×

Higher-order total variation approaches and generalisations. (English) Zbl 1453.92154

Summary: Over the last decades, the total variation (TV) has evolved to be one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. When first introduced as a regulariser, higher-order generalisations of TV were soon proposed and studied with increasing interest, which led to a variety of different approaches being available today. We review several of these approaches, discussing aspects ranging from functional-analytic foundations to regularisation theory for linear inverse problems in Banach space, and provide a unified framework concerning well-posedness and convergence for vanishing noise level for respective Tikhonov regularisation. This includes general higher orders of TV, additive and infimal-convolution multi-order total variation, total generalised variation, and beyond. Further, numerical optimisation algorithms are developed and discussed that are suitable for solving the Tikhonov minimisation problem for all presented models. Focus is laid in particular on covering the whole pipeline starting at the discretisation of the problem and ending at concrete, implementable iterative procedures. A major part of this review is finally concerned with presenting examples and applications where higher-order TV approaches turned out to be beneficial. These applications range from classical inverse problems in imaging such as denoising, deconvolution, compressed sensing, optical-flow estimation and decompression, to image reconstruction in medical imaging and beyond, including magnetic resonance imaging, computed tomography, magnetic-resonance positron emission tomography, and electron tomography.

MSC:

92C55 Biomedical imaging and signal processing

Software:

KITTI; UNLocBoX
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adams R A 1975 Sobolev Spaces (New York: Academic) · Zbl 0314.46030
[2] Adluru G, Awate S P, Tasdizen T, Whitaker R T and DiBella E V 2007 Temporally constrained reconstruction of dynamic cardiac perfusion MRI Magn. Reson. Med.57 1027-36 · doi:10.1002/mrm.21248
[3] Al-Aleef A, Alekseev A, MacLaren I and Cockshott P 2015 Electron tomography based on a total generalized variation minimization reconstruction technique 31st Picture Coding Symposium Cairns, Australia
[4] Alberti G, Bouchitté G and Dal Maso G 2003 The calibration method for the Mumford-Shah functional and free-discontinuity problems Calc. Var. PDE16 299-333 · Zbl 1015.49008 · doi:10.1007/s005260100152
[5] Alter F, Durand S and Froment J 2005 Adapted total variation for artifact free decompression of JPEG images J. Math. Imaging Vis.23 199-211 · Zbl 1446.68175 · doi:10.1007/s10851-005-6467-9
[6] Amar M, Cicco V D and Fusco N 2008 Lower semicontinuity and relaxation results in BV for integral functionals with BV integrants ESAIM: Contr. Optim. Calc. Var.14 456-77 · Zbl 1149.49016 · doi:10.1051/cocv:2007061
[7] Ambrosio L, Fusco N and Pallara D 2000 Functions of Bounded Variation and Free Discontinuity Problems (Oxford: Oxford University Press) · Zbl 0957.49001
[8] Attouch H and Brezis H 1986 Duality for the sum of convex functions in general Banach spaces Asp. Math. Appl.34 125-33 · Zbl 0642.46004 · doi:10.1016/s0924-6509(09)70252-1
[9] Bačák M 2014 Computing medians and means in Hadamard spaces SIAM J. Optim.24 1542-66 · Zbl 1306.49046 · doi:10.1137/140953393
[10] Bačák M, Bergmann R, Steidl G and Weinmann A 2016 A second order non-smooth variational model for restoring manifold-valued images SIAM J. Sci. Comput.38 A567-97 · Zbl 1382.94007 · doi:10.1137/15m101988x
[11] Baker S, Scharstein D, Lewis J P, Roth S, Black M J and Szeliski R 2011 A database and evaluation methodology for optical flow Int. J. Comput. Vis.92 1-31 · doi:10.1007/s11263-010-0390-2
[12] Basser P J and Pierpaoli C 1996 Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI J. Magn. Reson. B 111 209-19 · doi:10.1006/jmrb.1996.0086
[13] Bauschke H H and Combettes P L 2017 Convex Analysis and Monotone Operator Theory in Hilbert Spaces(CMS Books in Mathematics) (Berlin: Springer) · Zbl 1359.26003 · doi:10.1007/978-3-319-48311-5
[14] Bergmann R, Fitschen J H, Persch J and Steidl G 2017 Infimal convolution coupling of first and second order differences on manifold-valued images Scale Space and Variational Methods in Computer Vision ed F Lauze, Y Dong and A B Dahl (Berlin: Springer) pp 447-59 · Zbl 1489.65037 · doi:10.1007/978-3-319-58771-4_36
[15] Bergmann R, Fitschen J H, Persch J and Steidl G 2018 Priors with coupled first and second order differences for manifold-valued image processing J. Math. Imaging Vis.60 1459-81 · Zbl 1436.49040 · doi:10.1007/s10851-018-0840-y
[16] Bergounioux M 2016 Mathematical analysis of a inf-convolution model for image processing J. Optim. Theory Appl.168 1-21 · Zbl 1332.65030 · doi:10.1007/s10957-015-0734-8
