Abstract
This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called \(\mathrm {lex}\) and \(L\text {-}\mathrm {lex}\) minimal extensions are actually the same and call them minimal Lipschitz extensions. We prove that the minimizers of functionals involving grouped \(\ell _p\)-norms converge to these extensions as \(p\rightarrow \infty \). Further, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to \(\infty \)-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm proposed in [9] for finding the zero of the \(\infty \)-Laplacian is given.
Funding by the DFG within the Research Training Group 1932, and within project STE 571/13-1 is gratefully acknowledged.
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Hertrich, J., Bačák, M., Neumayer, S., Steidl, G. (2019). Minimal Lipschitz Extensions for Vector-Valued Functions on Finite Graphs. In: Lellmann, J., Burger, M., Modersitzki, J. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2019. Lecture Notes in Computer Science(), vol 11603. Springer, Cham. https://doi.org/10.1007/978-3-030-22368-7_15
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