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Wavelets in Image Compressionvictor/talks/idbfmvw.pdf · Ten Lectures on Wavelets. CBMS-NSF...
Transcript of Wavelets in Image Compressionvictor/talks/idbfmvw.pdf · Ten Lectures on Wavelets. CBMS-NSF...
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Wavelets in Image Compression
M. Victor WICKERHAUSER
Washington University in St. Louis, Missouri
http://www.math.wustl.edu/~victor
THEORY AND APPLICATIONS OF WAVELETS
A Workshop Honoring Ingrid Daubechies
Recipient of the 2011 Benjamin Franklin Medal
in Electrical Engineering
Villanova University, April 28th, 2011
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Great Men from the Eighteenth Century I
Benjamin Franklin, 1706–1790.
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Great Men from the Eighteenth Century II
Jean-Baptiste Joseph Fourier, 1768–1830.
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Joseph Fourier’s Construction
Theorem 1 Any function f = f(t) may be written as a sum of
sines and cosines, multiplied by numbers {an, bn} specific to f :
f(t) = a0 + a1 cos(t) + a2 cos(2t) + a3 cos(3t) + · · ·
+b1 sin(t) + b2 sin(2t) + b3 sin(2t) + · · ·
Key ideas:
• The building blocks are simple: sines and cosines.
• Data is simple: two numbers an, bn for each frequency n.
• The system is complete and efficient, an orthonormal basis.
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Application to Image Compression
• Images are functions.
• Functions are made of simple building blocks.
• Our senses are imperfect, so approximations suffice.
• Less accuracy requires less storage space.
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Example of Fourier’s Construction (Good)
Adding up just sines with bn ∼ 1/n2 to get a sawtooth.
Compression: just three terms b1, b3, b5 give the green curve.
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Pure and Applied Mathematicians
Adrien-Marie Legendre and Joseph Fourier.
Watercolor by Julien-Leopold Boilly, c.1820.
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Example of Fourier’s Construction (Not So Good)
Adding up just sines with bn ∼ 1/n to get a square wave.
Gibbs’ phenomenon: overshoots never go away.
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Problems with Fourier’s Construction
• In fact, not all functions f equal their Fourier series.
• Infinitely many numbers {an, bn} are needed to represent a
given function f , and some simple functions require very
many for a good approximation.
• Sines and cosines have no location and infinite duration.
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Time and Frequency Content Analyzed Together
T i m e
Frequency
Informationcells
Signal
∆x
(x,ξ)
∆ξ
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Waveforms Localized in Time and Frequency
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History B.D. [Before Daubechies]
• Fourier bases (1822, Paris)
• Haar bases (1910, Math. Annalen)
• Gabor functions (1946, J. IEE)
• Balian-Low theorem (1981, CRAS)
• Wilson bases (1987, Cornell)
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Ingrid Daubechies’ Construction
Theorem 2 Any function f = f(t) may be written as a sum of
wavelets wjk(t)def= w(2jt+k), multiplied by numbers cjk specific
to f :
f(t) =∑
j∈Z
∑
k∈Z
cjkwjk(t),
and the mother wavelet w = w(t) can be chosen with these three
properties:
Smoothness: w and its first d derivatives w′, w′′, . . . , w(d) are
continuous functions.
Compact support: w(t) is zero at all |t| > 5d.
Orthogonality: The set {wjk : j, k ∈ Z} is an orthonormal basis.
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Some Nice Wavelets
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120
Six dilations and translations, on an interval, of a particular
mother wavelet (9,7-biorthogonal symmetric).
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History A.D. [After Daubechies]
• Lapped orthogonal transforms (1990, IEEE ASSP)
• Biorthogonal wavelets, wavelet packets (1992, IEEE IT)
• WSQ fingerprint standard (1993, FBI)
• Wavelets on spheres (1995, ACM)
• The lifting implementation (1996, ACHA; 1998, JFAA; )
• JPEG-2000 compression (1999)
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Example Images
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Close Up of Correlated Pixels
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Two-Dimensional Waveforms I
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Two-Dimensional Waveforms II
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Two-Dimensional Waveforms III: JPEG vs. JPEG-2000
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Transform Coding Image Compression
Compression:
Scannedimage
Transform Quantize StorageCode
Decompression:
Storage UntransformUnquantizeDecode Restoredimage
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Parts Description
Compression:
Transform: convert pixels to amplitudes;
Quantize: round off the amplitudes to small numbers;
Code: remove redundancy from the small number sequence.
Decompression:
Decode: expand to recover the small number sequence;
Unquantize: insert an amplitude for each small number;
Untransform: recover pixels from approximate amplitudes.
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Wavelet Transform: Multiresolution Signal Splitting
x
hx g
hh gh
hhh ghh
ghhh
hhhh
gh
Split signal x into averages hx and details gx.
Replace x← hx and repeat
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Multiresolution Image Splitting
Picture (at top) becomes thumbnail (at bottom left) plus two
layers of saved details (highlighted).
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Storage of Multiresolution Image Data
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Custom Compression Algorithms
......
...
Best-basis search
Sampleimage
1
Sampleimage
2
Sampleimage
N
Sum
of
squares
1
2
N
∑
Training algorithm for a custom transform coding image
compression algorithm.
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Good Bases for Images I
Five-level wavelet basis, used in JPEG-2000.
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Good Bases for Images II
Five-level wavelet packet basis, used in WSQ.
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Compression Sometimes Improves Things
Rough Radiation Dose Approximation in 2D:
4 M particle simulation
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...By Eliminating the Rough Errors
Improved Approximation in 2D:
Compressed 4 M particle simulation
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...If the Right Amount of Compression is Done
0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.127
7.5
8
8.5
9
9.5
10
10.5
11 x 0.001
Threshold (ε )
RM
S e
rro
r
Deasy et al., Fig. 3
Reduction in RMS error by a rough approximation compressedtoward a smooth target, by wavelet threshold.
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...Which, Fortunately, is Easy to Find.
2 4 8 160
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Source electrons (millions)
Bes
t th
resh
old
(ε)
Deasy, et al., Fig. 4
Best wavelet thresholds for compression from a rough
approximation.
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Example: Rough Radiation Dose Approximation – 1D
4 M particle simulation — 1D cross-section, close up.
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Example: Compressed Approximation –1D
Compressed 4 M particle simulation — 1D cross-section, close
up.
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Some Notable Works
• Ingrid Daubechies. “Orthonormal Bases of Compactly
Supported Wavelets.” Comm. Pure Appl. Math.
41(1988),909–996.
• Ingrid Daubechies. Ten Lectures on Wavelets. CBMS-NSF
Regional Conference Series in Applied Mathematics. SIAM
Press, Philadelphia, 1992.
• Albert Cohen, Ingrid Daubechies and Jean-Christophe
Feauveau. “Biorthogonal Bases of Compactly Supported
Wavelets” Comm. Pure Appl. Math. 45(1992),485–500.
• Ingrid Daubechies and Wim Sweldens. “Factoring Wavelet
Transforms into Lifting Steps.” Fourier Anal. Appl.
4:3(1998),245–267.
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