[17] Bergounioux M and Piffet L 2010 A second-order model for image denoising Set-Valued Anal.18 277-306 · Zbl 1203.94006 · doi:10.1007/s11228-010-0156-6
[18] Bernstein M A, King K F and Zhou X J 2004 Handbook of MRI Pulse Sequences (Amsterdam: Elsevier)
[19] Biggs N 1993 Algebraic Graph Theory(Cambridge Mathematical Library vol 67) (Cambridge: Cambridge University Press)
[20] Bilgic B, Fan A P, Polimeni J R, Cauley S F, Bianciardi M, Adalsteinsson E, Wald L L and Setsompop K 2014 Fast quantitative susceptibility mapping with L1-regularization and automatic parameter selection Magn. Reson. Med.72 1444-59 · doi:10.1002/mrm.25029
[21] Block K T, Uecker M and Frahm J 2007 Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint Magn. Reson. Med.57 1086-98 · doi:10.1002/mrm.21236
[22] Block T 2008 Advanced methods for radial data sampling in Magnetic Resonance Imaging PhD Thesis Georg-August-Universität Göttingen
[23] Borwein J and Lewis A 1991 Convergence of best entropy estimates SIAM J. Optim.1 191-205 · Zbl 0756.41037 · doi:10.1137/0801014
[24] Borzi A, Ito K and Kunisch K 2003 Optimal control formulation for determining optical flow SIAM J. Sci. Comput.24 818-47 · Zbl 1041.68103 · doi:10.1137/s1064827501386481
[25] Boyd S, Parikh N, Chu E, Peleato B and Eckstein J 2011 Distributed optimization and statistical learning via the alternating direction method of multipliers Found. Trends Mach. Learn.3 1-122 · Zbl 1229.90122 · doi:10.1561/2200000016
[26] Boyer C, Chambolle A, Castro Y D, Duval V, De Gournay F and Weiss P 2019 On representer theorems and convex regularization SIAM J. Optim.29 1260-81 · Zbl 1423.49036 · doi:10.1137/18m1200750
[27] Bredies K 2014 Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty Efficient Algorithms for Global Optimization Methods in Computer Vision(Lecture Notes in Computer Science vol 8293) ed A Bruhn, T Pock and X C Tai (Berlin: Springer) pp 44-77 · doi:10.1007/978-3-642-54774-4_3
[28] Bredies K 2013 Symmetric tensor fields of bounded deformation Ann. Mat. Pura Appl.192 815-51 · Zbl 1288.46024 · doi:10.1007/s10231-011-0248-4
[29] Bredies K and Carioni M 2020 Sparsity of solutions for variational inverse problems with finite-dimensional data Calculus of Variations and Partial Differential Equations59 14 · Zbl 1430.49036 · doi:10.1007/s00526-019-1658-1
[30] Bredies K and Fanzon S 2019 An optimal transport approach for solving dynamic inverse problems in spaces of measures (arXiv:1901.10162)
[31] Bredies K and Holler M 2012 A total variation-based JPEG decompression model SIAM J. Imaging Sci.5 366-93 · Zbl 1258.94015 · doi:10.1137/110833531
[32] Bredies K and Holler M 2014 Regularization of linear inverse problems with total generalized variation J. Inverse Ill-Posed Problems22 871-913 · Zbl 1302.65167 · doi:10.1515/jip-2013-0068
[33] Bredies K and Holler M 2015 A TGV-based framework for variational image decompression, zooming and reconstruction. Part I: analytics SIAM J. Imaging Sci.8 2814-50 · Zbl 1333.94006 · doi:10.1137/15m1023865
[34] Bredies K and Holler M 2015 A TGV-based framework for variational image decompression, zooming and reconstruction. Part II: numerics SIAM J. Imaging Sci.8 2851-86 · Zbl 1333.94007 · doi:10.1137/15m1023877
[35] Bredies K and Holler M 2015 Artifact-free variational MPEG decompression Scale Space and Variational Methods in Computer Vision(Lecture Notes in Computer Science vol 9087) (Berlin: Springer) pp 216-28 · Zbl 1444.94005 · doi:10.1007/978-3-319-18461-6_18
[36] Bredies K, Holler M, Storath M and Weinmann A 2018 Total generalized variation for manifold-valued data SIAM J. Imaging Sci.11 1785-848 · Zbl 1401.94010 · doi:10.1137/17m1147597
[37] Bredies K, Kunisch K and Pock T 2010 Total generalized variation SIAM J. Imaging Sci.3 492-526 · Zbl 1195.49025 · doi:10.1137/090769521
[38] Bredies K, Kunisch K and Valkonen T 2013 Properties of L1-TGV2: the one-dimensional case J. Math. Anal. Appl.398 438-54 · Zbl 1253.49024 · doi:10.1016/j.jmaa.2012.08.053
[39] Bredies K and Lorenz D 2018 Mathematical Image Processing (Berlin: Springer) · Zbl 1418.94001 · doi:10.1007/978-3-030-01458-2
[40] Bredies K and Sun H 2015 Preconditioned Douglas-Rachford splitting methods for convex-concave saddle-point problems SIAM J. Numer. Anal.53 421-44 · Zbl 1314.65084 · doi:10.1137/140965028
[41] Bredies K and Sun H 2015 Preconditioned Douglas-Rachford algorithms for TV- and TGV-regularized variational imaging problems J. Math. Imaging Vis.52 317-44 · Zbl 1343.94003 · doi:10.1007/s10851-015-0564-1
[42] Bredies K and Sun H 2016 Accelerated Douglas-Rachford methods for the solution of convex-concave saddle-point problems (arXiv:1604.06282)
[43] Bredies K and Sun H 2017 A proximal point analysis of the preconditioned alternating direction method of multipliers J. Optim. Theory Appl.173 878-907 · Zbl 1380.65101 · doi:10.1007/s10957-017-1112-5
