TY - GEN
T1 - Fundamentals of non-local total variation spectral theory
AU - Aujol, Jean François
AU - Gilboa, Guy
AU - Papadakis, Nicolas
N1 - Publisher Copyright: © Springer International Publishing Switzerland 2015.
PY - 2015
Y1 - 2015
N2 - 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 λu ∈ ∂J(u) admits solutions, are studied. When u is the characteristic function of a set A, then it is called a calibrable set. If λ > 0 is a solution of the above problem, then 1/λ 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.
AB - 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 λu ∈ ∂J(u) admits solutions, are studied. When u is the characteristic function of a set A, then it is called a calibrable set. If λ > 0 is a solution of the above problem, then 1/λ 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.
KW - Calibrable sets
KW - Non-local
KW - Nonlinear eigenvalue problem
KW - Scale
KW - Total variation
UR - http://www.scopus.com/inward/record.url?scp=84931090402&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-18461-6_6
DO - 10.1007/978-3-319-18461-6_6
M3 - منشور من مؤتمر
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 66
EP - 77
BT - Scale Space and Variational Methods in Computer Vision - 5th International Conference, SSVM 2015, Proceedings
A2 - Nikolova, Mila
A2 - Aujol, Jean-François
A2 - Papadakis, Nicolas
T2 - 5th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2015
Y2 - 31 May 2015 through 4 June 2015
ER -