Article

Biorthogonal wavelets for subdivision volumes

Author:

Martin BertramAuthors Info & Claims

SMA '02: Proceedings of the seventh ACM symposium on Solid modeling and applications

Pages 72 - 82

https://doi.org/10.1145/566282.566296

Published: 17 June 2002 Publication History

Abstract

We present a biorthogonal wavelet construction based on Catmull-Clark-style subdivision volumes. Our wavelet transform is the three-dimensional extension of a previously developed construction of subdivision-surface wavelets that was used for multiresolution modeling of large-scale isosurfaces. Subdivision surfaces provide a flexible modeling tool for surfaces of arbitrary topology and for functions defined thereon. Wavelet representations add the ability to compactly represent large-scale geometries at multiple levels of detail. Our wavelet construction based on subdivision volumes extends these concepts to trivariate geometries, such as time-varying surfaces, free-form deformations, and solid models with non-uniform material properties. The domains of the repre-sented trivariate functions are defined by lattices composed of arbitrary polyhedral cells. These are recursively subdivided based on stationary rules converging to piecewise smooth limit-geometries. Sharp features and boundaries, defined by specific polygons, edges, and vertices of a lattice are explicitly represented using modified subdivision rules. Our wavelet transform provides the ability to reverse the subdivision process after a lattice has been re-shaped at a very fine level of detail, for example using an automatic fitting method. During this coarsening process all geometric detail is compactly stored in form of wavelet coefficients from which it can be reconstructed without loss.

References

[1]

C. Bajaj, J. Warren, and G. Xu, A smooth subdivision scheme for hexahedral meshes, The Visual Computer, special issue on subdivision (submitted), 2002, http://www.cs.rice.edu/ jwarren/

[2]

M. Bertram, D.E. Laney, M.A. Duchaineau, C.D. Hansen, B. Hamann, and K.I. Joy, Wavelet representation of contour sets, Proceedings of IEEE Visualization, Oct. 2001, pp. 303--310 & 566

Digital Library

[3]

M. Bertram, M.A. Duchaineau, B. Hamann, and K.I. Joy, Bicubic subdivision-surface wavelets for large-scale isosurface representation and visualization, Proceedings of IEEE Visualization, Oct. 2000, pp. 389--396 & 579

Digital Library

[4]

M. Bertram, Multiresolution Modeling for Scientific Visualization, Ph.D. Thesis, University of California at Davis, July 2000, http://daddi.informatik.uni-kl.de/bertram/

Digital Library

[5]

H. Biermann, A. Levin, and D. Zorin, Piecewise smooth subdivision surfaces with normal control, Computer Graphics, Proceedings of Siggraph 2000, ACM, pp. 113--120

Digital Library

[6]

G.-P. Bonneau, Multiresolution analysis on irregular surface meshes, IEEE Transactions on Visualization and Computer Graphics (TVCG), Vol. 4, No. 4, IEEE, Oct.-Dec. 1998, pp. 365--378

Digital Library

[7]

G.-P. Bonneau, Optimal triangular Haar bases for spherical data, Proceedings of Visualization '99, IEEE, 1999, pp. 279--284 & 534

Digital Library

[8]

E. Catmull and J. Clark, Recursively generated B-spline surfaces on arbitrary topological meshes, Computer-Aided Design, Vol. 10, No. 6, Nov. 1978, pp. 350--355

[9]

T. DeRose, M. Kass, and T. Truong, Subdivision surfaces in character animation, Computer Graphics, Proceedings of Siggraph '98, ACM, 1998, pp. 85--94

Digital Library

[10]

D. Doo and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer-Aided Design, Vol. 10, No. 6, Nov. 1978, pp. 356--360

[11]

N. Dyn, D. Levin, and J.A. Gregory, A butterfly subdivision scheme for surface interpolation with tension control, ACM Transactions on Graphics, Vol. 9, No. 2, April 1990, pp. 160--169

Digital Library

[12]

M. Halstead, M. Kass, and T. DeRose, Efficient, fair interpolation using Catmull-Clark surfaces, Computer Graphics, Proceedings of Siggraph '93, ACM, 1993, pp. 35--44

Digital Library

[13]

H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, and W. Stuetzle, Piecewise smooth surface reconstruction, Computer Graphics, Proceedings of Siggraph 94, ACM, 1994, pp. 295--302

Digital Library

[14]

