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 Qlum Qlum 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:

A similar compressed bit rate is produced by the 8×8 8 8 DCT when Qstep =5⁢ Qlum Qstep 5 Qlum .

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

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 mottled appearance.

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. Qstep Qstep 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.

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 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.

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.

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.

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This work is licensed by Nick Kingsbury under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Elizabeth Gregory on Jun 8, 2005 8:57 am -0500.