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<p><b>Abstract</b>—The current best bound on the number of comparison operations needed to compute the running maximum or minimum over a <it>p</it>-element sliding data window is approximately three comparisons per output sample [<ref rid="bibi05261" type="bib">1</ref>], [<ref rid="bibi05262" type="bib">2</ref>], [<ref rid="bibi05263" type="bib">3</ref>], [<ref rid="bibi05264" type="bib">4</ref>]. This bound is probabilistic for the algorithms in [<ref rid="bibi05262" type="bib">2</ref>], [<ref rid="bibi05263" type="bib">3</ref>], [<ref rid="bibi05264" type="bib">4</ref>] and is derived for their complexities on the average for independent, identically distributed (i.i.d.) input signals (uniformly i.i.d., in the case of the algorithm in [<ref rid="bibi05262" type="bib">2</ref>]). The worst-case complexities of these algorithms are <it>O</it>(<it>p</it>). The worst-case complexity <it>C</it><sub>1</sub> = 3 − 4 / <it>p</it> comparisons per output sample for 1D signals is achieved in the Gil-Werman algorithm [<ref rid="bibi05261" type="bib">1</ref>]. In this correspondence we propose a modification of the Gil-Werman algorithm with the same worst-case complexity but with a lower average complexity. A theoretical analysis shows that using the proposed modification the complexities of sliding Max or Min 1D and 2D filters over i.i.d. signals are reduced to <it>C</it><sub>1</sub> = 2.5 − 3.5 / <it>p</it> + 1 / <it>p</it><super>2</super> and <it>C</it><sub>2</sub> = 5 − 7 / <it>p</it> + 2 / <it>p</it><super>2</super> comparisons per output sample on the average, respectively. Simulations confirm the theoretical results. Moreover, experiments show that even for highly correlated data, namely, for real images the behavior of the algorithm remains the same as for i.i.d. signals.</p>
Running min and max, morphological filters, algorithms, computational complexity.

S. M. Atourian, D. Z. Gevorkian and J. T. Astola, "Improving Gil-Werman Algorithm for Running Min and Max Filters," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 19, no. , pp. 526-529, 1997.
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