Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-13T00:46:14.841Z Has data issue: false hasContentIssue false
  • English
  • Français

Modern regularization methods for inverse problems

Published online by Cambridge University Press:  04 May 2018

Martin Benning  and
Martin Burger
Martin Benning
Affiliation:
DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK E-mail: mb941@cam.ac.uk
Martin Burger
Affiliation:
Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, D-48149 Münster, Germany E-mail: martin.burger@wwu.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research.

In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.

Type
Research Article
Information
Acta Numerica , Volume 27 , 01 May 2018 , pp. 1 - 111
Copyright
© Cambridge University Press, 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES 2

Acar, R. and Vogel, C. R. (1994), ‘Analysis of bounded variation penalty methods for ill-posed problems’, Inverse Problems 10, 1217.CrossRefGoogle Scholar
Agapiou, S., Burger, M., Dashti, M. and Helin, T. (2018), ‘Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems’, Inverse Problems 34, 0450002.CrossRefGoogle Scholar
Aja-Fernandez, S., Alberola-Lopez, C. and Westin, C. F. (2008), ‘Noise and signal estimation in magnitude MRI and Rician distributed images: A LMMSE approach’, IEEE Trans. Image Process. 17, 1383–1398.Google Scholar
Allard, W. K. (2007), ‘Total variation regularization for image denoising, I: Geometric theory’, SIAM J. Math. Anal. 39, 1150–1190.Google Scholar
Ambrosio, L. and Tortorelli, V. M. (1990), ‘Approximation of functional depending on jumps by elliptic functional via t-convergence’, Commun. Pure Appl. Math. 43, 999–1036.Google Scholar
Ambrosio, L., Fusco, N. and Pallara, D. (2000), Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon Press, Oxford University Press.Google Scholar
Anderssen, R. (1986), The linear functional strategy for improperly posed problems. In Inverse Problems (Cannon, J. R. and Hornung, U., eds), Springer, pp. 11–30.Google Scholar
Attouch, H. and Bolte, J. (2009), ‘On the convergence of the proximal algorithm for nonsmooth functions involving analytic features’, Math. Program. 116, 5–16.Google Scholar
Attouch, H., Bolte, J. and Svaiter, B. F. (2013), ‘Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods’, Math. Program. 137, 91–129.Google Scholar
Attouch, H., Bolte, J., Redont, P. and Soubeyran, A. (2010), ‘Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka–Łojasiewicz inequality’, Math. Oper. Res. 35, 438–457.CrossRefGoogle Scholar
Aubert, G. and Aujol, J.-F. (2008), ‘A variational approach to removing multiplicative noise’, SIAM J. Appl. Math. 68, 925–946.CrossRefGoogle Scholar
Bachmayr, M. and Burger, M. (2009), ‘Iterative total variation schemes for nonlinear inverse problems’, Inverse Problems 25, 105004.Google Scholar
Backus, G. and Gilbert, F. (1968), ‘The resolving power of gross earth data’, Geophys. J. Internat. 16, 169–205.Google Scholar
Bakushinskii, A. B. (1967), ‘A general method of constructing regularizing algorithms for a linear incorrect equation in Hilbert space’, Zh. Vychisl. Mat. Mat. Fiz. 7, 672–677.Google Scholar
Bakushinskii, A. B. (1973), ‘On the proof of the “discrepancy principle”’, Differential and Integral Equations (Differents. i integr. un-niya), Izd-vo IGU, Irkutsk.Google Scholar
Bakushinskii, A. B. (1977), ‘Methods for solving monotonic variational inequalities, based on the principle of iterative regularization’, USSR Comput. Math. Math. Phys. 17, 12–24.CrossRefGoogle Scholar
Bakushinskii, A. B. (1979), ‘On the principle of iterative regularization’, USSR Comput. Math. Math. Phys. 19, 256–260.Google Scholar
Bakushinskii, A. B. (1984), ‘Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion’, USSR Comput. Math. Math. Phys. 24, 181–182.CrossRefGoogle Scholar
Banks, H. and Kunisch, K. (1989), Estimation Techniques for Distributed Parameter Systems, Birkhäuser.Google Scholar
Bates, D. M. and Wahba, G. (1983), A truncated singular value decomposition and other methods for generalized cross-validation. Technical report 715, Department of Statistics, University of Wisconsin.Google Scholar
Bauer, F., Hohage, T. and Munk, A. (2009), ‘Iteratively regularized Gauss–Newton method for nonlinear inverse problems with random noise’, SIAM J. Numer. Anal. 47, 1827–1846.CrossRefGoogle Scholar
Bauschke, H. H., Bolte, J. and Teboulle, M. (2016), ‘A descent lemma beyond Lipschitz gradient continuity: First-order methods revisited and applications’, Math. Oper. Res. 42, 330–348.Google Scholar
Bauschke, H. H., Borwein, J. M. and Combettes, P. L. (2001), ‘Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces’, Commun. Contemp. Math. 3, 615–647.CrossRefGoogle Scholar
Beck, A. and Teboulle, M. (2003), ‘Mirror descent and nonlinear projected subgradient methods for convex optimization’, Oper. Res. Lett. 31, 167–175.CrossRefGoogle Scholar
Benning, M. (2011), Singular regularization of inverse problems: Bregman distances and their applications to variational frameworks with singular regularization energies. PhD thesis, Westfälische Wilhelms-Universität Münster, Germany.Google Scholar
Benning, M. and Burger, M. (2013), ‘Ground states and singular vectors of convex variational regularization methods’, Methods Appl. Anal. 20, 295–334.CrossRefGoogle Scholar
Benning, M., Betcke, M. M., Ehrhardt, M. J. and Schönlieb, C.-B. (2017a), Choose your path wisely: Gradient descent in a Bregman distance framework. arXiv:1712.04045 Google Scholar
Benning, M., Betcke, M. M., Ehrhardt, M. J. and Schönlieb, C.-B. (2017b), Gradient descent in a generalised Bregman distance framework. In Geometric Numerical Integration and its Applications, (Quispel, G. R. W. et al. , eds), Vol. 74 of MI Lecture Notes series of Kyushu University, pp. 40–45.Google Scholar
Benning, M., Brune, C., Burger, M. and Müller, J. (2013), ‘Higher-order TV methods: Enhancement via Bregman iteration’, J. Sci. Comput. 54, 269–310.CrossRefGoogle Scholar
Benning, M., Gilboa, G. and Schönlieb, C.-B. (2016), ‘Learning parametrised regularisation functions via quotient minimisation’, Proc. Appl. Math. Mech. 16, 933–936.CrossRefGoogle Scholar
Benning, M., Gilboa, G., Grah, J. S. and Schönlieb, C.-B. (2017c), Learning filter functions in regularisers by minimising quotients. In SSVM 2017: Scale Space and Variational Methods in Computer Vision (Lauze, F. et al. , eds), Springer, pp. 511–523.Google Scholar
Benning, M., Gladden, L., Holland, D., Schönlieb, C.-B. and Valkonen, T. (2014), ‘Phase reconstruction from velocity-encoded MRI measurements: A survey of sparsity-promoting variational approaches’, J. Magnetic Resonance 238, 26–43.Google Scholar
Benning, M., Knoll, F., Schönlieb, C.-B. and Valkonen, T. (2015), Preconditioned ADMM with nonlinear operator constraint. In System Modeling and Optimization (Bociu, L. et al. , eds), Springer, pp. 117–126.Google Scholar
Benning, M., Möller, M., Nossek, R. Z., Burger, M., Cremers, D., Gilboa, G. and Schönlieb, C.-B. (2017d), Nonlinear spectral image fusion. In SSVM 2017: Scale Space and Variational Methods in Computer Vision (Lauze, F. et al. , eds), Springer, pp. 41–53.Google Scholar
Bergounioux, M. (2016), ‘Mathematical analysis of a inf-convolution model for image processing’, J. Optim. Theory Appl. 168, 1–21.Google Scholar
Bergounioux, M. and Papoutsellis, E. (2018), ‘An anisotropic inf-convolution BV type model for dynamic reconstruction’, SIAM J. Imaging Sci. 11, 129–163.Google Scholar
Bertero, M. and Boccacci, P. (1998), Introduction to Inverse Problems in Imaging, CRC press.Google Scholar
Bertsekas, D. P. (2011), Incremental gradient, subgradient, and proximal methods for convex optimization: A survey. In Optimization for Machine Learning (Sra, S. et al. , eds), MIT Press, pp. 85–120.Google Scholar
Biegler, L., Biros, G., Ghattas, O., Heinkenschloss, M., Keyes, D., Mallick, B., Tenorio, L., van Bloemen Waanders, B., Willcox, K. and Marzouk, Y. (2011), Large-Scale Inverse Problems and Quantification of Uncertainty, Wiley.Google Scholar
Bissantz, N., Hohage, T. and Munk, A. (2004), ‘Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise’, Inverse Problems 20, 1773.Google Scholar
