Optimal Pivot Selection in Fast Weighted Median Search

Weighted median filters are increasingly being used in signal processing applications and thus fast implementations are of importance. This paper introduces a fast algorithm to compute the weighted median (WM) of N samples which has linear time and space complexity as opposed to O(N log N) which is the time complexity of traditional sorting algorithms. A popular selection algorithm often used to find the WM in large data sets is Quickselect whose performance is highly dependent on how the pivots are chosen. We introduce an optimization based pivot selection strategy which results in significantly improved performance as well as a more consistent runtime compared to traditional approaches. The selected pivots are order statistics of subsets. In order to find the optimal order statistics as well as the optimal subset sizes, a set of cost functions are derived, which when minimized lead to optimal design parameters. We compare the complexity to Floyd and Rivest's algorithm SELECT which to date has been the fastest median finding algorithm and we show that the proposed algorithm compared with SELECT requires close to 30% fewer comparisons. It is also shown that the proposed selection algorithm is asymptotically optimal for large N.