Wavelet Methods For Computer Graphics

This thesis discusses how a wavelet basis can be used in the context of two computer graphics applications, realistic rendering and geometric modeling, to produce more e cient and exible algorithms. The goal of realistic rendering is to simulate the interre ection of light in some geometric environment to produce realistic images. Radiosity is a commonly used solution method for this problem. Recently Hanrahan et al. have introduced a hierarchical method that can solve radiosity problems in O(n) time instead of O(n2). This thesis explores how the hierarchical radiosity algorithm can be formally understood from the context of wavelet theory. When the radiosity problem is expressed with respect to a wavelet basis, the resulting linear system is sparse, with only O(n) signi cant terms. By casting the hierarchical method in this framework, a variety of wavelet basis functions can be used, resulting in more e cient radiosity methods. This thesis also discusses how wavelets can be used in the context of geometric modeling. Geometric modeling is the study of how geometric shapes can be represented and manipulated by a designer. This thesis explores the use of wavelets to represent parametric curves and surfaces within the context of geometric modeling interfaces. One intuitive modeling interface commonly used in geometric modeling allows the user to directly manipulate curves and surfaces. This manipulation de nes some set of constraints that the curve or surface must satisfy (such as interpolation and tangent constraints). Direct manipulation, however, usually leads to an underconstrained problem since there are, in general, many possible surfaces meeting some set of constraints. Therefore an optimization problem must be solved. iv This thesis discusses how geometric modeling optimization problems can be solved more e ciently by using a wavelet basis. Because the wavelet basis is hierarchical, iterative optimization methods converge rapidly. And because the wavelet coe cients indicate the degree of detail in the solution, they can be used to determine the number of basis functions needed to express the variational minimum, thus avoiding unnecessary computation. An implementation is discussed and experimental results are reported. v Acknowledgments None of the work discussed in this thesis would have been possible without the mostexcellent guidance of my advisor, Professor Michael Cohen. He has given me insight, guidance, and inspiration every step of the way. I also would like to thank the readers of this thesis, Professors Pat Hanrahan and Elisha Sacks for their important input to this thesis. The radiosity research described in this thesis was done in collaboration with Professor Hanrahan, and his comments at the fourth oor library meetings helped us focus on the important research issues. Professor Sacks' guidance during my early years at Princeton are sincerely appreciated. The radiosity research was done in close collaboration with Peter Schr oder. It was a delightful experience working with him. James Shaw implemented the user interface for the variational modeling tool. His e ort was a great help. I also enjoyed yapping with him and his wife Colleen. Chuck Rose and Dan Boneh rounded out the yapping crew, and made the department almost a fun place to be. Dan Boneh was also very patient with all my math questions. I would like to thank David Ohsie who always had good advice. My various o ce mates throughout the years have enriched my Princeton experience, but certainly none more than Ayellet Tal. It looks like our study meetings for general exams have nally paid o . I would like to thank my parents for their loving support. The research reported in this thesis was made possible in part through support provided by the NSF (Grant No. CCR92-96146) and DARPA (Grant No. DABT6392-C-0053.P00002). vi