Regularization of inverse problems via time discrete geodesics in image spaces. (English) Zbl 1448.94027
Summary: This paper addresses the solution of inverse problems in imaging given an additional reference image. We combine a modification of the discrete geodesic path model for image metamorphosis with a variational model, actually the \(L^2\)-\(TV\) model, for image reconstruction. We prove that the space continuous model has a minimizer which depends in a stable way from the input data. Two minimization procedures which alternate over the involved sequences of deformations and images in different ways are proposed. The updates with respect to the image sequence exploit recent algorithms from convex analysis to minimize the \(L^2\)-\(TV\) functional. For the numerical computation we apply a finite difference approach on staggered grids together with a multilevel strategy. We present proof-of-the-concept numerical results for sparse and limited angle computerized tomography as well as for superresolution demonstrating the power of the method.
MSC:
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 65J22 | Numerical solution to inverse problems in abstract spaces |
References:
| [1] | Alt H W 2002 Lineare Funktionalanalysis: Eine Anwendungsorientierte Einführung vol 6 (Berlin: Springer) · Zbl 1098.46500 |
| [2] | Ambrosio L, Fusco N and Pallara D 2000 Functions of Bounded Variation and Free Discontinuity Problems (Oxford: Oxford University Press) · Zbl 0957.49001 |
| [3] | Ball J M 1981 Global invertibility of Sobolev functions and the interpenetration of matter Proc. R. Soc. Edinburgh A 88 315-28 · Zbl 0478.46032 · doi:10.1017/S030821050002014X |
| [4] | Beck A 2017 First-Order Methods in Optimization vol 25 (Philadelphia, PA: SIAM) · Zbl 1384.65033 · doi:10.1137/1.9781611974997 |
| [5] | Beck A and Tetruashvili L 2013 On the convergence of block coordinate descent type methods SIAM J. Optim.23 2037-60 · Zbl 1297.90113 · doi:10.1137/120887679 |
| [6] | Beg M F, Miller M I, Trouvé A and Younes L 2005 Computing large deformation metric mappings via geodesic flows of diffeomorphisms Int. J. Comput. Vis.61 139-57 · Zbl 1477.68459 · doi:10.1023/B:VISI.0000043755.93987.aa |
| [7] | Berkels B, Effland A and Rumpf M 2015 Time discrete geodesic paths in the space of images SIAM J. Imaging Sci.8 1457-88 · Zbl 1325.65031 · doi:10.1137/140970719 |
| [8] | Bertolazzi E Splines toolbox https://github.com/ebertolazzi/Splines |
| [9] | Bolte J, Sabach S and Teboulle M 2014 Proximal alternating linearized minimization for nonconvex and nonsmooth problems Math. Program.146 459-94 · Zbl 1297.90125 · doi:10.1007/s10107-013-0701-9 |
| [10] | Borg L, Frikel J, Sauer-Jorgensen J and Quinto E T 2018 Full characterization of reconstruction artifacts from arbitrary incomplete x-ray CT data preprint (arXiv:1701.03055v3) · Zbl 1439.44004 |
| [11] | Bredies K and Lorenz D 2011 Mathematische Bildverarbeitung (Braunschweig: Vieweg) · Zbl 1220.68001 · doi:10.1007/978-3-8348-9814-2 |
| [12] | Burger M, Sawatzky A and Steidl G 2017 First order algorithms in variational image processing Operator Splittings and Alternating Direction Methods ed S O R Glowinski and W Yin (New York: Springer) |
| [13] | Candés E J, Romberg J 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 |
| [14] | 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 |
