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.
Mots-clés : 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
@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},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018009},
zbl = {1437.35680},
mrnumber = {4043861},
language = {en},
url = {http://archive.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 - http://archive.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 http://archive.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. http://archive.numdam.org/articles/10.1051/cocv/2018009/
[1] and , A generalized sampling theorem for stable reconstructions in arbitrary bases. J. Fourier Anal. Appl. 18 (2012) 685–716. | DOI | MR | Zbl
[2] and , Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon. Appl. Comput. Harmon. Anal. 32 (2012) 357–388. | DOI | MR | Zbl
[3] , and , On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate. Appl. Comput. Harmon. Anal. 36 (2014) 387–415. | DOI | MR | Zbl
[4] , and , Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications (2000). | DOI | MR | Zbl
[5] , and , Primal-dual strategy for optimal control problems. SIAM J. Control Optim. 37 (1999) 1176–1194. | DOI | MR | Zbl
[6] , and , Real Algebraic geometry. Springer (1998). | DOI | MR | Zbl
[7] , Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty. Lecture Notes Comput. Sci. 8293 (2014) 44–77. | DOI
[8] and , Inverse problems with second-order total generalized variation constraints. In Proceedings of SampTA 2011 – 9th International Conference on Sampling Theory and Applications. Singapore (2011).
[9] and , Regularization of linear inverse problems with Total Generalized Variation. J. Inverse Ill-posed Probl. 22 (2014) 871–913. | DOI | MR | Zbl
[10] , and , Total Generalized Variation. SIAM J. Imag. Sci. 3 (2010) 492–526. | DOI | MR | Zbl
[11] , and , TGV-based single-step QSM reconstruction with coil combination. In 4th International Workshop on MRI Phase Constrast and Quantitative Susceptibility Mapping. Medical University of Graz (2016).
[12] and , Towards a mathematical theory of super-resolution. Commun. Pure Appl. Math. 67 (2014) 906–956. | DOI | MR | Zbl
[13] , and , Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52 (2006) 489–509. | DOI | MR | Zbl
[14] , and , The discontinuity set of solutions of the TV denoising problem and some extensions. Multiscale Modeling Simul. 6 (2007) 879–894. | DOI | MR | Zbl
[15] and , A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imag. Vision 40 (2011) 120–145. | DOI | MR | Zbl
[16] , , and , Geometric properties of solutions to the total variation denoising problem. Inverse Probl. 33 (2016) 015002. | DOI | MR | Zbl
[17] , , , , , , , , , and , Single-step quantitative susceptibility mapping with variational penalties. NMR Biomed. (2016).
[18] , , and , Inverse problem in quantitative susceptibility mapping. SIAM J. Imag. Sci. 7 (2014) 1669–1689. | DOI | MR | Zbl
[19] , , and , Quantitative MR susceptibility mapping using piecewise constant regularized inversion of the magnetic field. Magn. Res. Med. 60 (2008) 1003–1009. | DOI
[20] and , On the structure of 𝒜-free measures and applications. Ann. Math. 184 (2016) 1017–1039. | DOI | MR | Zbl
[21] and , Exact support recovery for sparse spikes deconvolution. Found. Comput. Math. 15 (2015) 1315–1355. | DOI | MR | Zbl
[22] and , Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (1992). | MR | Zbl
[23] and , The split Bregman method for L1-regularized problems. SIAM J. Imag. Sci. 2 (2009) 323–343. | DOI | MR | Zbl
[24] , , and , Magnetic Resonance Imaging: Physical Principles and Sequence Design. John Wiley and Sons, New York (1999).
[25] , , , , , , , , and , Imaging iron stores in the brain using magnetic resonance imaging. Magn. Res. Imag. 23 (2005) 1–25. | DOI
[26] , and , The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2006) 865–888. | DOI | MR | Zbl
[27] , The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, 2nd edition. Springer Verlag, Berlin (1990). | MR | Zbl
[28] , , , , , , , , and , Fast Quantitative Susceptibility Mapping using 3D EPI and Total Generalized Variation. NeuroImage 111 (2015) 622–630. | DOI
[29] , and , TVAL3: TV minimization by Augmented Lagrangian and ALternating direction ALgorithms. Available at: http://www.caam.rice.edu/~optimization/L1/TVAL3/ (2019).
[30] , and , Optimal control problems of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33 (2006) 209–228. | DOI | MR | Zbl
[31] , Lie groups: an introduction through linear groups. Oxford University Press (2002). | DOI | MR | Zbl
[32] , and , Nonlinear total variation based noise removal algorithms. Physica D 60 (1992) 259–268. | DOI | MR | Zbl
[33] , , , and , An illustrated comparison of processing methods for phase MRI and QSM: removal of background field contributions from sources outside the region of interest. NMR Biomed. 30 (2017). | DOI
[34] , Optimal control of partial differential equations. Americal Mathematical Society (2010). | MR | Zbl
[35] and , Quantitative Susceptibility Mapping (QSM): Decoding MRI data for a tissue magnetic biomarker. Magn. Res. Med. 73 (2015) 82–101. | DOI
[36] , , and , YALL1: Your ALgorithms for L1. Available at: http://yall1.blogs.rice.edu/ (2019).
Cité par Sources :
