Big Data Optimization: Recent Developments and Challenges pp 97-131

Part of the Studies in Big Data book series (SBD, volume 18)

Optimising Big Images



We take a look at big data challenges in image processing. Real-life photographs and other images, such ones from medical imaging modalities, consist of tens of million data points. Mathematically based models for their improvement—due to noise, camera shake, physical and technical limitations, etc.—are moreover often highly non-smooth and increasingly often non-convex. This creates significant optimisation challenges for the application of the models in quasi-real-time software packages, as opposed to more ad hoc approaches whose reliability is not as easily proven as that of mathematically based variational models. After introducing a general framework for mathematical image processing, we take a look at the current state-of-the-art in optimisation methods for solving such problems, and discuss future possibilities and challenges.


  1. 1.
    Adcock, B., Hansen, A.C., Poon, C., Roman, B.: Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing. In: Proceedings SampTA 2013 (2013)Google Scholar
  2. 2.
    Alberti, G., Bouchitté, G., Dal Maso, G.: The calibration method for the mumford-shah functional and free-discontinuity problems. Calc. Var. Partial Differ. Equ. 16(3), 299–333 (2003). doi:10.1007/s005260100152 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000)Google Scholar
  4. 4.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990). doi:10.1002/cpa.3160430805
  5. 5.
    Arridge, S.R., Schotland, J.C.: Optical tomography: forward and inverse problems. Inverse Probl. 25(12), 123,010 (2009). doi:10.1088/0266-5611/25/12/123010
  6. 6.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2nd edn. Springer, New York (2006)MATHGoogle Scholar
  7. 7.
    Bachmayr, M., Burger, M.: Iterative total variation schemes for nonlinear inverse problems. Inverse Probl. 25(10) (2009). doi:10.1088/0266-5611/25/10/105004
  8. 8.
    Becker, S., Bobin, J., Candés, E.: Nesta: a fast and accurate first-order method for sparse recovery. SIAM J. Imaging Sci. 4(1), 1–39 (2011). doi:10.1137/090756855 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009). doi:10.1109/TIP.2009.2028250 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009). doi:10.1137/080716542 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Benning, M., Gladden, L., Holland, D., Schönlieb, C.B., Valkonen, T.: Phase reconstruction from velocity-encoded MRI measurements—a survey of sparsity-promoting variational approaches. J. Magn. Reson. 238, 26–43 (2014). doi:10.1016/j.jmr.2013.10.003 CrossRefGoogle Scholar
  12. 12.
    Bertozzi, A.L., Greer, J.B.: Low-curvature image simplifiers: global regularity of smooth solutions and Laplacian limiting schemes. Commun. Pure Appl. Math. 57(6), 764–790 (2004). doi:10.1002/cpa.20019 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bianchi, P., Hachem, W., Iutzeler, F.: A stochastic coordinate descent primal-dual algorithm and applications to large-scale composite optimization (2016). PreprintGoogle Scholar
  14. 14.
    Blaschke, B., Neubauer, A., Scherzer, O.: On convergence rates for the iteratively regularized Gauss-Newton method. IMA J. Numer. Anal. 17(3), 421–436 (1997)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Blum, R., Liu, Z.: Multi-Sensor Image Fusion and Its Applications. Signal Processing and Communications. Taylor & Francis (2005)Google Scholar
  16. 16.
    Bredies, K., Pock, T., Wirth, B.: A convex, lower semi-continuous approximation of euler’s elastica energy. SFB-Report 2013–013, University of Graz (2013)Google Scholar
  17. 17.
    Bredies, K., Sun, H.: Preconditioned Douglas-Rachford splitting methods saddle-point problems with applications to image denoising and deblurring. SFB-Report 2014–002, University of Graz (2014)Google Scholar
  18. 18.
