## Quadric-based simplification in any dimension (2005)

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Venue: | ACM Transactions on Graphics |

Citations: | 36 - 2 self |

### BibTeX

@ARTICLE{Garland05quadric-basedsimplification,

author = {Michael Garland and Yuan Zhou},

title = {Quadric-based simplification in any dimension},

journal = {ACM Transactions on Graphics},

year = {2005},

volume = {24},

pages = {209--239}

}

### OpenURL

### Abstract

We present a new method for simplifying simplicial complexes of any type embedded in Euclidean spaces of any dimension. At the heart of this system is a novel generalization of the quadric error metric used in surface simplification. 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.

### Citations

3921 |
Pattern Classification and Scene Analysis
- Duda, Hart
- 1973
(Show Context)
Citation Context ...t until the approximation meets some termination criteria. This is evidently a rather natural algorithm for curve approximation, since it was independently invented by a number of people [Ramer 1972; =-=Duda and Hart 1973-=-; Douglas and Peucker 1973; Baumgart 1974; Turner 1974; Pavlidis 1977; Ballard 1981]. Hershberger and Snoeyink [1994] discuss the efficient implementation of the Douglas–Peucker algorithm, specificall... |

1114 | Progressive meshes
- Hoppe
- 1996
(Show Context)
Citation Context ...on [Hoppe et al. 1993]. Of the numerous algorithms developed since, many of the most effective have been based on iterative edge contraction. Particularly relevant methods include progressive meshes [=-=Hoppe 1996-=-], quadric-based simplification [Garland and Heckbert 1997], and memoryless simplification [Lindstrom and Turk 1998]. Since typical graphics applications use additional non-geometric surface attribute... |

945 | P.: Surface simplification using quadric error metrics
- GARLAND, HECKBERT
- 1997
(Show Context)
Citation Context ...nguage, we have been able to produce a single templatized simplification code to handle all input types. At the heart of our approach is a novel generalization of the well-known quadric error metric [=-=Garland and Heckbert 1997-=-; Garland 1999b]. 2. BACKGROUND Unless noted otherwise, we assume that we are given a manifold simplicial complex M = (V, F ), composed of a set of d-simplices F and a set of vertices V embedded in th... |

704 |
Linear Algebra and Its Applications
- Strang
- 1980
(Show Context)
Citation Context ... the error of this optimal point will be: Q(x ∗ ) = b T x ∗ + c = −b T A −1 b + c (10) Note that these are both instances of well-known formulae for the minimum of a positive definite quadratic form [=-=Strang 1988-=-, pg. 347]. In the case when A is singular, it can be advantageous to perform singular value decomposition so as to project vertices onto the solution space [Lindstrom 2000b]. However, in the context ... |

569 | Decimation of triangle meshes
- Schroeder, Zarge, et al.
- 1992
(Show Context)
Citation Context ...ification. The earliest general methods for the automatic simplification of triangulated surfaces were vertex clustering [Rossignac and Borrel 1993], surface re-tiling [Turk 1992], vertex decimation [=-=Schroeder et al. 1992-=-], and mesh optimization [Hoppe et al. 1993]. Of the numerous algorithms developed since, many of the most effective have been based on iterative edge contraction. Particularly relevant methods includ... |

443 |
Algorithms for the reduction of the number of points required to represent a digitized line or its caricature
- Douglas, Peucker
- 1973
(Show Context)
Citation Context ...ation meets some termination criteria. This is evidently a rather natural algorithm for curve approximation, since it was independently invented by a number of people [Ramer 1972; Duda and Hart 1973; =-=Douglas and Peucker 1973-=-; Baumgart 1974; Turner 1974; Pavlidis 1977; Ballard 1981]. Hershberger and Snoeyink [1994] discuss the efficient implementation of the Douglas–Peucker algorithm, specifically with an eye towards redu... |

404 | Re-tiling polygonal surfaces
- Turk
- 1992
(Show Context)
Citation Context ...to our own. 2.1.1 Surface Simplification. The earliest general methods for the automatic simplification of triangulated surfaces were vertex clustering [Rossignac and Borrel 1993], surface re-tiling [=-=Turk 1992-=-], vertex decimation [Schroeder et al. 1992], and mesh optimization [Hoppe et al. 1993]. Of the numerous algorithms developed since, many of the most effective have been based on iterative edge contra... |

