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Minimal Lipschitz Extensions for Vector-Valued Functions on Finite Graphs

: Hertrich, J.; Bačák, M.; Neumayer, S.; Steidl, G.


Lellmann, J.:
Scale Space and Variational Methods in Computer Vision. 7th International Conference, SSVM 2019. Proceedings : Hofgeismar, Germany, June 30 - July 4, 2019
Cham: Springer Nature, 2019 (Lecture Notes in Computer Science 11603)
ISBN: 978-3-030-22367-0 (Print)
ISBN: 978-3-030-22368-7 (Online)
International Conference on Scale Space and Variational Methods in Computer Vision (SSVM) <7, 2019, Kassel>
Fraunhofer ITWM ()

This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called lex and L-lex minimal extensions are actually the same and call them minimal Lipschitz extensions. We prove that the minimizers of functionals involving grouped ℓp-norms converge to these extensions as p→∞. Further, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to ∞-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm proposed in [9] for finding the zero of the ∞-Laplacian is given.