    Bredies, K., Valkonen, T.: Inverse problems with second-order total generalized variation constraints. In: Proceedings SampTA 2011 (2011)Google Scholar
  19. 19.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3, 492–526 (2011). doi:10.1137/090769521 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Bredies, K., Kunisch, K., Valkonen, T.: Properties of \(L^1\)-\(\text{ TGV }^2\): the one-dimensional case. J. Math. Anal. Appl. 398, 438–454 (2013). doi:10.1016/j.jmaa.2012.08.053 MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: IEEE CVPR, vol. 2, pp. 60–65 (2005). doi:10.1109/CVPR.2005.38
  22. 22.
    Burger, M., Franek, M., Schönlieb, C.B.: Regularized regression and density estimation based on optimal transport. AMRX Appl. Math. Res. Express 2012(2), 209–253 (2012). doi:10.1093/amrx/abs007
  23. 23.
    Caselles, V., Chambolle, A., Novaga, M.: The discontinuity set of solutions of the TV denoising problem and some extensions. Multiscale Model. Simul. 6(3), 879–894 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Chambolle, A.: Convex representation for lower semicontinuous envelopes of functionals in \(l^1\). J. Convex Anal. 8(1), 149–170 (2001)MathSciNetMATHGoogle Scholar
  25. 25.
    Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011). doi:10.1007/s10851-010-0251-1 MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Chan, T.F., Kang, S.H., Shen, J.: Euler’s elastica and curvature-based inpainting. SIAM J. Appl. Math. pp. 564–592 (2002)Google Scholar
  28. 28.
    Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000). doi:10.1137/S1064827598344169 MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Chan, T.F., Esedoglu, S.: Aspects of total variation regularized \(L^1\) function approximation. SIAM J. Appl. Math. 65, 1817–1837 (2005)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Chen, K., Lorenz, D.A.: Image sequence interpolation based on optical flow, segmentation, and optimal control. IEEE Trans. Image Process. 21(3) (2012). doi:10.1109/TIP.2011.2179305
  31. 31.
    Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38(4), 1200–1216 (2001)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Cheney, M., Borden, B.: Problems in synthetic-aperture radar imaging. Inverse Probl. 25(12), 123,005 (2009). doi:10.1088/0266-5611/25/12/123005
  33. 33.
    Chen, K., Lorenz, D.A.: Image sequence interpolation using optimal control. J. Math. Imaging Vis. 41, 222–238 (2011). doi:10.1007/s10851-011-0274-2 MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Cremers, D., Pock, T., Kolev, K., Chambolle, A.: Convex relaxation techniques for segmentation, stereo and multiview reconstruction. In: Markov Random Fields for Vision and Image Processing. MIT Press (2011)Google Scholar
  35. 35.
    Csiba, D., Qu, Z., Richtárik, P.: Stochastic dual coordinate ascent with adaptive probabilities (2016). PreprintGoogle Scholar
  36. 36.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-d transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007). doi:10.1109/TIP.2007.901238
  37. 37.
    Dal Maso, G., Fonseca, I., Leoni, G., Morini, M.: A higher order model for image restoration: the one-dimensional case. SIAM J. Math. Anal. 40(6), 2351–2391 (2009). doi:10.1137/070697823 MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004). doi:10.1002/cpa.20042 MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    de Los Reyes, J.C., Schönlieb, C.B., Valkonen, T.: Optimal parameter learning for higher-order regularisation models (2014). In preparationGoogle Scholar
  40. 40.
    de Los Reyes, J.C., Schönlieb, C.B., Valkonen, T.: The structure of optimal parameters for image restoration problems (2015). Submitted
  41. 41.
    de Los Reyes, J.C., Schönlieb, C.B.: Image denoising: Learning noise distribution via PDE-constrained optimization. Inverse Probl. Imaging (2014). To appearGoogle Scholar
  42. 42.
    Didas, S., Weickert, J., Burgeth, B.: Properties of higher order nonlinear diffusion filtering. J. Math. Imaging Vis. 35(3), 208–226 (2009). doi:10.1007/s10851-009-0166-x MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82(2), 421–439 (1956)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Duval, V., Aujol, J.F., Gousseau, Y.: The TVL1 model: a geometric point of view. Multiscale Model. Simul. 8, 154–189 (2009)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Ehrhardt, M., Arridge, S.: Vector-valued image processing by parallel level sets. IEEE Trans. Image Process. 23(1), 9–18 (2014). doi:10.1109/TIP.2013.2277775 MathSciNetCrossRefGoogle Scholar
  46. 46.
    Engl, H., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and Its Applications. Springer, Netherlands (2000)MATHGoogle Scholar
  47. 47.
    Esser, E., Zhang, X., Chan, T.F.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3(4), 1015–1046 (2010). doi:10.1137/09076934X MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Estrada, F.J., Fleet, D.J., Jepson, A.D.: Stochastic image denoising. In: BMVC, pp. 1–11 (2009). See also for updated benchmarks
  49. 49.
    Fang, F., Li, F., Zeng, T.: Single image dehazing and denoising: a fast variational approach. SIAM J. Imaging Sci. 7(2), 969–996 (2014). doi:10.1137/130919696 MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Fercoq, O., Richtárik, P.: Accelerated, parallel and proximal coordinate descent (2013). PreprintGoogle Scholar
  51. 51.
    Fornasier, M., Langer, A., Schönlieb, C.B.: A convergent overlapping domain decomposition method for total variation minimization. Numer. Math. 116(4), 645–685 (2010). doi:10.1007/s00211-010-0314-7 MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Fornasier, M., Schönlieb, C.: Subspace correction methods for total variation and \(\ell _1\)-minimization. SIAM J. Numer. Anal. 47(5), 3397–3428 (2009). doi:10.1137/070710779 MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, Studies in Mathematics and its Applications, vol. 15, pp. 299–331. North-Holland, Amsterdam (1983)Google Scholar
  54. 54.
    Goldstein, T., Osher, S.: The split bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009). doi:10.1137/080725891 MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1 (2014).
  56. 56.
    Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: Blondel, V., Boyd, S., Kimura, H. (eds.) Recent Advances in Learning and Control. Lecture Notes in Control and Information Sciences, pp. 95–110. Springer, Verlag (2008)CrossRefGoogle Scholar
  57. 57.
    Haber, E., Horesh, L., Tenorio, L.: Numerical methods for experimental design of large-scale linear ill-posed inverse problems. Inverse Probl. 24(5), 055,012 (2008). doi:10.1088/0266-5611/24/5/055012
  58. 58.
    Hanke, M.: A regularizing levenberg-marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Probl. 13(1), 79 (1997). doi:10.1088/0266-5611/13/1/007 MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5(1), 119–149 (2012). doi:10.1137/100814494 MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Hintermüller, M., Langer, A.: Non-overlapping domain decomposition methods for dual total variation based image denoising. SFB-Report 2013–014, University of Graz (2013)Google Scholar
  61. 61.
    Hintermüller, M., Rautenberg, C.N., Hahn, J.: Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction. Inverse Probl. 30(5), 055,014 (2014). doi:10.1088/0266-5611/30/5/055014
  62. 62.
    Hintermüller, M., Valkonen, T., Wu, T.: Limiting aspects of non-convex \(\text{ TV }^\varphi \) models (2014). Submitted
  63. 63.
    Hintermüller, M., Langer, A.: Subspace correction methods for a class of nonsmooth and nonadditive convex variational problems with mixed \(l^1/l^2\) data-fidelity in image processing. SIAM J. Imaging Sci. 6(4), 2134–2173 (2013). doi:10.1137/120894130 MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Hintermüller, M., Stadler, G.: An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration. SIAM J. Sci. Comput. 28(1), 1–23 (2006)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Hintermüller, M., Wu, T.: Nonconvex TV\(^q\)-models in image restoration: analysis and a trust-region regularization-based superlinearly convergent solver. SIAM J. Imaging Sci. 6, 1385–1415 (2013)MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    Hintermüller, M., Wu, T.: A superlinearly convergent \(R\)-regularized Newton scheme for variational models with concave sparsity-promoting priors. Comput. Optim. Appl. 57, 1–25 (2014)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex analysis and minimization algorithms I-II. Springer (1993)Google Scholar
  68. 68.