| [15] | Chambolle A and Pock T 2016 An introduction to continuous optimization for imaging Acta Numer.25 161-319 · Zbl 1343.65064 · doi:10.1017/S096249291600009X |
| [16] | Chen C and Öktem O 2018 Indirect image registration with large diffeomorphic deformations SIAM J. Imaging Sci.11 575-617 · Zbl 1401.94013 · doi:10.1137/17M1134627 |
| [17] | Christensen G E, Rabbitt R D and Miller M I 1996 Deformable templates using large deformation kinematics IEEE Trans. Image Process.5 1435-47 · doi:10.1109/83.536892 |
| [18] | Colonna F, Easley G, Guo K and Labate D 2010 Radon transform inversion using the shearlet representation Appl. Comput. Harmon. Anal.29 232-50 · Zbl 1196.65206 · doi:10.1016/j.acha.2009.10.005 |
| [19] | Davison M E 1983 The ill-conditioned nature of the limited angle tomography problem SIAM J. Appl. Math.43 428-48 · Zbl 0526.44005 · doi:10.1137/0143028 |
| [20] | Donoho D 2006 Compressed sensing IEEE Trans. Inf. Theory52 1289-306 · Zbl 1288.94016 · doi:10.1109/TIT.2006.871582 |
| [21] | Dupuis P, Grenander U and Miller M I 1998 Variational problems on flows of diffeomorphisms for image matching Q. Appl. Math.56 587-600 · Zbl 0949.49002 · doi:10.1090/qam/1632326 |
| [22] | Effland A 2017 Discrete Riemannian calculus and a posteriori error control on shape spaces Dissertation University of Bonn |
| [23] | Ehrhardt J and Lorenz C 2013 4D Modeling and Estimation of Respiratory Motion for Radiation Therapy (New York: Springer) · doi:10.1007/978-3-642-36441-9 |
| [24] | Frikel J 2013 Sparse regularization in limited angle tomography Appl. Comput. Harmon. Anal.34 117-41 · Zbl 1253.92031 · doi:10.1016/j.acha.2012.03.005 |
| [25] | Frikel J and Quinto E T 2013 Characterization and reduction of artifacts in limited angle tomography Inverse Problems29 125007 · Zbl 1284.92044 · doi:10.1088/0266-5611/29/12/125007 |
| [26] | Frikel J and Quinto E T 2016 Limited data problems for the generalized radon transform in Rn SIAM J. Math. Anal.48 2301-18 · Zbl 1344.44002 · doi:10.1137/15M1045405 |
| [27] | Gigengack F, Jiang X, Dawood M and Schäfers K P 2015 Motion Correction in Thoracic Positron Emission Tomography (New York: Springer) · doi:10.1007/978-3-319-08392-6 |
| [28] | Gris B, Chen C and Öktem O 2018 Image reconstruction through metamorphosis HAL preprint hal-01773633v1 https://arxiv.org/abs/1806.01225 |
| [29] | Guerquin-Kern M, Lejeune L, Pruessmann K P and Unser M 2012 Realistic analytical phantoms for parallel magnetic resonance imaging IEEE Trans. Med. Imaging31 626-36 · doi:10.1109/TMI.2011.2174158 |
| [30] | Haber E and Modersitzki J 2006 A multilevel method for image registration SIAM J. Sci. Comput.27 1594-607 · Zbl 1141.65367 · doi:10.1137/040608106 |
| [31] | He H and Siu W C 2011 Single image super-resolution using Gaussian process regression IEEE Conf. on Computer Vision and Pattern Recognition pp 449-56 · doi:10.1109/CVPR.2011.5995713 |
| [32] | Herman G T 1980 Image Reconstructions from Projections. The Fundamentals of Computerized Tomography (New York: Academic) · Zbl 0538.92005 |
| [33] | Herman G T and Davidi R 2008 Image reconstruction from a small number of projections Inverse Problems24 045011 · Zbl 1161.68825 · doi:10.1088/0266-5611/24/4/045011 |
