| Issue |
ESAIM: COCV
Volume 25, 2019
|
|
|---|---|---|
| Article Number | 83 | |
| Number of page(s) | 19 | |
| DOI | https://doi.org/10.1051/cocv/2018009 | |
| Published online | 19 December 2019 | |
A perfect reconstruction property for PDE-constrained total-variation minimization with application in Quantitative Susceptibility Mapping*
Institute of Mathematics and Scientific Computing, University of Graz,
Heinrichstraße 36,
8010
Graz, Austria.
** Corresponding author: david.vicente154@gmail.com
Received:
23
January
2017
Accepted:
28
January
2018
We study the recovery of piecewise constant functions of finite bounded variation (BV) from their image under a linear partial differential operator with unknown boundary conditions. It is shown that minimizing the total variation (TV) semi-norm subject to the associated PDE-constraints yields perfect reconstruction up to a global constant under a mild geometric assumption on the jump set of the function to reconstruct. The proof bases on establishing a structural result about the jump set associated with BV-solutions of the homogeneous PDE. Furthermore, we show that the geometric assumption is satisfied up to a negligible set of orthonormal transformations. The results are then applied to Quantitative Susceptibility Mapping (QSM) which can be formulated as solving a two-dimensional wave equation with unknown boundary conditions. This yields in particular that total variation regularization is able to reconstruct piecewise constant susceptibility distributions, explaining the high-quality results obtained with TV-based techniques for QSM.
Mathematics Subject Classification: 35Q93 / 49Q20 / 35L67 / 92C55
Key words: Optimization with partial differential equations / total-variation minimization / perfect reconstruction property / piecewise constant functions of bounded variation / jump sets of BV-solutions / Quantitative Susceptibility Mapping
The Institute of Mathematics and Scientific Computing is a member of NAWI Graz (www.nawigraz.at) and BioTechMed Graz (www.biotechmedgraz.at).
© EDP Sciences, SMAI 2019
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