---
abstract: 'Having been studied since long by statisticians, multivariate median concepts found their way into the image processing literature in the course of the last decades, being used to construct robust and efficient denoising filters for multivariate images such as colour images but also matrix-valued images. Based on the similarities between image and geometric data as results of the sampling of continuous physical quantities, it can be expected that the understanding of multivariate median filters for images provides a starting point for the development of shape processing techniques. This paper presents an overview of multivariate median concepts relevant for image and shape processing. It focusses on their mathematical principles and discusses important properties especially in the context of image processing.'
author:
- |
    Martin Welk\
    Institute of Biomedical Image Analysis\
    UMIT – Private University for Health Sciences,\
    Biomedical Informatics and Technology\
    Eduard-Walln[ö]{}fer-Zentrum 1, 6060 Hall/Tyrol, Austria\
    [martin.welk@umit.at](martin.welk@umit.at)
date: 'October 31, 2019'
title: Multivariate Medians for Image and Shape Analysis
---

Introduction
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For almost half a century, the median filter has established itself as an efficient tool for filtering of univariate signals and images [@Tukey-Book71]. Despite its simplicity, it has favourable properties as a denoising method. Whereas average filters 