Selection of Windows for W-Operator Combination from Entropy Based Ranking

When training morphological operators that are locally defined with respect to a neighborhood window, one must deal with the trade off between window size and statistical precision of the learned operator. More precisely, too small windows result in large restriction errors due to the constrained operator space and, on the other hand, too large windows result in large variance error due to often insufficient number of samples. A two-level training method that combines a number of operators designed on distinct windows of moderate size is an effective way to mitigate this issue. However, in order to train combined operators, one must specify not only how many operators will be combined, but also the windows for each of them. To date, a genetic algorithm that searches for window combinations has produced the best results for this problem. In this work we propose an alternative approach that is computationally much more efficient. The proposed method consists in efficiently reducing the search space by ranking windows of a collection according to an entropy based measure estimated from input-output joint probabilities. Computational efficiency comes from the fact that only few operators need to be trained. Experimental results show that this method produces results that outperform the best results obtained with manually selected combinations and are competitive with results obtained with the genetic algorithm based solution. The proposed approach is, thus, a promising step towards fully automating the process of binary morphological operator design.