Introduction to Wavelets
(an intention for CG applications)
Jyun-Ming Chen
Spring 2001
Contents
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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/
– www.cc.gatech.edu/projects/large_
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:
weight of gantry:
7.5 meters
800 kilograms
Statistics about the scan
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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
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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
– 2nd 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 …
RoadMap
AP: multiresolution
curve
Subdivision
curve
B-spline basics
Semiorthogonal & spline
wavelet
Haar
Daubechies
AP: signal
compression
MRA &
orthogonal
wavelets
Subdivision surface &
biorthogonal wavelets
Two-dimensional
wavelet
Other Applications
lifting