Bissantz, N., Hohage, T., Munk, A. and Ruymgaart, F. (2007), ‘Convergence rates of general regularization methods for statistical inverse problems and applications’, SIAM J. Numer. Anal. 45, 2610–2636.Google Scholar
Bleyer, I. and Leitao, A. (2009), ‘On Tikhonov functionals penalized by Bregman distances’, CUBO 11, 99–115.Google Scholar
Blomgren, P. and Chan, T. F. (1998), ‘Color TV: Total variation methods for restoration of vector-valued images’, IEEE Trans. Image Process. 7, 304–309.Google Scholar
Bolte, J., Daniilidis, A. and Lewis, A. (2007), ‘The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems’, SIAM J. Optim. 17, 1205–1223.CrossRefGoogle Scholar
Bolte, J., Daniilidis, A., Ley, O. and Mazet, L. (2010), ‘Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity’, Trans. Amer. Math. Soc. 362, 3319–3363.Google Scholar
Bolte, J., Sabach, S. and Teboulle, M. (2014), ‘Proximal alternating linearized minimization for nonconvex and nonsmooth problems’, Math. Program. 146, 459–494.Google Scholar
Bolte, J., Sabach, S., Teboulle, M. and Vaisbourd, Y. (2017), First order methods beyond convexity and Lipschitz gradient continuity with applications to quadratic inverse problems. arXiv:1706.06461 Google Scholar
Bonettini, S., Loris, I., Porta, F. and Prato, M. (2016), ‘Variable metric inexact line-search based methods for nonsmooth optimization’, SIAM J. Optim. 26, 891–921.Google Scholar
Bonettini, S., Loris, I., Porta, F., Prato, M. and Rebegoldi, S. (2017), ‘On the convergence of a linesearch based proximal-gradient method for nonconvex optimization’, Inverse Problems 33, 055005.Google Scholar
Boţ, R. I. and Csetnek, E. R. (2017), ‘Proximal-gradient algorithms for fractional programming’, Optimization 66, 1383–1396.CrossRefGoogle ScholarPubMed
Bredies, K. and Holler, M. (2014), ‘Regularization of linear inverse problems with total generalized variation’, J. Inverse Ill-Posed Probl. 22, 871–913.Google Scholar
Bredies, K. and Holler, M. (2015a), ‘A TGV-based framework for variational image decompression, zooming and reconstruction, I: Analytics’, SIAM J. Imaging Sci. 8, 2814–2850.Google Scholar
Bredies, K. and Holler, M. (2015b), ‘A TGV-based framework for variational image decompression, zooming, and reconstruction, II: Numerics’, SIAM J. Imaging Sci. 8, 2851–2886.Google Scholar
Bredies, K. and Pikkarainen, H. K. (2013), ‘Inverse problems in spaces of measures’, ESAIM Control Optim. Calc. Var. 19, 190–218.Google Scholar
Bredies, K. and Valkonen, T. (2011), Inverse problems with second-order total generalized variation constraints. In Proceedings of SampTA 2011: 9th International Conference on Sampling Theory and Applications, Singapore.Google Scholar
Bredies, K., Kunisch, K. and Pock, T. (2010), ‘Total generalized variation’, SIAM J. Imaging Sci. 3, 492–526.Google Scholar
Bregman, L. (1967), ‘The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming’, USSR Comp. Math. Math. Phys. 7, 200–217.Google Scholar
Bresson, X. and Chan, T. F. (2008), ‘Fast dual minimization of the vectorial total variation norm and applications to color image processing’, Inverse Probl. Imaging 2, 455–484.CrossRefGoogle Scholar
Bresson, X., Laurent, T., Uminsky, D. and Brecht, J. V. (2012), Convergence and energy landscape for Cheeger cut clustering. In NIPS 2012: Advances in Neural Information Processing Systems 25 (Pereira, F. et al. , eds), Curran Associates, pp. 1385–1393.Google Scholar
Brinkmann, E.-M., Burger, M., Rasch, J. and Sutour, C. (2017), ‘Bias reduction in variational regularization’, J. Math. Imaging Vision 59, 534–566.Google Scholar
Brune, C., Sawatzky, A. and Burger, M. (2009), Bregman-EM-TV methods with application to optical nanoscopy. In SSVM 2009: Scale Space and Variational Methods in Computer Vision, (Tai, X.-C. et al. , eds), Vol. 5567 of Lecture Notes in Computer Science, Springer, pp. 235–246.CrossRefGoogle Scholar
Brune, C., Sawatzky, A. and Burger, M. (2009c), Primal and dual Bregman methods with application to optical nanoscopy. CAM Report 09-47, UCLA.Google Scholar
Brune, C., Sawatzky, A. and Burger, M. (2011), ‘Primal and dual Bregman methods with application to optical nanoscopy’, Int. J. Comput. Vis. 92, 211–229.Google Scholar
Bui-Thanh, T., Willcox, K. and Ghattas, O. (2008), ‘Model reduction for large-scale systems with high-dimensional parametric input space’, SIAM J. Sci. Comput. 30, 3270–3288.Google Scholar
Bungert, L., Coomes, D. A., Ehrhardt, M. J., Rasch, J., Reisenhofer, R. and Schönlieb, C.-B. (2018), ‘Blind image fusion for hyperspectral imaging with the directional total variation’, Inverse Problems 34, 044003.CrossRefGoogle Scholar
Burger, M. (2016), Bregman distances in inverse problems and partial differential equations. In Advances in Mathematical Modeling, Optimization and Optimal Control (Hiriart-Urruty, J.-B. et al. , eds), Springer, pp. 3–33.CrossRefGoogle Scholar
Burger, M. and Osher, S. (2004), ‘Convergence rates of convex variational regularization’, Inverse Problems 20, 1411.Google Scholar
Burger, M. and Osher, S. (2013), A guide to the TV zoo. In Level Set and PDE Based Reconstruction Methods in Imaging (Burger, M. et al. , eds), Springer, pp. 1–70.Google Scholar
Burger, M., Eckardt, L., Gilboa, G. and Moeller, M. (2015a), Spectral representations of one-homogeneous functionals. In SSVM 2015: Scale Space and Variational Methods in Computer Vision (Aujol, J.-F. et al. , eds), Springer, pp. 16–27.CrossRefGoogle Scholar
Burger, M., Flemming, J. and Hofmann, B. (2013a), ‘Convergence rates in $\ell ^{1}$ -regularization if the sparsity assumption fails’, Inverse Problems 29, 025013.Google Scholar
Burger, M., Frick, K., Osher, S. and Scherzer, O. (2007a), ‘Inverse total variation flow’, Multiscale Model. Simul. 6, 366–395.Google Scholar
Burger, M., Gilboa, G., Moeller, M., Eckardt, L. and Cremers, D. (2016a), ‘Spectral decompositions using one-homogeneous functionals’, SIAM J. Imaging Sci. 9, 1374–1408.Google Scholar
Burger, M., Gilboa, G., Osher, S. and Xu, J. et al. (2006), ‘Nonlinear inverse scale space methods’, Commun. Math. Sci. 4, 179–212.Google Scholar
Burger, M., Helin, T. and Kekkonen, H. (2016b), Large noise in variational regularization. arXiv:1602.00520 Google Scholar
Burger, M., Modersitzki, J. and Ruthotto, L. (2013b), ‘A hyperelastic regularization energy for image registration’, SIAM J. Sci. Comput. 35, B132–B148.Google Scholar
Burger, M., Moeller, M., Benning, M. and Osher, S. (2013c), ‘An adaptive inverse scale space method for compressed sensing’, 82, 269–299.Google Scholar
Burger, M., Müller, J., Papoutsellis, E. and Schönlieb, C.-B. (2014), ‘Total variation regularization in measurement and image space for PET reconstruction’, Inverse Problems 30, 105003.Google Scholar
Burger, M., Osher, S., Xu, J. and Gilboa, G. (2005), Nonlinear inverse scale space methods for image restoration. In VLSM 2005: Variational, Geometric, and Level Set Methods in Computer Vision (Paragios, N. et al. , eds), Springer, pp. 25–36.Google Scholar
Burger, M., Papafitsoros, K., Papoutsellis, E. and Schönlieb, C.-B. (2015b), System Modeling and Optimization, (Bociu, L. et al. , eds), Springer, pp. 169–179.Google Scholar
Burger, M., Papafitsoros, K., Papoutsellis, E. and Schönlieb, C.-B. (2016c), ‘Infimal convolution regularisation functionals of BV and $L^{p}$ spaces, I: The finite $p$ case’, J. Math. Imaging Vision 55, 343–369.Google Scholar
Burger, M., Resmerita, E. and He, L. (2007b), ‘Error estimation for Bregman iterations and inverse scale space methods in image restoration’, Computing 81, 109–135.Google Scholar
Cai, J.-F. and Osher, S. (2013), ‘Fast singular value thresholding without singular value decomposition’, Methods Appl. Anal. 20, 335–352.Google Scholar
Cai, J.-F., Candès, E. J. and Shen, Z. (2010), ‘A singular value thresholding algorithm for matrix completion’, SIAM J. Optim. 20, 1956–1982.Google Scholar
Cai, J.-F., Osher, S. and Shen, Z. (2009a), ‘Convergence of the linearized Bregman iteration for $\ell _{1}$ -norm minimization’, Math. Comp. 78, 2127–2136.Google Scholar
