedra, treated as a 3-manifold embedded in 6 dimensions to capture an associated vector field, was achieved in six minutes on a 1GHz processor. Online Computing Reviews Service

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cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 24, Issue 2

April 2005

193 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/1061347

Issue’s Table of Contents

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Efficient adaptive simplification of massive meshes

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Efficient adaptive simplification of massive meshes

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Feature preserved volume simplification

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