Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

2 Wavelets
2.5 Pyramidal wavelet transform
While the transformation process described in section 2.4 transforms a 1D-signal, in image
processing the main focus lies on 2D-data. To apply the wavelet transform on images first
the column vectors are transformed, then the row vectors are transformed - or vice versa.
This results in four subbands - an approximation subband and three detail subbands.
In the pyramidal wavelet transform only the approximation subband is decomposed further.
Thus, if repeating the decomposition step of the approximation subband again and again,
the result is a pyramidal structure, no matter what image is used as input. Figure 2.2(a)
shows such a pyramidal decomposition quadtree.
The motivation behind the pyramidal wavelet transform is the fact that in most natural images
the energy is concentrated in the approximation subband. Thus by decomposing the
approximation subband again and again the highest energies are contained within very few
coefficients since the approximation subband gets smaller and smaller with each decomposition
step. This is an important property for image compression for example.
But there are also images for which this decomposition structure is not optimal. If an image
for example has periodic elements the pyramidal transform is not able anymore to concentrate
the energy into one subband. A solution to this problem are wavelet packets.
2.6 Wavelet packets
Wavelet packets have been introduced by Coifman, Meyer and Wickerhauser as an extension
to multiresolution analysis and wavelets. In contrast to the pyramidal wavelet transform,
where only the approximation subband is decomposed further the wavelet packet transform
also allows further decomposition of detail subbands. This allows an isolation of other frequency
subbands containing high energy which is not possible in the pyramidal transform.
Due to the fact that any subband can now be decomposed further this results in a huge
number of possible bases. But depending on the image data and the field of application an
optimal basis has to be found. In the next sections some methods for basis selection are
presented.
2.6.1 Basis selection
When performing a full wavelet packet decomposition all subbands are decomposed recursively
until a maximum decomposition level is reached, no matter how much information is
contained within each subband.
The task of basis selection is used to optimize this process by selecting a subset of all possible
bases which fits as well as possible for a specific task.
Depending on whether the goal is compression of digital data or data classification for example
different basis selection algorithms are used. The reason for this is that there is currently
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