403 | Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator
- Shewchuk
- 1996
(Show Context)
Citation Context ...tical). Consider again the model shown in Figure 1. This is a planar triangle mesh of the interior of Lake Superior and contains 2894 vertices. It was made with the Triangle mesh generation software [=-=Shewchuk 1996-=-]. As we can clearly see in Figure 1, the triangle aspect ratios of the simplified mesh are quite extreme. However, we can easily remedy this by adding an additional vertex distribution quadric of the... |

354 | Mesh Optimization
- Hoppe, DeRose, et al.
- 1993
(Show Context)
Citation Context ... automatic simplification of triangulated surfaces were vertex clustering [Rossignac and Borrel 1993], surface re-tiling [Turk 1992], vertex decimation [Schroeder et al. 1992], and mesh optimization [=-=Hoppe et al. 1993-=-]. Of the numerous algorithms developed since, many of the most effective have been based on iterative edge contraction. Particularly relevant methods include progressive meshes [Hoppe 1996], quadric-... |

348 | Discrete differential-geometry operators for triangulated 2-manifolds, inVisualizationandMathematicsIII,H.-C.HegeandK.Polthier,eds
- Meyer, Desbrun, et al.
- 2003
(Show Context)
Citation Context ...nd yields little improvement in accuracy. It can also complicate our construction as the set of points closer to v than any other vertex may extend beyond its 1-ring in the presence of obtuse angles [=-=Meyer et al. 2003-=-]. Therefore, we adopt a much simpler approach. The fundamental quadric of a vertex is the weighted sum of the fundamental quadrics of the maximal simplices incident on it. Each simplex is divided equ... |

305 |
Multi-resolution 3D approximations for rendering complex scenes”, Geometric Modeling
- Rossignac, Borrel
- 1993
(Show Context)
Citation Context ...section, we review only the work most relevant to our own. 2.1.1 Surface Simplification. The earliest general methods for the automatic simplification of triangulated surfaces were vertex clustering [=-=Rossignac and Borrel 1993-=-], surface re-tiling [Turk 1992], vertex decimation [Schroeder et al. 1992], and mesh optimization [Hoppe et al. 1993]. Of the numerous algorithms developed since, many of the most effective have been... |

270 | Metro: Measuring Error on Simplified Surfaces
- Cignoni, Rocchini, et al.
- 1998
(Show Context)
Citation Context ... 2.1 Related Work Over the years, many methods have been developed for the automatic simplification of simplicial models. Details on many of them can be found in one of a number of available surveys [=-=Cignoni et al. 1998-=-; Garland 1999a; Luebke et al. 2002]. In this section, we review only the work most relevant to our own. 2.1.1 Surface Simplification. The earliest general methods for the automatic simplification of ... |

198 |
Level of Detail for 3D Graphics
- Luebke, Reddy, et al.
- 2003
(Show Context)
Citation Context ...ny methods have been developed for the automatic simplification of simplicial models. Details on many of them can be found in one of a number of available surveys [Cignoni et al. 1998; Garland 1999a; =-=Luebke et al. 2002-=-]. In this section, we review only the work most relevant to our own. 2.1.1 Surface Simplification. The earliest general methods for the automatic simplification of triangulated surfaces were vertex c... |

196 | Dual contouring of hermite data - Ju, Losasso, et al. - 2002 |

193 | M.: Survey of polygonal surface simplification algorithms - Heckbert, Garland - 1995 |

171 | Simplification envelopes
- Cohen, Varshney, et al.
- 1996
(Show Context)
Citation Context ...rantee that self-intersections are never created. In such cases, it is best to couple iterative contraction methods with envelope methods. In particular, we recommend either simplification envelopes [=-=Cohen et al. 1996-=-] or permission grids [Zelinka and Garland 2002]. 4. STANDARD ERROR METRIC We begin by briefly reviewing the derivation of the standard quadric error metric for triangulated 2-manifolds [Garland and H... |

169 | Simplifying surfaces with color and texture using quadric error metrics - Garland, Heckbert - 1998 |

