• $\begingroup$ Are you satisfied with the answers? If so, please consider accepting one of them $\endgroup$ – Laurent Duval Aug 20 '16 at 20:47

A median filter is most certainly not a "blur" filter, purely on the basis that it tends to preserve edges. Edges are abrupt transitions of brightness and therefore that information is encoded in the high frequencies of the spectrum. Incidentally those high frequencies are the ones that low-pass filters suppress the most, leading to that "blurry" appearance because only the form of the depicted objects is retained rather than their details.

Convolution can only be used to represent linear time invariant systems. That is, systems whose relationship between the input and the output is a linear combination and therefore proportional inputs produce proportional outputs. Furthermore, this relationship does not change with respect to time.

A median operator works by sorting the values of the $M \times N$ mask of pixels surrounding some $i,j$ pixel and then assigning their median (the pixel value that happens to lie at the midpoint of the range of values) to the $i,j$ pixel.

This operation, of sorting and then selecting the median, cannot be expressed as a linear combination of the pixel values within the $M \times N$ mask, hence the name "non-linear filter".

Hope this helps.

  • $\begingroup$ Well, for any given patch in the image, the median could be expressed as a linear combination of the mask pixels... But then the mask would be shift varying. :-). +1 anyway $\endgroup$ – Peter K. Jun 14 '16 at 18:21

There are linear and nonlinear filters.

Linear ones are naturally linked to standard convolution and frequency interpretation (linearity and Fourier are close concepts, since Fourier diagonalizes convolution). So a convolution filter is a term pretty related to linear filters. However, people often uses them, especially for images, to describe limited-size kernel (e.g. at Adobe), otherwise termed Finite Impulse Response (FIR) filters. Low-pass filters are a sub-species of linear filters, aimed a better preserving low-frequencies. In the Fourier domain, their response is often globally decreasing. But they can ripple (see averaging filters, with sinc-shape in frequency).

Blur filters are often taken in the linear class. Quite often, they have non negative coefficients (low-pass filters can have some negative coefficients, but their sum should be strictly positive).

But there exist nonlinear blurs. And although median better preserve edges, long enough median filters blur images as well, for instance with even-length medians, which take intermediate pixel values.

Median filters are nonlinear. Weighted median filters are a larger class, that somehow combine the properties of averaging and median operators. Although Fourier theory does not apply for them, close linear approximations may help provide a Spectral Design of Weighted Median Filters.


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