The Wilcoxon filter is a special case of a rankfilter, which is used for noise reduction in image processing [9]. Further, this filter approach can also be applied to digital signal processing problems. If *X* is a rank sorted vector and part of a signal with *N* elements and written as *X* = [*x*_{1}, *x*_{2}, …, *x*_{N}], the Wilcoxon filter is defined in [10], [11] as:

$${Y}_{i}=\text{median}(\frac{{x}_{j}+{x}_{k}}{2},\mathrm{\hspace{0.25em}1}\le j\le k\le N),$$(1)

The term $\frac{{x}_{j}+{x}_{k}}{2}$ is well known as Walsh average [12]. The Wilcoxon filter is effective to white noise reduction and edges or slopes were not preserved because of the averaging of all possible pairs [10]. Calculating the Walsh average for all the pairs causes problems in computational complexity because *N*(*N*+1)/2 values have to be sorted. The number of computations of the Walsh averages can be reduced using the inherent property of the Wilcoxon filter presented in [13]. As presented in [10] all computed Walsh averages can be written in a matrix what is exemplary done for *N* = 5 as follows:

$$\left(\begin{array}{cccccc}{x}_{1}\hfill & \hfill \frac{\left({x}_{1}+{x}_{2}\right)}{2}\hfill & \hfill \frac{\left({x}_{1}+{x}_{3}\right)}{2}\hfill & \hfill \frac{\left({x}_{1}+{x}_{4}\right)}{2}\hfill & \hfill \frac{\left({x}_{1}+{x}_{5}\right)}{2}\hfill & \\ & \hfill {x}_{2}\hfill & \hfill \frac{\left({x}_{2}+{x}_{3}\right)}{2}\hfill & \hfill \frac{\left({x}_{2}+{x}_{4}\right)}{2}\hfill & \hfill \frac{\left({x}_{2}+{x}_{5}\right)}{2}\hfill & \\ & & \hfill {x}_{3}\hfill & \hfill \frac{\left({x}_{3}+{x}_{4}\right)}{2}\hfill & \hfill \frac{\left({x}_{3}+{x}_{5}\right)}{2}\hfill & \\ & & & \hfill {x}_{4}\hfill & \hfill \frac{\left({x}_{4}+{x}_{5}\right)}{2}\hfill & \\ & & & & \hfill {x}_{5}\hfill & \end{array}\right)$$

The result of the median of eq. 1 is close to the median of the main diagonal with the elements colored in red [10], [13]. Therefore, the Wilcoxon filter for the example of *N* = 5 can be written as:

$${y}_{i}=\text{median}[{x}_{3},\frac{{x}_{2}+{x}_{4}}{2},\frac{{x}_{1}+{x}_{5}}{2}].$$(2)

Thus, the median is calculated from 3 instead of 15 elements. The computational complexity was reduced from *N*(*N*+1)/2 to (*N*+1)/2 . As shown in [13], this simplification can be applied for a larger filter lengths.

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