    Horn, B.K., Schunck, B.G.: Determining optical flow. Proc. SPIE 0281, 319–331 (1981). doi:10.1117/12.965761 CrossRefGoogle Scholar
  69. 69.
    Huang, J., Mumford, D.: Statistics of natural images and models. In: IEEE CVPR, vol. 1 (1999)Google Scholar
  70. 70.
    Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. No. 6 in Radon Series on Computational and Applied Mathematics. De Gruyter (2008)Google Scholar
  71. 71.
    Kluckner, S., Pock, T., Bischof, H.: Exploiting redundancy for aerial image fusion using convex optimization. In: Goesele, M., Roth, S., Kuijper, A., Schiele, B., Schindler, K. (eds.) Pattern Recognition, Lecture Notes in Computer Science, vol. 6376, pp. 303–312. Springer, Berlin Heidelberg (2010). doi:10.1007/978-3-642-15986-2_31
  72. 72.
    Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation (TGV) for MRI. Mag. Reson. Med. 65(2), 480–491 (2011). doi:10.1002/mrm.22595 CrossRefGoogle Scholar
  73. 73.
    Knoll, F., Clason, C., Bredies, K., Uecker, M., Stollberger, R.: Parallel imaging with nonlinear reconstruction using variational penalties. Mag. Reson. Med. 67(1), 34–41 (2012)CrossRefGoogle Scholar
  74. 74.
    Kolehmainen, V., Tarvainen, T., Arridge, S.R., Kaipio, J.P.: Marginalization of uninteresting distributed parameters in inverse problems-application to diffuse optical tomography. Int. J. Uncertain. Quantif. 1(1) (2011)Google Scholar
  75. 75.
    Konečný, J., Richtárik, P.: Semi-stochastic gradient descent methods (2013). PreprintGoogle Scholar
  76. 76.
    Kunisch, K., Hintermüller, M.: Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM J. Imaging Sci. 64(4), 1311–1333 (2004). doi:10.1137/S0036139903422784 MathSciNetMATHGoogle Scholar
  77. 77.
    Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM J. Imaging Sci. 6(2), 938–983 (2013)MathSciNetCrossRefMATHGoogle Scholar
  78. 78.
    Lellmann, J., Lorenz, D., Schönlieb, C.B., Valkonen, T.: Imaging with Kantorovich-Rubinstein discrepancy. SIAM J. Imaging Sci. 7, 2833–2859 (2014). doi:10.1137/140975528,
  79. 79.
    Lellmann, J., Strekalovskiy, E., Koetter, S., Cremers, D.: Total variation regularization for functions with values in a manifold. In: 2013 IEEE International Conference on Computer Vision (ICCV), pp. 2944–2951 (2013). doi:10.1109/ICCV.2013.366
  80. 80.
    Lellmann, J., Lellmann, B., Widmann, F., Schnörr, C.: Discrete and continuous models for partitioning problems. Int. J. Comput. Vis. 104(3), 241–269 (2013). doi:10.1007/s11263-013-0621-4 MathSciNetCrossRefMATHGoogle Scholar
  81. 81.
    Lewis, A.: The convex analysis of unitarily invariant matrix functions. J. Convex Anal. 2(1), 173–183 (1995)MathSciNetMATHGoogle Scholar
  82. 82.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)MathSciNetCrossRefMATHGoogle Scholar
  83. 83.
    Lorenz, D.A., Pock, T.: An accelerated forward-backward method for monotone inclusions (2014). PreprintGoogle Scholar
  84. 84.
    Loris, I., Verhoeven, C.: On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty. Inverse Probl. 27(12), 125,007 (2011). doi:10.1088/0266-5611/27/12/125007
  85. 85.
    Loris, I.: On the performance of algorithms for the minimization of \(\ell _1\)-penalized functionals. Inverse Probl. 25(3), 035,008 (2009). doi:10.1088/0266-5611/25/3/035008
  86. 86.
    Loris, I., Verhoeven, C.: Iterative algorithms for total variation-like reconstructions in seismic tomography. GEM Int. J. Geomath. 3(2), 179–208 (2012). doi:10.1007/s13137-012-0036-3 MathSciNetCrossRefMATHGoogle Scholar
  87. 87.
