4.4 Good 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:

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:

• 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 .

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, $$P(z)=\frac{1}{2}(z+2+z^{(-1)})=\frac{1}{2}(z+1)(1+z^{(-1)})=G_0(z)H_0(z)$$ 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:

If we allocate the factors of $P_t$ such that ( $1+Z$ ) gives $H_0$ and $(1+Z)(1+\alpha Z)$ gives $G_0$ , we get: