Introduction to Wavelets (an intention for CG applications) Jyun-Ming Chen Spring 2001

Contents Motivation Haar wavelets Daubechies wavelets Subdivision and MRA Two dimensional wavelets 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/ – models/ Sometimes produce huge model –# of triangles Implication: –Rendering time, storage, transmission

3D Models # of triangles: –Bunny: 750K –Budda: 9.2M –Lucy: 116M

Scanning the David (M.Levoy) height of gantry: 7.5 meters weight 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 wavelet to image processing B-spline basics (Farin, …) Subdivision curve/surface Wavelet construction (orthogonal, biorthogonal, semiorthogonal wavelets) Lifting –2 nd generation wavelets other wavelet topics (other: not strongly related to our main line of lecture) –Fourier transform primer –Continuous wavelet transform vs STFT –Advanced EZW –Musical sound experiment …

Tracks: Haar, Daub4, 2D, signal/image compression B-Spline, spline wavelets (examples of semiorthogonal wavelets; not necessarily orthogonal) Subdivision-related MRA (and all other background info in geometric modeling)

RoadMap Haar Daubechies MRA & orthogonal wavelets Subdivision curve Semiorthogonal & spline wavelet B-spline basics Subdivision surface & biorthogonal wavelets Other Applications Two-dimensional wavelet AP: signal compression AP: multiresolution curve lifting