Cai, J.-F., Osher, S. and Shen, Z. (2009b), ‘Linearized Bregman iterations for compressed sensing’, Math. Comp. 78, 1515–1536.Google Scholar
Cakoni, F. and Colton, D. (2005), ‘Open problems in the qualitative approach to inverse electromagnetic scattering theory’, European J. Appl. Math. 16, 411–425.Google Scholar
Calatroni, L., De los Reyes, J. C. and Schönlieb, C.-B. (2013), Dynamic sampling schemes for optimal noise learning under multiple nonsmooth constraints. In System Modeling and Optimization (Pötzsche, C. et al. , eds), Springer, pp. 85–95.Google Scholar
Calatroni, L., De los Reyes, J. C. and Schönlieb, C.-B. (2017), ‘Infimal convolution of data discrepancies for mixed noise removal’, SIAM J. Imaging Sci. 10, 1196–1233.Google Scholar
Callaghan, P. T. (1993), Principles of Nuclear Magnetic Resonance Microscopy, Oxford University Press.Google Scholar
Callaghan, P. T. (1999), ‘Rheo-NMR: Nuclear magnetic resonance and the rheology of complex fluids’, Rep. Progr. Phys. 62, 599.Google Scholar
Campisi, P. and Egiazarian, K. (2016), Blind Image Deconvolution: Theory and Applications, CRC press.Google Scholar
Candès, E. J. and Donoho, D. L. (2000a), Curvelets: A surprisingly effective nonadaptive representation for objects with edges. Technical report, Department of Statistics, Stanford University.Google Scholar
Candès, E. J. and Donoho, D. L. (2000b), Curvelets, multiresolution representation, and scaling laws. In SPIE Wavelet Applications in Signal and Image Processing VIII, pp. 1–12.Google Scholar
Candès, E. J. and Donoho, D. L. (2002), ‘Recovering edges in ill-posed inverse problems: Optimality of curvelet frames’, Ann. Statist. 30, 784–842.Google Scholar
Candès, E. J. and Fernandez-Granda, C. (2013), ‘Super-resolution from noisy data’, J. Fourier Anal. Appl. 19, 1229–1254.Google Scholar
Candès, E. J. and Fernandez-Granda, C. (2014), ‘Towards a mathematical theory of super-resolution’, Commun. Pure Appl. Math. 67, 906–956.Google Scholar
Candès, E. J. and Recht, B. (2009), ‘Exact matrix completion via convex optimization’, Found. Comput. Math. 9, 717.CrossRefGoogle Scholar
Candès, E. J. and Romberg, J. (2007), ‘Sparsity and incoherence in compressive sampling’, Inverse Problems 23, 969.Google Scholar
Candès, E. J. and Tao, T. (2004a), ‘Decoding by linear programming’, IEEE Trans. Inform. Theory 51, 4203–4215.Google Scholar
Candès, E. J. and Tao, T. (2004b), ‘Near-optimal signal recovery from random projections: Universal encoding strategies’, IEEE Trans. Inform. Theory 52, 5406–5425.Google Scholar
Candès, E. J., Li, X., Ma, Y. and Wright, J. (2011), ‘Robust principal component analysis?’, J. Assoc. Comput. Mach. 58, 11.Google Scholar
Candès, E. J., Romberg, J. and Tao, T. (2006), ‘Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information’, IEEE Trans. Inform. Theory 52, 489–509.Google Scholar
Caselles, V., Chambolle, A. and Novaga, M. (2007), ‘The discontinuity set of solutions of the TV denoising problem and some extensions’, Multiscale Model. Simul. 6, 879–894.Google Scholar
Castillo, I. and Nickl, R. et al. (2014), ‘On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures’, Ann. Statist. 42, 1941–1969.Google Scholar
Cavalier, L. (2008), ‘Nonparametric statistical inverse problems’, Inverse Problems 24, 034004.CrossRefGoogle Scholar
Censor, Y. and Zenios, S. A. (1992), ‘Proximal minimization algorithm withd-functions’, J. Optim. Theory Appl. 73, 451–464.Google Scholar
Chadan, K., Colton, D., Päivärinta, L. and Rundell, W. (1997), An Introduction to Inverse Scattering and Inverse Spectral Problems, SIAM.Google Scholar
Chambolle, A. (2004), ‘An algorithm for total variation minimization and applications’, J. Math. Imaging Vision 20, 89–97.Google Scholar
Chambolle, A. and Lions, P.-L. (1997), ‘Image recovery via total variation minimization and related problems’, Numer. Math. 76, 167–188.Google Scholar
Chambolle, A. and Pock, T. (2011), ‘A first-order primal–dual algorithm for convex problems with applications to imaging’, J. Math. Imaging Vision 40, 120–145.Google Scholar
Chambolle, A. and Pock, T. (2016), An introduction to continuous optimization for imaging. In Acta Numerica, Vol. 25, Cambridge University Press, pp. 161–319.Google Scholar
Chambolle, A., Caselles, V., Cremers, D., Novaga, M. and Pock, T. (2010), An introduction to total variation for image analysis. In Theoretical Foundations and Numerical Methods for Sparse Recovery, (Fornasier, M., ed.), Vol. 9 of Radon Series on Computational and Applied Mathematics, De Gruyter, pp. 263–340.Google Scholar
Chan, T. F. and Shen, J. (2005), Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM.Google Scholar
Chan, T. F., Esedoglu, S. and Park, F. (2010), A fourth order dual method for staircase reduction in texture extraction and image restoration problems. In ICIP 2010: 17th IEEE International Conference on Image Processing, pp. 4137–4140.Google Scholar
Chan, T. F., Golub, G. H. and Mulet, P. (1999), ‘A nonlinear primal–dual method for total variation-based image restoration’, SIAM J. Sci. Comput. 20, 1964–1977.Google Scholar
Chaux, C., Combettes, P. L., Pesquet, J.-C. and Wajs, V. R. (2007), ‘A variational formulation for frame-based inverse problems’, Inverse Problems 23, 1495.Google Scholar
Chavent, G. and Kunisch, K. (1997), ‘Regularization of linear least squares problems by total bounded variation’, ESAIM Control Optim. Calc. Var. 2, 359–376.Google Scholar
Chen, Y. and Pock, T. (2017), ‘Trainable nonlinear reaction diffusion: A flexible framework for fast and effective image restoration’, IEEE Trans. Pattern Anal. Machine Intell. 39, 1256–1272.Google Scholar
Chen, Y., Pock, T. and Bischof, H. (2014a), Learning $\ell ^{1}$ -based analysis and synthesis sparsity priors using bi-level optimization. arXiv:1401.4105 Google Scholar
Chen, Y., Pock, T., Ranftl, R. and Bischof, H. (2013), Revisiting loss-specific training of filter-based MRFs for image restoration. In GCPR 2013: German Conference on Pattern Recognition, (Weickert, J. et al. , eds), Vol. 8142 of Lecture Notes in Computer Science, Springer, pp. 271–281.Google Scholar
Chen, Y., Ranftl, R. and Pock, T. (2014b), ‘Insights into analysis operator learning: From patch-based sparse models to higher order MRFs’, IEEE Trans. Image Process. 23, 1060–1072.Google Scholar
Chen, Y., Yu, W. and Pock, T. (2015), On learning optimized reaction diffusion processes for effective image restoration. In CVPR 2015: IEEE Conference on Computer Vision and Pattern Recognition, pp. 5261–5269.Google Scholar
Christensen, O. (2003), An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Springer.CrossRefGoogle Scholar
Chung, C. V., De los Reyes, J. C. and Schönlieb, C.-B. (2017), ‘Learning optimal spatially-dependent regularization parameters in total variation image denoising’, Inverse Problems 33, 074005.Google Scholar
Chung, J., Chung, M. and O’Leary, D. P. (2011), ‘Designing optimal spectral filters for inverse problems’, SIAM J. Sci. Comput. 33, 3132–3152.Google Scholar
Chung, J., Espanol, M. I. and Nguyen, T. (2014), Optimal regularization parameters for general-form Tikhonov regularization. arXiv:1407.1911 Google Scholar
Colonna, F., Easley, G., Guo, K. and Labate, D. (2010), ‘Radon transform inversion using the shearlet representation’, Appl. Comput. Harmon. Anal. 29, 232–250.Google Scholar
Colton, D. and Kress, R. (2012), Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 of Applied Mathematical Sciences, Springer.Google Scholar
Colton, D. and Monk, P. (1988), ‘The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium’, Quart. J. Mech Appl. Math. 41, 97–125.Google Scholar
Colton, D., Engl, H., Louis, A. K., Mclaughlin, J. and Rundell, W. (2012), Surveys on Solution Methods for Inverse Problems, Springer.Google Scholar
Colton, D., Ewing, R. E. and Rundell, W. et al. (1990), Inverse Problems in Partial Differential Equations, SIAM.Google Scholar
Cotter, S. F., Rao, B. D., Engan, K. and Kreutz-Delgado, K. (2005), ‘Sparse solutions to linear inverse problems with multiple measurement vectors’, IEEE Trans. Signal Process. 53, 2477–2488.Google Scholar
