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The Multi-level Haar Transform

Module by: Nick Kingsbury. E-mail the author

Summary: This module introduces the multi-level Haar transform.

(a) of Figure 1 shows the result of applying the Haar transform to the Lo-Lo subimage of this previous figure and Figure 2 shows the probabilities pi pi and entropies hi hi for the 4 new subimages.

The level 2 column of the figure Cumulative Entropies of Subimages for Qstep=15 shows how the total bit rate can be reduced by transforming the level 1 Lo-Lo subimage into four level 2 subimages. The process can be repeated by transforming the final Lo-Lo subimage again and again, giving the subimages in (b) of Figure 1 and (c) of Figure 1 and the histograms in Figure 3 and Figure 4. The levels 3 and 4 columns of the figure Cumulative Entropies of Subimages for Qstep=15 show that little is gained by transforming to more than 4 levels.

However a total compression ratio of 4 bit/pel : 1.61 bit/pel = 2.45 : 1 has been achieved (in theory).

Figure 1: Levels 2(a), 3(b), and 4(c) Haar transforms of Lenna; and at all of levels 1 to 4(d).
Figure 1 (figure7.png)
Figure 2: The probabilities pi pi and entropies hi hi for the 4 subimages at level 2.
Figure 2 (figure8.png)
Figure 3: The probabilities pi pi and entropies hi hi for the 4 subimages at level 3.
Figure 3 (figure9.png)
Figure 4: The probabilities pi pi and entropies hi hi for the 4 subimages at level 4.
Figure 4 (figure10.png)
Figure 5: Images reconstructed from (a) the original Lenna, and (b) the 4-level Haar transform, each quantised with Qstep =15 Qstep 15 . The rms error of (a) = 4.3513, and of (b) = 3.5343.
Figure 5 (figure11.png)

Note the following features of the 4-level Haar transform:

  • (d) of Figure 1 shows the subimages from all 4 levels of the transform and illustrates the transform's multi-scale nature. It also shows that all the subimages occupy the same total area as the original and hence that the total number of transform output samples (coefficients) equals the number of input pels - there is no redundancy.
  • From the Lo-Lo subimage histograms of the figure Haar Transform, Level 1 energies, and entropies for Qstep=15, Figure 2, Figure 3 and Figure 4, we see the magnitudes of the Lo-Lo subimage samples increasing with transform level. This is because energy is being conserved and most of it is being concentrated in fewer and fewer Lo-Lo samples. (The DC gain of the Lo-Lo filter of this previous equation is 2.)
  • We may reconstruct the image from the transform samples ((d) of Figure 1), quantised to Qstep =15 Qstep 15 , by inverting the transform, using the right hand part of this equation. We then get the image in (b) of Figure 5. Contrast this with (a) of Figure 5, obtained by quantising the pels of the original directly to Qstep =15 Qstep 15 , in which contour artifacts are much more visible. Thus the transform provides improved subjective quality as well as significant data compression. The improved quality arises mainly from the high amplitude of the low frequency transform samples, which means that they are quantised to many more levels than the basic pels would be for a given Qstep Qstep .
  • If Qstep Qstep is doubled to 30, then the entropies of all the subimages are reduced as shown in Figure 6 (compare this with the figure, Cumulative Entropies of Subimages for Qstep=15 in which Qstep =15 Qstep 15 ). The mean bit rate with the 4-level Haar transform drops from 1.61 to 0.97 bit/pel. However the reconstructed image quality drops to that shown in (b) of Figure 7. For comparison, (a) of Figure 7 shows the quality if xx is directly quantised with Qstep =30 Qstep 30 .

Figure 6: Mean bit rate for the original Lenna image and for the Haar transforms of the image after 1 to 4 levels, using a quantiser step size Qstep =30 Qstep 30 .
Figure 6 (figure12.png)
Figure 7: Images reconstructed from (a) the original Lenna, and (b) the 4-level Haar transform, each quantised with Qstep =30 Qstep 30 . The rms error of (a) = 8.6219, and of (b) = 5.8781.
Figure 7 (figure13.png)

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