Iterative design of morphological binary image operators
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Iterating (composing) a sequence of window operators results in an operator defined over the window determined by the dilation of the component windows. Although a statistically designed iterative operator uses potentially all of the variables in the large dilated window, the design of each component operator requires design only over a much smaller component window, thereby resulting in a reduced estimation error. This means that designed iterative operators can perform better than designed estimates of fully optimal operators over the same effective window. While the optimal iterative and fully optimal operators may differ substantially in their logical structure, they may be probabilistically very close as operators on the random image processes under consideration. Thus, a precisely designed iterative operator can be closer to optimal than a less precisely designed unconstrained operator. We present three measures by which to compare iterative operators, with main interest focusing on the difference in their mean-absolute errors (MAEs), and discuss iterative design procedures and relationships between MAEs occurring from various procedures. A key aspect of design is the dependency on sample size. Increasing the number of iterations may in theory produce a better filter but, like using large windows, increasing the number of iterations increases the amount of data required for precise design. We pay particular attention to this issue. Using both restoration and recognition operators, we consider the best number of iterations and window size. We also consider the manner in which the training data should be split when designing the individual component operators. Iteration number, window size, and training method are all dependent on the filtering task, image characteristics, and amount of training data available.