4.6 Compression properties of wavelets

We now look at how well the various wavelet filters perform in practice. We have used them in place of the Haar transform discussed earlier, and have measured the entropies and reconstructed theimages from quantised coefficients.

In order to allow a fair comparison with the JPEG DCT results, we have modified the DWT quantising strategy to take advantageof the reduced visibility of the higher frequency wavelets. This approximately matches the effects achieved by the JPEG $Q_{\mathrm{lum}}$ matrix of this previous equation . To achieve a high degree of compression we have used the following allocation ofquantiser step sizes to the 4-level DWT bands:

A similar compressed bit rate is produced by the $8\times 8$ DCT when $Q_{\mathrm{step}}=5Q_{\mathrm{lum}}$ .

For reference, compares the DCT and Haar transforms using these two quantisers. The rmserrors between the reconstructed images and the original are virtually the same at 10.49 and 10.61 respectively, but the DCTentropy of 0.2910 bit/pel is significantly lower than the Haar entropy of 0.3820 bit/pel. Both images display significant blocking artefacts at this compression level.

shows the reconstructed images for the following four DWTs using the quantiser of :

• The LeGall 3,5-tap filters: $H_0$ , $H_1$ and $G_0$ , $G_1$ from these previous equations and these previous equations from our discussion of good filters/wavelets.
• The inverse-LeGall 5,3-tap filters: these previous equations and these previous equations from our discussion of good filters/wavelets with $H_0$ , $H_1$ and $G_0$ , $G_1$ swapped.
• The near-balanced 5,7-tap filters: substituting $Z=\frac{1}{2}(z+z^{(-1)})$ into this equation from Good Filter/Wavelets.
• The near-balanced 13,19-tap filters: substituting this previous equation into this equation .

The near-balanced 5,7-tap filters ( (c)) produce a relatively good image but there are still a few bright or dark point-artefacts produced by the sharppeaks in the wavelets (shown in this previous figure ). The smoother 13,19-tap wavelets ( see this figure ) eliminate these, but their longer impulse responses tend to cause the image to have a slightly blotchy or mottled appearance.

shows the entropies (with RLC) of the separate subimages of the 4-level DWT for the Haarfilter set and the four filter sets of . $Q_{\mathrm{step}}$ is defined by and it is particularly noticeable how the higher step sizes atlevels 1 and 2 substantially reduce the entropy required to code these levels (compare with this previous figure ). In fact the Hi-Hi band at level 1 is not coded at all! The reduction ofentropy with increasing filter smoothness is also apparent.

Measurements at many more step sizes can be taken in order to give more compete rate-distortion curves if required.

The good performance of the 13,19-tap filters is clear, but the inverse-LeGall filters do surprisingly well - showing that thepoor smoothness of the analysis filters does not seem to matter. Correct ways to characterise unbalanced filter sets toaccount properly for this phenomenon are still the subject of current research.

Finally, in these tests, the assessments of subjective image quality approximately match the assessments based on rmserrors. However this is not always true and one must be careful to backup any conclusions from rms error measurements with atleast some subjective tests.