Introduction to Wavelets (an intention for CG applications)

Jyun-Ming ChenSpring 2001

Contents

• Motivation• Haar wavelets• Daubechies wavelets• Subdivision and MRA• Two dimensional wav

elets

• Other applications– Signal compression– Image compression

• Relation with Fourier transform– Frequency domain

thoughts

Geometric Modeling• Indexedfaceset

– Topology/geometry• Where the model come from:

– Laser scanning (Cyberware)– www-graphics.stanford.edu/data/– www.cc.gatech.edu/projects/large_mo

dels/

• Sometimes produce huge model– # of triangles

• Implication:– Rendering time, storage, trans

mission

3D Models• # of triangles:

– Bunny: 750K– Budda: 9.2M– Lucy: 116M

Scanning the David (M.Levoy)

height of gantry: 7.5 metersweight of gantry: 800 kilograms

Statistics about the scan

• 480 individually aimed scans• 2 billion polygons• 7,000 color images• 32 gigabytes• 30 nights of scanning• 22 people

Polygonal Simplification

• Used in level of detail• Various approaches• Yet duplicated effort for

storage/transmission• Wavelet seems to be a mathematically

elegant tool for it

What wavelet is like (approximately)

• Idea similar to filter banks in signal processing

General Concepts

• A way of representing function in different basis such that the “effective” terms can be reduced (i.e. ignore the terms with small coefficient)– This can be potentially useful in information

compression• The choice of basis is not fixed (can be designed

to suit your need)– This is different from Fourier transform

• The decomposition process can be applied iteratively (until a global average is obtained)

After we’ve got that

• recognition, synthesis, …• progressive transmission• multiresolution editing• feature recognition• … (whatever you may want to pursue)

Hence,

• We need to get a hold of the theory behind

Yet, Wavelet is also related to signals and images

• 1D: signal compression• 2D: image compression• It is therefore necessary that we cover some

of these in class• Be aware. Lots of books are math intensive.

I’ll try to make the course as simple as possible mathematically.

Contents• 1D Haar wavelets

– In great detail (with numbers)

– To illustrate concepts • 2D: ways to apply Haar w

avelet to image processing• B-spline basics (Farin, …)• Subdivision curve/surface• Wavelet construction (orth

ogonal, biorthogonal, semiorthogonal wavelets)

• Lifting– 2nd generation wavelets

• other wavelet topics (other: not strongly related to our main line of lecture)– Fourier transform primer– Continuous wavelet transfo

rm vs STFT– Advanced EZW– Musical sound experiment

…

RoadMap

Haar

Daubechies

MRA & orthogonal wavelets

Subdivision curve

Semiorthogonal & spline wavelet

B-spline basics

Subdivision surface & biorthogonal wavelets

Other Applications

Two-dimensional waveletAP: signal

compression

AP: multiresolution curve

lifting