| [34] | Hewett R J, Jermyn I, Heath M and Kamalabadi F 2012 A phase field method for tomographic reconstruction from limited data Proc. of the British Machine Vision Conf. pp 1-11 (BMVA Press) · doi:10.5244/C.26.120 |
| [35] | Karlsson J and Ringh A 2017 Generalized Sinkhorn iterations for regularizing inverse problems using optimal mass transport SIAM J. Imaging Sci.10 1935-62 · Zbl 1401.49049 · doi:10.1137/17M111208X |
| [36] | Katsevich A I 1997 Local tomography for the limited-angle problem J. Math. Anal. Appl.213 160-82 · Zbl 0894.65065 · doi:10.1006/jmaa.1997.5412 |
| [37] | Kuchment P 2014 The Radon Transform and Medical Imaging (Philadelphia, PA: SIAM) · Zbl 1282.92001 |
| [38] | Louis A K 1986 Incomplete data problems in x-ray computerized tomography Numer. Math.48 251-62 · Zbl 0578.65131 · doi:10.1007/BF01389474 |
| [39] | Maas J, Rumpf M, Schönlieb C and Simon S 2015 A generalized model for optimal transport of images including dissipation and density modulation ESAIM: Math. Modelling Numer. Anal.49 1745-69 · Zbl 1348.94009 · doi:10.1051/m2an/2015043 |
| [40] | Miller M I, Trouvé A and Younes L 2002 On the metrics and Euler-Lagrange equations of computational anatomy Annu. Rev. Biomed. Eng.4 375-405 · doi:10.1146/annurev.bioeng.4.092101.125733 |
| [41] | Miller M I, Trouvé A and Younes L 2015 Hamiltonian systems and optimal control in computational anatomy: 100 years since d’arcy thompson Annu. Rev. Biomed. Eng.17 447-509 · doi:10.1146/annurev-bioeng-071114-040601 |
| [42] | Miller M I and Younes L 2001 Group actions, homeomorphisms, and matching: a general framework Int. J. Comput. Vis.41 61-84 · Zbl 1012.68714 · doi:10.1023/A:1011161132514 |
| [43] | Modersitzki J 2004 Numerical Methods for Image Registration (Oxford: Oxford University Press) · Zbl 1055.68140 |
| [44] | Modersitzki J 2009 FAIR: Flexible Algorithms for Image Registration (Philadelphia, PA: SIAM) · Zbl 1183.68695 · doi:10.1137/1.9780898718843 |
| [45] | Natterer F 2001 The Mathematics of Computerized Tomography(Classics in Applied Mathematics) (Philadelphia, PA: SIAM) · Zbl 0973.92020 · doi:10.1137/1.9780898719284 |
| [46] | Neumayer S, Persch J and Steidl G 2018 Morphing of manifold-valued images inspired by discrete geodesics in image spaces SIAM J. Imaging Sci.11 1898-930 · Zbl 1429.65043 · doi:10.1137/17M1150906 |
| [47] | Nguyen L H, Stoter S K F, Baum T, Kirschke J S, Ruess M, Yosibash Z and Schillinger1 D 2017 Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of ct-based bone structures Int. J. Numer. Methods Biomed. Eng.33 1-34 · doi:10.1002/cnm.2880 |
| [48] | Nguyen L V 2015 How strong are streak artifacs in limited angle computed tomography? Inverse Problems31 055003 · Zbl 1364.65287 · doi:10.1088/0266-5611/31/5/055003 |
| [49] | Nirenberg L 1966 An extended interpolation inequality Ann. Della Scuola Normale Super. Pisa-Cl. Sci.20 733-7 · Zbl 0163.29905 |
| [50] | Nogueira M A, Abreu P H, Martins P, Machado P, Duarte H and Santos J 2017 Image descriptors in radiology images: a systematic review Artif. Intell. Rev.47 531-59 · doi:10.1007/s10462-016-9492-8 |
| [51] | Öktem O, Chen C, Domaniç N O, Ravikumar P and Bajaj C 2017 Shape-based image reconstruction using linearized deformations Inverse Problems33 035004 · Zbl 1372.35369 · doi:10.1088/1361-6420/aa55af |
| [52] | Palenstijn W, Batenburg K and Sijbers J 2011 Performance improvements for iterative electron tomography reconstruction using graphics processing units (gpus) J. Struct. Biol.176 250-3 · doi:10.1016/j.jsb.2011.07.017 |