Darbon, J. and Osher, S. (2007), Fast discrete optimization for sparse approximations and deconvolutions. UCLA CAM Report preprint.Google Scholar
Dashti, M., Law, K. J. H., Stuart, A. M. and Voss, J. (2013), ‘MAP estimators and their consistency in Bayesian nonparametric inverse problems’, Inverse Problems 29, 095017.Google Scholar
De los Reyes, J. C. and Schönlieb, C.-B. (2013), ‘Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization’, Inverse Probl. Imaging 7, 1183–1214.Google Scholar
De los Reyes, J. C., Schönlieb, C.-B. and Valkonen, T. (2016), ‘The structure of optimal parameters for image restoration problems’, J. Math. Anal. Appl. 434, 464–500.Google Scholar
De los Reyes, J. C., Schönlieb, C.-B. and Valkonen, T. (2017), ‘Bilevel parameter learning for higher-order total variation regularisation models’, J. Math. Imaging Vision 57, 1–25.Google Scholar
Defazio, A., Bach, F. and Lacoste-Julien, S. (2014), SAGA: A fast incremental gradient method with support for non-strongly convex composite objectives. In NIPS 2014: Advances in Neural Information Processing Systems 27 (Ghahramani, Z. et al. , eds), Curran Associates, pp. 1–12.Google Scholar
Deledalle, C.-A., Papadakis, N. and Salmon, J. (2015), On debiasing restoration algorithms: Applications to total-variation and nonlocal-means. In SSVM 2015: Scale Space and Variational Methods in Computer Vision (Aujol, J.-F. et al. , eds), Springer, pp. 129–141.Google Scholar
Deledalle, C.-A., Papadakis, N., Salmon, J. and Vaiter, S. (2017), ‘CLEAR: Covariant least-square refitting with applications to image restoration’, SIAM J. Imaging Sci. 10, 243–284.Google Scholar
Denoyelle, Q., Duval, V. and Peyré, G. (2017), ‘Support recovery for sparse super-resolution of positive measures’, J. Fourier Anal. Appl. 23, 1153–1194.Google Scholar
Domke, J. (2012), Generic methods for optimization-based modeling. In Fifteenth International Conference on Artificial Intelligence and Statistics, (Lawrence, N. D. and Girolami, M., eds), PMLR, pp. 318–326.Google Scholar
Donoho, D. L. (1992), ‘Superresolution via sparsity constraints’, SIAM J. Math. Anal. 23, 1309–1331.Google Scholar
Donoho, D. L. (2006), ‘Compressed sensing’, IEEE Trans. Inform. Theory 52, 1289–1306.Google Scholar
Donoho, D. L. and Johnstone, I. M. (1995), ‘Adapting to unknown smoothness via wavelet shrinkage’, J. Amer. Statist. Assoc. 90(432), 1200–1224.Google Scholar
Donoho, D. L., Elad, M. and Temlyakov, V. N. (2006), ‘Stable recovery of sparse overcomplete representations in the presence of noise’, IEEE Trans. Inform. Theory 52, 6–18.Google Scholar
Droske, M., Rumpf, M. and Schaller, C. (2003), Nonrigid morphological image registration & its practical issues. In ICIP 2003: IEEE International Conference on Image Processing, pp. II–699.Google Scholar
Drusvyatskiy, D., Ioffe, A. D. and Lewis, A. S. (2016), Nonsmooth optimization using Taylor-like models: Error bounds, convergence, and termination criteria. arXiv:1610.03446 Google Scholar
Duarte, M. F., Sarvotham, S., Wakin, M. B., Baron, D. and Baraniuk, R. G. (2005), Joint sparsity models for distributed compressed sensing. In Proceedings of the Workshop on Signal Processing with Adaptive Sparse Structured Representations, IEEE.Google Scholar
Duval, V. and Peyré, G. (2017a), ‘Sparse regularization on thin grids, I: The Lasso’, Inverse Problems 33, 055008.Google Scholar
Duval, V. and Peyré, G. (2017b), ‘Sparse spikes deconvolution on thin grids, II: The continuous basis pursuit’, Inverse Problems 33, 095008.Google Scholar
Eckstein, J. (1993), ‘Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming’, Math. Oper. Res. 18, 202–226.Google Scholar
Eggermont, P. P. B. (1993), ‘Maximum entropy regularization for Fredholm integral equations of the first kind’, SIAM J. Math. Anal. 24, 1557–1576.Google Scholar
Ehrhardt, M. J. and Arridge, S. R. (2014), ‘Vector-valued image processing by parallel level sets’, IEEE Trans. Image Process. 23, 9–18.Google Scholar
Ehrhardt, M. J. and Betcke, M. M. (2016), ‘Multicontrast MRI reconstruction with structure-guided total variation’, SIAM J. Imaging Sci. 9, 1084–1106.Google Scholar
Ehrhardt, M. J., Markiewicz, P., Liljeroth, M., Barnes, A., Kolehmainen, V., Duncan, J. S., Pizarro, L., Atkinson, D., Hutton, B. F. and Ourselin, S. (2016), ‘PET reconstruction with an anatomical MRI prior using parallel level sets’, IEEE Trans. Medical Imaging 35, 2189–2199.Google Scholar
Ehrhardt, M. J., Thielemans, K., Pizarro, L., Atkinson, D., Ourselin, S., Hutton, B. F. and Arridge, S. R. (2014), ‘Joint reconstruction of PET-MRI by exploiting structural similarity’, Inverse Problems 31, 015001.Google Scholar
Eicke, B. (1992), ‘Iteration methods for convexly constrained ill-posed problems in Hilbert space’, Numer. Funct. Anal. Optim. 13, 413–429.CrossRefGoogle Scholar
Ekeland, I. and Temam, R. (1999), Convex Analysis and Variational Problems, corrected reprint edition, SIAM.Google Scholar
Elad, M., Milanfar, P. and Rubinstein, R. (2007), ‘Analysis versus synthesis in signal priors’, Inverse Problems 23, 947.Google Scholar
Eldén, L. (1977), ‘Algorithms for the regularization of ill-conditioned least squares problems’, BIT Numer. Math. 17, 134–145.Google Scholar
Engl, H. W. (1987a), ‘Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates’, J. Optim. Theory Appl. 52, 209–215.Google Scholar
Engl, H. W. (1987b), ‘On the choice of the regularization parameter for iterated Tikhonov regularization of ill-posed problems’, J. Approx. Theory 49, 55–63.Google Scholar
Engl, H. W. and Gfrerer, H. (1988), ‘ A posteriori parameter choice for general regularization methods for solving linear ill-posed problems’, Appl. Numer. Math. 4, 395–417.Google Scholar
Engl, H. W. and Landl, G. (1993), ‘Convergence rates for maximum entropy regularization’, SIAM J. Numer. Anal. 30, 1509–1536.Google Scholar
Engl, H. W. and Neubauer, A. (1985), Optimal discrepancy principles for the Tikhonov regularization of integral equations of the first kind. In Constructive Methods for the Practical Treatment of Integral Equations (Hämmerlin, G. and Hoffmann, K.-H., eds), Springer, pp. 120–141.Google Scholar
Engl, H. W. and Neubauer, A. (1987), Optimal parameter choice for ordinary and iterated Tikhonov regularization. In Inverse and Ill-Posed Problems (Engl, H. W. and Groetsch, C. W., eds), Elsevier, pp. 97–125.Google Scholar
Engl, H. W., Hanke, M. and Neubauer, A. (1996), Regularization of Inverse Problems, Mathematics and Its Applications, Springer.Google Scholar
Engl, H. W., Kunisch, K. and Neubauer, A. (1989), ‘Convergence rates for Tikhonov regularisation of non-linear ill-posed problems’, Inverse Problems 5, 523.Google Scholar
Engl, H. W., Louis, A. K. & Rundell, W. (Eds) (2012), Inverse Problems in Medical Imaging and Nondestructive Testing, Springer.Google Scholar
Esser, E., Zhang, X. and Chan, T. F. (2010), ‘A general framework for a class of first order primal–dual algorithms for convex optimization in imaging science’, SIAM J. Imaging Sci. 3, 1015–1046.Google Scholar
Evans, L. and Gariepy, R. (1992), Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press.Google Scholar
Flemming, J. (2013), ‘Variational smoothness assumptions in convergence rate theory: An overview’, J. Inverse Ill-Posed Probl. 21, 395–409.Google Scholar
Flemming, J. (2017a), A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations. arXiv:1712.01499 Google Scholar
Flemming, J. (2017b), ‘Existence of variational source conditions for nonlinear inverse problems in Banach spaces’, J. Inverse Ill-Posed Probl. doi:10.1515/jiip-2017-0092 Google Scholar
Flemming, J. and Gerth, D. (2017), ‘Injectivity and weak*-to-weak continuity suffice for convergence rates in $\ell ^{1}$ -regularization’, J. Inverse Ill-Posed Probl. 26, 85–94.Google Scholar
Flemming, J. and Hofmann, B. (2010), ‘A new approach to source conditions in regularization with general residual term’, Numer. Funct. Anal. Optim. 31, 254–284.Google Scholar
Flemming, J., Hofmann, B. and Veselić, I. (2015), ‘On $\ell ^{1}$ -regularization in light of Nashed’s ill-posedness concept’, Comput. Methods Appl. Math. 15, 279–289.Google Scholar
