Theoretical Analysis of Flows Estimating Eigenfunctions of One-homogeneous Functionals

Abstract : Nonlinear eigenfunctions, induced by subgradients of one-homogeneous functionals (such as the 1-Laplacian), have shown to be instrumental in segmentation, clustering and image decomposition. We present a class of flows for finding such eigenfunctions, generalizing a method recently suggested by Nossek and Gilboa. We analyze the flows on grids and graphs in the time-continuous and time-discrete settings. For a specific type of flow within this class, we prove convergence of the numerical iterations procedure and prove existence and uniqueness of the time-continuous case. Several toy examples are provided for illustrating the theoretical results, showing how such flows can be used on images and graphs.
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Jean-François Aujol, Guy Gilboa, Nicolas Papadakis. Theoretical Analysis of Flows Estimating Eigenfunctions of One-homogeneous Functionals . SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2018, 11 (2), pp.1416-1440. ⟨hal-01563922⟩

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