Wavelets and Filter Banks 4C8 Integrated Systems Design.
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Transcript of Wavelets and Filter Banks 4C8 Integrated Systems Design.
![Page 1: Wavelets and Filter Banks 4C8 Integrated Systems Design.](https://reader036.fdocuments.in/reader036/viewer/2022081422/5517f6d1550346a2228b4866/html5/thumbnails/1.jpg)
Wavelets and Filter Banks
4C8 Integrated Systems Design
![Page 2: Wavelets and Filter Banks 4C8 Integrated Systems Design.](https://reader036.fdocuments.in/reader036/viewer/2022081422/5517f6d1550346a2228b4866/html5/thumbnails/2.jpg)
Recall the 1D Haar Xform
x y
2
1
2
1
2
1
2
1
x
yTT
)1(x )2(x )3(x )4(x
)1(y )2(y
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Now consider as filtering
]1[2
1][
2
1][0 nxnxny
][2
1]1[
2
1][1 nxnxny
x
y
)1(x )2(x )3(x )4(x
)1(y )2(y
FIR Filter H0
FIR Filter H1
Downsample by 2 2
ba
ba
b
a
q
p
2
1
11
11
2
1
ba a b
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Hence Analysis Filter Bank
)(]1[2
1)(
]1[2
1][
2
1][
1
1
zXzzY
nxnxny
)(]1[2
1)()()(
][2
1]1[
2
1][
100
0
zXzzXzHzY
nxnxny
)(]1[2
1)()()(
][2
1]1[
2
1][
11
1
zXzzXzHzY
nxnxny
Low Pass Filter
High Pass Filter
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0][0 ny 0 0 0 0 0
00 0 0 0 0Upsampling means that there are zeros at odd n when compared to their values before downsampling in the analysis stage.
][1 ny
Reconstruction
To do the inverse transform to apply the satges in reverse
1. Upsampling2. Filtering (the filters are not
necessarily the same as before)
![Page 6: Wavelets and Filter Banks 4C8 Integrated Systems Design.](https://reader036.fdocuments.in/reader036/viewer/2022081422/5517f6d1550346a2228b4866/html5/thumbnails/6.jpg)
So combine into single equation
00 0 0 0 0
00 0 0 0 0y0 and y1 are zero at odd nNot the same as y0 and y1 output from analysis stageBecause they have 0’s in them!
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To avoid confusion….
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So how is this modeled?
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Hence 2 band filter bank
0
0
Normal filter outputs Downsample by 2 then upsample by 2 by putting 0’s inbetween
0
00
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Perfect Reconstruction
• We want the output from the reconstruction to be the same as the input i.e. a Perfect Reconstruction Filterbank so …
)()(ˆ zXzX
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PR
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PR
• H are analysis filters• G are synthesis/reconstruction filters
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Can now extend analysis to more stages .. A binary tree
Lo
Not that Hi
Not quite so HiQuite Hi
HiLevel 1 Level 2 Level 3 Level 4
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2D Wavelet Transform
LoLo
LoHi
HiHi
HiLo
Downsample Rows
Downsample Columns
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The Multilevel 2D Discrete Wavelet Xform
Downsample Rows
Downsample Columns
Downsample Rows
Downsample Columns
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2D DWT of Lena
COARSE Levels
Fine Levels
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What does this do to a signal?
• Need to work out the impulse response of each equivalent filter output
• Can do this by shifting the downsample operation to the output of each stage
Not that Hi
Not quite so Hi
Quite Hi
Hi
Level 1 Level 2 Level 3 Level 4
Lo
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Multirate Theory
≡
𝐻 (𝑧 )=1+𝑧− 1+ 𝑧−2 𝐻 (𝑧 2 )=1+𝑧−2+𝑧−4
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What does this do to a signal?
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So now we can examine impulse responses
• Process of creating y1, y01 etc is the Wavelet Transform
• “Wavelet” refers to the impulse response of the cascade of filters
• Shape of impulse response similar at each level .. Derived from something called a “Mother wavelet”
• Low pass Impulse response to level k is called the “scaling function at level k”
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Good wavelets for compression
• There are better filters than the “haar” filters
• Want PR because energy compaction stages should be reversible
• Wavelet filter design is art and science– Won’t go into this at all in this course– You will just be exposed to a couple of wavelets that are used in
the literature
• There are very many wavelets! Only some are good for compression and others for analysis
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Le Gall 3,5 Tap Filter Set
• Note how filter outputs (H1,G1) shifted by z, z-1
• So implement by filtering without shift but select ODD outputs
• (H0,G0) select EVEN outputs
A TRICKY THING!
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Le Gall 3,5 Tap Filter Set
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Le Gall Filters
• Pretty good for image processing because of the smooth nature of the analysis filters and they are symmetric
• But reconstruction filters not smooth .. bummer
It turns out that you can swap the analysis and reconstruction filters around
Known as the LeGall 5,3 wavelet or inverse LeGall wavelet
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Near-Balanced Wavelets (5,7)
Analysis Filters Reconstruction Filters
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Near-Balanced Wavelets (13,19)
Analysis Filters Reconstruction Filters
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2D Impulse responses of the separable filters
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Coding with Wavelets
• Quantise the Coarse levels more finely than the Fine levels
• Large Qstep at Fine levels and Small Qstep at low levels
DCTHAAR
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Coding with Wavelets
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Entropies with RLC
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Rate-Distortion Curves
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Wavelets for Analysis: Noise Reduction
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Wavelets for Analysis: Noise Reduction
• Note that true image detail is represented by Large value Coefficients
• So perform noise reduction by setting small coefficients to 0.
• What is small?• Wavelet Coring
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Wavelets for Analysis: Coring
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Wavelet Noise Reduction
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Noise Reduction
• Important in video for compression efficiency
• Important for image quality• SONY, Philips, Snell and Wilcox, Foundry,
Digital Vision all use wavelet noise reduction of some kind
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The price for decimation• Is aliasing
• Wavelets work because of the very clever filter frequency response designs that cancel aliasing by the end of reconstruction
High Pass output is aliased!
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Shift Variant Wavelets
• This means that decimated wavelets are shift variant!
• If you move the signal the DWT coefficients change!
• This means that they are not so good for analysis .. And definitely not good for motion estimation
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A tricky example..
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Can get around this …
• By NOT downsampling .. “Algorithme a-trous”
• Yields loads of data• OR use Nick Kingsbury’s Complex
Wavelets
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Summary
• Matlab has a good wavelet package .. Useful for development
• Wavelets have made their way into compression
• Powerful idea for analysis but data explosion is a problem
• JPEG200, MPEG4 define methods for using DWT in compression