[44] Bredies K and Vicente D 2019 A perfect reconstruction property for PDE-constrained total-variation minimization with application in quantitative susceptibility mapping ESAIM: Contr. Optim. Calc. Var.25 83 · Zbl 1437.35680 · doi:10.1051/cocv/2018009
[45] Briceño-Arias L and Combettes P 2011 A monotone+skew splitting model for composite monotone inclusions in duality SIAM J. Optim.21 1230-50 · Zbl 1239.47053 · doi:10.1137/10081602x
[46] Brown R W, Cheng Y C N, Haacke E M, Thompson M R and Venkatesan R 2014 Magnetic Resonance Imaging: Physical Principles and Sequence Design (New York: Wiley) · doi:10.1002/9781118633953
[47] Brox T, Bruhn A, Papenberg N and Weickert J 2004 High accuracy optical flow estimation based on a theory for warping Proc. ECCV (Berlin: Springer) pp 25-36 · Zbl 1098.68736 · doi:10.1007/978-3-540-24673-2_3
[48] Burger M and Osher S 2004 Convergence rates of convex variational regularization Inverse Problems20 1411-21 · Zbl 1068.65085 · doi:10.1088/0266-5611/20/5/005
[49] Calatroni L, Cao C, De Los Reyes J C, Schönlieb C B and Valkonen T 2016 Bilevel approaches for learning of variational imaging models Variational Methods in Imaging and Geometric Control(Radon Series on Computational and Applied Mathematics vol 18) ((Berlin: Walter de Gruyter)) pp 252-90 · Zbl 1468.94014
[50] Calderón A P and Zygmund A 1952 On the existence of certain singular integrals Acta Math.88 85-139 · Zbl 0047.10201 · doi:10.1007/bf02392130
[51] Callaghan P T 1991 Principles of Nuclear Magnetic Resonance Microscopy (Clarendon: Oxford)
[52] Candès E J, Romberg J and Tao T 2006 Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information IEEE Trans. Inf. Theory52 489-509 · Zbl 1231.94017 · doi:10.1109/tit.2005.862083
[53] Candès E J, Romberg J K and Tao T 2006 Stable signal recovery from incomplete and inaccurate measurements Commun. Pure Appl. Math.59 1207-23 · Zbl 1098.94009 · doi:10.1002/cpa.20124
[54] Caselles V, Chambolle A and Novaga M 2007 The discontinuity set of solutions of the TV denoising problem and some extensions Multiscale Modelling Simul.6 879-94 · Zbl 1145.49024 · doi:10.1137/070683003
[55] Chambolle A 2001 Convex representation for lower semicontinuous envelopes of functionals in L1 J. Convex Anal.8 149-70 · Zbl 0977.49007
[56] Chambolle A 2004 An algorithm for total variation minimization and applications J. Math. Imaging Vis.20 89-97 · Zbl 1366.94048 · doi:10.1023/b:jmiv.0000011320.81911.38
[57] Chambolle A, Caselles V, Cremers D, Novaga M and Pock T 2010 An introduction to total variation for image analysis Theor. Found. Numer. Methods Sparse Recovery9 227 · doi:10.1515/9783110226157.263
[58] Chambolle A, Duval V, Peyré G and Poon C 2016 Geometric properties of solutions to the total variation denoising problem Inverse Problems33 015002 · Zbl 1369.94020 · doi:10.1088/0266-5611/33/1/015002
[59] Chambolle A and Lions P L 1997 Image recovery via total variation minimization and related problems Numer. Math.76 167-88 · Zbl 0874.68299 · doi:10.1007/s002110050258
[60] Chambolle A and Pock T 2011 A first-order primal-dual algorithm for convex problems with applications to imaging J. Math. Imaging Vis.40 120-45 · Zbl 1255.68217 · doi:10.1007/s10851-010-0251-1
[61] Chambolle A and Pock T 2016 On the ergodic convergence rates of a first-order primal-dual algorithm Math. Program.159 253-87 · Zbl 1350.49035 · doi:10.1007/s10107-015-0957-3
[62] Chan T F and Esedōlu S 2005 Aspects of total variation regularized L1 function approximation SIAM J. Appl. Math.65 1817-37 · Zbl 1096.94004 · doi:10.1137/040604297
[63] Chatnuntawech I et al 2017 Single-step quantitative susceptibility mapping with variational penalties NMR Biomed.30 e3570 · doi:10.1002/nbm.3570
[64] Chen K and Lorenz D A 2011 Image sequence interpolation using optimal control J. Math. Imaging Vis.41 222-38 · Zbl 1255.68220 · doi:10.1007/s10851-011-0274-2
[65] Cohen A, Daubechies I and Feauveau J C 1992 Biorthogonal bases of compactly supported wavelets Commun. Pure Appl. Math.45 485-560 · Zbl 0776.42020 · doi:10.1002/cpa.3160450502
[66] Combettes P L and Pesquet J C 2011 Proximal splitting methods in signal processing Fixed-Point Algorithms for Inverse Problems in Science and Engineering (Berlin: Springer) pp 185-212 · Zbl 1242.90160 · doi:10.1007/978-1-4419-9569-8_10
[67] Cory D G and Garroway A N 1990 Measurement of translational displacement probabilities by NMR: an indicator of compartmentation Magn. Reson. Med.14 435-44 · doi:10.1002/mrm.1910140303
[68] Cremers D and Strekalovskiy E 2013 Total cyclic variation and generalizations J. Math. Imaging Vis.47 258-77 · Zbl 1298.94011 · doi:10.1007/s10851-012-0396-1
[69] Daubechies I, Defrise M and De Mol C 2004 An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Commun. Pure Appl. Math.57 1413-57 · Zbl 1077.65055 · doi:10.1002/cpa.20042
[70] Davoli E and Liu P 2018 One dimensional fractional order TGV: gamma-convergence and bilevel training scheme Commun. Math. Sci.16 213-37 · Zbl 1441.94007 · doi:10.4310/cms.2018.v16.n1.a10
