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Subdivision for Modeling and Animation
 SIGGRAPH ’99 Courses, no. 37. ACM SIGGRAPH
, 1999
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Interactive Multiresolution Mesh Editing
"... We describe a multiresolution representation for meshes based on subdivision. Subdivision is a natural extension of the existing patchbased surface representations. At the same time subdivision algorithms can be viewed as operating directly on polygonal meshes, which makes them a useful tool for me ..."
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Cited by 208 (18 self)
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We describe a multiresolution representation for meshes based on subdivision. Subdivision is a natural extension of the existing patchbased surface representations. At the same time subdivision algorithms can be viewed as operating directly on polygonal meshes, which makes them a useful tool for mesh manipulation. Combination of subdivision and smoothing algorithms of Taubin [26] allows us to construct a set of algorithms for interactive multiresolution editing of complex meshes of arbitrary topology. Simplicity of the essential algorithms for re nement and coarsi cation allows to make them local and adaptive, considerably improving their efficiency. We have built a scalable interactive multiresolution editing system based on such algorithms.
Geometry clipmaps: terrain rendering using nested regular grids
 In SIGGRAPH ’04: ACM SIGGRAPH 2004 Papers
, 2004
"... Illustration using a coarse geometry clipmap (size n=31) View of the 216,000×93,600 U.S. dataset near Grand Canyon (n=255) Figure 1:Terrains rendered using geometry clipmaps, showing clipmap levels (size n×n) and transition regions (in blue on right). Rendering throughput has reached a level that en ..."
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Cited by 146 (2 self)
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Illustration using a coarse geometry clipmap (size n=31) View of the 216,000×93,600 U.S. dataset near Grand Canyon (n=255) Figure 1:Terrains rendered using geometry clipmaps, showing clipmap levels (size n×n) and transition regions (in blue on right). Rendering throughput has reached a level that enables a novel approach to levelofdetail (LOD) control in terrain rendering. We introduce the geometry clipmap, which caches the terrain in a set of nested regular grids centered about the viewer. The grids are stored as vertex buffers in fast video memory, and are incrementally refilled as the viewpoint moves. This simple framework provides visual continuity, uniform frame rate, complexity throttling, and graceful degradation. Moreover it allows two new exciting realtime functionalities: decompression and synthesis. Our main dataset is a 40GB height map of the United States. A compressed image pyramid reduces the size by a remarkable factor of 100, so that it fits entirely in memory. This compressed data also contributes normal maps for shading. As the viewer approaches the surface, we synthesize grid levels finer than the stored terrain using fractal noise displacement. Decompression, synthesis, and normalmap computations are incremental, thereby allowing interactive flight at 60 frames/sec.
√3subdivision
 IN PROCEEDINGS OF ACM SIGGRAPH
, 2000
"... A new stationary subdivision scheme is presented which performs slower topological refinement than the usual dyadic split operation. The number of triangles increases in every step by a factor of 3 instead of 4. Applying the subdivision operator twice causes a uniform refinement with trisection of ..."
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Cited by 138 (4 self)
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A new stationary subdivision scheme is presented which performs slower topological refinement than the usual dyadic split operation. The number of triangles increases in every step by a factor of 3 instead of 4. Applying the subdivision operator twice causes a uniform refinement with trisection of every original edge (hence the name 3subdivision) while two dyadic splits would quadsect every original edge. Besides the finer gradation of the hierarchy levels, the new scheme has several important properties: The stencils for the subdivision rules have minimum size and maximum symmetry. The smoothness of the limit surface is C2 everywhere except for the extraordinary points where it is C1. The convergence analysis of the scheme is presented based on a new general technique which also applies to the analysis of other subdivision schemes. The new splitting operation enables locally adaptive refinement under builtin preservation of the mesh consistency without temporary crackfixing between neighboring faces from different refinement levels. The size of the surrounding mesh area which is affected by selective refinement is smaller than for the dyadic split operation. We further present a simple extension of the new subdivision scheme which makes it applicable to meshes with boundary and allows us to generate sharp feature lines.
Discrete Fairing
 In Proceedings of the Seventh IMA Conference on the Mathematics of Surfaces
, 1997
"... We address the general problem of, given a triangular net of arbitrary topology in IR 3 , find a refined net which contains the original vertices and yields an improved approximation of a smooth and fair interpolating surface. The (topological) mesh refinement is performed by uniform subdivision o ..."
