Formulation of an accurate discrete theory of median shifts

Authors:
E. R. Davies
Affiliations:
Machine Vision Group, Department of Physics, Royal Holloway, University of London, Egham Hill, Egham, Surrey, TW20 0EX, UK
Venue:
Signal Processing
Year:
2003

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Abstract

This paper presents an accurate discrete theory of median shifts. It shows that a quadratic law exists for low curvature values, and that this gives way only gradually to the linear law predicted by the earlier continuum model as the curvature increases. As a result, the variation is almost entirely quadratic for 3 × 3 neighbourhoods, but approaches linearity for 7 × 7 and larger neighbourhoods. In all cases the median shifts exhibit substantial angular variations, though the isotropy can be improved by truncating square neighbourhoods to make them more circular. Tests on circular objects show very good agreement between the new discrete theory and the observed results, with minor exceptions that are explainable by simple geometry. Thus the prime aim of this paper--to achieve a full understanding of the detailed mechanisms of the median filter including the way in which it produces shifts--is achieved.