[71] De los Reyes J C, Schönlieb C B and Valkonen T 2016 Bilevel parameter learning for higher-order total variation regularisation models J. Math. Imaging Vis.57 1-25 · Zbl 1425.94010 · doi:10.1007/s10851-016-0662-8
[72] Deistung A, Schweser F and Reichenbach J R 2017 Overview of quantitative susceptibility mapping NMR Biomed.30 e3569 · doi:10.1002/nbm.3569
[73] Demengel F 1984 Fonctions à hessien borné Ann. Inst. Fourier34 155-90 (in French) · Zbl 0525.46020 · doi:10.5802/aif.969
[74] Deng W and Yin W 2016 On the global and linear convergence of the generalized alternating direction method of multipliers J. Sci. Comput.66 889-916 · Zbl 1379.65036 · doi:10.1007/s10915-015-0048-x
[75] Deslauriers G, Dubuc S and Lemire D 1999 Une famille d’ondelettes biorthogonales sur l’intervalle obtenue par un schéma d’interpolation itérative Ann. Sci. Math. Quebec23 37-48 (in French) · Zbl 1100.42502
[76] Di Nezza E, Palatucci G and Valdinoci E 2012 Hitchhiker’s guide to the fractional Sobolev spaces Bull. Sci. Math.136 521-73 · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004
[77] Donoho D L 2006 Compressed sensing IEEE Trans. Inf. Theory52 1289-306 · Zbl 1288.94016 · doi:10.1109/tit.2006.871582
[78] Duarte M F, Davenport M A, Takhar D, Laska J, Sun T, Kelly K and Baraniuk R G 2008 Single-pixel imaging via compressive sampling IEEE Signal Process. Mag.25 83-91 · doi:10.1109/msp.2007.914730
[79] Duran J, Möller M, Sbert C and Cremers D 2016 Collaborative total variation: a general framework for vectorial TV models SIAM J. Imaging Sci.9 116-51 · Zbl 1381.94016 · doi:10.1137/15m102873x
[80] Eckstein J and Bertsekas D P 1992 On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators Math. Program.55 293-318 · Zbl 0765.90073 · doi:10.1007/bf01581204
[81] Ehrhardt M et al 2016 PET reconstruction with an anatomical MRI prior using parallel level sets IEEE Trans. Med. Imaging35 2189-99 · doi:10.1109/tmi.2016.2549601
[82] Ehrhardt M, Thielemans K, Pizarro L, Atkinson D, Ourselin S, Hutton B and Arridge S 2015 Joint reconstruction of PET-MRI by exploiting structural similarity Inverse Problems31 015001 · Zbl 1320.92057 · doi:10.1088/0266-5611/31/1/015001
[83] Ekeland I and Témam R 1999 Convex Analysis and Variational Problems (Philadelphia, PA: SIAM) · Zbl 0939.49002 · doi:10.1137/1.9781611971088
[84] Engl H W, Hanke M and Neubauer A 2000 Regularization of Inverse Problems(Mathematics and Its Applications vol 375) (Berlin: Springer)
[85] Evans L C and Gariepy R F 1992 Measure Theory and Fine Properties of Functions(Mathematical Chemistry Series) (London: Taylor and Francis) · Zbl 0804.28001
[86] Feng L, Srichai M B, Lim R P, Harrison A, King W, Adluru G, Dibella E V, Sodickson D K, Otazo R and Kim D 2013 Highly accelerated real-time cardiac cine MRI using k-t SPARSE-SENSE Magn. Reson. Med.70 64-74 · doi:10.1002/mrm.24440
[87] Fessler J and Sutton B 2003 Nonuniform fast Fourier transforms using min-max interpolation IEEE Trans. Signal Process.51 560-74 · Zbl 1369.94048 · doi:10.1109/tsp.2002.807005
[88] Flohr T G et al 2006 First performance evaluation of a dual-source CT (DSCT) system Eur. Radiol.16 1405 · doi:10.1007/s00330-006-0158-9
[89] Gabay D 1983 Applications of the method of multipliers to variational inequalities Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems(Studies in Mathematics and Its Applications vol 15) ed M Fortin and R Glowinski (Berlin: Elsevier) pp 299-331 · Zbl 0525.65045 · doi:10.1016/S0168-2024(08)70034-1
[90] Gao Y and Bredies K 2018 Infimal convolution of oscillation total generalized variation for the recovery of images with structured texture SIAM J. Imaging Sci.11 2021-63 · Zbl 1419.94004 · doi:10.1137/17m1153960
[91] Geiger A, Lenz P and Urtasun R 2012 Are we ready for autonomous driving? The KITTI vision benchmark suite IEEE Conf. on Computer Vision and Pattern Recognition pp 3354-61 · doi:10.1109/CVPR.2012.6248074
[92] Gheorghita I P 2020 Alina’s eye https://flickr.com/photos/angel_ina/3201337190 CC BY 2.0
[93] Gilbert P 1972 Iterative methods for the three-dimensional reconstruction of an object from projections J. Theor. Biol.36 105-17 · doi:10.1016/0022-5193(72)90180-4
[94] Gilboa G and Osher S 2008 Nonlocal operators with applications to image processing Multiscale Modelling Simul.7 1005-28 · Zbl 1181.35006 · doi:10.1137/070698592
[95] Glowinski R and Marroco A 1975 Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires ESAIM: Math. Modelling Numer. Anal.9 41-76 (in French) · Zbl 0368.65053 · doi:10.1051/m2an/197509r200411
[96] Goris B, Van den Broek W, Batenburg K J, Heidari Mezerji H and Bals S 2012 Electron tomography based on a total variation minimization reconstruction technique Ultramicroscopy113 120-30 · doi:10.1016/j.ultramic.2011.11.004
