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2
Agenda
Why Wavelet Transform Continuous & Discrete Wavelet
Transform Haar Wavelet Transform Application of wavelet transform is
JPEG2000: EZW coding
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Introduction Multimedia Transformations are applied
to signals to obtain further information. Most of the signals in practice, are
time-domain signals in their raw format.
Not always the best representation of the signal.
The most distinguished information is hidden in the frequency content.
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Fourier Transform The frequency spectrum of the signal shows
what frequencies exist in the signal FT
Frequency domain Temporal domain
No frequency information is available in time-domain
No time information is available in frequency-domain signal
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Stationary Signalsx(t)=cos(2π*10t)+cos(2 π *25t)+cos(2 π *50t)+cos(2 π *100t)
FT
Four spectral components corresponding to the frequencies 10, 25, 50, 100 Hz
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Non-stationary Signals
FT
Four different frequency components at four different time intervals
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Comparison of two examples Two spectrums are similar! Four spectral components at exactly the same
frequencies The corresponding time domain signals are not even
close
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What is wavelet transform? Provides time-frequency representation Wavelet transform decomposes a signal
into a set of basis functions (wavelets) Wavelets are obtained from a single
prototype wavelet Ψ(t) called mother wavelet by dilations and shifting:
where a is the scaling parameter and b is the shifting parameter
)(1
)(, a
bt
atba
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Wavelet Transform
Continuous Wavelet Transform (CWT)
Discrete Wavelet Transform (DWT)
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CWT
Continuous wavelet transform (CWT) of 1D signal is defined as
The a,b is computed from the mother wavelet by translation and dilation
dxxxfbfW baa )()()( ,
abx
axba
1)(,
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Separates the high and low-frequency portions of a signal through the use of filters One level of transform:
Signal is passed through G & H filters. Down sample by a factor of two
Multiple levels (scales) are made by repeating the filtering and decimation process on lowpass outputs
1D Discrete Wavelet Transform
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Haar Wavelet Transform Find the average of each pair of samples Find the difference between the average and
sample Fill the first half with averages Fill the second half with differences Repeat the process on the first half Step 1:
[3 5 4 8 13 7 5 3]
[4 6 10 4 -1 -2 3 1]
Averaging
Differencing
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Haar Wavelet Transform Step 2
[4 6 10 4 -1 -2 3 1]
[5 7 -1 3 -1 -2 3 1]
ex. (4 + 6)/2 = 5 4 - 5 = -1
Averaging Differencing
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Haar Wavelet Transform Step 3
[5 7 -1 3 -1 -2 3 1]
[6 -1 -1 3 -1 -2 3 1]
ex. (5 + 7)/2 = 6 5 - 6 = -1
Averaging Differencing
row average
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Image representation
[33 32 33 32 31 -29 27 -25][32.5 32.5 0.5 0.5 31 -29 27 -25][32.5 0 0.5 0.5 31 -29 27 -25]
163624559588
5610115352141549
4818194544222341
2539382829353432
3331303637272640
2442432120494717
165051131254559
577660613264
A
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Applying on rows
3129272550500532
2321191750500532
151311950500532
753150500532
135750500532
911131550500532
1719212350500532
2527293150500532
...
...
...
...
...
...
...
...
row averagedetail coefficients
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Applying on columns
Choose a threshold δ
272523215500
11975505000
579115500
21232527505000
44440000
44440000
00000000
0000000532
..
..
..
..
.
δ = 5
272523210000
119700000
079110000
212325270000
00000000
00000000
00000000
0000000532.
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Decompressing
apply the inverse of the averaging the differencing operations
555595575759555553511
559555755755559511553
521543523541539525532532
543521541523525539532532
532532525539541523521543
532532539525523541543521
511553555595755755955
553511595555575755559
........
........
........
........
........
........
........
........
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Result
Decompressed ImageOriginal Image
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2-D DWT Step 1: replace each row with its 1-D DWT. Step 2: Replace each column with its 1-D DWT Step 3: Repeat steps 1 & 2 on the lowest
subband for the next scale. Step 4: Repeat step 3 until as many scales as
desired
original
L HLH HH
HLLL
LH HH
HL
One scale two scales
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Discrete Wavelet Transform
LL2 HL2
LH2 HH2HL1
LH1 HH1
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JPEG2000 (J2K) is an emerging standard for image compression Achieves low bit rate compression
Not only better efficiency, but also more functionality
Lossless and lossy compression
JPEG2000
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JPEG2000 v.s. JPEG
low bit-rate performance
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Embedded Zero Tree Wavelet Coding
The era of modern lossy wavelet coding began in 1993 when Jerry Shapiro introduced EZW coding
Improved performance at low bit rates relative to the existing JPEG standard.
Much of the energy in the wavelet transform is concentrated into the LLk band.
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Significance map An indication of whether a particular
coefficient is zero or nonzero relative to a given quantization level.
EZW determined a very efficient way to code significance maps.
A wavelet coefficient is insignificant if |x| < T.
By coding the location of zeros.
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EZW algorithm If a wavelet coefficient at a coarse scale
is insignificant, then all wavelet coefficients of the same orientation in the same spatial location at finer scales are likely to be insignificant.
Tree Structure: Recognizing the coefficients of the same spatial location
Zero tree: set of insignificant coefficients
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DWT for Image decomposition
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Zero Tree
A coefficient is part of a zero tree if it’s zero and all of its descendents are zero
Efficient for coding: by declaring only one coefficient a zero tree root, all descendants are known to be zero
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Implementation
Implementing 2D DWT image compression algorithm
A JPEG2000 like implementation: EZW coding Haar wavelet transform
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Question
