In todays show this is going to be brutal

1
The Story of WaveletsTheory and Engineering
Applications
Torture
Jamboree 8 March 7, 2001

• In todays show (this is going to be brutal!!!)
• Review of last lecture
• Filter implementation of the Haar wavelet
• Multiresolution approximation in general
• Filter implementation of DWT
• I wanted to do more, but..

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Filter Implementation of Haar Wavelet
We can show that a(k,n) can be obtained from
a(k-1,n) through filtering by using a filter

followed by a downsampling operation (drop every
other sample). Similarly, d(k,n) can also be obta
ined from a(k-1,n) using the filter gn
followed by down sampling by 2
This is called decomposition in the wavelet
jargon. The reverse is also true, that is we can
obtain a(k-1,n) from a(k,n) and d(k,n) using an
other set of filters, which is called
Reconstruction.
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Decomposition Filters

• If we take the FT of hn and gn

LPF
HPF
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Decomposition / Reconstruction Filters

• We can obtain the coarser level coefficients
  a(k,n) or d(k,n) by filtering a(k-1,n) with hn
  or gn, respectively, followed by downsampling
  by 2.
• Would any LPF and HPF work? No! There are certain
  requirements that the filters need to satisfy. In
  fact, the filters are obtained from scaling and
  wavelet functions using dilation (two-scale)
  equations (coming soon)
• Can we go the other way? That is, obtain the next
  finer level coefficients a(k-1,n) from a(k,n) and
  d(k,n)? Yes !
  
• Upsample a(k,n) and d(k,n) by 2 (inser zeros
  between every sample) and use filters hn?n
  ?n-1 and gn ?n- ?n-1. Add the filter
  outputs !

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The Discrete Wavelet Transform
a(k,n)
a(k,n)
d(k1,n)
a(k1,n)
d(k2,n)
a(k1,n)
a(k2,n)
Decomposition
Reconstruction
We have only shown the above implementation for
the Haar Wavelet, however, as we will
see later, this implementation subband coding
is applicable in general.
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Multiresolution Analysis (MRA)

• A vector space V is called a MRA if it satisfies
  the following
  
• 
  where Ak is
  the projection operation onto space Vk
  
• is dense in
  L2(R)
  
• Among all functions that can be used to
  approximate f(t), there exists a unique function
  such that
  
• 
  constitutes an orthonormal basis
  for Vk. This function is called the scaling
  function.

So what is this good for?
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MRA

• The 5th property tells us that we can approximate
  any function in L2(R) as a weighted sum of
  scaling functions at various approximation levels
  k
• where the weights akn can be computed as
• The detail lost at level k approximation lies in
  another vector space, Wk, which is an orthogonal
  complement to the space Vk. This is denoted as

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MRA

• Suppose the approximation at level k-1 contains
  all the necessary information to compute
  approximation at level k. Then
  
• where Dk denotes the projection operation
  onto space Wk. This space also has a unique
  function ?(t), whose dilations and translations
  constitute an orthonormal basis for Wk
• This function is called the wavelet which
  allows us to represent the detail lost at any
  level k as a weighted sum

where
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MRA

• Since Wk is the space where all detail
  information lie for the kth resolution, then the
  orthogonal sum of all Wk should give all the
  details there are at all resolutions.
  Collectively, Wk would then form the space of all
  square integrable functions, L2(R)
• We can now compute the approximation of f(t) at
  any resolution k-1 as the sum of approximation at
  resolution k-1 and the detail lost in going to
  resolution k-1 from resolution k

Wavelet Synthesis
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MRA on Discrete Functions

• Lets suppose that the function f(t) is sampled
  at N points to give the sequence fn, and
  further suppose that kth resolution is the
  highest resolution (we will compute
  approximations at k1, k2, etc. Then
• Multiplying and integrating and
  by

1
2
1
2
hn
gn
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From MRA to Filters

• This substitution gives us level k1
  approximation and detail coefficients in terms of
  level k coefficients
  
• we can put the above expressions in
  convolution (filter) form as

H
a(k,n)
a(k1,n)
1-level of DWT decomposition

G
d(k1,n)


hnh-n, and gng-n
So where do these filters really come from?
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MRA Filters Dilation / Two-scale Equations

• Two scale (dilation) equations for the scaling
  and wavelet functions determine the filters
  associated with these functions. In particular
  
• The coefficients c(n) can be obtained as
• Note that hnc(n)/2. In some books, hn
  c(n)/v2. Then the two-scale equation becomes
  
• Similarly, the two-scale equation for the wavelet
  function
  
• Then

or in more general


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Two-Scale Equations

• These two equations determine the coefficients of
  all 4 filters
  
• hn Reconstruction, lowpass filter
• gn Reconstruction, highpass filter
• hn Decomposition, lowpass filter
• gn Decomposition, highpass filter
• The following observations can therefore be made

Note H(jw) H(jw)
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Quadrature Mirror Filters

• It can be shown that
• that is, h and g filters are related to
  each other
  
• in fact, that
  is, h and g are mirrors of each other, with
  every other coefficient negated. Such filters
  are called quadrature mirror filters. For
  example, Daubechies wavelets with 4 vanishing
  moments..

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DB-4 Wavelets

• h -0.0106 0.0329 0.0308 -0.1870
  -0.0280 0.6309 0.7148 0.2304
  
• g -0.0106 -0.0329 0.0308 0.1870
  -0.0280 -0.6309 0.7148 -0.2304
  
• h 0.2304 0.7148 0.6309 -0.0280
  -0.1870 0.0308 0.0329 -0.0106
  
• g -0.0106 -0.0329 0.0308 0.1870
  -0.0280 -0.6309 0.7148 -0.2304

L filter length (8, in this case)
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DWT implementationSubband Coding
xn
xn




Decomposition
Reconstruction
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DWT Decomposition
xn
Length 512 B 0 ?
gn
hn
Length 256 B 0 ?/2 Hz
Length 256 B ?/2 ? Hz
G(jw)
d1 Level 1 DWT Coeff.
gn
hn
Length 128 B 0 ? /4 Hz
w
Length 128 B ?/4 ?/2 Hz
-?
?/2
-?/2
?
d2 Level 2 DWT Coeff.
gn
hn
2
Length 64 B 0 ?/8 Hz
Length 64 B ?/8 ?/4 Hz
.
d3 Level 3 DWT Coeff.
18
.

• You must be brain dead by now. Thats itEnough
  torture for one lecture.Go home !