[97] Griswold M A, Jakob P M, Heidemann R M, Nittka M, Jellus V, Wang J, Kiefer B and Haase A 2002 Generalized autocalibrating partially parallel acquisitions (GRAPPA) Magn. Reson. Med.47 1202-10 · doi:10.1002/mrm.10171
[98] Grohs P and Sprecher M 2016 Total variation regularization on Riemannian manifolds by iteratively reweighted minimization Inf. Inference5 353-78 · Zbl 1382.94013 · doi:10.1093/imaiai/iaw011
[99] Hackbusch W 2012 Tensor Spaces and Numerical Tensor Calculus (Berlin: Springer) · Zbl 1244.65061 · doi:10.1007/978-3-642-28027-6
[100] Bottou L, Bengio Y, Haffner P, Howard P G, LeCun Y and Simard P 1998 High quality document image compression with DjVu J. Electron. Imaging7 410-25 · doi:10.1117/1.482609
[101] Hammernik K, Klatzer T, Kobler E, Recht M P, Sodickson D K, Pock T and Knoll F 2018 Learning a variational network for reconstruction of accelerated MRI data Magn. Reson. Med.79 3055-71 · doi:10.1002/mrm.26977
[102] He B and Yuan X 2012 Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective SIAM J. Imaging Sci.5 119-49 · Zbl 1250.90066 · doi:10.1137/100814494
[103] Hinterberger W and Scherzer O 2001 Models for image interpolation based on the optical flow Computing66 231-47 · Zbl 1004.49029 · doi:10.1007/s006070170023
[104] Hinterberger W and Scherzer O 2006 Variational methods on the space of functions of bounded Hessian for convexification and denoising Computing76 109-33 · Zbl 1098.49022 · doi:10.1007/s00607-005-0119-1
[105] Hintermüller M, Holler M and Papafitsoros K 2018 A function space framework for structural total variation regularization with applications in inverse problems Inverse Problems34 064002 · Zbl 1436.65070 · doi:10.1088/1361-6420/aab586
[106] Hofmann B, Kaltenbacher B, Pöschl C and Scherzer O 2007 A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators Inverse Problems23 987-1010 · Zbl 1131.65046 · doi:10.1088/0266-5611/23/3/009
[107] Holler M, Huber R and Knoll F 2018 Coupled regularization with multiple data discrepancies Inverse Problems34 084003 · Zbl 1439.65072 · doi:10.1088/1361-6420/aac539
[108] Holler M and Kazimierski K S 2018 Variational decompression of image data from DjVu encoded files IEEE Trans. Image Process.27 490-9 · Zbl 1409.94227 · doi:10.1109/tip.2017.2760513
[109] 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 2258-300 · Zbl 1308.94019 · doi:10.1137/130948793
[110] Horn B K P and Schunck B G 1981 Determining optical flow Artif. Intell.17 185-203 · Zbl 1497.68488 · doi:10.1016/0004-3702(81)90024-2
[111] Huber R, Haberfehlner G, Holler M, Kothleitner G and Bredies K 2019 Total generalized variation regularization for multi-modal electron tomography Nanoscale11 5617-32 · doi:10.1039/c8nr09058k
[112] Iglesias J A, Mercier G and Scherzer O 2018 A note on convergence of solutions of total variation regularized linear inverse problems Inverse Problems34 055011 · Zbl 1516.35514 · doi:10.1088/1361-6420/aab92a
[113] Ito K and Jin B 2014 Inverse Problems: Tikhonov Theory and Algorithms(Series On Applied Mathematics vol 22) (Singapore: World Scientific) · Zbl 1306.65210 · doi:10.1142/9120
[114] Ivanov V K 1962 On linear problems which are not well-posed Dokl. Akad. Nauk SSSR145 270-2 · Zbl 0201.46701
[115] Johnson T R C et al 2007 Material differentiation by dual energy CT: initial experience Eur. Radiol.17 1510-7 · doi:10.1007/s00330-006-0517-6
[116] Jonsson E, Huang S C and Chan T 1998 Total variation regularization in positron emission tomography CAM Report 98-48 (UCLA)
[117] Jung H, Sung K, Nayak K S, Kim E Y and Ye J C 2009 k-t FOCUSS: a general compressed sensing framework for high resolution dynamic MRI Magn. Reson. Med.61 103-16 · doi:10.1002/mrm.21757
[118] Keeling S L and Ring W 2005 Medical image registration and interpolation by optical flow with maximal rigidity J. Math. Imaging Vis.23 47-65 · Zbl 1478.94053 · doi:10.1007/s10851-005-4967-2
[119] Kheyfets A, Miller W A and Newton G A 2000 Schild’s ladder parallel transport procedure for an arbitrary connection Int. J. Theor. Phys.39 2891-8 · Zbl 0979.83008 · doi:10.1023/a:1026473418439
[120] Kikuchi N and Oden J 1988 Contact Problems in Elasticity (Philadelphia, PA: SIAM) · Zbl 0685.73002 · doi:10.1137/1.9781611970845
[121] Knoll F, Bredies K, Pock T and Stollberger R 2011 Second order total generalized variation (TGV) for MRI Magn. Reson. Med.65 480-91 · doi:10.1002/mrm.22595
[122] 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. Med. Imaging36 1-16 · doi:10.1109/tmi.2016.2564989
[123] Knoll F, Holler M, Koesters T and Sodickson D K 2015 Joint MR-PET reconstruction using vector valued total generalized variation Proc. Int. Society for Magnetic Resonance in Medicine vol 23 p 3424
[124] Komodakis N and Pesquet J C 2015 Playing with duality: an overview of recent primal-dual approaches for solving large-scale optimization problems IEEE Signal Process. Mag.32 31-54 · doi:10.1109/msp.2014.2377273
