<title>Design of large-window binary filters via iteration</title>

Abstract

Rather than design an optimal filter over a large window, which may be computationally impossible or require unacceptable computation time, one can design an iterative filter, each stage of which is designed over a small window with acceptable design time. Theoretically, a two-stage iterative filter with each stage optimally designed over a window is suboptimal in comparison to the optimal filter over the larger window formed as the dilation of the small window with itself. In practice, however, filters are designed from realizations and lack of precision for design over a large window can result in a directly estimate optimal filter over a large window that performs worse than an iteratively designed filter. Using image-noise models, this paper considers there cases: (1) the designed filters are good estimates of the theoretically optimal filters, the two-stage iterative filter is close to optimal, and as further iterations are considered for both the small and large windows, the performance difference becomes small; (2) the designed filter over the large window is a poor estimate of the theoretically optimal filter and the iteratively designed filter outperforms the directly designed filter; (3) iteration cannot do well because the iteration window is too small for the image-noise model. We will see that, while in terms of logic there may be a significant difference between a noniterative and an approximating iterative filter, their probabilistic difference as operators on random processes can be negligible.