158 | Progressive simplicial complexes
- Popovic, Hoppe
- 1997
(Show Context)
Citation Context ...faces, and tetrahedralized volumes. However, comparatively little effort has been directed towards algorithmic techniques that can successfully simplify all such data. And with one notable exception [=-=Popović and Hoppe 1997-=-], the problem of simplifying objects of mixed dimension (e.g., containing both lines and triangles) has remained unaddressed. In this paper, we present a new simplification method that can produce ap... |

145 | A comparison of mesh simplification algorithms
- Cignoni, Montani, et al.
- 1998
(Show Context)
Citation Context ... 2.1 Related Work Over the years, many methods have been developed for the automatic simplification of simplicial models. Details on many of them can be found in one of a number of available surveys [=-=Cignoni et al. 1998-=-; Garland 1999a; Luebke et al. 2002]. In this section, we review only the work most relevant to our own. 2.1.1 Surface Simplification. The earliest general methods for the automatic simplification of ... |

138 | Fast and memory efficient polygonal simplification
- Lindstrom, Turk
- 1998
(Show Context)
Citation Context ...en based on iterative edge contraction. Particularly relevant methods include progressive meshes [Hoppe 1996], quadric-based simplification [Garland and Heckbert 1997], and memoryless simplification [=-=Lindstrom and Turk 1998-=-]. Since typical graphics applications use additional non-geometric surface attributes, especially color and texture, methods have also been developed for preserving these attributes during the simpli... |

136 |
Structural Pattern Recognition
- Pavlidis
- 1977
(Show Context)
Citation Context ...ly a rather natural algorithm for curve approximation, since it was independently invented by a number of people [Ramer 1972; Duda and Hart 1973; Douglas and Peucker 1973; Baumgart 1974; Turner 1974; =-=Pavlidis 1977-=-; Ballard 1981]. Hershberger and Snoeyink [1994] discuss the efficient implementation of the Douglas–Peucker algorithm, specifically with an eye towards reducing its worst case complexity. Decimation ... |

135 |
Geometric modeling for computer vision
- Baumgart
- 1974
(Show Context)
Citation Context ...on criteria. This is evidently a rather natural algorithm for curve approximation, since it was independently invented by a number of people [Ramer 1972; Duda and Hart 1973; Douglas and Peucker 1973; =-=Baumgart 1974-=-; Turner 1974; Pavlidis 1977; Ballard 1981]. Hershberger and Snoeyink [1994] discuss the efficient implementation of the Douglas–Peucker algorithm, specifically with an eye towards reducing its worst ... |

133 | Out-of-core simplification of large polygonal models - LINDSTROM |

133 |
Full-range approximation of triangulated polyhedra
- Ronfard, Rossignac
- 1996
(Show Context)
Citation Context ...ximations. Preventing inversion by explicit orientation checks is quite common. It has been used for planar graphs [Ciarlet and Lamour 1996], triangular meshes [Garland and Heckbert 1997; Hoppe 1996; =-=Ronfard and Rossignac 1996-=-], and tetrahedral volumes [Staadt and Gross 1998; Gelder et al. 1999; Trotts et al. 1999; Cignoni et al. 2000; Chopra and Meyer 2002]. In our quadric-based system, it is in planar regions where mesh ... |

132 | Appearance-Preserving Simplification - Cohen, Olano, et al. |

118 | Multiresolution modeling: Survey & future opportunities
- Garland
- 1999
(Show Context)
Citation Context ...o produce a single templatized simplification code to handle all input types. At the heart of our approach is a novel generalization of the well-known quadric error metric [Garland and Heckbert 1997; =-=Garland 1999-=-b]. 2. BACKGROUND Unless noted otherwise, we assume that we are given a manifold simplicial complex M = (V, F ), composed of a set of d-simplices F and a set of vertices V embedded in the Euclidean sp... |

114 |
An iterative procedure for the polygonal approximation of plane curves
- Ramer
- 1972
(Show Context)
Citation Context ...rsively split until the approximation meets some termination criteria. This is evidently a rather natural algorithm for curve approximation, since it was independently invented by a number of people [=-=Ramer 1972-=-; Duda and Hart 1973; Douglas and Peucker 1973; Baumgart 1974; Turner 1974; Pavlidis 1977; Ballard 1981]. Hershberger and Snoeyink [1994] discuss the efficient implementation of the Douglas–Peucker al... |

98 | ªNew Quadric Metric for Simplifying Meshes with Appearance Attributes,º
- Hoppe
- 1999
(Show Context)
Citation Context ...r the kinds of data encountered in practice, this is not a problem, but it would be useful to explore the possibility of generalizing Hoppe’s scheme for reducing the number of necessary coefficients [=-=Hoppe 1999-=-]. Even more dramatic memory reduction could be achieved by implementing a multiphase system [Garland and Shaffer 2002] built on top of our generalized metric. Both the standard quadric metric and our... |

94 | Surface approximation and geometric partitions
- Agarwal, Suri
- 1998
(Show Context)
Citation Context ...6; Cignoni et al. 1998]. All existing surface simplification are fundamentally heuristic in nature. Optimal height field approximation, and by extension surface approximation, is known to be NP-Hard [=-=Agarwal and Suri 1994-=-]. However, Heckbert and Garland [1999] have shown that, in the limit of infinitesimal triangulations, their quadric-based method produces triangles with the optimal aspect ratio required for L2-optim... |

88 | Multiresolution decimation based on global error
- Ciampalini, Cignoni, et al.
- 1996
(Show Context)
Citation Context ...n levels, it also produces the best results. The next two algorithms below mesh optimization are the the simplification envelopes algorithm [Cohen et al. 1996], and the JADE vertex decimation system [=-=Ciampalini et al. 1997-=-]. By comparison to mesh optimization, these algorithms provide a significant savings in running time, but produce approximations with higher error at higher levels of simplification. The final two al... |

76 | Quadric-Based Polygonal Surface Simplification
- Garland
- 1999
(Show Context)
Citation Context ...lation; they do not enforce the strict angle guarantees provided by Triangle. Positional error terms of this sort are quite natural and have been previously proposed in a number of contexts. Garland [=-=Garland 1999-=-b] recommended their use for preserving important feature points (e.g., the tips of horns) and they are a key component of the user-guided simplification system of Kho and Garland [2003]. Lindstrom [L... |

63 | Progressive Tetrahedralizations
- Staadt, Gross
- 1998
(Show Context)
Citation Context ...n checks is quite common. It has been used for planar graphs [Ciarlet and Lamour 1996], triangular meshes [Garland and Heckbert 1997; Hoppe 1996; Ronfard and Rossignac 1996], and tetrahedral volumes [=-=Staadt and Gross 1998-=-; Gelder et al. 1999; Trotts et al. 1999; Cignoni et al. 2000; Chopra and Meyer 2002]. In our quadric-based system, it is in planar regions where mesh inversions are most likely to occur. Figure 1 dem... |

62 | Topology preserving edge contraction - Dey, Edelsbrunner, et al. - 1999 |

57 | Surface simplification with variable tolerance - Guéziec - 1995 |

54 |
Strip trees: A hierarchical representation for curves
- Ballard
- 1981
(Show Context)
Citation Context ...ural algorithm for curve approximation, since it was independently invented by a number of people [Ramer 1972; Duda and Hart 1973; Douglas and Peucker 1973; Baumgart 1974; Turner 1974; Pavlidis 1977; =-=Ballard 1981-=-]. Hershberger and Snoeyink [1994] discuss the efficient implementation of the Douglas–Peucker algorithm, specifically with an eye towards reducing its worst case complexity. Decimation algorithms, wh... |

53 |
On the Difficulty of Triangulating Three-Dimensional Nonconvex Polyhedra
- Ruppert, Seidel
- 1992
(Show Context)
Citation Context ... by the fact that the hole left by a vertex removal is not necessarily tetrahedralizable without adding Steiner Technical Report No. UIUCDCS-R-2004-2450, June 2004.s4 · M. Garland and Y. Zhou points [=-=Ruppert and Seidel 1992-=-]. Van Gelder et al. [1999] implemented vertex removal via half-edge contractions and suggested a local density metric to guide the selection of vertices to remove. Trotts et al. [1998; 1999], Cignoni... |

