Image Processing Reference
In-Depth Information
•
Accurate prototypes are needed to yield accurate boundaries, whereas
•
Accurate boundaries are precisely what is needed to compute accurate proto-
types.
Therefore, the prototypes and the boundaries cannot be simultaneously determined
[227] to yield high accuracy. The presence of noise in the feature space, mainly pro-
duced by modeling errors (inappropriate features), is another source of noise which
contributes to a class overlap.
The variance of a signal decreases when it is lowpass filtered, i.e., averaged.
Because of this, multiple resolutions are crucial in reducing the uncertainty. At lower
resolutions the class prototype estimates are better defined than high resolutions.
A Gaussian pyramid can be built, up to a predefined level in which each level is
obtained by smoothing the preceding finer level. Accordingly, each element of the
pixel value tensor, a
feature image
, will have its own pyramid. At the coarsest level,
the amount of noise in the features therefore decreases significantly, allowing the
feature prototypes to be determined more accurately, but at the expense of the region
boundary resolution.
Class uncertainty reduction via pyramids is illustrated in Fig. 16.1 [195, 228]. At
the finest level the histogram is unimodal, even if it is possible to see that distinct
classes are present. At a lower resolution the noise has been smoothed out signifi-
cantly, making possible the detection of the two classes (the histogram is bimodal).
It is at a low resolution level that a good opportunity to partition the feature space
into its constituent
N
c
classes appears. Partitioning can be seen as finding
N
c
subsets
of feature vectors gathered around their respective prototypes (or class centers) au-
tomatically. This can be obtained by applying a clustering algorithm in the
smoothed
feature space
(obtained at the coarsest level). For most of these algorithms to work,
the number of classes
N
c
, or an equivalent information, has to be available a priori.
Image segmentation experiments using automatic clustering indicate that the results
do not critically depend on the choice of a clustering algorithm, provided that the
classes are separable by means of the provided features.
Next, isolated points as well as isolated and scattered small classes are eliminated
by reassigning them to a class in the spatial vicinity to obtain a spatial connectivity.
It is, however, now, when a segmentation at the coarsest level is available, that it
becomes evident that the cost of good class separation is bad boundaries.
The last but not the least step is a boundary estimation procedure that gradually
improves the class boundaries by traversing down the pyramid. First, at the resolution
level where the clustering is performed, the crude boundaries are identified. The
children of the
boundary points
define a
boundary region
at the next higher resolution
because every pixel at a certain level represents several pixels at the lower resolution.
The nonboundary nodes at the children level are given the same labels and properties
as their parents. The class uncertainty within the boundary region is high and has to
be reduced before reassignment of the boundary vectors. Now that the approximate
boundaries and thereby the boundary directions are available, orientation-adaptive
filters are used to smooth the boundary regions. For each dominant local direction,
a butterflylike averaging filter (see Fig. 16.2 and Sect. 16.6) reduces the influence of
