Image Processing Reference
In-Depth Information
feature-vectors across the boundary while decreasing the within-class variance along
the boundary. The two halves of the filter (applied separately) produce two responses.
The distances between the resulting vectors and the two prototypes, associated to
the classes defining the boundary, are computed and a reclassification is performed.
The procedure, which will be discussed further in the subsequent sections, can be
summarized as follows:
1. Build a multiresolution pyramid up to a certain level. The noise in the feature
space is reduced, increasing the separation between the classes at the expense of
the spatial resolution.
2. Cluster the data in the smoothed feature space by using algorithms at suitable
levels of the pyramid. Reassign isolated pixels as well as small and scattered
classes to enforce spatial continuity.
3. Gradually improve the spatial resolution by projecting down the labels and refine
the boundaries using orientation-adaptive filters.
16.2 Pyramid Building
Noise reduction can be done by means of smoothing, which can efficiently be imple-
mented in image pyramids [43, 92]. Lower resolution levels are obtained by taking
the (weighted) average of small neighborhoods to the next coarser level, as discussed
in Sect. 9.5. In the discussion here, an octave pyramid , i.e., one in which the image
width and heights are halved between subsequent levels, is assumed. Let f ( r k ,l ) be
the feature vector having N dimensions (layers) at image location r k and level l of a
feature pyramid constructed from the highest resolution (lowest level) as follows:
f ( r k ,l +1)=
r p
g ( r p ) f (2 r k r p ,l )
(16.1)
The value of a parent node is the weighted mean of its children within the support
of a filter g , such as a Gaussian. The different levels are computed using Eq. (16.1)
in a bottom-up manner, starting from level l =0up to a predefined level l = L .
The height and width of the image decrease with a factor of 2 l at level l . A pyramid
is constructed for each feature separately. Therefore, the data structure can be seen
as a set of pyramids or as a single one composed of M
1 vectors at each image
point. The choice of the number of levels is important, as this determines the mini-
mum connected region size in the grouping. If L is too small, the uncertainty is not
reduced sufficiently, whereas if L is too high, small regions will disappear. Pyramids
also reduce the computational cost by progressively reducing the number of feature
vectors on which a clustering algorithm will be applied. Assume, for instance, that
the feature images have the size 256
×
256 and that L =3. Then the number of fea-
tures is reduced from 65 , 536 to 256 at the level L , a reduction with factor 4 L =64.
The level L should, however, be chosen as a function of the noise rather than a func-
tion of the size. If the number of feature vectors remains prohibitively large for a
clustering algorithm, it still can be reduced by taking them from random image sites.
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