Stationary Subdivision and Multiresolution Surface Representations

Creators: Zorin, Denis N.

Abstract

Stationary subdivision is an important tool for generating smooth free-form surfaces used in CAGD and computer graphics. One of the challenges in the construction of subdivision schemes for arbitrary meshes is to guarantee that the surfaces produced by the algorithm are Ck-continuous. First results in this direction were obtained only recently. In this thesis we derive necessary and sufficient criteria for Ck-continuity that generalize and extend most known conditions. We present a new method for analysis of smoothness of subdivision which allows us to analyze subdivision schemes which do not generate surfaces admitting closed-form parameterization on regular meshes, such as the Butterfly scheme and schemes with modified rules for tagged edges. The theoretical basis for analysis of subdivision that we develop allows us to suggest methods for constructing new subdivision schemes with improved behavior. We present a new interpolating subdivision scheme based on the Butterfly scheme, which generates Ck-continuous surfaces from arbitrary meshes. We describe a multiresolution representation for meshes based on subdivision. Combining subdivision and the smoothing algorithms of Taubin [61] allows us to construct a set of algorithms for interactive multiresolution editing of complex hierarchical meshes of arbitrary topology.

Additional Information

© 1998 Denis N. Zorin, California Institute of Technology. (Submitted September 23, 1997) I would like to thank my advisors, Alan H. Barr and Peter Schr¨oder, and all people of the Caltech Computer Graphics Group: Cindy Ball, Dave Breen, Bena Currin, Dave Felt, Dan Fain, Louise Foucher, Kurt Fleischer, David Laidlaw, Alf Mikula, Mark Montague, Preston Pfarner, Eric Winfree and all the other people who made the lab a great place to work. Special thanks go to Andrei Khodakovsky and GaryWu for their help with implementing the mulitresolution editing system. I am grateful to Wim Sweldens for his contribution to the papers that we have coauthored and for the opportunity to spend a summer at Bell Laboratories. I am also grateful to Jeff Lagarias for his suggestions. I would like to thank Tom Duchamp, who has suggested to me some of the ideas presented in this thesis, and has provided a lot of valuable feedback. I would like to thank Venkat Krishnamurthy for providing the Armadillo dataset, and Hughes Hoppe for the mannequin head. And, finally, I would like to thank Isabella Goldmints; without her, my life would be very boring. The research was supported in part through grants from Pixar, the Intel Corporation, Microsoft, the Charles Lee Powell Foundation, the Sloan Foundation, an NSF CAREER award (ASC-96-24957), the NSF STC for Computer Graphics and Scientific Visualization (ASC-89-20219), the Office of the Director, Defense Research and Engineering, the Air Force Office of Scientific Research (F49620-96-1-0471), as part of the MURI program.

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Submitted - postscript.ps

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