Partial Differential Equations of Bivariate Median Filters
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- Welk M. (2015) Partial Differential Equations of Bivariate Median Filters. In: Aujol JF., Nikolova M., Papadakis N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science, vol 9087. Springer, Cham
Multivariate median filters have been proposed as generalisations of the well-established median filter for grey-value images to multi-channel images. As multivariate median, most of the recent approaches use the \(L^1\) median, i.e. the minimiser of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalise to such a filter. However, the famous result by Guichard and Morel about approximation of the mean curvature motion PDE by median filtering does not have a comparably simple counterpart for \(L^1\) multivariate median filtering. We discuss the affine equivariant Oja median as an alternative to \(L^1\) median filtering. We derive the PDE approximated by Oja median filtering in the bivariate case, and demonstrate its validity by a numerical experiment.
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