Multiresolution curves

ABSTRACT

We describe a multiresolution curve representation, based on wavelets, that conveniently supports a variety of operations: smoothing a curve; editing the overall form of a curve while preserving its details; and approximating a curve within any given error tolerance for scan conversion. We present methods to support continuous levels of smoothing as well as direct manipulation of an arbitrary portion of the curve; the control points, as well as the discrete nature of the underlying hierarchical representation, can be hidden from the user. The multiresolution representation requires no extra storage beyond that of the original control points, and the algorithms using the representation are both simple and fast.

References

1. 1.PostScript Language Reference Manual. Addison-Wesley Publishing Company, Inc., 1985.
2. 2.M. J. Banks and E. Cohen. Realtime spline curves from interactively sketched data. Computer Graphics, 24(2):99-107, 1990.
3. 3.R. Bartels and J. Beatty. A technique for the direct manipulation of spline curves. In Proceedings of the 1989 Graphics Interface Confer-ence, pages 33-39, London, Ontario, Canada, June 1989.
4. 4.R. Bartels, J. Beatty, and B. Barsky. An Introduction to Splines for Use in Computer Graphics and Geometric Modeling. Morgan Kaufmann, 1987.
5. 5.Berman, J. Bartell, and D. Salesin. Multiresolution painting and ompositing. Proceedings of SIGGRAPH 94. In Computer Graphics, nnual Conference Series, 1994.
6. 6.Beylkin, R. Coifman, and V. Rokhlin. Fast wavelet transforms and merical algorithms I. Communications on Pure and Applied Math-atics, 44:141-183, 1991.
7. 7.Celniker and D. Gossard. Deformable curve and surface finite el-ents for free-form shape design. Computer Graphics, 25(4):257- , July 1991.
8. 8.C. K. Chui. An Introduction to Wavelets. AcademicPress, Inc., Boston, 1992.
9. 9.C. K. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods in Approximation Theory, volume 9, pages 53-75. Birkhauser Verlag, Basel, 1992.
10. 10.M. D~hlen and T. Lyche. Decomposition of splines. In T. Lyche and L. L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design II, pages 135-160. Academic Press, New York, 1992.
11. 11.R. DeVore, B. Jawerth, and B. Lucier. Image compression through wavelet transform coding. IEEE Transactions on Information Theory, 38(2):719-746, March 1992.
12. 12.G. Farin. Curves and Surfaces for Computer Aided Geometric Design. Academic Press, third edition, 1992.
13. 13.D. Forsey and R. Bartels. Hierarchical B-spline refinement. Computer Graphics, 22(4):205-212, 1988.
14. 14.D. Forsey and R. Bartels. Tensor products and hierarchical fitting. In Curves and Surfaces in Computer Vision and Graphics II, SPIE Pro-ceedings Vol. 1610, pages 88-96, 1991.
15. 15.B. Fowler. Geometric manipulation of tensor product surfaces. In Pro-ceedings of the 1992 Symposium on Interactive 3D Graphics, March 1992. Available as Computer Graphics, Vol. 26, No. 2.
16. 16.J. Hoschek and D. Lasser. Fundamentals of Computer Aided Geomet-ric Design. A K Peters, Ltd., Wellesley, Massachusetts, third edition, 1992.
17. 17.S. Hsu and I. Lee. Skeletal strokes. Proceedings of SIGGRAPH 94. In Computer Graphics, Annual Conference Series, 1994.
18. 18.W. M. Hsu, J. F. Hughes, and H. Kaufman. Direct manipulation of free-form deformations. Computer Graphics, 26(2):177-184, 1992.
19. 19.M. Lounsbery, T. DeRose, and J. Warren. Multiresolution surfaces of arbitrary topological type. Technical Report 93-10-05B, University of Washington, Department of Computer Science and Engineering, Jan-uary 1994.
20. 20.T. Lyche and K. M~rken. Knot removal for parametric B-spline curves and surfaces. Computer Aided Geometric Design, 4(3):217-230, 1987.
21. 21.T. Lyche and K. M~rken. Spline-wavelets of minimal support. In D. Braess and L. L. Schumaker, editors, Numerical Methodsin Approx-imation Theory, volume 9, pages 177-194. Birkhauser Verlag, Basel, 1992.
22. 22.S. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674-693, July 1989. curves from interactively
23. 23.M. Plass and M. Stone. Curve-fitting with piecewise parametric cubics. Computer Graphics, 17(3):229-239, July 1983. 107, 1990. direct manipulation of Graphics Interface Confer-ence, 1989. to Splines for Use Morgan Kaufmann, Multiresolution painting and Computer Graphics, transforms and Applied Math-atics,
24. 24.W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Fetterling. Nu-merical Recipes. Cambridge University Press, Cambridge, second edi-tion, 1992.
25. 25.E. Quak and N. Weyrich. Decomposition and reconstruction algo-rithms for spline wavelets on a bounded interval. CAT Report 294, Center for Approximation Theory, Texas A&MUniversity, April 1993.
26. 26.M. P. Salisbury, S. E. Anderson, R. Barzel, and D. H. Salesin. In-teractive pen-and-ink illustration. Proceedings of SIGGRAPH 94. In Computer Graphics, Annual Conference Series, 1994.
27. 27.P. J. Schneider. Phoenix: An interactive curve design system based on the automatic fitting of hand-sketched curves. Master's thesis, Depart-ment of Computer Science and Engineering, University of Washing-ton, 1988. and surface finite el-ents Graphics, 25(4):257-
28. 28.W. Welch and A. Witkin. Variational surface modeling. Computer Graphics, 26(2):157-166, 1992.

Index Terms

• Computing methodologies

  • Computer graphics

    • Graphics systems and interfaces

    • Shape modeling

      • Parametric curve and surface models

      • Volumetric models

• Human-centered computing

  • Human computer interaction (HCI)

    • Interaction devices

      • Graphics input devices