Mathematics Colloquium

Thursday, February 8, 1996, 4:00pm

102 Bradley Hall

Professor Wim Sweldens

AT & T Bell Labs

speaks on

The lifting scheme: Building wavelets at home

Abstract. In this talk we present the lifting scheme, a new philosophy in wavelet constructions. Its main difference with classical constructions is that it does not rely on the Fourier transform. It has the following features:

  1. Lifting allows the construction of wavelets with very little mathematical machinery. It therefore serves as an educational tool to introduce wavelets to people without a strong math background. The only thing one needs is polynomial interpolation, therefore allowing "at home" constructions.
  2. Lifting allows a faster calculation of the classical wavelet transform. Also, using a partial bit reversal the wavelet transform can be calculated fully in-place, i.e. without allocating extra memory.
  3. Lifting allows the construction of wavelets in situations were no Fourier transform is available. These wavelets are referred to as "Second Generation Wavelets". Typical examples are wavelets adapted to weighted inner products, wavelets adapted to irregular sample locations, irregular meshes, and wavelets on curves, surfaces, and manifolds. More particularly we consider wavelets on the sphere.
  4. Lifting bridges the gap between classical wavelets and finite elements.
  5. Lifting allows custom-designed wavelets. Examples are adaptive wavelets, h-p wavelets, non-linear wavelets, and wavelet probing.

These features will be illustrated with several examples and applications.

This material includes joint work with Peter Schroeder (Caltech).

Tea. High tea will be served at 3:30pm in the Lounge.
Emmy's. Certain refreshments will be available at the Emmy's after the talk.
Host. Geoff Davis is the host. Anybody who is interested in having dinner with the speaker should contact Geoff at 646-1618.