, Volume 79, Issue 2–4, pp 197–209 | Cite as

Multiresolution morphing for planar curves

  • S. HahmannEmail author
  • G.-P. Bonneau
  • B. Caramiaux
  • M. Cornillac


We present a multiresolution morphing algorithm using ``as-rigid-as-possible'' shape interpolation combined with an angle-length based multiresolution decomposition of simple 2D piecewise curves. This novel multiresolution representation is defined intrinsically and has the advantage that the details' orientation follows any deformation naturally. The multiresolution morphing algorithm consists of transforming separately the coarse and detail coefficients of the multiresolution decomposition. Thus all LoD (level of detail) applications like LoD display, compression, LoD editing etc. can be applied directly to all morphs without any extra computation. Furthermore, the algorithm can robustly morph between very large size polygons with many local details as illustrated in numerous figures. The intermediate morphs behave natural and least-distorting due to the particular intrinsic multiresolution representation.

AMS Subject Classifications

68U07 68U05 65D05 65D08 65D17 


Geometric modeling curves multiresolution analysis morphing interpolation vertex correspondence 


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  1. Adams, J. A. 1975The intrinsic method for curve definitionComput. Aided Des.7243249CrossRefGoogle Scholar
  2. Alexa, M. 2002Recent advances in mesh morphingComput. Graph. Forum21173196CrossRefGoogle Scholar
  3. Alexa, M., Cohen-Or, D., Levin, D.: As-rigid-as-possible shape interpolation. In: SIGGRAPH '00: Proc. 27th Annual Conf. on Computer Graphics and Interactive Techniques. New York: ACM Press/Addison-Wesley 2000, pp. 157–164.Google Scholar
  4. Bertram, M. 2005Single-knot wavelets for nonuniform b-splinesComput. Aided Geom.22848864MathSciNetGoogle Scholar
  5. Carmo, M. D.: Differential geometry of curves and surfaces. Prentice Hall 1976.Google Scholar
  6. Chui, C., Quak, E. 1992Wavelets on a bounded intervalBraess, D.Schumaker, L. eds. Numerical methods of approximation theoryBirkhäuser Basel124Google Scholar
  7. Elber, G., Gotsman, C.: Multiresolution control for nonuniform bspline curve editing. In: 3rd Pacific Graphics Conf. on Computer Graphics and Applications, Seoul, Korea, pp. 267–278, August 1995.Google Scholar
  8. Finkelstein, A., Salesin, D. H.: Multiresolution curves. Computer Graphics Proc. (SIGGRAPH 94), pp. 261–268 (1994).Google Scholar
  9. Floater, M. S., Gotsman, C. 1999How to morph tilings injectivelyJ. Comput. Appl. Math.101117129zbMATHCrossRefMathSciNetGoogle Scholar
  10. Goldstein, E., Gotsman, C.: Polygon morphing using a multiresolution representation. In: Graphics Interface '95. Canadian Inf. Process. Soc., pp. 247–254 (1995).Google Scholar
  11. Hahmann, S., Bonneau, G.-P., Sauvage, B. 2005Area preserving deformation of multiresolution curvesComput. Aided Geom.22249267MathSciNetGoogle Scholar
  12. Lee, A. W. F., Dobkin, D., Sweldens, W., Schröder, P.: Multiresolution mesh morphing. Computer Graphics Proc. (SIGGRAPH 99), 343–350, (1999).Google Scholar
  13. Lipman, Y., Sorkine, O., Levin, D., Cohen-Or, D. 2005Linear rotation-invariant coordinates for meshesACM Trans. Graph.24479487CrossRefGoogle Scholar
  14. Mallat, S. 1989A theory for multiresolution signal decomposition: The wavelet representationIEEE Trans. Pattern Anal. Mach. Intell.11674693zbMATHCrossRefGoogle Scholar
  15. Meyers, D. 1994Multiresolution tilingComput. Graph. Forum13325340CrossRefMathSciNetGoogle Scholar
  16. Sederberg, T., Gao, P., Wang, G., Mu, H. 19932-D shape blending: An intrinsic solution to the vertex path problemComput. Graph. (SIGGRAPH 93 Proc.)271518CrossRefGoogle Scholar
  17. Sederberg, T. W., Greenwood, E. 1992A physical based approach to 2-D shape bendingComput. Graph. (SIGGRAPH '92 Proc.)262534CrossRefGoogle Scholar
  18. Shapira, M., Rappoport, A. 1995Shape blending using the star-skeleton representationIEEE Comput. Graph. Appl.154450CrossRefGoogle Scholar
  19. Stollnitz, E., DeRose, T., Salesin, D.: Wavelets for computer graphics: Theory and applications. Morgan Kaufmann 1996.Google Scholar
  20. Surazhsky, V., Gotsman, C. 2003Intrinsic morphing of compatible triangulationsInt. J. Shape Model9191201zbMATHGoogle Scholar
  21. Sweldens, W. 1997The lifting scheme, A construction of second generation waveletsSIAM J. Math. Anal.29511546CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 2007

Authors and Affiliations

  • S. Hahmann
    • 1
    Email author
  • G.-P. Bonneau
    • 2
  • B. Caramiaux
    • 1
  • M. Cornillac
    • 1
  1. 1.Laboratoire Jean KuntzmannInstitut National Polytechnique de GrenboleGrenoble Cedex 9France
  2. 2.Laboratoire Jean KuntzmannUniveristé Joseph Fourier, INRIA Rhône-AlpesMontbonnotFrance

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