Efficient dilation, erosion, opening, and closing algorithms

We propose an efficient and deterministic algorithm for computing the one-dimensional dilation and erosion (max and min) sliding window filters. For a p-element sliding window, our algorithm computes the 1D filter using 1.5 + o(1) comparisons per sample point. Our algorithm constitutes a deterministic improvement over the best previously known such algorithm, independently developed by van Herk (1992) and by Gil and Werman (1993) (the HGW algorithm). Also, the results presented in this paper constitute an improvement over the Gevorkian et al. (1997) (GAA) variant of the HGW algorithm. The improvement over the GAA variant is also in the computation model. The GAA algorithm makes the assumption that the input is independently and identically distributed (the i.i.d. assumption), whereas our main result is deterministic. We also deal with the problem of computing the dilation and erosion filters simultaneously, as required, e.g., for computing the unbiased morphological edge. In the case of i.i.d. inputs, we show that this simultaneous computation can be done more efficiently then separately computing each. We then turn to the opening filter, defined as the application of the min filter to the max filter and give an efficient algorithm for its computation. Specifically, this algorithm is only slightly slower than the computation of just the max filter. The improved algorithms are readily generalized to two dimensions (for a rectangular window), as well as to any higher finite dimension (for a hyperbox window), with the number of comparisons per window remaining constant. For the sake of concreteness, we also make a few comments on implementation considerations in a contemporary programming language.