introduction to wavelets (an intention for cg applications) jyun-ming chen spring 2001
DESCRIPTION
Geometric Modeling Indexedfaceset –Topology/geometry Where the model come from: –Laser scanning (Cyberware) –www-graphics.stanford.edu/data/ –www.cc.gatech.edu/projects/large_ models/ Sometimes produce huge model –# of triangles Implication: –Rendering time, storage, transmissionTRANSCRIPT
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