This module introduces the multi-level Haar transform.
(a) of
shows the result of
applying the Haar transform to the Lo-Lo subimage of this previous
figure and
shows the probabilities
and entropies
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 2subimages. The process can be repeated by transforming the final
Lo-Lo subimage again and again, giving the subimages in (b) of
and (c) of
and the histograms in
and
. 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).
Levels 2(a), 3(b), and 4(c) Haar transforms of Lenna; and at
all of levels 1 to 4(d).The probabilities
and entropies
for the 4 subimages at level 2.The probabilities
and entropies
for the 4 subimages at level 3.The probabilities
and entropies
for the 4 subimages at level 4.Images reconstructed from (a) the original Lenna, and (b) the
4-level Haar transform, each quantised with
. The rms error of (a) = 4.3513, and of (b) = 3.5343.
Note the following features of the 4-level Haar transform:
(d) of
shows the
subimages from all 4 levels of the transform and illustratesthe transform's
multi-scale nature. It
also shows that all the subimages occupy the same total areaas the original and hence that the total number of transform
output samples (coefficients) equals the number of inputpels - there is
no redundancy .
From the Lo-Lo subimage histograms of the figure
Haar Transform, Level 1 energies, and entropies for Qstep=15 ,
,
and
, we see the
magnitudes of the Lo-Lo subimage samples increasing withtransform level. This is because energy is being conserved
and most of it is being concentrated in fewer and fewerLo-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
), quantised to
, by inverting the transform, using the right hand part of this
equation . We then get
the image in (b) of
. Contrast this with (a) of
, obtained by quantising the pels of the
original directly to
, in which contour artifacts are much more
visible. Thus the transform provides improved subjectivequality as well as significant data compression. The
improved quality arises mainly from the high amplitude ofthe low frequency transform samples, which means that they
are quantised to many more levels than the basic pels wouldbe for a given
.
If
is doubled to 30, then the entropies of all the
subimages are reduced as shown in
(compare this with the figure,
Cumulative Entropies of Subimages for Qstep=15 in which
). The mean bit rate with the 4-level Haar
transform drops from 1.61 to 0.97 bit/pel. However thereconstructed image quality drops to that shown in (b) of
. For comparison, (a)
of
shows the quality
if
is directly quantised
with
.
Mean bit rate for the original Lenna image and for the Haar
transforms of the image after 1 to 4 levels, using a quantiserstep size
.Images reconstructed from (a) the original Lenna, and (b) the
4-level Haar transform, each quantised with
. The rms error of (a) = 8.6219, and of (b) = 5.8781.
