0.5 The scaling function and scaling coefficients, wavelet and  (Page 10/13)

Parameterization of the scaling coefficients

The case where $\phi \left(t\right)$ and $h\left(n\right)$ 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 $h\left(n\right)$ 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 $h\left(n\right)$ , must be even and $h\left(n\right)$ must satisfy the linear constraint of [link] and the $\frac{N}{2}$ bilinear constraints of [link] . This leaves $\frac{N}{2}-1$ degrees of freedom in choosing $h\left(n\right)$ that will still guarantee the existence of $\phi \left(t\right)$ and a set of essentially orthogonal basis functions generated from $\phi \left(t\right)$ .

Length-2 scaling coefficient vector

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

and

which are uniquely satisfied by

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:

and

Letting the parameter be the angle $\alpha$ , the coefficients become

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

Length-6 scaling coefficient vector

For a length-6 coefficient sequence $h\left(n\right)$ , the two parameters are defined as $\alpha$ and $\beta$ and the resulting coefficients are

Here the Haar coefficients are generated for any $\alpha =\beta$ and the length-4 coefficients [link] result if $\beta =0$ with $\alpha$ being the free parameter. The length-4 Daubechies coefficients are calculatedfor $\alpha =\pi /3$ and $\beta =0$ . The length-6 Daubechies coefficients result from $\alpha =1.35980373244182\text{and}\beta =-0.78210638474440$ .

The inverse of these formulas which will give $\alpha$ and $\beta$ from the allowed $h\left(n\right)$ are

As $\alpha$ and $\beta$ range over $-\pi$ to $\pi$ all possible $h\left(n\right)$ 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 $h\left(n\right)$ , $h_1\left(n\right)$ , $\phi \left(t\right)$ , and $\psi \left(t\right)$ to be approximately centered around the origin. This is achieved by having the initial nonzero scaling coefficient start at $n=-\frac{N}{2}+1$ rather than zero. We prefer to have the origin at $n=t=0$ .