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half-space median, the median of uˆ yields the me- dianof theoriginaldataby the inverse transform. As the PDE system of Lemma 2 is identical to that for the Oja median in [14], the calculations from [14, Eqs. (25)–(26)] for the transform step apply verba- tim,and yield theclaimofour proposition. 4.SummaryandOutlook In thiswork,wehavestudied thecontinuous limit of half-space median filtering, one of the possible generalisations of median filtering of grey-value im- ages to multi-channel images, in the bivariate case. We have proven a result already conjectured in [15] stating theapproximationofaparticularPDEbythis filter. The result is embedded in the context of pre- vious work on PDE approximation by multivariate median filters, see [14], and is a step on the way to adeeperunderstandingofmultivariatemedianfilters for signals and images. Aninteresting fact is thatdespitecleardifferences in the practical outcome of the corresponding filters on discrete images (see Figure 1), the affine equiv- ariant Oja median and half-space median filter ap- proximatethesamePDE.This indicates that theycan be seen as different discrete realisations of one un- derlying fundamental multivariate median filter, de- spite the substantial differences in their underlying discrete concepts (see thediscussion in [15]). As mentioned earlier, the focus of our work was in the theoretical domain. Further study of the prac- tical applicability of half-space median filtering is a subject of ongoing work. In particular, algorithmic efficiency issues will require further investigation. Moreover, bivariate images as considered here are a rare exception in practice (with two-dimensional optic flow fields being the most relevant case, see [14]). A much greater role is played by images with three (such as RGB colour images or tensor fields in two dimensions) or even more channels (multispec- tral images, tensor fields in three dimensions). Ex- tension of the theoretical investigation to three and morechannels is thereforeanother importantgoalfor future research. References [1] J. Astola, P. Haavisto, and Y. Neuvo. Vector me- dian filters. Proceedings of the IEEE, 78(4):678– 689,1990. [2] T. L. Austin. An approximation to the point of min- imumaggregate distance. Metron, 19:10–21, 1959. [3] V. Barnett. The ordering of multivariate data. Jour- nal of the Royal Statistical Society A, 139(3):318– 355,1976. [4] C.GiniandL.Galvani. Di taluneestensionideicon- cetti di media ai caratteri qualitativi. Metron, 8:3– 209,1929. [5] F. Guichard and J.-M. Morel. Partial differential equations and image iterative filtering. In I. S. Duff andG.A.Watson,editors,TheStateof theArt inNu- mericalAnalysis,number63inIMAConferenceSe- ries (New Series), pages 525–562. Clarendon Press, Oxford,1997. [6] R.Y.Liu.Onanotionofdatadepthbasedonrandom simplices. The Annals of Statistics, 18(1):405–414, 1990. [7] H. Oja. Descriptive statistics for multivariate distri- butions. Statistics and Probability Letters, 1:327– 332,1983. [8] A. H. Seheult, P. J. Diggle, and D. A. Evans. Dis- cussionofpaperbyV.Barnett. Journalof theRoyal StatisticalSocietyA, 139(3):351–352,1976. [9] C. Spence and C. Fancourt. An iterative method for vectormedianfiltering. InProc.2007IEEEInterna- tional Conference on Image Processing, volume 5, pages 265–268,2007. [10] J. W. Tukey. Exploratory Data Analysis. Addison– Wesley,MenloPark, 1971. [11] J. W. Tukey. Mathematics and the picturing of data. InProc.of the InternationalCongressofMathemat- ics1974, pages523–532,Vancouver,Canada,1975. [12] A. Weber. U¨ber den Standort der Industrien. Mohr, Tu¨bingen,1909. [13] E.Weiszfeld. Sur lepointpour lequel la sommedes distances denpoints donne´s est minimum. Toˆhoku MathematicsJournal, 43:355–386, 1937. [14] M.Welk. Multivariatemedianfiltersandpartialdif- ferential equations. Journal of Mathematical Imag- ing andVision, 56:320–351,2016. [15] M.Welk. Multivariatemedians for imageandshape analysis. Technical Report 1911.00143 [eess.IV], arXiv.org,2019. [16] M. Welk and M. Breuß. The convex-hull-stripping median approximates affine curvature motion. In M. Burger, J. Lellmann, and J. Modersitzki, editors, Scale Space and Variational Methods in Computer Vision, volume11603ofLectureNotes inComputer Science, pages 199–210.Springer,Cham,2019. [17] M. Welk, C. Feddern, B. Burgeth, and J. Weick- ert. Median filtering of tensor-valued images. In B. Michaelis and G. Krell, editors, Pattern Recog- nition, volume 2781 of Lecture Notes in Computer Science, pages 17–24.Springer,Berlin, 2003. 156