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A median filter is a non-linear and lossy process, so it doesn't have a closed form frequency response as would a FIR filter (say a box filter of the same length) in an LTI system.

  • But how closely can something similar to a frequency response of a median filter be approximated?
  • How would this scale with the length of a median filter?
  • Under what conditions or for what class of signals might this approximation be ballpark "close"?
  • For what class of signals might this approximation be very inaccurate?
  • What kinds of frequency domain distortion or additive noise does a median filter produce?
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  • $\begingroup$ Well it's definitely a low-pass filter, right? Is there any scenario in which it amplifies high spatial frequencies? $\endgroup$
    – endolith
    Mar 6, 2013 at 14:42

1 Answer 1

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For a start, any non-linear system will not have an easily-identifiable frequency response. So, it's really a nonsensical question. I intend no offense; nonsensical questions are often the most enlightening!

However one way to try to answer your question is to assume that the LTI filter involved is the mean (rather than the median) of the windowed data.

Then your question:

Under what conditions or for what class of signals might this approximation be ballpark "close"?

becomes:

Under what conditions or for what class of signals might the mean be ballpark "close" to the median.

In that case, for a purely stochastic signal, the mean and median are similar when the probability density function (PDF) of the signal is symmetric about the mean.

For what class of signals might this approximation be very inaccurate?

When the signal's PDF is "very" asymmetric.

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  • $\begingroup$ Ah yes, that might make sense, a very asymmetrical PDF (say, with some outliers), would have a median within the non-out-liears, as well as a mean within the non-out-liers as well. $\endgroup$
    – Spacey
    Feb 11, 2013 at 23:23
  • $\begingroup$ A very asymmetrical PDF with just one extreme value could have its mean far from the bulk of the data. Typical example (ala NNTaleb) is Elon Musk walks into a generic room of 100 people, and the average net worth of the occupants can suddenly jump to over 100X above the median. $\endgroup$
    – hotpaw2
    Jan 15 at 23:06
  • $\begingroup$ @hotpaw2 Absolutely! It's like the "Lake Wobegon" statement "all the children are above average." It just takes one below-average to almost make it true. :-) $\endgroup$
    – Peter K.
    Jan 15 at 23:59

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