[125] Kongskov R D and Dong Y 2018 Tomographic reconstruction methods for decomposing directional components Inverse Problems Imaging12 1429-42 · Zbl 1404.49024 · doi:10.3934/ipi.2018060
[126] Langkammer C, Bredies K, Poser B A, Barth M, Reishofer G, Fan A P, Bilgic B, Fazekas F, Mainero C and Ropele S 2015 Fast quantitative susceptibility mapping using 3D EPI and total generalized variation NeuroImage111 622-30 · doi:10.1016/j.neuroimage.2015.02.041
[127] Lebrun M, Colom M, Buades A and Morel J M 2012 Secrets of image denoising cuisine Acta Numer.21 475-576 · Zbl 1260.94016 · doi:10.1017/s0962492912000062
[128] Lefkimmiatis S, Ward J and Unser M 2013 Hessian Schatten-norm regularization for linear inverse problems IEEE Trans. Image Process.22 1873-88 · Zbl 1373.94229 · doi:10.1109/tip.2013.2237919
[129] Lellmann J, Strekalovskiy E, Koetter S and Cremers D 2013 Total variation regularization for functions with values in a manifold Int. Conf. on Computer Vision (IEEE) pp 2944-51 · doi:10.1109/ICCV.2013.366
[130] Lions P L and Mercier B 1979 Splitting algorithms for the sum of two nonlinear operators SIAM J. Numer. Anal.16 964-79 · Zbl 0426.65050 · doi:10.1137/0716071
[131] Lu G M, Wu S Y, Yeh B M and Zhang L J 2010 Dual-energy computed tomography in pulmonary embolism Br. J. Radiol.83 707-18 · doi:10.1259/bjr/16337436
[132] Lustig M, Donoho D L and Pauly J M 2007 Sparse MRI: the application of compressed sensing for rapid MR imaging Magn. Reson. Med.58 1182-95 · doi:10.1002/mrm.21391
[133] Ma J, März M, Funk S, Schulz-Menger J, Kutyniok G, Schaeffter T and Kolbitsch C 2018 Shearlet-based compressed sensing for fast 3D cardiac MR imaging using iterative reweighting Phys. Med. Biol.63 235004 · doi:10.1088/1361-6560/aaea04
[134] Markoe A 2006 Analytic Tomography(Encyclopedia of Mathematics and Its Applications) (Cambridge: Cambridge University Press) · Zbl 1341.44001 · doi:10.1017/CBO9780511530012
[135] Möllenhoff T, Laude E, Moeller M, Lellmann J and Cremers D 2016 Sublabel-accurate relaxation of nonconvex energies The IEEE Conf. on Computer Vision and Pattern Recognition (CVPR) pp 3948-56
[136] Morozov V 1967 Choice of parameter for the solution of functional equations by the regularization method Dokl. Akad. Nauk SSSR175 1225-8 · Zbl 0189.47501
[137] Müller T, Rabe C, Rannacher J, Franke U and Mester R 2011 Illumination-robust dense optical flow using census signatures Pattern Recognition ed R Mester and M Felsberg (Berlin: Springer) pp 236-45 · doi:10.1007/978-3-642-23123-0_24
[138] Nesterov Y E 2004 Introductory Lectures on Convex Optimization(Applied Optimization vol 87) (Berlin: Springer) · Zbl 1086.90045 · doi:10.1007/978-1-4419-8853-9
[139] Nikolova M 2000 Local strong homogeneity of a regularized estimator SIAM J. Appl. Math.61 633-58 · Zbl 0991.94015 · doi:10.1137/s0036139997327794
[140] Oldham K and Spanier J 1974 The Fractional Calculus (Amsterdam: Elsevier Science) · Zbl 0292.26011
[141] Otazo R, Candès E and Sodickson D K 2014 Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components Magn. Reson. Med.73 1125-36 · doi:10.1002/mrm.25240
[142] Papafitsoros K and Schönlieb C B 2014 A combined first and second order variational approach for image reconstruction J. Math. Imaging Vis.48 308-38 · Zbl 1362.94009 · doi:10.1007/s10851-013-0445-4
[143] Pock T and Chambolle A 2011 Diagonal preconditioning for first order primal-dual algorithms in convex optimization IEEE Int. Conf. on Computer Vision (ICCV) pp 1762-9 · doi:10.1109/ICCV.2011.6126441
[144] Pock T, Cremers D, Bischof H and Chambolle A 2009 An algorithm for minimizing the Mumford-Shah functional IEEE Int. Conf. on Computer Vision (ICCV) pp 1133-40 · doi:10.1109/ICCV.2009.5459348
[145] Pock T, Cremers D, Bischof H and Chambolle A 2010 Global solutions of variational models with convex regularization SIAM J. Imaging Sci.3 1122-45 · Zbl 1202.49031 · doi:10.1137/090757617
[146] Podlubny I 1998 Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (Amsterdam: Elsevier Science) · Zbl 0922.45001
[147] Poser B, Koopmans P, Witzel T, Wald L and Barth M 2010 Three dimensional echo-planar imaging at 7 Tesla NeuroImage51 261-6 · doi:10.1016/j.neuroimage.2010.01.108
[148] Pruessmann K P, Weiger M, Boernert P and Boesiger P 2001 Advances in sensitivity encoding with arbitrary k-space trajectories Magn. Reson. Med.46 638-51 · doi:10.1002/mrm.1241
[149] Pruessmann K P, Weiger M, Scheidegger M B and Boesiger P 1999 SENSE: sensitivity encoding for fast MRI Magn. Reson. Med.42 952-62 · doi:10.1002/(sici)1522-2594(199911)42:5<952::aid-mrm16>3.0.co;2-s
[150] Qin C, Schlemper J, Caballero J, Price A N, Hajnal J V and Rueckert D 2019 Convolutional recurrent neural networks for dynamic MR image reconstruction IEEE Trans. Med. Imaging38 280-90 · doi:10.1109/tmi.2018.2863670
[151] Rahmim A, Qi J and Sossi V 2013 Resolution modeling in PET imaging: theory, practice, benefits, and pitfalls Med. Phys.40 064301 · doi:10.1118/1.4800806
