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This module focus on search for better filters/wavelets.

Our main aim now is to search for better filters / wavelets which result in compression performance that rivals or beats theDCT.

We assume that perfect reconstruction is a prime requirement, so that the only image degradations are caused by coefficientquantisation, and may be made as small as we wish by increasing bit rate.

We start our search with the two PR identities from our discussion of Perfect Reconstruction , which we repeat here:

G 0 z H 0 z G 1 z H 1 z 2
and
G 0 z H 0 z G 1 z H 1 z 0
The usual way of satisfying the anti-aliasing condition ( ), while permitting H 0 and G 0 to have lowpass responses (passband where z 0 ) and H 1 and G 1 to have highpass responses (passband where z 0 ), is with the following relations:
H 1 z z k G 0 z
and
G 1 z z k H 0 z
where k must be odd so that: G 0 z H 0 z G 1 z H 1 z G 0 z H 0 z z k H 0 z z k G 0 z 0 Now define the lowpass product filter:
P z H 0 z G 0 z
and substitute relations and into identity to get:
G 0 z H 0 z G 1 z H 1 z G 0 z H 0 z H 0 z G 0 z P z P z 2
This requires all P z terms in even powers of z to be zero, except the z 0 term which should be 1. The P z terms in odd powers of z may take any desired values since they cancel out in .

A further constraint on P z is that it should be zero phase, in order to minimise the visibility of any distortions due to the high-band beingquantised to zero. Hence P z should be of the form:

P z p 5 z 5 p 3 z 3 p 1 z 1 p 1 z -1 p 3 z -3 p 5 z -5
The design of a set of PR filters H 0 , H 1 and G 0 , G 1 can now be summarised as:
  • Choose a set of coefficients p 1 , p 3 , p 5 …to give a zero-phase lowpass product filter P z with desirable characteristics. (This is non-trivial and is discussed below.)
  • Factorize P z into H 0 z and G 0 z , preferably so that the two filters have similar lowpass frequency responses.
  • Calculate H 1 z and G 1 z from and .
It can help to simplify the tasks of choosing P z and factorising it if, based on the zero-phase requirement, we transform P z into P t Z such that:
P z P t Z 1 p t , 1 Z p t , 3 Z 3 p t , 5 Z 5
where Z 1 2 z z . To calculate the frequency response of P t , let z ω T s : therefore,
Z 1 2 ω T s ω T s ω T s
This is a purely real function of ω , varying from 1 at ω 0 to -1 at ω T s (half the sampling frequency).

A Belgian mathematician, Ingrid Daubechies, did much pioneering work on wavelets in the 1980s. She discovered that to achievesmooth wavelets after many levels of the binary tree, the lowpass filters H 0 z and G 0 z must both have a number of zeros at half the sampling frequency (at z -1 ). These will also be zeros of P z , and so P t z will have zeros at Z -1 .

The simplest case is a single zero at Z -1 , so that P t z 1 Z . Therefore, P z 1 2 z 2 z 1 2 z 1 1 z G 0 z H 0 z which gives the familiar Haar filters.

As we have seen, the Haar wavelets have significant discontinuities so we need to add more zeros at Z -1 . However to maintain PR, we must also ensure that all terms in even powers of Z are zero, so the next more complicated P t must be of the form:

P t z 1 Z 2 1 α Z 1 2 α Z 1 2 α Z 2 α Z 3
if α 1 2 to suppress the term in Z 2 , P t z 1 3 2 Z 1 2 Z 3

If we allocate the factors of P t such that ( 1 Z ) gives H 0 and 1 Z 1 α Z gives G 0 , we get:

H 0 z 1 2 z 2 z
G 0 z 1 8 z 2 z z 4 z 1 8 z 2 2 z 6 2 z z -2
Using and with k 1 , the corresponding highpass filters then become:
G 1 z z H 0 z 1 2 z z 2 z
H 1 z z G 0 z 1 8 z z 2 2 z 6 2 z z -2
This is often known as the LeGall 3,5-tap filter set , since it was first published in the context of 2-band filter banks by Didier LeGall in 1988.

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Source:  OpenStax, Image coding. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10206/1.3
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