0.6 Regularity, moments, and wavelet system design

We now look at a particular way to use the remaining $\frac{N}{2}-1$ degrees of freedom to design the $N$ values of $h\left(n\right)$ after satisfying [link] and [link] , which insure the existence and orthogonality (or property of being a tight frame) of the scaling function and wavelets [link] , [link] , [link] .

One of the interesting characteristics of the scaling functions and wavelets is that while satisfying [link] and [link] will guarantee the existence of an integrable scaling function, it may be extraordinarilyirregular, even fractal in nature. This may be an advantage in analyzing rough or fractal signals but it is likely to be a disadvantage for mostsignals and images.

We will see in this section that the number of vanishing moments of $h_1\left(n\right)$ and $\psi \left(t\right)$ are related to the smoothness or differentiability of $\phi \left(t\right)$ and $\psi \left(t\right)$ . Unfortunately, smoothness is difficult to determine directly because, unlike with differential equations, thedefining recursion [link] does not involve derivatives.

We also see that the representation and approximation of polynomials are related to the number of vanishing or minimized wavelet moments. Sincepolynomials are often a good model for certain signals and images, this property is both interesting and important.

The number of zero scaling function moments is related to the “goodness" of the approximation of high-resolution scaling coefficients by samples of thesignal. They also affect the symmetry and concentration of the scaling function and wavelets.

This section will consider the basic 2-band or multiplier-2 case defined in [link] . The more general M-band or multiplier-M case is discussed in Section: Multiplicity-M (M-Band) Scaling Functions and Wavelets .

K-regular scaling filters

Here we start by defining a unitary scaling filter to be an FIR filter with coefficients $h\left(n\right)$ from the basic recursive [link] satisfying the admissibility conditions from [link] and orthogonality conditions from [link] as

The term “scaling filter" comes from Mallat's algorithm, and the relation to filter banks discussed in Chapter: Filter Banks and the Discrete Wavelet Transform . The term “unitary" comesfrom the orthogonality conditions expressed in filter bank language, which is explained in Chapter: Filter Banks and Transmultiplexers .

A unitary scaling filter is said to be $K$ - regular if its z-transform has $K$ zeros at $z=e^{i\pi }$ . This looks like

where $H\left(z\right)={\sum }_{n}h\left(n\right)z^{-n}$ is the z-transform of the scaling coefficients $h\left(n\right)$ and $Q\left(z\right)$ has no poles or zeros at $z=e^{i\pi }$ . Note that we are presenting a definition of regularity of $h\left(n\right)$ , not of the scaling function $\phi \left(t\right)$ or wavelet $\psi \left(t\right)$ . They are related but not the same. Note also from [link] that any unitary scaling filter is at least $K=1$ regular.

The length of the scaling filter is $N$ which means $H\left(z\right)$ is an $N-1$ degree polynomial. Since the multiple zero at $z=-1$ is order $K$ , the polynomial $Q\left(z\right)$ is degree $N-1-K$ . The existence of $\phi \left(t\right)$ requires the zero ^th moment be $\sqrt{2}$ which is the result of the linear condition in [link] . Satisfying the conditions for orthogonality requires $N/2$ conditions which are the quadratic equations in [link] . This means the degree of regularity is limited by