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 the images from quantised coefficients.

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

TABLE 1

Levels	Qstep
All bands at levels 3 and 4:	50
Hi-Lo and Lo-Hi bands at level 2:	50
Hi-Hi band at level 2:	100
Hi-Lo and Lo-Hi bands at level 1:	100
Hi-Hi band at level 1:	200

A similar compressed bit rate is produced by the 8×8 DCT when Q_step=5Q_lum.

For reference, Figure 1 compares the DCT and Haar transforms using these two quantisers. The rms errors between the reconstructed images and the original are virtually the same at 10.49 and 10.61 respectively, but the DCT entropy 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.

Figure 1: Reconstructions after coding using the 8×8 DCT (a) with Q_step=5Q_lum, and (b) using the Haar transform with Q_step from Table 1.

Figure 2 shows the reconstructed images for the following four DWTs using the quantiser of Table 1:

• The LeGall 3,5-tap filters: H₀, H₁ and G₀, G₁ 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 H₀, H₁ and G₀, G₁ swapped.
• The near-balanced 5,7-tap filters: substituting Z=

  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.

We see that the LeGall 3,5-tap filters (Figure 2(a)) produce a poor image, whereas the other three images are all significantly better. The poor image is caused by the roughness of the LeGall 5,3-tap filters (shown in this previous figure) which are used for reconstructing he image when the 3,5-tap filters are used for analysing the image. When these filters are swapped, so that the reconstruction filters are the 3,5-tap ones of this figure, the quality is greatly improved (Figure 2(b)).

The near-balanced 5,7-tap filters (Figure 2(c)) produce a relatively good image but there are still a few bright or dark point-artefacts produced by the sharp peaks 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 mottledappearance.

Figure 2: Reconstructions after coding using Q_step from Table 1 with (a) the LeGall 3,5-tap filters, (b) the inverse-LeGall 5,3-tap filters, (c) the near-balanced 5,7-tap filters, and (d) the near-balanced 13,19-tap filters.

Figure 3 shows the entropies (with RLC) of the separate subimages of the 4-level DWT for the Haar filter set and the four filter sets of Figure 2. Q_step is defined by Table 1 and it is particularly noticeable how the higher step sizes at levels 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 of entropy with increasing filter smoothness is also apparent.

Figure 3: Entropies of 4-level DWT subimages using Q_step defined by Table 1, for five different wavelet filter pairs.

NOTE: 

We see that we have now been able to reduce the bit rate to around 0.3 bit/pel.

However measurement of entropy is not the whole story, as it is the tradeoff of entropy vs quantising error which is important. Figure 4 attempts to show this trade-off by plotting rms quantising error (obtained by subtracting the reconstructed image from the original) versus the entropy for the 8×8 DCT and the five DWTs. To show the slope of the curves, the measurements are repeated with an 80% lower quantiser step-size, giving lower rms errors and higher entropies. The pair of points for each configuration are jointed by lines which indicate the slope of the rate-distortion curve.

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

Figure 4: RMS error vs. entropy for the 8×8 DCT and five wavelet filter pairs. For the DCT, Q_step=5Q_lumand 4Q_lum; for the DWT, Q_step is 1.0 and 0.8 of the values in (Table 1).

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

NOTE: 

What is clear is that when filters are unbalanced between analysis and reconstruction, the ones which give smoother wavelets must be used for reconstruction.

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