4.4 Good filters / wavelets  (Page 2/3)

The wavelets of the LeGall 3,5-tap filters, $H_0$ and $H_1$ above, and their frequency responses are shown in . The scaling function (bottom left) converges to a pure triangular pulse and thewavelets are the superposition of two triangular pulses.

The triangular scaling function produces linear interpolation between consecutive lowband coefficients and also causes thewavelets to be linear interpolations of the coefficients of the $H_1$ filter, -1, -2, 6, -2, -1 (scaled appropriately).

These wavelets have quite desirable properties for image compression (note the absence of waveform discontinuities andthe much lower sidelobes of the frequency responses), and they represent probably the simplest useful waveletdesign. Unfortunately there is one drawback -- the inverse wavelets are not very good. These are formed from the LeGall5,3-tap filter pair, $G_0$ and $G_1$ above, whose wavelets and frequency responses are shown in .

The main problem is that the wavelets do not converge after many levels to a smooth function and hence the frequency responseshave large unwanted sidelobes. The jaggedness of the scaling function and wavelets causes highly visible coding artefacts ifthese filters are used for reconstruction of a compressed image.

However the allocation of the factors of $P_t(Z)$ to $H_0$ and $G_0$ is a free design choice, so we may swap the factors (and hence swap $G$ and $H$ ) in order that the smoother 3,5-tap filters become $G_0$ , $G_1$ and are used for reconstruction. We shall show later that this leads to a good low-complexity solution for imagecompression and that the jaggedness of the analysis filters is not critical.

Unbalance between analysis and reconstruction filters / wavelets is nevertheless often regarded as being undesirable,particularly as it prevents the filtering process from being represented as an orthonormal transformation of the input signal(since an orthonormally transformed signal may be reconstructed simply by transposing the transform matrix). An unbalanced PRfilter system is often termed a bi-orthogonal transformation .

We now consider ways to reduce this unbalance.

Filters with balanced h and g frequency responses (but non-linear phase responses) - daubechies wavelets:

In the above analysis, we used the factorisation of $P_t(Z)$ to give us $H_0$ and $G_0$ . This always gives unbalanced factors if terms of $P_t$ in even powers of $Z$ are zero.

However each of these factors in $Z$ may itself be factorised into a pair of factors in $z$ , since:

For each factor of $P_t(Z)$ , we may allocate one of its $z$ subfactors to $H_0(z)$ and the other to $G_0(z)$ . Where roots of $P_t(Z)$ are complex, the subfactors must be allocated in conjugate pairs so that $H_0$ and $G_0$ remain purely real.

Since the subfactors occur in reciprocal pairs (roots at $z=\alpha$ and $\alpha ^{(-1)}$ ), we find that