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Cited by 96 (17 self)
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We address the general problem of, given a triangular net of arbitrary topology in IR 3 , find a refined net which contains the original vertices and yields an improved approximation of a smooth and fair interpolating surface. The (topological) mesh refinement is performed by uniform subdivision of the original triangles while the (geometric) position of the newly inserted vertices is determined by variational methods, i.e., by the minimization of a functional measuring a discrete approximation of bending energy. The major problem in this approach is to find an appropriate parameterization for the refined net's vertices such that second divided differences (derivatives) tightly approximate intrinsic curvatures. We prove the existence of a unique optimal solution for the minimization of discrete functionals that involve squared second order derivatives. Finally, we address the efficient computation of fair nets. 1 Introduction One of the main problems in geometric modeling is the gen...
Multiresolution shape deformations for meshes with dynamic connectivity
 In Computer Graphics Forum (Proc. Eurographics 2000
"... Multiresolution shape representation is a very effective way to decompose surface geometry into several levels of detail. Geometric modeling with such representations enables flexible modifications of the global shape while preserving the detail information. Many schemes for modeling with multiresol ..."
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Cited by 89 (10 self)
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Multiresolution shape representation is a very effective way to decompose surface geometry into several levels of detail. Geometric modeling with such representations enables flexible modifications of the global shape while preserving the detail information. Many schemes for modeling with multiresolution decompositions based on splines, polygonal meshes and subdivision surfaces have been proposed recently. In this paper we modify the classical concept of multiresolution representation by no longer requiring a global hierarchical structure that links the different levels of detail. Instead we represent the detail information implicitly by the geometric difference between independent meshes. The detail function is evaluated by shooting rays in normal direction from one surface to the other without assuming a consistent tesselation. In the context of multiresolution shape deformation, we propose a dynamic mesh representation which adapts the connectivity during the modification in order to maintain a prescribed mesh quality. Combining the two techniques leads to an efficient mechanism which enables extreme deformations of the global shape while preventing the mesh from degenerating. During the deformation, the detail is reconstructed in a natural and robust way. The key to the intuitive detail preservation is a transformation map which associates points on the original and the modified geometry with minimum distortion. We show several examples which demonstrate the effectiveness and robustness of our approach including the editing of multiresolution models and models with texture. 1.
4–8 Subdivision
, 2000
"... In this paper we introduce 4–8 subdivision, a new scheme that generalizes the fourdirectional box spline of class � � to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more c ..."
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Cited by 66 (6 self)
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In this paper we introduce 4–8 subdivision, a new scheme that generalizes the fourdirectional box spline of class � � to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more commonly used face or vertex splits. In the uniform case, bisection refinement results in doubling, rather than quadrupling of the number of faces in a mesh. Adaptive bisection refinement, automatically generates conforming variableresolution meshes in contrast to face and vertex split methods which require an postprocessing step to make an adaptively refined mesh conforming. The fact that the size of faces decreases more gradually with refinement allows one to have greater control over the resolution of a refined mesh. It also makes it possible to achieve higher smoothness while using small stencils (the size of the stencils used by our scheme is similar to Loop subdivision). We show that the subdivision surfaces produced by the 4–8 scheme are � � continuous almost everywhere, except at extraordinary vertices where they are is �continuous.
Nonuniform recursive subdivision surfaces
 Proceedings of SIGGRAPH
"... DooSabin and CatmullClark subdivision surfaces are based on the notion of repeated knot insertion of uniform tensor product Bspline surfaces. This paper develops rules for nonuniform DooSabin and CatmullClark surfaces that generalize nonuniform tensor product Bspline surfaces to arbitrary to ..."
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Cited by 63 (9 self)
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DooSabin and CatmullClark subdivision surfaces are based on the notion of repeated knot insertion of uniform tensor product Bspline surfaces. This paper develops rules for nonuniform DooSabin and CatmullClark surfaces that generalize nonuniform tensor product Bspline surfaces to arbitrary topologies. This added flexibility allows, among other things, the natural introduction of features such as cusps, creases, and darts, while elsewhere maintaining the same order of continuity as their uniform counterparts.
Wavelets on Irregular Point Sets
 Phil. Trans. R. Soc. Lond. A
, 1999
"... this article we review techniques for building and analyzing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular we focus on subdivision schemes and commutation rules. Several examples are included. ..."
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Cited by 49 (0 self)
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this article we review techniques for building and analyzing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular we focus on subdivision schemes and commutation rules. Several examples are included.