    Lysaker, M., Lundervold, A., Tai, X.C.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003). doi:10.1109/TIP.2003.819229 CrossRefMATHGoogle Scholar
  88. 88.
    Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. AMS (2001)Google Scholar
  89. 89.
    Möllenhoff, T., Strekalovskiy, E., Möller, M., Cremers, D.: The primal-dual hybrid gradient method for semiconvex splittings (2014). arXiv preprint arXiv:1407.1723
  90. 90.
    Möller, M., Burger, M., Dieterich, P., Schwab, A.: A framework for automated cell tracking in phase contrast microscopic videos based on normal velocities. J. Vis. Commun. Image Represent. 25(2), 396–409 (2014). doi:10.1016/j.jvcir.2013.12.002,
  91. 91.
    Morozov, V.A.: On the solution of functional equations by the method of regularization. Soviet Math. Dokldy 7, 414–417 (1966)MathSciNetMATHGoogle Scholar
  92. 92.
    Mueller, J.L., Siltanen, S.: Linear and Nonlinear Inverse Problems with Practical Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2012). doi:10.1137/1.9781611972344
  93. 93.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989). doi:10.1002/cpa.3160420503 MathSciNetCrossRefMATHGoogle Scholar
  94. 94.
    Nesterov, Y.: A method of solving a convex programming problem with convergence rate \(O(1/k^2)\). Soviet Math. Doklady 27(2), 372–376 (1983)MATHGoogle Scholar
  95. 95.
    Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. 103(1), 127–152 (2005). doi:10.1007/s10107-004-0552-5 MathSciNetCrossRefMATHGoogle Scholar
  96. 96.
    Nesterov, Y.: Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM J. Optim. 22(2), 341–362 (2012). doi:10.1137/100802001 MathSciNetCrossRefMATHGoogle Scholar
  97. 97.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer (2006)Google Scholar
  98. 98.
    Ochs, P., Chen, Y., Brox, T., Pock, T.: iPiano: Inertial proximal algorithm for non-convex optimization (2014). arXiv preprint arXiv:1404.4805
  99. 99.
    Ochs, P., Dosovitskiy, A., Brox, T., Pock, T.: An iterated 11 algorithm for non-smooth non-convex optimization in computer vision. In: IEEE CVPR (2013)Google Scholar
  100. 100.
    Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005). doi:10.1137/040605412 MathSciNetCrossRefMATHGoogle Scholar
  101. 101.
    Pan, X., Sidky, E.Y., Vannier, M.: Why do commercial ct scanners still employ traditional, filtered back-projection for image reconstruction? Inverse Probl. 25(12), 123,009 (2009). doi:10.1088/0266-5611/25/12/123009
  102. 102.
    Papafitsoros, K., Bredies, K.: A study of the one dimensional total generalised variation regularisation problem (2013). PreprintGoogle Scholar
  103. 103.
    Papafitsoros, K., Valkonen, T.: Asymptotic behaviour of total generalised variation. In: Fifth International Conference on Scale Space and Variational Methods in Computer Vision (SSVM) (2015)., Accepted
  104. 104.
    Papafitsoros, K., Schönlieb, C.B.: A combined first and second order variational approach for image reconstruction. J. Math. Imaging Vis. 48(2), 308–338 (2014). doi:10.1007/s10851-013-0445-4 MathSciNetCrossRefMATHGoogle Scholar
  105. 105.
    Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in hilbert space. J. Math. Anal Appl. 72(2), 383–390 (1979). doi:10.1016/0022-247X(79)90234-8 MathSciNetCrossRefMATHGoogle Scholar
  106. 106.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE TPAMI 12(7), 629–639 (1990). doi:10.1109/34.56205 CrossRefGoogle Scholar
  107. 107.
    Pock, T., Chambolle, A., Cremers, D., Bischof, H.: A convex relaxation approach for computing minimal partitions. In: IEEE CVPR, pp. 810–817 (2009). doi:10.1109/CVPR.2009.5206604
  108. 108.
