Figure 6 Surface manipulation at different levels of detail: The original surface(a) is changed at a narrow scale(b), an intermediate scale (c), and a broad scale(d). Figure 7 Surface approximation using subdivision surface wavelets: (a) the original surface, (b)an intermediate approximation, and(c)coarse approximation. plained enough of the fundamentals for interested readers to explore [5] Ingrid Daubechies. Orthonormal bases of compactly supported both the construction of wavelets and their application to problems wavelets. Communications on Pure and Applied Mathematics in graphics and beyond. We present a more thorough discussion in 41(7):909-996, October1988 a forthcoming monograph [151 阿6 Eck, Tony DeRose, To mer Stuetzle Acknowledgments New York, 1995. [7 Gerald Farin Curves and surfaces for Computer Aided Geometric De We wish to thank Adam Finkelstein, Michael Lounsbery, and Sean sign. Academic Press, Boston, third edition, 199 Anderson for help with several of the figures in this paper. Thanks [8] Adam Fin [8] Adam Finkelstein and David H Salesin. Multiresolution curves. In also go to Ronen Barzel, Steven Gortler, Michael Shantzis, and the Proceedings of SIGGRAPH 94, pages 261-268. ACM, New York anonymous reviewers for their many helpful comments. This work was supported by NSF Presidential and National Young Investiga- 9] Zicheng tor awards(CCR-8957323 and CCR-9357790), by NSF grant CDA 9123308, by an NSF Graduate Research Fellowship, by the Univer- spcm. New ror. 194 roceedings of S/GGRAPH 94, pages sity of Washington Royalty Research Fund (65-9731), and by indus- [10] Michael Lounsbery, Tony DeRose, and Joe Warren. Multiresolutie trial gifts from Adobe, Aldus, Microsoft, and Xerox. surfaces of arbitrary topological type. ACM Transactions on Graphics, [11] Stephane Mallat. A theory for multiresolution signal decomposition References The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(7): 674-693, July 1989 [0 R. Bartels, J. Beatty,and B. Barsky. An Introduction to Splines for Use [12] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. San Francisco. 1987. Fetterling. Numerical Recipes. Cambridge University Press, New 2 Charles K Chui. An overview of wavelets In Charles K Chui, editor, [13] Ewald Quak and Norman Weyrich. Decomposition and reconstruc- Approximation Theory and Functional Analysis pages 47-71.Aca- ounded interval. pplied and demic Press, Boston, 1991 omputational Harmonic Analysis 1(3): 217-231, June 199 3 Charles K. Chui. An Introduction to Wavelets Academic Press [14] Michael P. Salisbury, Sean E. Anderson, Ronen Barzel, and David H oston, 1992 [4 Charles K Chui and Ewald Quak. Wavelets on a bounded interval GRAPH 94, pages 101-108. ACM, New York, 1994 In D. Braess and L. L. Schumaker, editors, Numerical Methods in Ap- proximation Theory, volume 9, pages 53-75. Birkhauser Verlag, Basel, 1992. San Francisco, 1996( to appear)(a) (b) (c) (d) Figure 6 Surface manipulation at different levels of detail: The original surface (a) is changed at a narrow scale (b), an intermediate scale (c), and a broad scale (d). (a) (b) (c) Figure 7 Surface approximation using subdivision surface wavelets: (a) the original surface, (b) an intermediate approximation, and (c) a coarse approximation. plained enough of the fundamentals for interested readers to explore both the construction of wavelets and their application to problems in graphics and beyond. We present a more thorough discussion in a forthcoming monograph [15]. Acknowledgments We wish to thank Adam Finkelstein, Michael Lounsbery, and Sean Anderson for help with several of the figures in this paper. Thanks also go to Ronen Barzel, Steven Gortler, Michael Shantzis, and the anonymous reviewers for their many helpful comments. This work was supported by NSF Presidential and National Young Investiga￾tor awards (CCR-8957323 and CCR-9357790), by NSF grant CDA- 9123308, by an NSF Graduate Research Fellowship, by the Univer￾sity of Washington Royalty Research Fund (65-9731), and by indus￾trial gifts from Adobe, Aldus, Microsoft, and Xerox. References [1] R. Bartels, J. Beatty, and B. Barsky. An Introduction to Splines for Use in Computer Graphics and Geometric Modeling. Morgan Kaufmann, San Francisco, 1987. [2] Charles K. Chui. An overview of wavelets. In Charles K. Chui, editor, Approximation Theory and Functional Analysis, pages 47–71. Aca￾demic Press, Boston, 1991. [3] Charles K. Chui. An Introduction to Wavelets. Academic Press, Boston, 1992. [4] Charles K. Chui and Ewald Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors,Numerical Methods in Ap￾proximation Theory, volume 9, pages 53–75. Birkhauser Verlag, Basel, 1992. [5] Ingrid Daubechies. Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics, 41(7):909–996, October 1988. [6] Matthias Eck, Tony DeRose, Tom Duchamp, Hugues Hoppe, Michael Lounsbery, and Werner Stuetzle. Multiresolution analysis of arbitrary meshes. In Proceedings of SIGGRAPH 95, pages 173–182. ACM, New York, 1995. [7] Gerald Farin. Curves and Surfaces for Computer Aided Geometric De￾sign. Academic Press, Boston, third edition, 1993. [8] Adam Finkelstein and David H. Salesin. Multiresolution curves. In Proceedings of SIGGRAPH 94, pages 261–268. ACM, New York, 1994. [9] Zicheng Liu, Steven J. Gortler, and Michael F. Cohen. Hierarchical spacetime control. In Proceedings of SIGGRAPH 94, pages 35–42. ACM, New York, 1994. [10] Michael Lounsbery, Tony DeRose, and Joe Warren. Multiresolution surfaces of arbitrary topological type.ACM Transactions on Graphics, 1996 (to appear). [11] Stephane Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674–693, July 1989. [12] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Fetterling. Numerical Recipes. Cambridge University Press, New York, second edition, 1992. [13] Ewald Quak and Norman Weyrich. Decomposition and reconstruc￾tion algorithms for spline wavelets on a bounded interval.Applied and Computational Harmonic Analysis, 1(3):217–231, June 1994. [14] Michael P. Salisbury, Sean E. Anderson, Ronen Barzel, and David H. Salesin. Interactive pen and ink illustration. InProceedings of SIG￾GRAPH 94, pages 101–108. ACM, New York, 1994. [15] Eric J. Stollnitz, Tony D. DeRose, and David H. Salesin. Wavelets for Computer Graphics: Theory and Applications. Morgan Kaufmann, San Francisco, 1996 (to appear). 6