Filter implementation of the Haar wavelet

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The Story of WaveletsTheory and Engineering
Applications

• Filter implementation of the Haar wavelet
• Multiresolution approximation in general
• Filter implementation of DWT
• Applications - Compression

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DWT Using Filtering
Note that at the next finer level, intervals are
half as long, so you need 2k to get the same
interval.
hn
yj-1,k
aj-1,k
Approx. coefficients at any level j can be
obtained by filtering coef. at level j-1 (next
finer level) by hn and downsampling by 2
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Filter Implementation of Haar Wavelet
We showed that aj,k can be obtained from aj-1,k
through filtering by using a filter
followed by a downsampling operation (drop every
other sample). Similarly, dj,k can also be
obtained from aj-1,k using the filter gn
followed by down sampling by 2
Detail coefficients at any level j can be
obtained by filtering approximation coefficients
at level j-1 (next finer level) by gn and
downsampling by 2
This is called decomposition in the wavelet
jargon.
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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 aj,k
  or dj,k by filtering aj-1,k 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? Can we obtain aj-1,k
  from aj,k and dj,k from a set of filters. YES!.
  This process is called reconstruction.
• Upsample a(k,n) and d(k,n) by 2 (insert 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
aj,k
aj,k
dj1,k
aj1,k
aj1,k
dj2,k
aj2,k
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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????

• We see that app. and detail coefficients can be
  obtained through filtering operations, but where
  do scaling and wavelet functions appear in the
  subband coding DWT implementation?
• Clearly, these functions are somehow hidden in
  the filter coefficients, but how?
• To find out, we need to know little bit more
  about these scaling and wavelet functions

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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

1
2
hk
gk
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From MRA to Filters

• This substitution gives us level j1
  approximation and detail coefficients in terms of
  level j coefficients
• we can put the above expressions in
  convolution (filter) form as

H
aj1,k
aj,k
1-level of DWT decomposition

G
dj1,k


hnh-n, and gng-n
So where do these filters really come from?
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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
• Recall that
• In some books, hk c(k)/v2. Then the two-scale
  equation becomes

or more generally
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Dilation / Two-scale Equations

• Similarly, the two-scale equation for the wavelet
  function
• Then

In some books, gk b(k)/v2. Then the two-scale
equation becomes
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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 -0.2304 0.7148 -0.6309 -0.0280
  0.1870 0.0308 -0.0329 -0.0106
• 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)
Matlab command wfilters() ? Use freqz() to see
its freq. response
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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.
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Applications
Detect discontinuities
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Applications
Detect hidden discontinuities
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Applications
Simple denoising
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Compression

• DWT is commonly used for compression, since most
  DWT are very small, can be zeroed-out!

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Compression
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Compression
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Compression - ECG
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ECG - Compression
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ECG- Compression