Wavelets Research Peter Schröder Assistant Professor of Computer Science California Institute of Technology Second Generation Wavelets Wavelets have proven to be powerful tools for many signal processing tasks as well as numerical computations. However, classical constructions have been limited to simple domains and regular settings (e.g., regularly spaced samples and product domains). Many practical applications in computer graphics-and engineering in general-require more flexible constructions. These need to accomodate Irregular subdivisions to facilitate optimal hierarchical representations of complex geometry; Adaptive subdivisions to support flexible decomposition of operators and optimal non-linear approximation of functions; Weighted measures to account for complex geometry and to remove singularities; Geometry dependent constraints such as domain boundaries, edges, and corners; Data dependent constraints such as discontinuities, locally exact reconstruction, and algebraic singularities. Classical construction methods for wavelets fail in these settings and new techniques such as lifting need to be employed. Aside from the practical aspects of data structures and fast algorithms many deep mathematical questions need to be answered before these techniques will become widely availalble. The mathematical foundations and algorithms which are being developed as part of this research will be of fundamental importance in flexible, adaptive, and efficient curve and surface representations. Applications range from interactive curve editing to processing of laser rangescanner acquired geometry for interactive display and efficient transmission over networks. The ability of wavelet representations to capture important mathematical features will form the foundation of a new class of handwriting and gesture recognition algorithms. Wavelet Methods for Integral Equation Solvers Solving partial differential equations over three dimensional domains is notoriously resource intensive both in compute time and in memory requirements. After discretizing the original equation large sparse linear systems result whose memory requirements increase rapidly as more accurate solutions are needed. In many cases finding the solution to the partial differential equation can be turned into a problem of finding a solution to a boundary integral equation. These are often referred to as potential problems. In this case the problem only needs to be solved over the boundary of the domain, which leads to a much smaller linear system and significantly reduced memory requirements. A drawback of this method is that the linear system is now dense. However, using wavelets this linear system can be turned into a sparse linear system once again, leading to very fast algorithms for the solution of potential problems over complex three dimensional domains. The algorithms which are being developed in this research project will have wide applicability in science and engineering. A main focus of the work is on overall scalability to large problem sizes, i.e., very complex domains, robustness, generality, i.e., no custom solutions for particular geometries, and well understood error behavior to guarantee the quality of the computed solutions.

Home	Research	Outreach	Televideo	Admin	Education