Efficient Dilation, Erosion, Opening and Closing Algorithms

Abstract

We propose an efficient algorithm for computing the dilation and erosion 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 improvements over the best previously known such algorithm by Gil and Werman [5]. The previous improvement on [5] offered by Gevorkian, Astola and Atourian [2] was in better expected performance for random signals. Our result improves on [5] result without assuming any distribution of the input. Further, a randomized version of our algorithm gives an expected number of 1.25 + o(1) comparisons per sample point, for any input distribution. We deal with the problem of computing the dilation and the erosion filters simultaneously, and again improve the Gil-Werman algorithm in this case for independently distributed inputs. 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 rectangular structuring element, as well as to any higher finite dimension for a hyper-box structuring element, with the number of comparisons per window remaining constant.