Flemming, J., Hofmann, B. and Veselić, I. (2016), ‘A unified approach to convergence rates for $\ell ^{1}$ -regularization and lacking sparsity’, J. Inverse Ill-Posed Probl. 24, 139–148.Google Scholar
Fornasier, M. and Rauhut, H. (2008), ‘Recovery algorithms for vector-valued data with joint sparsity constraints’, SIAM J. Numer. Anal. 46, 577–613.Google Scholar
Gao, Y. and Bredies, K. (2017), Infimal convolution of oscillation total generalized variation for the recovery of images with structured texture. arXiv:1710.11591 Google Scholar
Gatehouse, P. D., Keegan, J., Crowe, L. A., Masood, S., Mohiaddin, R. H., Kreitner, K.-F. and Firmin, D. N. (2005), ‘Applications of phase-contrast flow and velocity imaging in cardiovascular MRI’, European Radiology 15, 2172–2184.Google Scholar
Gfrerer, H. (1987), ‘An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates’, Math. Comp. 49(180), 507–522.Google Scholar
Gholami, A. and Siahkoohi, H. (2010), ‘Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints’, Geophys. J. Internat. 180, 871–882.Google Scholar
Gilboa, G. (2014a), Nonlinear band-pass filtering using the TV transform. In EUSIPCO 2014: 22nd European Signal Processing Conference, IEEE, pp. 1696–1700.Google Scholar
Gilboa, G. (2014b), ‘A total variation spectral framework for scale and texture analysis’, SIAM J. Imaging Sci. 7, 1937–1961.Google Scholar
Gilboa, G., Moeller, M. and Burger, M. (2016), ‘Nonlinear spectral analysis via one-homogeneous functionals: Overview and future prospects’, J. Math. Imaging Vision 56, 300–319.Google Scholar
Giné, E. and Nickl, R. (2015), Mathematical Foundations of Infinite-Dimensional Statistical Models, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.Google Scholar
Grah, J. S. (2017) Mathematical imaging tools in cancer research: From mitosis analysis to sparse regularisation. PhD thesis, University of Cambridge.Google Scholar
Grasmair, M. (2011), ‘Linear convergence rates for Tikhonov regularization with positively homogeneous functionals’, Inverse Problems 27, 075014.Google Scholar
Grasmair, M. (2013), ‘Variational inequalities and higher order convergence rates for Tikhonov regularisation on Banach spaces’, J. Inverse Ill-Posed Probl. 21, 379–394.Google Scholar
Grasmair, M. and Lenzen, F. (2010), ‘Anisotropic total variation filtering’, Appl. Math. Optim. 62, 323–339.Google Scholar
Grasmair, M., Scherzer, O. and Haltmeier, M. (2011), ‘Necessary and sufficient conditions for linear convergence of $\ell ^{1}$ -regularization’, Commun. Pure Appl. Math. 64, 161–182.Google Scholar
Groetsch, C. W. (1977), ‘Sequential regularization of ill-posed problems involving unbounded operators’, Comment. Math. Univ. Carolin. 18, 489–498.Google Scholar
Groetsch, C. W. (1993), Inverse Problems in the Mathematical Sciences, Vieweg Mathematics for Scientists and Engineers, Vieweg.Google Scholar
Groetsch, C. W. and King, J. T. (1979), ‘Extrapolation and the method of regularization for generalized inverses’, J. Approx. Theory 25, 233–247.Google Scholar
Guo, K. and Labate, D. (2007), ‘Optimally sparse multidimensional representation using shearlets’, SIAM J. Math. Anal. 39, 298–318.Google Scholar
Haber, E. and Tenorio, L. (2003), ‘Learning regularization functionals: A supervised training approach’, Inverse Problems 19, 611.Google Scholar
Haber, E., Horesh, L. and Tenorio, L. (2009), ‘Numerical methods for the design of large-scale nonlinear discrete ill-posed inverse problems’, Inverse Problems 26, 025002.Google Scholar
Hadamard, J. (1902), ‘Sur les problèmes aux dérivées partielles et leur signification physique’, Princeton University Bulletin 13, 49–52.Google Scholar
Hadamard, J. (1923), Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press.Google Scholar
Hammernik, K., Klatzer, T., Kobler, E., Recht, M. P., Sodickson, D. K., Pock, T. and Knoll, F. (2017), ‘Learning a variational network for reconstruction of accelerated MRI data’, Magn. Reson. Med. 79, 3055–3071.Google Scholar
Hanke, M., Neubauer, A. and Scherzer, O. (1995), ‘A convergence analysis of the Landweber iteration for nonlinear ill-posed problems’, Numer. Math. 72, 21–37.Google Scholar
Hansen, P. C. (1987), ‘The truncated SVD as a method for regularization’, BIT Numer. Math. 27, 534–553.Google Scholar
Hansen, P. C. (1992), ‘Analysis of discrete ill-posed problems by means of the L-curve’, SIAM Review 34, 561–580.Google Scholar
Hein, M. and Bühler, T. (2010), An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA. In NIPS 2010: Advances in Neural Information Processing Systems 23 (Lafferty, J. D. et al. , eds), Curran Associates, pp. 847–855.Google Scholar
Heins, P. (2014) Reconstruction using local sparsity: A novel regularization technique and an asymptotic analysis of spatial sparsity priors. PhD thesis, Westfälische Wilhelms-Universität Münster, Germany.Google Scholar
Heins, P., Moeller, M. and Burger, M. (2015), ‘Locally sparse reconstruction using the $\ell ^{1,\infty }$ -norm’, Inverse Probl. Imaging 9, 1093–1137.Google Scholar
Helin, T. and Burger, M. (2015), ‘Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems’, Inverse Problems 31, 085009.Google Scholar
Helin, T. and Lassas, M. (2011), ‘Hierarchical models in statistical inverse problems and the Mumford–Shah functional’, Inverse Problems 27, 015008.Google Scholar
Hinterberger, W. and Scherzer, O. (2006), ‘Variational methods on the space of functions of bounded Hessian for convexification and denoising’, Computing 76, 109–133.Google Scholar
Hinterberger, W., Scherzer, O., Schnörr, C. and Weickert, J. (2002), ‘Analysis of optical flow models in the framework of the calculus of variations’, Numer. Funct. Anal. Optim. 23, 69–89.Google Scholar
Hintermüller, M. and Wu, T. (2015), ‘Bilevel optimization for calibrating point spread functions in blind deconvolution’, Inverse Probl. Imaging 9, 1139–1169.Google Scholar
Hintermüller, M., Holler, M. and Papafitsoros, K. (2017), A function space framework for structural total variation regularization with applications in inverse problems. arXiv:1710.01527 Google Scholar
Hoerl, A. E. (1959), ‘Optimum solution of many variables equations’, Chem. Engrg Progr. 55, 69–78.Google Scholar
Hoerl, A. E. and Kennard, R. W. (1970), ‘Ridge regression: Biased estimation for nonorthogonal problems’, Technometrics 12, 55–67.Google Scholar
Hohage, T. (1997), ‘Logarithmic convergence rates of the iteratively regularized Gauss–Newton method for an inverse potential and an inverse scattering problem’, Inverse Problems 13, 1279.Google Scholar
Hohage, T. and Weidling, F. (2017), ‘Characterizations of variational source conditions, converse results, and maxisets of spectral regularization methods’, SIAM J. Numer. Anal. 55, 598–620.Google Scholar
Hohage, T. and Werner, F. (2013), ‘Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data’, Numer. Math. 123, 745–779.Google Scholar
Hohage, T. and Werner, F. (2016), ‘Inverse problems with Poisson data: Statistical regularization theory, applications and algorithms’, Inverse Problems 32, 093001.Google Scholar
Holland, D., Malioutov, D., Blake, A., Sederman, A. and Gladden, L. (2010), ‘Reducing data acquisition times in phase-encoded velocity imaging using compressed sensing’, J. Magnetic Resonance 203, 236–246.Google Scholar
Holland, D., Müller, C., Dennis, J., Gladden, L. and Sederman, A. (2008), ‘Spatially resolved measurement of anisotropic granular temperature in gas-fluidized beds’, Powder Technology 182, 171–181.Google Scholar
Holler, M. and Kunisch, K. (2014), ‘On infimal convolution of TV-type functionals and applications to video and image reconstruction’, SIAM J. Imaging Sci. 7, 2258–2300.Google Scholar
Hu, Y. and Jacob, M. (2012), ‘Higher degree total variation (HDTV) regularization for image recovery’, IEEE Trans. Image Process. 21, 2559–2571.Google Scholar
Huang, J. and Mumford, D. (1999), Statistics of natural images and models. In CVPR 1999: IEEE Computer Society Conference On Computer Vision and Pattern Recognition, pp. 541–547.Google Scholar
Isakov, V. (2006), Inverse Problems for Partial Differential Equations, Vol. 127 of Applied Mathematical Sciences, Springer.Google Scholar
Isakov, V. (2008), ‘On inverse problems in secondary oil recovery’, European J. Appl. Math. 19, 459–478.Google Scholar
