The jump set under geometric regularisation. Part 2: Higher-order approaches

Type
Article

Authors
Valkonen, Tuomo

KAUST Grant Number
KUK-I1-007-43

Date
2017-09

Abstract
In Part 1, we developed a new technique based on Lipschitz pushforwards for proving the jump set containment property Hm−1(Ju∖Jf)=0 of solutions u to total variation denoising. We demonstrated that the technique also applies to Huber-regularised TV. Now, in this Part 2, we extend the technique to higher-order regularisers. We are not quite able to prove the property for total generalised variation (TGV) based on the symmetrised gradient for the second-order term. We show that the property holds under three conditions: First, the solution u is locally bounded. Second, the second-order variable is of locally bounded variation, w∈BVloc(Ω;Rm), instead of just bounded deformation, w∈BD(Ω). Third, w does not jump on Ju parallel to it. The second condition can be achieved for non-symmetric TGV. Both the second and third condition can be achieved if we change the Radon (or L1) norm of the symmetrised gradient Ew into an Lπ norm, p>1, in which case Korn's inequality holds. On the positive side, we verify the jump set containment property for second-order infimal convolution TV (ICTV) in dimension m=2. We also study the limiting behaviour of the singular part of Du, as the second parameter of TGV2 goes to zero. Unsurprisingly, it vanishes, but in numerical discretisations the situation looks quite different. Finally, our work additionally includes a result on TGV-strict approximation in BV(Ω).

Citation
Valkonen, T. (2017). The jump set under geometric regularisation. Part 2: Higher-order approaches. Journal of Mathematical Analysis and Applications, 453(2), 1044–1085. doi:10.1016/j.jmaa.2017.04.037

Acknowledgements
This manuscript has been prepared over the course of several years, exploiting funding from various short-term projects. While the author was at the Institute for Mathematics and Scientific Computing at the University of Graz, this work was financially supported by the SFB research program F32 “Mathematical Optimization and Applications in Biomedical Sciences” of the Austrian Science Fund (FWF). While at the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, this work was financially supported by the King Abdullah University of Science and Technology (KAUST) Award No. KUK-I1-007-43 as well as the EPSRC / Isaac Newton Trust Small Grant “Non-smooth geometric reconstruction for high resolution MRI imaging of fluid transport in bed reactors”, and the EPSRC first grant Nr. EP/J009539/1 “Sparse & Higher-order Image Restoration”. At the Research Center on Mathematical Modeling (Modemat) at the Escuela Politécnica Nacional de Quito, where this work was finalised, it was supported by the Prometeo initiative of the Senescyt.

Publisher
Elsevier BV

Journal
Journal of Mathematical Analysis and Applications

DOI
10.1016/j.jmaa.2017.04.037

arXiv
1407.2334

Additional Links
https://linkinghub.elsevier.com/retrieve/pii/S0022247X17303918

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