to the discrete median filtering process, whereas their actual occurrence depends on the interaction between small-scale image structures and sliding windows, and can be considered as filtering artifacts related to the resolution limit of the discrete image. Worst-case examples are a checkerboard pattern (say, all pixels $(i,j)$ with $i+j$ even are black, and all others white) which is stationary under whatever $(2\ell+1)\times(2\ell+1)$ median filter, and a stripe pattern (say, all pixels $(i,j)$ with $i$ even are black, and all others white) which is inverted in each iteration.

Space-Continuous Median Filtering {#ssec-meddf}
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Whereas practical image processing always has to deal with discrete images, a continuous viewpoint is helpful for the theoretical understanding of image processing. Image acquisition creates discrete images by sampling spatially continuous physical quantities, and involves the choice of a sampling grid, which is, in essence, arbitrary. For image processing methods to be meaningful, they need to be consistent with a continuous reality, and depend as little as possible on sampling.

To support theoretical analysis of median filtering, it is therefore of interest to consider a median filtering process for space-continuous images. Thus, we interpret a (planar) grey-value image as a bounded function from a suitable function space 