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. If a twostage iterative filter is designed with each stage optimally designed over a window, then the iterative filter is a suboptimal approximation to the optimal filter over the larger window formed as the dilation of the small window with itself. Using image-noise models, three basic questions regarding optimal increasing iterative filters are statistically addressed: How many iterations are required before there is only a negligible increase in filter performance? As the number of iterations increases, how good is filter performance in comparison with large-window noniterative filters? What are the logical and probabilistic relations between noniterative and approximating interative filters? Iterative-filter performance is seen to be only slightly suboptimal, in return for tractable design. A key conclusion is that, while in terms of logic there may be a significant difference between a noniterative and approximating iterative filter, their probabilistic difference as operators on random sets can be neglible. This fundamental point is discussed in the context of application to digital document processes, which form a key area of application.