| [53] | Persch J, Pierre F and Steidl G 2017 Exemplar-based face colorization using image morphing J. Imaging3 48 · doi:10.3390/jimaging3040048 |
| [54] | Pock T, Chambolle A, Cremers D and Bischof H 2009 A convex relaxation approach for computing minimal partitions IEEE Conf. on Computer Vision and Pattern Recognition pp 810-7 · doi:10.1109/CVPR.2009.5206604 |
| [55] | Pock T and Sabach S 2016 Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems SIAM J. Imaging Sci.9 1756-87 · Zbl 1358.90109 · doi:10.1137/16M1064064 |
| [56] | Rudin L, 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 |
| [57] | Rumpf M and Wirth B 2013 Discrete geodesic calculus in shape space and applications in the space of viscous fluidic objects SIAM J. Imaging Sci.6 2581-602 · Zbl 1279.68337 · doi:10.1137/120870864 |
| [58] | Rumpf M and Wirth B 2015 Variational time discretization of geodesic calculus IMA J. Numer. Anal.35 1011-46 · Zbl 1329.65133 · doi:10.1093/imanum/dru027 |
| [59] | Scherzer O, Grasmair M, Grossauer H, Haltmeier M and Lenzen F 2009 Variational Methods in Imaging (New York: Springer) · Zbl 1177.68245 |
| [60] | Schumacher H, Modersitzki J and Fischer B 2009 Combined reconstruction and motion correction in spect imaging IEEE Trans. Nucl. Sci.56 73-80 · doi:10.1109/TNS.2008.2007907 |
| [61] | Swierczynski P, Papiez B W, Schnabel J A and Macdonald C 2018 A level-set approach to joint image segmentation and registration with application to ct lung imaging Comput. Med. Imaging Graph.65 58-68 · doi:10.1016/j.compmedimag.2017.06.003 |
| [62] | Trouvé A 1995 An infinite dimensional group approach for physics based models in pattern recognition Int. J. Comput. Vis.37 977-987 · doi:10.1109/TMI.2018.2790962 |
| [63] | Trouvé A 1998 Diffeomorphisms groups and pattern matching in image analysis Int. J. Comput. Vis.28 213-21 · doi:10.1023/A:1008001603737 |
| [64] | Trouvé A and Younes L 2005 Local geometry of deformable templates SIAM J. Math. Anal.37 17-59 · Zbl 1090.58008 · doi:10.1137/S0036141002404838 |
| [65] | Trouvé A and Younes L 2005 Metamorphoses through Lie group action Found. Comput. Math.5 173-98 · Zbl 1099.68116 · doi:10.1007/s10208-004-0128-z |
| [66] | van Aarle W, Palenstijn W J, Cant J, Janssens E, Bleichrodt F, Dabravolski A, De Beenhouwer J, Batenburg K J and Sijbers J 2016 Fast and flexible x-ray tomography using the ASTRA toolbox Opt. Express24 25129-47 · doi:10.1364/OE.24.025129 |
| [67] | van Aarle W, Palenstijn W J, De Beenhouwer J, Altantzis T, Bals S, Batenburg K J and Sijbers J 2015 The ASTRA toolbox: A platform for advanced algorithm development in electron tomography Ultramicroscopy157 35-47 · doi:10.1016/j.ultramic.2015.05.002 |
| [68] | Wheeden R L 2015 Measure and Integral: an Introduction to Real Analysis vol 308 (Boca Raton, FL: CRC Press) · Zbl 1326.26007 · doi:10.1201/b18361 |
| [69] | Yang W, Zhong L, Lin L, Chen Y, Lu Z, Liu S, Wu Y, Feng Q and Chen W Predicting ct image from mri data through feature matching with learned nonlinear local descriptors IEEE Trans. Med. Imaging (in print) |
| [70] | Younes L 2010 Shapes and Diffeomorphisms (Berlin: Springer) · Zbl 1205.68355 · doi:10.1007/978-3-642-12055-8 |
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