Seite - 151 - in Joint Austrian Computer Vision and Robotics Workshop 2020

Asymptotic AnalysisofBivariateHalf-Space MedianFiltering MartinWelk UMIT–PrivateUniversity forHealthSciences,Medical InformaticsandTechnology Hall/Tyrol,Austria martin.welk@umit.at Abstract. Median filtering is well established in signal and image processing as an efficient and ro- bustdenoisingfilterwith favourableedge-preserving properties, and capable of denoising some types of heavy-tailed noise such as impulse noise. For multi- channel images such as colour images, flow fields or diffusion tensor fields, multivariate median filters have been considered in the literature. Whereas the L1median filter so far dominates in image process- ing applications, other multivariate concepts from statistics may be used such as the half-space median which in the focusof thiswork. In the understanding of discrete image filters a central question is always how these relate to the space-continuous physical reality underlying dis- crete images. Fortheunivariatemedianfilter,amile- stone inanswering thisquestion isanasymptoticap- proximation result that links median filtering to the meancurvaturemotionevolution. Wewillpresentan analogous result for half-space median filtering in the bivariate (two-channel) case, which contributes to the theoretical understanding of multivariate me- dian filtering and provides the basis for further gen- eralisations in futurework. 1. Introduction Median filtering [10] is a well-established proce- dure in signal and image processing. For grey-value images it is known as an efficient and robust denois- ing method with favourable edge-preserving proper- ties. In standard median filtering, a pixel mask (for example, a (2m+ 1)× (2m+ 1) square, or a dis- crete approximation of a disc) is moved as a sliding windowacross the image. Ateachpixel location, the mask is used to select grey-values of the input im- age; themedianof thesegrey-values is thenassigned tothecentralpixelas itsnewgrey-valuein theoutput image. This filter can also be iterated, which is then called iteratedmedianfiltering. Continuousmedianfiltering. Thus,medianfilter- ing is designed in the first place as a discrete pro- cedure. An important question regarding its valid- ity for images is therefore whether it is in a sound relationship to the underlying continuous nature of images. This is indeed thecase: Firstly, it is straight- forward to conceive mathematically a median filter for space-continuous images: Given an image as a function over a planar domain, one can cut out a neighbourhood around each location in the plane (say, a square or disc centered at the reference point) and determine the median of the (continuous) distri- bution of image values within this neighbourhood. Discrete median filtering of a sampled image ap- proximates this concept. Secondly, assuming disc- shaped neighbourhoods (of radius %) in this pro- cess, it has been proven in [5] that iterated space- continuous median filtering approximates a partial differential equation (PDE) as %→ 0 in the sense that one space-continuous median filter step asymp- totically approximates a time step of size%2/6 of an explicit timediscretisationofthemeancurvaturemo- tion PDEut = |∇u|div (∇u/|∇u|) for the planar imageuevolving in time. Multivariate medians. Due to the success of me- dian filtering for grey-value images, researchers have proposed generalisations of the median filter to multi-channel images (such as colour images, op- tic flow fields, diffusion tensor fields). After early attempts such as the vector median filters from [1] whichfocussedonmethods toselectonevector from a given set of input vectors as its median, attention turned soon to multivariate median concepts known from the statistical literature in which the median 151