Caltech Multi-Res Modeling Group

The main research focus of our group is the exploration and advancement of fundamentally new representations and algorithms to meet the demands of complex computer graphics environments (hundreds of thousands to millions of primitives). This is motivated by the observation that classical representations, for example, patches used in geometric modeling, often lead to algorithms which do not scale well, lack robustness, and require complex data structures and constraint management. The result are implementations with a tendency to "cave in" under their own complexity.

Adaptive, multiresolution representation of a machine part; with exactly resolved features (edges, corners); top: 5% error; middle: 1% error; finest level mesh

Multiresolution representations (see "What is Multiresolution?") have proven to be extraordinarily powerful tools in meeting these challenges. One class of these multiresolution representations is based on radical generalizations of classical wavelet constructions to the settings encountered in computer graphics. A key element of this strategy is the use of "Subdivision" and the "Lifting Scheme" to build wavelets in the so called "2nd Generation" setting. Another, more recent class of multiresolution representations is based on sequential simplification procedures. We are engaged in exploring both approaches for computer graphics simulation, modeling, and rendering.

Key features that we look for in these new approaches are

Solid mathematical foundations: Smoothness, approximation properties, and error analyses are required to make performance guarantees.
Robust numerical algorithms: As model sizes grow, issues of numerical robustness become paramount and algorithms need to deal with them gracefully.
Generality and uniformity: The basic algorithms should be applicable in the arbitrary topology setting and be uniform across topologies, constraints, and scale.
Scalability: The asymptotic complexity of the basic computational kernels should ideally be linear in time and space.
Simple data structures and algorithms: The basic kernels should be simple so that specializations and enhancements do not lead to an explosion of software complexity.
Conciseness of representation and control: To keep large models manageable representation must be concise and support control "knobs" at different scales for effective control.

Multiresolution representations meet these challenges and we are using them in a wide variety of projects.