Image Processing Reference
In-Depth Information
It is possible for a stack filter to be designed to use a different binary filter
t
for
every threshold level. Care must be taken that the stacking property is not violated,
so a constraint must be placed on the filters so that the ordering
……
(7.7)
m
1
t
+1
t
1
is preserved for every possible input. Dougherty calls this property
consistency
.
5
The use of different filters at each threshold level can theoretically result in im-
proved results since each filter can more closely model the required behavior at
each level. However, if the total amount of training data is insufficient to be divided
among the design of many filters, it can lead to worse results overall because of in-
creased estimation error. Using the same filter at every level may be a compromise
in terms of the different effects required at each level. However, the larger training
set resulting from aggregating the data over all threshold levels can lead to a filter
with a lower estimation error and better overall performance.
Stack filters can give excellent results for certain types of problems. Figure 7.8(a)
shows a training set containing a noisy astronomical image and an ideal version. A
stack filter was trained on these images and then applied to the image shown in
Fig. 7.8(b). The noise is very severe in this type of data, and the stack filter does a
good job of removing it. For comparison, Fig. 7.8(c) shows the two noisy images af-
ter filtering with Paintshop Pro, version 7 using the despeckle option. It can be seen
that it makes little impression on the speckle. This is hardly surprising because it is
operating without the benefit of an ideal image and is hence producing a general-pur-
pose despeckle filter.
As can be seen above, given the right problem, stack filters can produce excel-
lent results. They do, however, have strict limitations. Their processing structure
treats the signal at each threshold level as an independent entity, and there is no
communication path between levels. A training set with a brightness or contrast dif-
ference between the noisy and ideal image would confuse the filter and lead to poor
results. Stack filters cannot detect objects or shapes because they are increasing fil-
ters, neither can they shift brightness levels. For more difficult problems it is neces-
sary to link the threshold levels and use more complex filters. This is covered in the
following sections.
7.2 Grayscale Morphology
All types of stack filters may be implemented through mathematical morphology
and result in grayscale structuring elements that have vertical sides and flat tops.
The process therefore acts on each threshold level independently. A more general
type of filtering allows the structuring elements to take a shape of varying
cross-sections such as a triangle or cone. This has the effect of linking the process-
ing across the threshold levels.