[152] Ranftl R, Gehrig S, Pock T and Bischof H 2012 Pushing the limits of stereo using variational stereo estimation IEEE Intelligent Vehicles Symp. 401-7 · doi:10.1109/IVS.2012.6232171
[153] Ranftl R 2013 Alternating minimization + image-driven TGV and census-based data term KITTI Vision Benchmark Suite http://www.cvlibs.net/datasets/kitti/eval_stereo_flow_detail.php?benchmark=stereo&error=3&eval=all&result=a7b77e8fddfa90949754d01a1d09b6d8bbabdbb7 CC BY-NC-SA 3.0
[154] Ranftl R, Bredies K and Pock T 2014 Non-local total generalized variation for optical flow estimation Computer Vision—ECCV ed D Fleet, T Pajdla, B Schiele and T Tuytelaars (Berlin: Springer) pp 439-54 · doi:10.1007/978-3-319-10590-1_29
[155] Ranftl R, Pock T and Bischof H 2013 Minimizing TGV-based variational models with non-convex data terms Scale Space and Variational Methods in Computer Vision (Berlin: Springer) pp 282-93 · Zbl 1362.68014 · doi:10.1007/978-3-642-38267-3_24
[156] Recht B, Fazel M and Parrilo P A 2010 Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization SIAM Rev.52 471-501 · Zbl 1198.90321 · doi:10.1137/070697835
[157] Rice Single-Pixel Camera Project 2007 Compressive-sensing camera https://web.archive.org/web/20161231160210/http://dsp.rice.edu/cscamera“>http://dsp.rice.edu/cscamera”>https://web.archive.org/web/20161231160210/http://dsp.rice.edu/cscamera archived
[158] Ring W 2000 Structural properties of solutions to total variation regularization problems ESAIM: Math. Modelling Numer. Anal.34 799-810 · Zbl 1018.49021 · doi:10.1051/m2an:2000104
[159] Robinson S D, Bredies K, Khabipova D, Dymerska B, Marques J P and Schweser F 2017 An illustrated comparison of processing methods for MR phase imaging and QSM: combining array coil signals and phase unwrapping NMR Biomed.30 e3601 · doi:10.1002/nbm.3601
[160] Rockafellar R T 1997 Convex Analysis Princeton Landmarks in Mathematics (Princeton, NJ: Princeton University Press) · Zbl 0932.90001
[161] Rockafellar R T 1976 Monotone operators and the proximal point algorithm SIAM J. Control Optim.14 877-98 · Zbl 0358.90053 · doi:10.1137/0314056
[162] Rudin L I, Osher S and Fatemi E 1992 Nonlinear total variation based noise removal algorithms Physica D 60 259-68 · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-f
[163] Rudin W 1987 Real and Complex Analysis (New Delhi: Tata McGraw-Hill) · Zbl 0925.00005
[164] Sapiro G and Ringach D L 1996 Anisotropic diffusion of multivalued images with applications to color filtering IEEE Trans. Image Process.5 1582-6 · doi:10.1109/83.541429
[165] Sawatzky A, Brune C, Kösters T, Wübbeling F and Burger M 2013 EM-TV methods for inverse problems with Poisson noise Level Set and PDE Based Reconstruction Methods in Imaging (Berlin: Springer) pp 71-142 · Zbl 1342.94026 · doi:10.1007/978-3-319-01712-9_2
[166] Schloegl M, Holler M, Bredies K and Stollberger R 2015 A variational approach for coil-sensitivity estimation for undersampled phase-sensitive dynamic MRI reconstruction Proc. Int. Society for Magnetic Resonance in Medicine vol 23 p 3692
[167] Schloegl M, Holler M, Schwarzl A, Bredies K and Stollberger R 2017 nfimal convolution of total generalized variation functionals for dynamic MRI Magn. Reson. Med.78 142-55 · doi:10.1002/mrm.26352
[168] Schofield M A and Zhu Y 2003 Fast phase unwrapping algorithm for interferometric applications Opt. Lett.28 1194-6 · doi:10.1364/ol.28.001194
[169] Schramm G, Holler M, Rezaei A, Vunckx K, Knoll F, Bredies K, Boada F and Nuyts J 2017 Evaluation of parallel level sets and Bowsher’s method as segmentation-free anatomical priors for time-of-flight PET reconstruction IEEE Trans. Med. Imaging37 590-603 · doi:10.1109/tmi.2017.2767940
[170] Schuster T, Kaltenbacher B, Hofmann B and Kazimierski K S 2012 Regularization Methods in Banach Spaces (Berlin: Walter de Gruyter) · Zbl 1259.65087 · doi:10.1515/9783110255720
[171] Schwarzl A and Schloegl M 2019 Accelerated Variational Dynamic MRI Reconstruction (AVIONIC) (Version v1.0) Zenodo (http://doi.org/10.5281/zenodo.3193409)
[172] Schweser F, Deistung A and Reichenbach J R 2016 Foundations of MRI phase imaging and processing for quantitative susceptibility mapping (QSM) Z. Med. Phys.26 6-34 · doi:10.1016/j.zemedi.2015.10.002
[173] Schweser F, Robinson S D, de Rochefort L, Li W and Bredies K 2017 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 e3604 · doi:10.1002/nbm.3604
[174] Setzer S and Steidl G 2008 Variational methods with higher order derivatives in image processing Approximation XII (Brentwood, TN: Nashboro Press) pp 360-86 · Zbl 1175.68520
[175] Setzer S, Steidl G and Teuber T 2011 Infimal convolution regularizations with discrete ℓ1-type functionals Commun. Math. Sci.9 797-827 · Zbl 1269.49063 · doi:10.4310/cms.2011.v9.n3.a7
[176] Shepp L and Vardi Y 1982 Maximum likelihood reconstruction for emission tomography IEEE Trans. Med. Imaging1 113-22 · doi:10.1109/tmi.1982.4307558
