# 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:

- 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.
- 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.
- 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.
- Lifting bridges the gap between classical wavelets and finite elements.
- 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.