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.
Keywords: 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
Bredies, Kristian 1 ; Vicente, David 1
@article{COCV_2019__25__A83_0,
author = {Bredies, Kristian and Vicente, David},
title = {A perfect reconstruction property for {PDE-constrained} total-variation minimization with application in {Quantitative} {Susceptibility} {Mapping}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2018009},
zbl = {1437.35680},
mrnumber = {4043861},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2018009/}
}
TY - JOUR AU - Bredies, Kristian AU - Vicente, David TI - A perfect reconstruction property for PDE-constrained total-variation minimization with application in Quantitative Susceptibility Mapping JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2018009/ DO - 10.1051/cocv/2018009 LA - en ID - COCV_2019__25__A83_0 ER -
%0 Journal Article %A Bredies, Kristian %A Vicente, David %T A perfect reconstruction property for PDE-constrained total-variation minimization with application in Quantitative Susceptibility Mapping %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2018009/ %R 10.1051/cocv/2018009 %G en %F COCV_2019__25__A83_0
Bredies, Kristian; Vicente, David. A perfect reconstruction property for PDE-constrained total-variation minimization with application in Quantitative Susceptibility Mapping. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 83. doi: 10.1051/cocv/2018009
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