(MATH) L. Kobbelt, Sqrt(3)-subdivision Computer Graphics, Proceedings of Siggraph 2000, ACM, 2000, pp. 103--112

Digital Library

[15]

L.P. Kobbelt, J. Vorsatz, U. Labsik, and H.-P. Seidel, A shrink wrapping approach to remeshing polygonal surfaces, Proceedings of Eurographics '99, Computer Graphics Forum, Vol. 18, Blackwell Publishers, 1999, pp. 119--129

[16]

L. Kobbelt, Interpolatory subdivision on open quadrilateral nets with arbitrary topology, Proceedings of Eurographics '96, Computer Graphics Forum Vol. 15, Blackwell Publishers, 1996, pp. 409--420

[17]

A.W.F. Lee, W. Sweldens, P. Schröder, L. Cowsar, and D. Dobkin, MAPS: multiresolution adaptive parameterization of surfaces, Computer Graphics, Proceedings of Siggraph '98, ACM, 1998, pp. 95--104

Digital Library

[18]

N. Litke, A. Levin, and P. Schröder, Fitting Subdivision Surfaces, Proceedings of IEEE Visualization 2001, pp. 319--324 & 568

Digital Library

[19]

C.T. Loop, Smooth subdivision surfaces based on triangles, M.S. thesis, Department of Mathematics, University of Utah, 1987

[20]

J.M. Lounsbery, Multiresolution analysis for surfaces of arbitrary topological type, Ph.D. thesis, Department of Mathematics, University of Washington, 1994

Digital Library

[21]

M. Lounsbery, T. DeRose, and J. Warren, Multiresolution analysis for surfaces of arbitrary topological type, ACM Transactions on Graphics, Vol. 16, No. 1, ACM, Jan. 1997, pp. 34--73

Digital Library

[22]

R. MacCracken and K.I. Joy, Free-form deformations with lattices of arbitrary topology, Computer Graphics, Proceedings of Siggraph '96, ACM, 1996, pp. 181--188

Digital Library

[23]

K.T. McDonnell and H. Qin, FEM-based subdivision solids for dynamic and haptic interaction, Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, pp. 312--313, June 2001

Digital Library

[24]

A. Moffat, R.M. Neal, and I.H. Witten, Arithmetic coding revisited, ACM Transactions on Information Systems, Vol. 16, No. 3, July 1998, pp. 256--294

Digital Library

[25]

G.M. Nielson, I.-H. Jung, and J. Sung, Haar wavelets over triangular domains with applications to multiresolution models for flow over a sphere, Proceedings of Visualization '97, IEEE, 1997, pp. 143--150

Digital Library

[26]

V. Pascucci, Slow-growing subdivision in any dimension: towards Removing the curse of dimensionality, Technical Report, Lawrence Livermore National Laboratory, July 2001

[27]

J. Peters and U. Reif, The simplest subdivision scheme for smoothing polyhedra, ACM Transactions on Graphics, Vol. 16, No. 4, 1997, pp. 420--431

Digital Library

[28]

J. Peters and U. Reif, Analysis of algorithms generalizing B-spline subdivision, SIAM Journal on Numerical Analysis, Vol. 13, No. 2, April 1998, pp. 728--748

Digital Library

[29]

H. Prautzsch, Analysis of Ck-subdivision surfaces at extraordinary points Technical Report, University of Karlsruhe (Germany), 1995

[30]

U.A. Reif, A unified approach to subdivision algorithms near extraordinary vertices, Computer-Aided Geometric Design, Vol. 12, No. 2, Elsevier, March 1995, pp. 153--174

Digital Library

[31]

P. Schröder and W. Sweldens, Spherical wavelets: efficiently representing functions on the sphere, Computer Graphics, Proceedings of Siggraph '95, ACM, 1995, pp. 161--172

Digital Library

[32]

T.W. Sederberg, D. Sewell, and M. Sabin, Non-uniform recursive subdivision surfaces, Computer Graphics, Proceedings of Siggraph '98, ACM, 1998, pp. 287--394

Digital Library

[33]

J. Stam, Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values, Computer Graphics, Proceedings of Siggraph '98, ACM, 1998, pp. 395--404

Digital Library

[34]

W. Sweldens, The lifting scheme: a custom-design construction of biorthogonal wavelets, Applied and Computational Harmonic Analysis, Vol. 3, No. 2, pp. 186--200, 1996

[35]

D. Zorin, P. Schröder, and W. Sweldens, Interpolating subdivision for meshes with arbitrary topology, Computer Graphics, Proceedings of Siggraph '96, ACM, 1996, pp. 189--192