Ivanov, V. K. (1962), ‘On linear problems which are not well-posed’, Soviet Math. Dokl. 3, 981–983.Google Scholar
Jalalzai, K. (2016), ‘Some remarks on the staircasing phenomenon in total variation-based image denoising’, J. Math. Imaging Vision 54, 256–268.Google Scholar
John, F. (1960), ‘Continuous dependence on data for solutions of partial differential equations with a prescribed bound’, Commun. Pure Appl. Math. 13, 551–585.Google Scholar
Johnson, R. and Zhang, T. (2013), Accelerating stochastic gradient descent using predictive variance reduction. In NIPS 2013: Advances in Neural Information Processing Systems 26 (Burges, C. J. C. et al. , eds), Curran Associates, pp. 315–323.Google Scholar
Kaipio, J. P. and Somersalo, E. (2006), Statistical and Computational Inverse Problems, Applied Mathematical Sciences, Springer.Google Scholar
Kaipio, J. P., Kolehmainen, V., Vauhkonen, M. and Somersalo, E. (1999), ‘Inverse problems with structural prior information’, Inverse Problems 15, 713.Google Scholar
Kaltenbacher, B. (1997), ‘Some Newton-type methods for the regularization of nonlinear ill-posed problems’, Inverse Problems 13, 729.Google Scholar
Kaltenbacher, B. (2008), ‘A note on logarithmic convergence rates for nonlinear Tikhonov regularization’, J. Inverse Ill-Posed Probl. 16, 79–88.Google Scholar
Kaltenbacher, B., Schöpfer, F. and Schuster, T. (2009), ‘Iterative methods for nonlinear ill-posed problems in Banach spaces: Convergence and applications to parameter identification problems’, Inverse Problems 25, 065003.Google Scholar
Kekkonen, H., Lassas, M. and Siltanen, S. (2014), ‘Analysis of regularized inversion of data corrupted by white Gaussian noise’, Inverse Problems 30, 045009.Google Scholar
Kekkonen, H., Lassas, M. and Siltanen, S. (2016), ‘Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators’, Inverse Problems 32, 085005.Google Scholar
Kirisits, C. and Scherzer, O. (2017), ‘Convergence rates for regularization functionals with polyconvex integrands’, Inverse Problems 33, 085008.Google Scholar
Kiwiel, K. C. (1997), ‘Proximal minimization methods with generalized Bregman functions’, SIAM J. Control Optim. 35, 1142–1168.Google Scholar
Klann, E. and Ramlau, R. (2013), ‘Regularization properties of Mumford–Shah-type functionals with perimeter and norm constraints for linear ill-posed problems’, SIAM J. Imaging Sci. 6, 413–436.Google Scholar
Klann, E., Ramlau, R. and Ring, W. (2011), ‘A Mumford–Shah level-set approach for the inversion and segmentation of SPECT/CT data’, Inverse Probl. Imaging 5, 137–166.Google Scholar
Klatzer, T., Soukup, D., Kobler, E., Hammernik, K. and Pock, T. (2017), Trainable regularization for multi-frame superresolution. In GCPR 2017: German Conference on Pattern Recognition, (Roth, V. and Vetter, T., eds), Vol. 10496 of Lecture Notes in Computer Science, Springer, pp. 90–100.Google Scholar
Knoll, F., Bredies, K., Pock, T. and Stollberger, R. (2011), ‘Second order total generalized variation (TGV) for MRI’, Magnetic Resonance Medicine 65, 480–491.Google Scholar
Knoll, F., Holler, M., Koesters, T., Otazo, R., Bredies, K. and Sodickson, D. K. (2017), ‘Joint MR-PET reconstruction using a multi-channel image regularizer’, IEEE Trans. Medical Imaging 36, 1–16.Google Scholar
Kobler, E., Klatzer, T., Hammernik, K. and Pock, T. (2017), Variational networks: connecting variational methods and deep learning. In GCPR 2017: German Conference on Pattern Recognition, (Roth, V. and Vetter, T., eds), Vol. 10496 of Lecture Notes in Computer Science, Springer, pp. 281–293.Google Scholar
Kolehmainen, V., Lassas, M., Niinimäki, K. and Siltanen, S. (2012), ‘Sparsity-promoting Bayesian inversion’, Inverse Problems 28, 025005.Google Scholar
Krause, M., Alles, R. M., Burgeth, B. and Weickert, J. (2016), ‘Fast retinal vessel analysis’, J. Real-Time Image Processing 11, 413–422.Google Scholar
Kravaris, C. and Seinfeld, J. H. (1985), ‘Identification of parameters in distributed parameter systems by regularization’, SIAM J. Control Optim. 23, 217–241.CrossRefGoogle Scholar
Kryanev, A. (1974), ‘An iterative method for solving incorrectly posed problems’, USSR Comput. Math. Math. Phys. 14, 24–35.Google Scholar
Kundur, D. and Hatzinakos, D. (1996), ‘Blind image deconvolution’, IEEE Signal Processing Magazine 13, 43.Google Scholar
Kunisch, K. and Hintermüller, M. (2004), ‘Total bounded variation regularization as a bilaterally constrained optimization problem’, SIAM J. Appl. Math. 64, 1311–1333.Google Scholar
Kunisch, K. and Pock, T. (2013), ‘A bilevel optimization approach for parameter learning in variational models’, SIAM J. Imaging Sci. 6, 938–983.Google Scholar
Kurdyka, K. (1998), ‘On gradients of functions definable in o-minimal structures’, Annales de l’Institut Fourier (Chartres) 48, 769–784.Google Scholar
Kutyniok, G. and Labate, D. (2012), Introduction to shearlets. In Shearlets: Multiscale Analysis for Multivariate Data, Applied and Numerical Harmonic Analysis, Springer, pp. 1–38.Google Scholar
Labate, D., Lim, W.-Q., Kutyniok, G. and Weiss, G. (2005), Sparse multidimensional representation using shearlets. In Optics and Photonics 2005, Proceedings Vol. 5914, SPIE, 59140U.Google Scholar
Landweber, L. (1951), ‘An iteration formula for Fredholm integral equations of the first kind’, Amer. J. Math. 73, 615–624.Google Scholar
Lassas, M., Saksman, E. and Siltanen, S. (2009), ‘Discretization-invariant Bayesian inversion and Besov space priors’, Inverse Probl. Imaging 3, 87–122.Google Scholar
Lattès, R. and Lions, J.-L. (1967), ‘Méthode de quasi-réversibilité et applications’.Google Scholar
Laurent, T., von Brecht, J., Bresson, X. and Szlam, A. (2016), The product cut. In NIPS 2016: Advances in Neural Information Processing Systems 29 (Lee, D. D. et al. , eds), Curran Associates, pp. 3792–3800.Google Scholar
LeCun, Y., Bengio, Y. and Hinton, G. (2015), ‘Deep learning’, Nature 521(7553), 436–444.Google Scholar
Lederer, J. (2013), Trust, but verify: Benefits and pitfalls of least-squares refitting in high dimensions. arXiv:1306.0113 Google Scholar
Lee, O., Kim, J. M., Bresler, Y. and Ye, J. C. (2011), ‘Compressive diffuse optical tomography: Noniterative exact reconstruction using joint sparsity’, IEEE Trans. Medical Imaging 30, 1129–1142.Google Scholar
Lenzen, F., Becker, F. and Lellmann, J. (2013), Adaptive second-order total variation: An approach aware of slope discontinuities. In SSVM 2015: Scale Space and Variational Methods in Computer Vision (Aujol, J.-F. et al. , eds), Springer, pp. 61–73.Google Scholar
Levenberg, K. (1944), ‘A method for the solution of certain non-linear problems in least squares’, Quart. Appl. Math. 2, 164–168.Google Scholar
Li, G. and Pong, T. K. (2015), ‘Global convergence of splitting methods for nonconvex composite optimization’, SIAM J. Optim. 25, 2434–2460.Google Scholar
Lie, H. C. and Sullivan, T. (2017), Equivalence of weak and strong modes of measures on topological vector spaces. arXiv:1708.02516 Google Scholar
Lions, P.-L. and Mercier, B. (1979), ‘Splitting algorithms for the sum of two nonlinear operators’, SIAM J. Numer. Anal. 16, 964–979.Google Scholar
Lojasiewicz, S. (1963), ‘Une propriété topologique des sous-ensembles analytiques réels’, Les Équations aux Dérivées Partielles 117, 87–89.Google Scholar
Louis, A. (1996), ‘Approximate inverse for linear and some nonlinear problems’, Inverse Problems 12, 175.Google Scholar
Lustig, M., Donoho, D. and Pauly, J. M. (2007), ‘Sparse MRI: The application of compressed sensing for rapid MR imaging’, Magnetic Resonance Medicine 58, 1182–1195.Google Scholar
Mallat, S. (2008), A Wavelet Tour of Signal Processing: The Sparse Way, Academic Press.Google Scholar
Mallat, S. and Zhang, Z. (1993), ‘Matching pursuits with time-frequency dictionaries’, IEEE Trans. Signal Process. 12, 3397–3415.Google Scholar
Marquardt, D. W. (1963), ‘An algorithm for least-squares estimation of nonlinear parameters’, J. Soc. Indust. Appl. Math. 11, 431–441.Google Scholar
Marquina, A. and Osher, S. J. (2008), ‘Image super-resolution by TV-regularization and Bregman iteration’, J. Sci. Comput. 37, 367–382.Google Scholar
