Fundamentals of Non-Local Total Variation Spectral Theory

  • Jean-François Aujol
  • Guy Gilboa
  • Nicolas Papadakis
Conference paper

DOI: 10.1007/978-3-319-18461-6_6

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)
Cite this paper as:
Aujol JF., Gilboa G., Papadakis N. (2015) Fundamentals of Non-Local Total Variation Spectral Theory. In: Aujol JF., Nikolova M., Papadakis N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science, vol 9087. Springer, Cham

Abstract

Eigenvalue analysis based on linear operators has been extensively used in signal and image processing to solve a variety of problems such as segmentation, dimensionality reduction and more. Recently, nonlinear spectral approaches, based on the total variation functional have been proposed. In this context, functions for which the nonlinear eigenvalue problem \( \lambda u \in \partial J(u)\) admits solutions, are studied. When \(u\) is the characteristic function of a set \(A\), then it is called a calibrable set. If \(\lambda >0\) is a solution of the above problem, then \(1/\lambda \) can be interpreted as the scale of \(A\). However, this notion of scale remains local, and it may not be adapted for non-local features. For this we introduce in this paper the definition of non-local scale related to the non-local total variation functional. In particular, we investigate sets that evolve with constant speed under the non-local total variation flow. We prove that non-local calibrable sets have this property. We propose an onion peel construction to build such sets. We eventually confirm our mathematical analysis with some simple numerical experiments.

Keywords

Non-local Total variation Calibrable sets Scale Nonlinear eigenvalue problem 
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jean-François Aujol
    • 1
    • 2
  • Guy Gilboa
    • 3
  • Nicolas Papadakis
    • 1
    • 2
  1. 1.University of Bordeaux, IMB, UMR 5251TalenceFrance
  2. 2.CNRS, IMB, UMR 5251TalenceFrance
  3. 3.Electrical Engineering DepartmentTechnion - IITHaifaIsrael

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