52 |
A Texture-Mapping Approach for the Compression of Colored 3D Triangulations
- Soucy, Godin, et al.
- 1996
(Show Context)
Citation Context ...ocused on minimizing parametric distortion during simplification. Others have pursued solutions based on resampling of attributes from the original surface onto the output approximation [Maruya 1995; =-=Soucy et al. 1996-=-; Cignoni et al. 1998]. All existing surface simplification are fundamentally heuristic in nature. Optimal height field approximation, and by extension surface approximation, is known to be NP-Hard [A... |

51 | Simplification Of Tetrahedral Meshes With Accurate Error Evaluation
- Cignoni, Costanza, et al.
- 2000
(Show Context)
Citation Context ...iarlet and Lamour 1996], triangular meshes [Garland and Heckbert 1997; Hoppe 1996; Ronfard and Rossignac 1996], and tetrahedral volumes [Staadt and Gross 1998; Gelder et al. 1999; Trotts et al. 1999; =-=Cignoni et al. 2000-=-; Chopra and Meyer 2002]. In our quadric-based system, it is in planar regions where mesh inversions are most likely to occur. Figure 1 demonstrates what can happen if mesh inversions are not prevente... |

49 | Optimal triangulation and quadric-based surface simplification - Heckbert, Garland - 1999 |

46 | M.: Efficient adaptive simplification of massive meshes
- SHAFFER, GARLAND
(Show Context)
Citation Context ... along the way. Fortunately, our generalized metric is equivalent in form to the standard quadric error metric. Therefore, it can trivially form the basis of an efficient clustering [Lindstrom 2000b; =-=Shaffer and Garland 2001-=-] or multiphase [Garland and Shaffer 2002] simplification system. 8.3 Implementation We believe that one of the great strengths of our error formulation is its uniformity. A single simple summation (E... |

39 | Surface simplification inside a tolerance volume - Guéziec - 1996 |

36 | Simplification Of Tetrahedral Meshes - Trotts, Hamann, et al. - 1998 |

34 | A general method for preserving attribute values on simplified meshes
- Garland, Cignoni, et al.
- 1998
(Show Context)
Citation Context ... 2.1 Related Work Over the years, many methods have been developed for the automatic simplification of simplicial models. Details on many of them can be found in one of a number of available surveys [=-=Cignoni et al. 1998-=-; Garland 1999a; Luebke et al. 2002]. In this section, we review only the work most relevant to our own. 2.1.1 Surface Simplification. The earliest general methods for the automatic simplification of ... |

32 | Selective Refinement Queries for Volume Visualization of Unstructured Tetrahedral Meshes - Cignoni, Floriani, et al. |

31 | Polygonal approximations of a curve (formulations and algorithms - Imai, Iri - 1988 |

31 |
Simplification Of Tetrahedral Meshes With Error Bounds
- Trotts, Hamann, et al.
- 1999
(Show Context)
Citation Context ...for planar graphs [Ciarlet and Lamour 1996], triangular meshes [Garland and Heckbert 1997; Hoppe 1996; Ronfard and Rossignac 1996], and tetrahedral volumes [Staadt and Gross 1998; Gelder et al. 1999; =-=Trotts et al. 1999-=-; Cignoni et al. 2000; Chopra and Meyer 2002]. In our quadric-based system, it is in planar regions where mesh inversions are most likely to occur. Figure 1 demonstrates what can happen if mesh invers... |

29 | GAPS: general and automatic polygonal simplification - Erikson, Manocha - 2002 |

27 | Modeling realistic virtual hairstyles - YU - 2001 |

26 |
A multiphase approach to efficient surface simplification
- Garland, Shaffer
- 2002
(Show Context)
Citation Context ...zed metric is equivalent in form to the standard quadric error metric. Therefore, it can trivially form the basis of an efficient clustering [Lindstrom 2000b; Shaffer and Garland 2001] or multiphase [=-=Garland and Shaffer 2002-=-] simplification system. 8.3 Implementation We believe that one of the great strengths of our error formulation is its uniformity. A single simple summation (Equation 15) is used to construct quadrics... |

26 | Generalized Unstructured Decimation - Renze, Oliver - 1996 |