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This link contains the following statement:

In contrast to the frequency domain, it is possible to implement non-linear filters in the spatial domain. In this case, the summations in the convolution function are replaced with some kind of non-linear operator.

Why is it not possible to perform non-linear filtering in frequency domain ?

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  • $\begingroup$ You could, by adding spectral components in just the right amounts, but it would be far more complicated than just doing it in the time domain. Likewise, non-linear frequency processing is much easier to do in the frequency domain than the time domain. $\endgroup$ – endolith May 6 '13 at 23:57
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Linear filters in the spatial domain have a direct equivalent in the frequency domain, so you can transform your data and filter between spatial<->frequency domains and get equivalent results. However the transforms that we use for converting between spatial<->frequency domains are only valid for linear systems - a non-linear filter such as a median filter therefore has no frequency domain equivalent, so we can't perform e.g. median filtering in the frequency domain.

Note that this does not mean that there are no non-linear operations that we can perform in the frequency domain, e.g. we could reduce all frequency components below a certain threshold value to zero, it's just that there are no counterparts between the two domains for non-linear operations, so the frequency domain threshold operation described would have no equivalent operation in the spatial domain.

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