Image Processing Reference
In-Depth Information
ideal constraint is one that limits the search space so that a filter may be found with
a finite training set, but allows sufficient flexibility to find an accurate solution to
the problem. This is the point where intelligent human intervention is required in
the process, rather than at the pixel level.
The error resulting from the design of a constrained filter has two components,
constraint error and estimation error. The constraint error occurs as a result of limit-
ing the filter complexity. The filter has fewer options available and so an increase in
error may occur. The estimation error occurs as a result of the trained filter not hav-
ing converged to its final value. For a fixed-sized training set, the estimation error
gets smaller with a reduction in filter complexity.
The introduction of a constraint therefore increases the constraint error but re-
duces the estimation error. As with all engineering design problems, a trade off is
involved in minimizing the overall error. In practice the estimation error can be
very severe, even resulting in filtered images that are worse than the original. It is
much easier to reduce the estimation error by adding a constraint (at the expense of
an increased constraint error) than it is to do so by increasing the size of the training
set.
Filter constraints can take many different forms. The earlier examples in this
book assumed that the output for each input combination was estimated independ-
ently, which is reasonable for small windows. For larger windows independent es-
timation is almost impossible, therefore assumptions must be made about some
inputs by considering others. This is equivalent to fitting a function in linear filter-
ing. The simplest constraint on the function involves limiting the filter to increasing
functions. Once a filter has been designed for a specific task, its performance can be
evaluated a number of ways, such as by viewing the filtered output or analyzing the
MAE figures. However, an interesting insight into the behavior of the resulting fil-
ter can be found by use of Boolean logic reduction techniques. The final optimized
output function of the filter can be minimized into a sum-of-products form. This
can be viewed as a set processing masks (consisting of black, white, and don't-care
terms) to show how the patterns of pixels are changed by the processing.
Where the resulting function is an increasing filter, the processing masks corre-
spond to morphological structuring elements. The sum-of-products expression is
therefore equivalent to a union of erosions in morphology. In many applications of
morphology, the structuring elements used are arrived at by heuristics, and it is un-
likely that they are optimal in these circumstances. The statistical approach used in
this topic is ideal for producing optimum structuring elements for a given task.
Where the resulting function is nonincreasing, it is not sufficient to test if the
structuring elements “fit” the foreground of the image. The hit-or-miss transform
must be used to determine if corresponding conditions are met for the background.
This would be the case for target recognition or OCR, for example. In any case, the
optimal structuring elements are produced by this approach.
Several well-known filters may be expressed in the framework which may be
used to design filters by this approach. Among them are weighted-order statistics
filters including rank-order filters, the median, and its variants. Each of these filters
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