Vector Median Filters, Inf-Sup Operations, and Coupled PDE's: Theoretical Connections

Authors:
Vicent Caselles;Guillermo Sapiro;Do Hyun Chung
Affiliations:
Department of Informatics and Mathematics, University of the Illes Balears, 07071 Palma de Mallorca, Spain;Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA. guille@ece.umn.edu;Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Venue:
Journal of Mathematical Imaging and Vision
Year:
2000

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Abstract

In this paper, we formally connect between vectormedian filters, inf-sup morphological operations, and geometricpartial differential equations. Considering a lexicographic order,which permits to define an order between vectorsin R^N, we first show that the vector median filterof a vector-valued image is equivalent to a collection ofinfimum-supremum morphological operations. We then proceed and studythe asymptotic behavior of this filter. We also provide aninterpretation of the infinitesimal iteration of this vectorialmedian filter in terms of systems of coupled geometric partialdifferential equations. The main component of the vector evolvesaccording to curvature motion, while, intuitively, the othersregularly deform their level-sets toward those of this maincomponent. These results extend to the vector case classicalconnections between scalar median filters, mathematical morphology,and mean curvature motion.