The output distribution of important LULU-operators

Two procedures to compute the output distribution phi_S of certain stack filters S (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields phi_S for some important LULU-operators S. Properties of phi_S can be used to characterize smoothing properties of S. One of the methods discussed also allows for the calculation of the reliability polynomial of any positive Boolean function (e.g. one derived from a connected graph).Comment: 20 pages, up to trivial differences this is the final version to be published in Quaestiones Mathematicae 201

Theory of and parallel algorithm for stack filters

Stack filters are easily implemented nonlinear filters which include all rank-order operators and all compositions of the morphological operations known as openings and closings. Stack filters have been shown very effective at the task of image recovery. Therefore, faster algorithms are needed in order to implement them more efficiently. Not only is the algorithm developed here suitable for hardware implementation, but also is suitable for software simulation after slight revision. The minimum time-area complexity of this algorithm is O(log\sb2\omega) compared with $O(B)$, for both hierarchical threshold decomposition and binary search algorithms, where $\omega$ is window size and $B$ is the number of bits required to represent the input signal. As we apply it to computer simulation, the maximum speedup is about 24 for real input signal and 8 for integer input signals running on a Sun 3/60 workstation with a Motorola 68881 floating point processor. The probability that the output of $F$ is the i\sp{th} largest element in a window is \alpha\sb{i}(F). The probability that the output of $F$ is in the i\sp{th} position of a window is \beta\sb{i}(F). \alpha\sb{i}(F) and \beta\sb{i}(F) are helpful in understanding some characteristics of a stack filter $F$. Also, we associate them with a stack filter lattice and derive several interesting formulas. Circular stack filters, a subset of stack filters, are proposed. The number of the circular stack filters is far less than that of stack filters. This reduces the time to find an optimal circular stack filter. Like rank order filters, and collection of circular stack filters forms a sublattice of a stack filter lattice. That is to say, all the properties of stack filter lattices can be applied to circular stack filter lattices. Circular stack filters are useful in designing a stack filter, such that part of pixels in a window have the same \beta\sb{i}\u27s. Many other properties of circular stack filters are derived