Multivariate Median Filters and Partial Differenti

Authors: Martin Welk

Publish Date: 2016/03/21

Volume: 56, Issue: 2, Pages: 320-351

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Abstract

Multivariate median filters have been proposed as generalizations of the wellestablished median filter for grayvalue images to multichannel images As multivariate median most of the recent approaches use the L1 median ie the minimizer of an objective function that is the sum of distances to all input points Many properties of univariate median filters generalize 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 L1 multivariate median filtering We discuss the affine equivariant Oja median and the affine equivariant transformation–retransformation L1 median as alternatives to L1 median filtering We analyze multivariate median filters in a spacecontinuous setting including the formulation of a spacecontinuous version of the transformation–retransformation L1 median and derive PDEs approximated by these filters in the cases of bivariate planar images threechannel volume images and threechannel planar images The PDEs for the affine equivariant filters can be interpreted geometrically as combinations of a diffusion and a principalcomponentwise curvature motion contribution with a crosseffect term based on torsions of principal components Numerical experiments are presented which demonstrate the validity of the approximation resultsFor each triangle MAB the negative gradient of its area as function of M is a force vector tfrac12 F MAB where F MAB is perpendicular to AB with a length proportional to the length AB see Fig 10 Assuming that MAB is positively oriented this vector equals v 2v 1u 2+u 1The factor 1 / 4 in front of the integral 53 combines the factor 1 / 2 from the force vector mentioned above with another factor 1 / 2 to compensate that each triangle MAB enters the integral twice once as MAB and once as MBA where the orientation factor cancels by squaring Note that in 31 the integral was stated differently integrating only over the triangles with positive orientationIntegration bounds densities integrals J ku and resulting coefficients G0 omega H0 omega of the expansions 62 63 for omega in alpha 1beta 1delta 1alpha 2beta 2delta 2 J 1u and H0 omega are always zero and therefore omitted

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