Digital Library

[36]

D. Zorin, Smoothness of subdivision on irregular meshes, Constructive Approximation, Vol. 16, No. 3, 2000, pp. 359--397

Digital Library

Cited By

Phan ARaffin RDaniel M(2016)A Review of Two Approaches for Joining 3D MeshesNature of Computation and Communication10.1007/978-3-319-46909-6_9(82-96)Online publication date: 26-Oct-2016
https://doi.org/10.1007/978-3-319-46909-6_9
Phan ARaffin RDaniel MTrong GHluchy LQuyet TCastelli EDuc KChi MTran V(2012)Mesh connection with RBF local interpolation and wavelet transformProceedings of the 3rd Symposium on Information and Communication Technology10.1145/2350716.2350731(81-90)Online publication date: 23-Aug-2012
https://dl.acm.org/doi/10.1145/2350716.2350731
McDonnell KQin H(2007)A novel framework for physically based sculpting and animation of free-form solidsThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-007-0096-923:4(285-296)Online publication date: 13-Mar-2007
https://dl.acm.org/doi/10.1007/s00371-007-0096-9
Show More Cited By

Index Terms

Biorthogonal wavelets for subdivision volumes
1. Information systems
  1. Data management systems
    1. Data structures
      1. Data layout
        Data compression
2. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Approximation algorithms
  2. Mathematical analysis
    1. Numerical analysis
      1. Computation of transforms

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Information & Contributors

Information

Published In

cover image ACM Conferences

SMA '02: Proceedings of the seventh ACM symposium on Solid modeling and applications

June 2002

424 pages

ISBN:1581135068

DOI:10.1145/566282

Conference Chairs:
Hans-Peter Seidel
Max-Planck-Institut für Informatik, Saarbrücken, Germany
,
Vadim Shapiro
University of Wisconsin, Madison, U.S.A.
,
Program Chairs:
Kunwoo Lee
Seoul National University, Seoul, Korea
,
Nick Patrikalakis
Massachsetts Institute of Technology, Cambridge, U.S.A.

Copyright © 2002 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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SIGGRAPH: ACM Special Interest Group on Computer Graphics and Interactive Techniques

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New York, NY, United States

Publication History

Published: 17 June 2002

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SM02: ACM Symposium on Solid Modeling and Applications 2002

June 17 - 21, 2002

Saarbrücken, Germany

Acceptance Rates

SMA '02 Paper Acceptance Rate 43 of 93 submissions, 46%;

Overall Acceptance Rate 86 of 173 submissions, 50%

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Cited By

Phan ARaffin RDaniel M(2016)A Review of Two Approaches for Joining 3D MeshesNature of Computation and Communication10.1007/978-3-319-46909-6_9(82-96)Online publication date: 26-Oct-2016
https://doi.org/10.1007/978-3-319-46909-6_9
Phan ARaffin RDaniel MTrong GHluchy LQuyet TCastelli EDuc KChi MTran V(2012)Mesh connection with RBF local interpolation and wavelet transformProceedings of the 3rd Symposium on Information and Communication Technology10.1145/2350716.2350731(81-90)Online publication date: 23-Aug-2012
https://dl.acm.org/doi/10.1145/2350716.2350731
McDonnell KQin H(2007)A novel framework for physically based sculpting and animation of free-form solidsThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-007-0096-923:4(285-296)Online publication date: 13-Mar-2007
https://dl.acm.org/doi/10.1007/s00371-007-0096-9
Kim SLee KHong TKim MJung MSong YKobbelt LShapiro V(2005)An integrated approach to realize multi-resolution of B-rep modelProceedings of the 2005 ACM symposium on Solid and physical modeling10.1145/1060244.1060262(153-162)Online publication date: 13-Jun-2005
https://dl.acm.org/doi/10.1145/1060244.1060262
McDonnell KChang YQin H(2005)DigitalSculptureGraphical Models10.1016/j.gmod.2004.11.00167:4(347-369)Online publication date: 1-Jul-2005
https://dl.acm.org/doi/10.1016/j.gmod.2004.11.001
McDonnell KChang YQin H(2004)Interpolatory, solid subdivision of unstructured hexahedral meshesThe Visual Computer10.1007/s00371-004-0246-220:6(418-436)Online publication date: 1-Aug-2004
https://doi.org/10.1007/s00371-004-0246-2

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