    Pock, T., Chambolle, A.: Diagonal preconditioning for first order primal-dual algorithms in convex optimization. In: 2011 IEEE International Conference on Computer Vision (ICCV), pp. 1762–1769 (2011). doi:10.1109/ICCV.2011.6126441
  109. 109.
    Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the mumford-shah functional. In: 2009 IEEE 12th International Conference on Computer Vision, pp. 1133–1140 (2009). doi:10.1109/ICCV.2009.5459348
  110. 110.
    Pock, T., Cremers, D., Bischof, H., Chambolle, A.: Global solutions of variational models with convex regularization. SIAM J. Imaging Sci. 3(4), 1122–1145 (2010). doi:10.1137/090757617 MathSciNetCrossRefMATHGoogle Scholar
  111. 111.
    Qi, L., Sun, J.: A nonsmooth version of newton’s method. Math. Program. 58(1–3), 353–367 (1993). doi:10.1007/BF01581275 MathSciNetCrossRefMATHGoogle Scholar
  112. 112.
    Ranftl, R., Pock, T., Bischof, H.: Minimizing tgv-based variational models with non-convex data terms. In: Kuijper, A., Bredies, K., Pock, T., Bischof, H. (eds.) Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, vol. 7893, pp. 282–293. Springer, Berlin Heidelberg (2013). doi:10.1007/978-3-642-38267-3_24
  113. 113.
    Richtárik, P., Takáč, M.: Parallel coordinate descent methods for big data optimization. Math. Program. pp. 1–52 (2015). doi:10.1007/s10107-015-0901-6
  114. 114.
    Ring, W.: Structural properties of solutions to total variation regularization problems. ESAIM Math. Model. Numer. Anal. 34, 799–810 (2000). doi:10.1051/m2an:2000104 MathSciNetCrossRefMATHGoogle Scholar
  115. 115.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer (1998)Google Scholar
  116. 116.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press (1972)Google Scholar
  117. 117.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Optim. 14(5), 877–898 (1976). doi:10.1137/0314056 MathSciNetCrossRefMATHGoogle Scholar
  118. 118.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)MathSciNetCrossRefMATHGoogle Scholar
  119. 119.
    Sawatzky, A., Brune, C., Möller, J., Burger, M.: Total variation processing of images with poisson statistics. In: Jiang, X., Petkov, N. (eds.) Computer Analysis of Images and Patterns, Lecture Notes in Computer Science, vol. 5702, pp. 533–540. Springer, Berlin Heidelberg (2009). doi:10.1007/978-3-642-03767-2_65
  120. 120.
    Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.: Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics. De Gruyter (2012)Google Scholar
  121. 121.
    Setzer, S.: Operator splittings, bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis. 92(3), 265–280 (2011). doi:10.1007/s11263-010-0357-3 MathSciNetCrossRefMATHGoogle Scholar
  122. 122.
    Shen, J., Kang, S., Chan, T.: Euler’s elastica and curvature-based inpainting. SIAM J. Appl. Math. 63(2), 564–592 (2003). doi:10.1137/S0036139901390088 MathSciNetCrossRefMATHGoogle Scholar
  123. 123.
    Stathaki, T.: Image Fusion: Algorithms and Applications. Elsevier Science (2011)Google Scholar
  124. 124.
    Strang, G., Nguyen, T.: Wavelets and filter banks. Wellesley Cambridge Press (1996)Google Scholar
  125. 125.
    Sun, D., Han, J.: Newton and Quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7(2), 463–480 (1997). doi:10.1137/S1052623494274970 MathSciNetCrossRefMATHGoogle Scholar
  126. 126.
    Suzuki, T.: Stochastic dual coordinate ascent with alternating direction multiplier method (2013). PreprintGoogle Scholar
  127. 127.
    Tournier, J.D., Mori, S., Leemans, A.: Diffusion tensor imaging and beyond. Mag. Reson. Med. 65(6), 1532–1556 (2011). doi:10.1002/mrm.22924 CrossRefGoogle Scholar
  128. 128.
    Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim. 29(1), 119–138 (1991). doi:10.1137/0329006 MathSciNetCrossRefMATHGoogle Scholar
  129. 129.
    Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Program. 117(1–2), 387–423 (2009). doi:10.1007/s10107-007-0170-0 MathSciNetCrossRefMATHGoogle Scholar
  130. 130.
    Valkonen, T.: A method for weighted projections to the positive definite cone. Optim. (2014). doi:10.1080/02331934.2014.929680, Published online 24 Jun
  131. 131.
    Valkonen, T.: A primal-dual hybrid gradient method for non-linear operators with applications to MRI. Inverse Probl. 30(5), 055,012 (2014). doi:10.1088/0266-5611/30/5/055012
  132. 132.
    Valkonen, T.: The jump set under geometric regularisation. Part 1: Basic technique and first-order denoising. SIAM J. Math. Anal. 47(4), 2587–2629 (2015). doi:10.1137/140976248,
  133. 133.
    Valkonen, T.: The jump set under geometric regularisation. Part 2: Higher-order approaches (2014). SubmittedGoogle Scholar
  134. 134.
    Valkonen, T.: Transport equation and image interpolation with SBD velocity fields. J. Math. Pures Appl. 95, 459–494 (2011). doi:10.1016/j.matpur.2010.10.010 MathSciNetCrossRefMATHGoogle Scholar
  135. 135.
    Valkonen, T., Bredies, K., Knoll, F.: Total generalised variation in diffusion tensor imaging. SIAM J. Imaging Sci. 6(1), 487–525 (2013). doi:10.1137/120867172 MathSciNetCrossRefMATHGoogle Scholar
  136. 136.
    Valkonen, T., Knoll, F., Bredies, K.: TGV for diffusion tensors: A comparison of fidelity functions. J. Inverse Ill-Posed Probl. 21(355–377), 2012 (2013). doi:10.1515/jip-2013-0005, Special issue for IP:M&S, Antalya, Turkey
  137. 137.
    Valkonen, T., Liebmann, M.: GPU-accelerated regularisation of large diffusion tensor volumes. Computing 95(771–784), 2012 (2013). doi:10.1007/s00607-012-0277-x, Special issue for ESCO, Pilsen, Czech Republic
  138. 138.
    Vese, L.A., Chan, T.F.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50(3), 271–293 (2002). doi:10.1023/A:1020874308076 CrossRefMATHGoogle Scholar
  139. 139.
    Vese, L.A., Osher, S.J.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003)MathSciNetCrossRefMATHGoogle Scholar
  140. 140.
    Vogel, C., Oman, M.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17(1), 227–238 (1996). doi:10.1137/0917016 MathSciNetCrossRefMATHGoogle Scholar
  141. 141.
    Wernick, M., Aarsvold, J.: Emission Tomography: The Fundamentals of PET and SPECT. Elsevier Science (2004)Google Scholar
  142. 142.
    Wright, S.: Coordinate descent algorithms. Math. Program. 151(1), 3–34 (2015). doi:10.1007/s10107-015-0892-3 MathSciNetCrossRefMATHGoogle Scholar
  143. 143.
    Yilmaz, A., Javed, O., Shah, M.: Object tracking: a survey. ACM Comput. Surv. (CSUR) 38(4), 13 (2006)CrossRefGoogle Scholar
  144. 144.
    Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for \(\ell _1\)-minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1(1), 143–168 (2008). doi:10.1137/070703983 MathSciNetCrossRefMATHGoogle Scholar
  145. 145.
    Zhao, P., Zhang, T.: Stochastic optimization with importance sampling (2014). PreprintGoogle Scholar
  146. 146.
    Zhao, F., Noll, D., Nielsen, J.F., Fessler, J.: Separate magnitude and phase regularization via compressed sensing. IEEE Trans. Med. Imaging 31(9), 1713–1723 (2012). doi:10.1109/TMI.2012.2196707 CrossRefGoogle Scholar
  147. 147.
    Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. UCLA CAM Report (2008)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambrigeCambrigeUK

Personalised recommendations