Modersitzki, J. (2004), Numerical Methods for Image Registration, Numerical Mathematics and Scientific Computation, Oxford University Press.Google Scholar
Moeller, M. (2012), Multiscale methods for polyhedral regularizations and applications in high dimensional imaging. PhD thesis, University of Münster, Germany.Google Scholar
Moeller, M. and Burger, M. (2013), ‘Multiscale methods for polyhedral regularizations’, SIAM J. Optim. 23, 1424–1456.Google Scholar
Moeller, M., Benning, M., Schönlieb, C. and Cremers, D. (2015), ‘Variational depth from focus reconstruction’, IEEE Trans. Image Process. 24, 5369–5378.Google Scholar
Moeller, M., Brinkmann, E., Burger, M. and Seybold, T. (2014), ‘Color Bregman TV’, SIAM J. Imaging Sci. 7, 2771–2806.Google Scholar
Moeller, M., Wittman, T., Bertozzi, A. and Burger, M. (2012), ‘A variational approach for sharpening high dimensional images’, SIAM J. Imaging Sci. 5, 150–178.Google Scholar
Morozov, V. A. (1966), ‘Regularization of incorrectly posed problems and the choice of regularization parameter’, USSR Comput. Math. Math. Phys. 6, 242–251.Google Scholar
Müller, J. (2013), Advanced image reconstruction and denoising: Bregmanized (higher order) total variation and application in PET. PhD thesis, Westfälische Wilhelms-Universität Münster, Germany.Google Scholar
Müller, J., Brune, C., Sawatzky, A., Kösters, T., Schäfers, K. P. and Burger, M. (2011), Reconstruction of short time PET scans using Bregman iterations. In NSS/MIC 2011: IEEE Nuclear Science Symposium and Medical Imaging Conference, pp. 2383–2385.Google Scholar
Mumford, D. and Shah, J. (1989), ‘Optimal approximations by piecewise smooth functions and associated variational problems’, Commun. Pure Appl. Math. 42, 577–685.Google Scholar
Nair, V. and Hinton, G. E. (2010), Rectified linear units improve restricted Boltzmann machines. In ICML’10: 27th International Conference on Machine Learning, pp. 807–814.Google Scholar
Nashed, M. Z. and Wahba, G. (1974a), ‘Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind’, Math. Comp. 28(125), 69–80.Google Scholar
Nashed, M. Z. and Wahba, G. (1974b), ‘Generalized inverses in reproducing kernel spaces: An approach to regularization of linear operator equations’, SIAM J. Math. Anal. 5, 974–987.Google Scholar
Nashed, M. Z. and Wahba, G. (1974c), ‘Regularization and approximation of linear operator equations in reproducing kernel spaces’, Bull. Amer. Math. Soc. 80, 1213–1218.Google Scholar
Natterer, F. (1984), ‘Error bounds for Tikhonov regularization in Hilbert scales’, Appl. Anal. 18, 29–37.Google Scholar
Natterer, F. (2001), The Mathematics of Computerized Tomography, SIAM Monographs on Mathematical Modeling and Computation, SIAM.Google Scholar
Natterer, F. and Wübbeling, F. (2001), Mathematical Methods in Image Reconstruction, SIAM.Google Scholar
Nemirovskii, A. and Yudin, D. B. (1983), Problem Complexity and Method Efficiency in Optimization, Wiley-Interscience Series in Discrete Mathematics, Wiley.Google Scholar
Neubauer, A. (1988a), ‘An a posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates’, SIAM J. Numer. Anal. 25, 1313–1326.Google Scholar
Neubauer, A. (1988b), ‘Tikhonov-regularization of ill-posed linear operator equations on closed convex sets’, J. Approx. Theory 53, 304–320.Google Scholar
Neubauer, A. and Pikkarainen, H. K. (2008), ‘Convergence results for the Bayesian inversion theory’, J. Inverse Ill-Posed Probl. 16, 601–613.Google Scholar
Nickl, R. and Söhl, J. et al. (2017), ‘Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions’, Ann. Statist. 45, 1664–1693.Google Scholar
Nikolova, M. and Tan, P. (2017), Alternating proximal gradient descent for nonconvex regularised problems with multiconvex coupling terms. arXiv:hal-01492846v2 Google Scholar
Ochs, P., Chen, Y., Brox, T. and Pock, T. (2014), ‘iPiano: Inertial proximal algorithm for nonconvex optimization’, SIAM J. Imaging Sci. 7, 1388–1419.Google Scholar
Ochs, P., Fadili, J. and Brox, T. (2017) Non-smooth non-convex Bregman minimization: Unification and new algorithms. arXiv:1707.02278 Google Scholar
Ochs, P., Ranftl, R., Brox, T. and Pock, T. (2015), Bilevel optimization with nonsmooth lower level problems. In SSVM 2015: Scale Space and Variational Methods in Computer Vision (Aujol, J.-F. et al. , eds), Springer, pp. 654–665.Google Scholar
Osher, S., Burger, M., Goldfarb, D., Xu, J. and Yin, W. (2005), ‘An iterative regularization method for total variation-based image restoration’, Multiscale Model. Simul. 4, 460–489.Google Scholar
Otazo, R., Candès, E. and Sodickson, D. K. (2015), ‘Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components’, Magnetic Resonance Medicine 73, 1125–1136.Google Scholar
Papafitsoros, K. and Schönlieb, C.-B. (2014), ‘A combined first and second order variational approach for image reconstruction’, J. Math. Imaging Vision 48, 308–338.Google Scholar
Parikh, N. and Boyd, S. (2014), ‘Proximal algorithms’, Found. Trends Optim. 1, 127–239.Google Scholar
Payne, L. E. (1975), Improperly Posed Problems in Partial Differential Equations, Vol. 22 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.Google Scholar
Phillips, D. L. (1962), ‘A technique for the numerical solution of certain integral equations of the first kind’, J. Assoc. Comput. Mach. 9, 84–97.Google Scholar
Pock, T. and Sabach, S. (2016), ‘Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems’, SIAM J. Imaging Sci. 9, 1756–1787.Google Scholar
Pock, T., Cremers, D., Bischof, H. and Chambolle, A. (2009), An algorithm for minimizing the Mumford–Shah functional. In ICCV 2009: IEEE 12th International Conference on Computer Vision, pp. 1133–1140.Google Scholar
Prato, M., Bonettini, S., Loris, I., Porta, F. and Rebegoldi, S. (2016), ‘On the constrained minimization of smooth Kurdyka–Łojasiewicz functions with the scaled gradient projection method’, J. Phys. Conf. Ser. 756, 012001.Google Scholar
Ranftl, R., Pock, T. and Bischof, H. (2013), Minimizing TGV-based variational models with non-convex data terms. In SSVM 2013: Scale Space and Variational Methods in Computer Vision (Kuijper, A. et al. , eds), Springer, pp. 282–293.Google Scholar
Rasch, J., Brinkmann, E.-M. and Burger, M. (2018), ‘Joint reconstruction via coupled Bregman iterations with applications to PET-MR imaging’, Inverse Problems 34, 014001.Google Scholar
Rasch, J., Kolehmainen, V., Nivajärvi, R., Kettunen, M., Gröhn, O., Burger, M. and Brinkmann, E.-M. (2017), Dynamic MRI reconstruction from undersampled data with an anatomical prescan. arXiv:1712.00099 Google Scholar
Raus, T. (1984), ‘Residue principle for ill-posed problems’, Acta et Comment. Univ. Tartuensis 672, 16–26.Google Scholar
Raus, T. (1992), ‘About regularization parameter choice in case of approximately given error bounds of data’, Acta et Comment. Univ. Tartuensis 937, 77–89.Google Scholar
Reader, A. J., Matthews, J., Sureau, F. C., Comtat, C., Trébossen, R. and Buvat, I. (2007), Fully 4D image reconstruction by estimation of an input function and spectral coefficients. In IEEE Nuclear Science Symposium Conference, pp. 3260–3267.Google Scholar
Recht, B., Fazel, M. and Parrilo, P. A. (2010), ‘Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization’, SIAM Review 52, 471–501.Google Scholar
Reed, M. and Simon, B. (1978), Methods of Mathematical Physics IV: Analysis of Operators, Elsevier.Google Scholar
Resmerita, E. (2005), ‘Regularization of ill-posed problems in Banach spaces: Convergence rates’, Inverse Problems 21, 1303.Google Scholar
Resmerita, E. and Scherzer, O. (2006), ‘Error estimates for non-quadratic regularization and the relation to enhancement’, Inverse Problems 22, 801.Google Scholar
Ring, W. (2000), ‘Structural properties of solutions to total variation regularization problems’, ESAIM Math. Model. Numer. Anal. 34, 799–810.Google Scholar
Rockafellar, R. (1972), Convex Analysis, Princeton Mathematical Series, Princeton University Press.Google Scholar
Romano, Y., Elad, M. and Milanfar, P. (2017), ‘The little engine that could: Regularization by denoising (RED)’, SIAM J. Imaging Sci. 10, 1804–1844.Google Scholar
