Spherical Wavelets: Efficiently Representing Functions on the Sphere
Peter Schr\"oder
Dept. of Mathematics
University of South Carolina
Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Traditional constructions have been limited to simple domains such as intervals and rectangles. In this talk I will present a wavelet construction for scalar functions defined on surfaces and more particularly the sphere. Treating these bases in the fully biorthogonal case I will explain how bases with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. The resulting algorithms have been implemented on workstation class machines in an interactive application. I will give examples of functions defined on the sphere, such as topographic data, bi-directional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.
This is joint work with Wim Sweldens of the Department of Mathematics at the University of South Carolina.
Peter Schr\"oder received his PhD in Computer Science from Princeton University under Pat Hanrahan. Prior to Princeton he received a Master's degree from the MIT Media Lab, and did his undergraduate work in Mathematics at the Technical University of Berlin. His research interests are in the area of the application of numerical analysis insights to graphics problems. He has worked on Newtonian dynamics simulation for interactive environments, scientific visualization, parallel graphics algorithms, and most of all illumination problems. Since this past fall he has been a postdoctoral fellow at the University of South Carolina in Columbia, where he is pursuing wavelet techniques further.