Classifying image data
An image is represented as a twodimentional array of
coefficients, each coefficient representing the
brightness level in that point. When looking from a
higher perspective, we can't differentiate between
coefficients as more important ones, and lesser important
ones. But thinking more intuitively, we can. Most natural
images have smooth colour variations, with the fine
details being represented as sharp edges in between the
smooth variations. Technically, the smooth variations in
colour can be termed as low frequency variations and the
sharp variations as high frequency variations.
The low frequency components (smooth variations)
constitute the base of an image, and the high frequency
components (the edges which give the detail) add upon
them to refine the image, thereby giving a detailed
image. Hence, the smooth variations are demanding more
importance than the details.
Separating the smooth variations and details of the
image can be done in many ways. One such way is the
decomposition of the image using a Discrete Wavelet
Transform (DWT).
The DWT of an image
The procedure goes like this. A low pass filter and a
high pass filter are chosen, such that they exactly halve
the frequency range between themselves. This filter pair
is called the Analysis Filter pair. First, the low pass
filter is applied for each row of data, thereby getting
the low frequency components of the row. But since the
lpf is a half band filter, the output data contains
frequencies only in the first half of the original
frequency range. So, by Shannon's Sampling Theorem, they
can be subsampled by two, so that the output data now
contains only half the original number of samples. Now,
the high pass filter is applied for the same row of data,
and similarly the high pass components are separated, and
placed by the side of the low pass components. This
procedure is done for all rows.
Next, the filtering is done for each
column of the intermediate data. The resulting
twodimensional array of coefficients contains four bands
of data, each labelled as LL (lowlow), HL (highlow), LH
(lowhigh) and HH (highhigh). The LL band can be
decomposed once again in the same manner, thereby
producing even more subbands. This can be done upto any
level, thereby resulting in a pyramidal decomposition as
shown below.
Fig 1. Pyramidal Decomposition of an
Image
As mentioned above, the LL band at the
highest level can be classified as most important, and
the other 'detail' bands can be classified as of lesser
importance, with the degree of importance decreasing from
the top of the pyramid to the bands at the bottom.
Fig 2. The three layer decomposition of
the 'Lena' image.
The Inverse DWT of an image
Just as a forward transform to used to separate the
image data into various classes of importance, a reverse
transform is used to reassemble the various classes of
data into a reconstructed image. A pair of high pass and
low pass filters are used here also. This filter pair is
called the Synthesis Filter pair. The filtering procedure
is just the opposite  we start from the topmost level,
apply the filters columnwise first and then rowwise, and
proceed to the next level, till we reach the first level.
Wavelet resources on the web
If wavelets seem to be interesting, you may follow the
links given below to learn more.
The wavelet digest page at www.wavelet.org
Wavelab matlab software at wwwstat.stanford.edu/~wavelab/
Wavelet papers at www.mathsoft.com/wavelets.html
Amara's wavelet page at www.amara.com/current/wavelet.html
Engineer's Guide to Wavelet Analysis by Polikar at users.rowan.edu/~polikar/WAVELETS/WTtutorial.html
Copyright (c) Satish Kumar. S 20012003. Last Modified  22 Oct 2001
Suggestions/Broken links/queries? Write to satish@debugmode.com