Rondi, L. (2008), ‘Reconstruction in the inverse crack problem by variational methods’, European J. Appl. Math. 19, 635–660.Google Scholar
Roth, S. and Black, M. J. (2005), Fields of experts: A framework for learning image priors. In CVPR 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 860–867.Google Scholar
Rudin, L., Lions, P.-L. and Osher, S. (2003), Multiplicative denoising and deblurring: Theory and algorithms. In Geometric Level Set Methods in Imaging, Vision, and Graphics (Osher, S. and Paragios, N., eds), Springer, pp. 103–119.Google Scholar
Rudin, L., Osher, S. and Fatemi, E. (1992), ‘Nonlinear total variation based noise removal algorithms’, Phys. D: Nonlinear Phenomena 60, 259–268.Google Scholar
Rudin, W. (2006), Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill.Google Scholar
Sawatzky, A., Brune, C., Kösters, T., Wübbeling, F. and Burger, M. (2013), EM-TV methods for inverse problems with Poisson noise. In Level Set and PDE Based Reconstruction Methods in Imaging, (Burger, M. and Osher, S., eds), Vol 2090 of Lecture Notes in Mathematics, Springer, pp. 71–142.Google Scholar
Scherzer, O. (1993), ‘Convergence rates of iterated Tikhonov regularized solutions of nonlinear ill-posed problems’, Numer. Math. 66, 259–279.Google Scholar
Scherzer, O. (1998), ‘Denoising with higher order derivatives of bounded variation and an application to parameter estimation’, Computing 60, 1–27.Google Scholar
Schmidt, M. F., Benning, M. and Schönlieb, C.-B. (2018), ‘Inverse scale space decomposition’, Inverse Problems 34, 045008.Google Scholar
Schmidt, U. and Roth, S. (2014), Shrinkage fields for effective image restoration. In CVPR 2014: IEEE Conference on Computer Vision and Pattern Recognition, pp. 2774–2781.Google Scholar
Schock, E. (1985), Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence. In Constructive Methods for the Practical Treatment of Integral Equations, (Hämmerlin, G. and Hoffmann, K. H., eds), Vol. 73 of International Series of Numerical Mathematics, Springer, pp. 234–243.Google Scholar
Schöpfer, F., Louis, A. K. and Schuster, T. (2006), ‘Nonlinear iterative methods for linear ill-posed problems in Banach spaces’, Inverse Problems 22, 311.Google Scholar
Schuster, T., Kaltenbacher, B., Hofmann, B. and Kazimierski, K. (2012), Regularization Methods in Banach Spaces, De Gruyter.Google Scholar
Sederman, A., Johns, M., Alexander, P. and Gladden, L. (1998), ‘Structure-flow correlations in packed beds’, Chem. Engrg Sci. 53, 2117–2128.Google Scholar
Seidman, T. I. and Vogel, C. R. (1989), ‘Well posedness and convergence of some regularisation methods for non-linear ill posed problems’, Inverse Problems 5, 227.Google Scholar
Setzer, S., Steidl, G. and Teuber, T. (2011), ‘Infimal convolution regularizations with discrete $\ell _{1}$ -type functionals’, Comm. Math. Sci 9, 797–872.Google Scholar
Starck, J.-L., Murtagh, F. and Fadili, J. M. (2010), Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity, Cambridge University Press.Google Scholar
Strong, D. and Chan, T. (2003), ‘Edge-preserving and scale-dependent properties of total variation regularization’, Inverse Problems 19, S165.Google Scholar
Strong, D. and Chan, T. et al. (1996), Exact solutions to total variation regularization problems. CAM Report 96-41, UCLA.Google Scholar
Stuart, A. M. (2010), Inverse problems: A Bayesian perspective. In Acta Numerica, Vol. 19, Cambridge University Press, pp. 451–559.Google Scholar
Stück, R., Burger, M. and Hohage, T. (2011), ‘The iteratively regularized Gauss–Newton method with convex constraints and applications in 4Pi microscopy’, Inverse Problems 28, 015012.Google Scholar
Tappen, M. F. (2007), Utilizing variational optimization to learn Markov random fields. In CVPR 2007: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8.Google Scholar
Tarantola, A. (2005), Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM.Google Scholar
Tarantola, A. and Valette, B. (1982), ‘Inverse problems = quest for information’, J. Geophys. 50, 150–170.Google Scholar
Tayler, A. B., Holland, D. J., Sederman, A. J. and Gladden, L. F. (2012), ‘Exploring the origins of turbulence in multiphase flow using compressed sensing MRI’, Phys. Rev. Lett. 108, 264505.Google Scholar
Teboulle, M. (1992), ‘Entropic proximal mappings with applications to nonlinear programming’, Math. Oper. Res. 17, 670–690.Google Scholar
Teschke, G. and Ramlau, R. (2007), ‘An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector-valued regimes and an application to color image inpainting’, Inverse Problems 23, 1851.Google Scholar
Thomas King, J. and Chillingworth, D. (1979), ‘Approximation of generalized inverses by iterated regularization’, Numer. Funct. Anal. Optim. 1, 499–513.Google Scholar
Thompson, A. M., Brown, J. C., Kay, J. W. and Titterington, D. M. (1991), ‘A study of methods of choosing the smoothing parameter in image restoration by regularization’, IEEE Trans. Pattern Anal. Machine Intell. 13, 326–339.Google Scholar
Tikhonov, A. N. (1943), ‘On the stability of inverse problems’, Dokl. Akad. Nauk SSSR 39, 195–198.Google Scholar
Tikhonov, A. N. (1963), ‘Solution of incorrectly formulated problems and the regularization method’, Soviet Meth. Dokl. 4, 1035–1038.Google Scholar
Tikhonov, A. N. (1966), ‘On the stability of the functional optimization problem’, USSR Comput. Math. Math. Phys. 6, 28–33.Google Scholar
Tikhonov, A. N. and Arsenin, V. Y. (1977), Solutions of Ill-Posed Problems, Winston & Sons.Google Scholar
Tikhonov, A. N., Goncharsky, A. and Bloch, M. (1987), Ill-Posed Problems in the Natural Sciences, Mir.Google Scholar
Vaiter, S., Deledalle, C.-A., Peyré, G., Dossal, C. and Fadili, J. (2013a), ‘Local behavior of sparse analysis regularization: Applications to risk estimation’, Appl. Comput. Harmon. Anal. 35, 433–451.Google Scholar
Vaiter, S., Peyré, G., Dossal, C. and Fadili, J. (2013b), ‘Robust sparse analysis regularization’, IEEE Trans. Inform. Theory 59, 2001–2016.Google Scholar
Valkonen, T. (2014), ‘A primal–dual hybrid gradient method for nonlinear operators with applications to MRI’, Inverse Problems 30, 055012.Google Scholar
Vogel, C. (2002), Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, SIAM.Google Scholar
Wahba, G. (1977), ‘Practical approximate solutions to linear operator equations when the data are noisy’, SIAM J. Numer. Anal. 14, 651–667.Google Scholar
Wang, Z., Bovik, A. C., Sheikh, H. R. and Simoncelli, E. P. (2004), ‘Image quality assessment: From error visibility to structural similarity’, IEEE Trans. Image Process. 13, 600–612.Google Scholar
Xu, Y. and Yin, W. (2013), ‘A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion’, SIAM J. Imaging Sci. 6, 1758–1789.Google Scholar
Xu, Y. and Yin, W. (2017), ‘A globally convergent algorithm for nonconvex optimization based on block coordinate update’, J. Sci. Comput. 72, 700–734.Google Scholar
Yang, Y., Ma, J. and Osher, S. (2013), ‘Seismic data reconstruction via matrix completion’, Inverse Probl. Imaging 7, 1379–1392.Google Scholar
Yin, W. (2010), ‘Analysis and generalizations of the linearized Bregman method’, SIAM J. Imaging Sci. 3, 856–877.Google Scholar
Yin, W., Osher, S., Goldfarb, D. and Darbon, J. (2008), ‘Bregman iterative algorithms for $\ell _{1}$ -minimization with applications to compressed sensing’, SIAM J. Imaging Sci. 1, 143–168.Google Scholar
Zach, C., Pock, T. and Bischof, H. (2007), A duality based approach for realtime TV-L 1 optical flow, Pattern Recognition: 29th DAGM Symposium, (Hamprecht, F. A. et al. , eds), Vol. 4713 of Lecture Notes in Computer Science, Springer, pp. 214–223.Google Scholar
Zeune, L., van Dalum, G., Terstappen, L. W., van Gils, S. A. and Brune, C. (2017), ‘Multiscale segmentation via Bregman distances and nonlinear spectral analysis’, SIAM J. Imaging Sci. 10, 111–146.Google Scholar
Zhao, F., Noll, D. C., Nielsen, J.-F. and Fessler, J. A. (2012), ‘Separate magnitude and phase regularization via compressed sensing’, IEEE Trans. Medical Imaging 31, 1713–1723.Google Scholar
Zhu, M. and Chan, T. (2008), An efficient primal–dual hybrid gradient algorithm for total variation image restoration. CAM Report 08-34, UCLA.Google Scholar