Page - 152 - in Joint Austrian Computer Vision and Robotics Workshop 2020

Figure1.Fromleft to right: Synthetic test image(30×30pixels) inorange–bluecolourspace. –Componentwisemedian filtering. –L1 median filtering. – Oja median filtering. – Half-space median filtering. For all median filters the sliding windowwasadiscretediscof radius √ 5, andone iteration wasapplied. of multivariate data (such as points in the plane or space) is not restricted to be one of the input data. TheL1median [12]was thefirstconceptof thiskind discussed in the statistical literature [2, 4, 13] and also in image processing [9, 17]. Shortcomings of this concept, especially its lack of affine equivari- ancewhichcontrasts to theverygeneralmonotonous equivariance of the classical univariate median, led statisticians to alternative concepts such as Oja me- dian [7], half-space median [6, 11] and convex-hull- strippingmedian [3,8]. All of these multivariate medians are defined in the first place as discrete concepts: Given a set of pointsx1, .. . ,xm inRn, they yield a medianµ ∈ Rn. Algorithmically, their application to multivari- ate images is straightforward; however, the validity of such a procedure again depends on the question whether it approximates a suitable filter for space- continuous images. Furthermore, the question arises whetheraPDEcanbestated that is approximatedby such a space-continuous multivariate median filter. For theL1 median and Oja median, these questions havebeenanswered in [14]: Thedefinitionof space- continuous variants of these filters is more or less straightforward, and PDE limits could be stated for images with values inR2 andR3. For the half-space median,aspace-continuouscounterparthasbeende- scribed in [15] but the PDE limit (inR2) was stated onlyasaconjecture,withoutproof. Wemention that for theconvex-hull-strippingmedianstatingaspace- continuous filtering procedure is already a difficult task in itself, see [16]. Our contribution. The purpose of this work is to advance the theoretical understanding of half- space median filtering as a multivariate image fil- ter. We will derive the PDE approximated by space- continuous half-space median filtering of bivariate images, therebyprovingtheconjecturestatedin[15]. Aspects of practical application are not in the fore- groundat thepresent stageof research;examplesare presented just for illustrating the properties of multi- variatemedianfilters, andare restricted to thebivari- atecase(notwithstandingthegreaterpractical impor- tanceof three-channel colour images). Structure of the paper. After shortly demonstrat- ing the effect of multivariate median filters, we will recall thedefinitionof thehalf-spacemedian fordis- crete data and its space-continuous analogue in Sec- tion 2. In Section 3 we will prove the PDE approx- imation result as conjectured in [15]. A short sum- maryandoutlook inSection4concludes thepaper. 2.MultivariateMedianFiltering The univariate median filter excels as an edge- preserving denoising filter for images that can deal well with types of noise such as impulse noise. Un- fortunately, for multi-channel images a straightfor- wardgeneralisationbyusingthemedianjust foreach channelseparatelydoesnot leadtoreasonableresults as we demonstrate by a small synthetic example in Figure 1. For simplicity, and since our theoretical work presented in the next section is currently re- strictedtothebivariatecase,weuseatest imagewith just two colour channels (yellow and blue) which is degraded by pepper noise (impulsive noise con- sisting of black noise pixels). Whereas componen- twise median filtering removes noise pixels in ho- mogeneous colour regions, it even amplifies noise near colour edges. A more plausible filtering re- sult is achieved by multivariate median filters three of which are demonstrated in the figure: theL1me- dian filter (see e.g. [9]), the Oja median filter (see e.g. [14]) and the half-space median filter which is in the focus of the present paper. As can be seen, the multivariate median filters lead to some interpo- lation between the two colours near edges but don’t 152