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Parameterization of the scaling coefficients

The case where φ ( t ) and h ( n ) have compact support is very important. It aids in the time localization properties of the DWT andoften reduces the computational requirements of calculating the DWT. If h ( n ) has compact support, then the filters described in  Chapter: Filter Banks and the Discrete Wavelet Transform are simple FIR filters. We have stated that N , the length of the sequence h ( n ) , must be even and h ( n ) must satisfy the linear constraint of [link] and the N 2 bilinear constraints of [link] . This leaves N 2 - 1 degrees of freedom in choosing h ( n ) that will still guarantee the existence of φ ( t ) and a set of essentially orthogonal basis functions generated from φ ( t ) .

Length-2 scaling coefficient vector

For a length-2 h ( n ) , there are no degrees of freedom left after satisfying the required conditions in [link] and [link] . These requirements are

h ( 0 ) + h ( 1 ) = 2

and

h 2 ( 0 ) + h 2 ( 1 ) = 1

which are uniquely satisfied by

h D 2 = h ( 0 ) , h ( 1 ) = 1 2 , 1 2 .

These are the Haar scaling functions coefficients which are also the length-2 Daubechies coefficients [link] used as an example in Chapter: A multiresolution formulation of Wavelet Systems and discussed later in this book.

Length-4 scaling coefficient vector

For the length-4 coefficient sequence, there is one degree of freedom or one parameter that gives all the coefficients that satisfy the requiredconditions:

h ( 0 ) + h ( 1 ) + h ( 2 ) + h ( 3 ) = 2 ,
h 2 ( 0 ) + h 2 ( 1 ) + h 2 ( 2 ) + h 2 ( 3 ) = 1

and

h ( 0 ) h ( 2 ) + h ( 1 ) h ( 3 ) = 0

Letting the parameter be the angle α , the coefficients become

h ( 0 ) = ( 1 - cos ( α ) + sin ( α ) ) / ( 2 2 ) h ( 1 ) = ( 1 + cos ( α ) + sin ( α ) ) / ( 2 2 ) h ( 2 ) = ( 1 + cos ( α ) - sin ( α ) ) / ( 2 2 ) h ( 3 ) = ( 1 - cos ( α ) - sin ( α ) ) / ( 2 2 ) .

These equations also give the length-2 Haar coefficients [link] for α = 0 , π / 2 , 3 π / 2 and a degenerate condition for α = π . We get the Daubechies coefficients (discussed later in this book) for α = π / 3 . These Daubechies-4 coefficients have a particularly clean form,

h D 4 = 1 + 3 4 2 , 3 + 3 4 2 , 3 - 3 4 2 , 1 - 3 4 2

Length-6 scaling coefficient vector

For a length-6 coefficient sequence h ( n ) , the two parameters are defined as α and β and the resulting coefficients are

h ( 0 ) = [ ( 1 + cos ( α ) + sin ( α ) ) ( 1 - cos ( β ) - sin ( β ) ) + 2 sin ( β ) cos ( α ) ] / ( 4 2 ) h ( 1 ) = [ ( 1 - cos ( α ) + sin ( α ) ) ( 1 + cos ( β ) - sin ( β ) ) - 2 sin ( β ) cos ( α ) ] / ( 4 2 ) h ( 2 ) = [ 1 + cos ( α - β ) + sin ( α - β ) ] / ( 2 2 ) h ( 3 ) = [ 1 + cos ( α - β ) - sin ( α - β ) ] / ( 2 2 ) h ( 4 ) = 1 / 2 - h ( 0 ) - h ( 2 ) h ( 5 ) = 1 / 2 - h ( 1 ) - h ( 3 )

Here the Haar coefficients are generated for any α = β and the length-4 coefficients [link] result if β = 0 with α being the free parameter. The length-4 Daubechies coefficients are calculatedfor α = π / 3 and β = 0 . The length-6 Daubechies coefficients result from α = 1 . 35980373244182 and β = - 0 . 78210638474440 .

The inverse of these formulas which will give α and β from the allowed h ( n ) are

α = arctan 2 ( h ( 0 ) 2 + h ( 1 ) 2 ) - 1 + ( h ( 2 ) + h ( 3 ) ) / 2 2 ( h ( 1 ) h ( 2 ) - h ( 0 ) h ( 3 ) ) + 2 ( h ( 0 ) - h ( 1 ) )
β = α - arctan h ( 2 ) - h ( 3 ) h ( 2 ) + h ( 3 ) - 1 / 2

As α and β range over - π to π all possible h ( n ) are generated. This allows informative experimentation to better see whatthese compactly supported wavelets look like. This parameterization is implemented in the Matlab programs in Appendix C and in the Aware, Inc. software, UltraWave [link] .

Since the scaling functions and wavelets are used with integer translations, the location of their support is not important, only thesize of the support. Some authors shift h ( n ) , h 1 ( n ) , φ ( t ) , and ψ ( t ) to be approximately centered around the origin. This is achieved by having the initial nonzero scaling coefficient start at n = - N 2 + 1 rather than zero. We prefer to have the origin at n = t = 0 .

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Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
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