Papers › Median Filtering is Equivalent to Sorting
Median Filtering is Equivalent to Sorting
Jukka Suomela
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This work shows that the following problems are equivalent, both in theory and in practice: - median filtering: given an n-element vector, compute the sliding window median with window size k, - piecewise sorting: given an n-element vector, divide it in n/k blocks of length k and sort each block. By prior work, median filtering is known to be at least as hard as piecewise sorting: with a single median filter operation we can sort Θ(n/k) blocks of length Θ(k). The present work shows that median filtering is also as easy as piecewise sorting: we can do median filtering with one piecewise sorting operation and linear-time postprocessing. In particular, median filtering can directly benefit from the vast literature on sorting algorithms---for example, adaptive sorting algorithms imply adaptive median filtering algorithms. The reduction is very efficient in practice---for random inputs the performance of the new sorting-based algorithm is on a par with the fastest heap-based algorithms, and for benign data distributions it typically outperforms prior algorithms. The key technical idea is that we can represent the sliding window with a pair of sorted doubly-linked lists: we delete items from one list and add items to the other list. Deletions are easy; additions can be done efficiently if we reverse the time twice: First we construct the full list and delete the items in the reverse order. Then we undo each deletion with Knuth's dancing links technique.
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