[177] Shmueli K, de Zwart J A, van Gelderen P, Li T Q, Dodd S J and Duyn J H 2009 Magnetic susceptibility mapping of brain tissue in vivo using MRI phase data Magn. Reson. Med.62 1510-22 · doi:10.1002/mrm.22135
[178] Showalter R E 1997 Monotone Operators in Banach Space and Nonlinear Partial Differential Equations(Mathematical Surveys and Monographs vol 49) (Providence, RI: American Mathematical Society) · Zbl 0870.35004
[179] Sodickson D K and Manning W J 1997 Simultaneous acquisition of spatial harmonics (SMASH): fast imaging with radiofrequency coil arrays Magn. Reson. Med.38 591-603 · doi:10.1002/mrm.1910380414
[180] Stejskal E O and Tanner J E 1965 Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient J. Chem. Phys.42 288-92 · doi:10.1063/1.1695690
[181] Strecke M and Goldluecke B 2019 Sublabel-accurate convex relaxation with total generalized variation regularization Pattern Recognition ed T Brox, A Bruhn and M Fritz (Berlin: Springer) pp 263-77 · doi:10.1007/978-3-030-12939-2_19
[182] Tikhonov A N, Leonov A S and Yagola A G 1998 Nonlinear Ill-Posed Problems (London: Chapman and Hall) · Zbl 0920.65038 · doi:10.1007/978-94-017-5167-4
[183] Torrey H C 1956 Bloch equations with diffusion terms Phys. Rev.104 563-5 · doi:10.1103/physrev.104.563
[184] Tuch D S 2004 Q-ball imaging Magn. Reson. Med.52 1358-72 · doi:10.1002/mrm.20279
[185] Uecker M, Lai P, Murphy M J, Virtue P, Elad M, Pauly J M, Vasanawala S S and Lustig M 2014 ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA Magn. Reson. Med.71 990-1001 · doi:10.1002/mrm.24751
[186] Valkonen T, Bredies K and Knoll F 2013 TGV for diffusion tensors: a comparison of fidelity functions J. Inverse Ill-Posed Problems21 355-77 · Zbl 1301.92046 · doi:10.1515/jip-2013-0005
[187] Valkonen T, Bredies K and Knoll F 2013 Total generalized variation in diffusion tensor imaging SIAM J. Imaging Sci.6 487-525 · Zbl 1322.94024 · doi:10.1137/120867172
[188] Vogel C, Roth S and Schindler K 2013 An evaluation of data costs for optical flow GCPR (Berlin: Springer) pp 343-53 · doi:10.1007/978-3-642-40602-7_37
[189] Vunckx K, Atre A, Baete K, Reilhac A, Deroose C, Laere K V and Nuyts J 2012 Evaluation of three MRI-based anatomical priors for quantitative PET brain imaging IEEE Tran. Med. Imaging31 599-612 · doi:10.1109/tmi.2011.2173766
[190] Walsh D O, Gmitro A F and Marcellin M W 2000 Adaptive reconstruction of phased array mr imagery Magn. Reson. Med.43 682-90 · doi:10.1002/(sici)1522-2594(200005)43:5<682::aid-mrm10>3.0.co;2-g
[191] Wang J and Lucier B J 2011 Error bounds for finite-difference methods for Rudin-Osher-Fatemi image smoothing SIAM J. Numer. Anal.49 845-68 · Zbl 1339.94013 · doi:10.1137/090769594
[192] Weinmann A, Demaret L and Storath M 2014 Total variation regularization for manifold-valued data SIAM J. Imaging Sci.7 2226-57 · Zbl 1309.65071 · doi:10.1137/130951075
[193] Werlberger M 2012 Convex approaches for high performance video processing PhD Thesis Graz University of Technology
[194] Williams B M, Zhang J and Chen K 2016 A new image deconvolution method with fractional regularisation J. Algorithms Comput. Technol.10 265-76 · doi:10.1177/1748301816660439
[195] Zabih R and Woodfill J 1994 Non-parametric local transforms for computing visual correspondence Computer Vision—ECCV ’94 ed J O Eklundh (Berlin: Springer) pp 151-8 · doi:10.1007/BFb0028345
[196] Zach C, Pock T and Bischof H 2007 A duality based approach for realtime TV-L1 optical flow Pattern Recognition ed F A Hamprecht, C Schnörr and B Jähne (Berlin: Springer) pp 214-23 · doi:10.1007/978-3-540-74936-3_22
[197] Zaitsev M, Maclaren J and Herbst M 2015 Motion artifacts in MRI: a complex problem with many partial solutions J. Magn. Reson. Imaging42 887-901 · doi:10.1002/jmri.24850
[198] Zalinescu C 2002 Convex Analysis in General Vector Spaces (Singapore: World Scientific) · Zbl 1023.46003 · doi:10.1142/5021
[199] Zhang J and Chen K 2015 A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution SIAM J. Imaging Sci.8 2487-518 · Zbl 1327.62388 · doi:10.1137/14097121x
[200] Zhong S 1997 Image coding with optimal reconstruction Int. Conf. on Image Processingvol 1 pp 161-4 · doi:10.1109/ICIP.1997.647412
[201] Zhong Z, Palenstijn W J, Adler J and Batenburg K J 2018 EDS tomographic reconstruction regularized by total nuclear variation joined with HAADF-STEM tomography Ultramicroscopy191 34-43 · doi:10.1016/j.ultramic.2018.04.011
[202] Zhong Z, Palenstijn W J, Viganò N R and Batenburg K J 2018 Numerical methods for low-dose EDS tomography Ultramicroscopy194 133-42 · doi:10.1016/j.ultramic.2018.08.003
[203] Zhu M and Chan T 2008 An efficient primal-dual hybrid gradient algorithm for total variation image restoration CAM Report 08-34 (UCLA)
[204] Ziemer W P 2012 Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation Graduate Texts in Mathematics (New York: Springer)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.