(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).
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
.
