Variational Methods

  • Kristian Bredies
  • Dirk Lorenz
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


To motivate the general approach and possibilities of variational methods in mathematical imaging, we begin with several examples.


  1. 1.
    R. Acar, C.R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10(6), 1217–1229 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    R.A. Adams, J.J.F. Fournier, Sobolev Spaces. Pure and Applied Mathematics, vol. 140, 2nd edn. (Elsevier, Amsterdam, 2003)Google Scholar
  3. 5.
    L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (Oxford University Press, Oxford, 2000)zbMATHGoogle Scholar
  4. 6.
    L. Ambrosio, N. Fusco, J.E. Hutchinson, Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford-Shah functional. Calc. Var. Partial Differ. Equ. 16(2), 187–215 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 7.
    K.J. Arrow, L. Hurwicz, H. Uzawa, Studies in Linear and Non-linear Programming. Stanford Mathematical Studies in the Social Sciences, 1st edn. (Stanford University Press, Palo Alto, 1958)Google Scholar
  6. 8.
    G. Aubert, P. Kornprobst, Mathematical Problems in Image Processing (Springer, New York, 2002)zbMATHCrossRefGoogle Scholar
  7. 9.
    G. Aubert, L. Blanc-Féraud, R. March, An approximation of the Mumford-Shah energy by a family of discrete edge-preserving functionals. Nonlinear Anal. Theory Methods Appl. Int. Multidiscip. J. Ser. A Theory Methods 64(9), 1908–1930 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 10.
    J.-F. Aujol, A. Chambolle, Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005)zbMATHCrossRefGoogle Scholar
  9. 14.
    M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998)zbMATHCrossRefGoogle Scholar
  10. 16.
    J.M. Borwein, A.S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples. CMS Books in Mathematics, vol. 3, 2nd edn. (Springer, New York, 2006)zbMATHCrossRefGoogle Scholar
  11. 17.
    A. Borzí, K. Ito, K. Kunisch, Optimal control formulation for determining optical flow. SIAM J. Sci. Comput. 24, 818–847 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 18.
    B. Bourdin, A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85(4), 609–646 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 19.
    K. Bredies, K. Kunisch, T. Pock, Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 21.
    H. Brézis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, vol. 5. Notas de Matemática (50) (North-Holland, Amsterdam; Elsevier, New York, 1973).Google Scholar
  15. 24.
    A. Bruhn, J. Weickert, C. Schnörr, Lucas/Kanade meets Horn/Schunck: combining local and global optical flow methods. Int. J. Comput. Vis. 61(3), 211–231 (2005)CrossRefGoogle Scholar
  16. 26.
    M. Burger, O. Scherzer, Regularization methods for blind deconvolution and blind source separation problems. Math. Control Signals Syst. 14, 358–383 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 31.
    A. Chambolle, P.-L. Lions, Image recovery via Total Variation minimization and related problems. Numer. Math. 76, 167–188 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 33.
    A. Chambolle, G.D. Maso, Discrete approximation of the Mumford-Shah functional in dimension two. Math. Model. Numer. Anal. 33(4), 651–672 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 34.
    A. Chambolle, T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 36.
    T.F. Chan, S. Esedoglu, Aspects of total variation regularized L 1 function approximation. SIAM J. Appl. Math. 65, 1817 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 37.
    T.F. Chan, J. Shen, Image Processing And Analysis: Variational, PDE, Wavelet, and Stochastic Methods (Society for Industrial and Applied Mathematics, Philadelphia, 2005)zbMATHCrossRefGoogle Scholar
  22. 38.
    T.F. Chan, L.A. Vese, Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)zbMATHCrossRefGoogle Scholar
  23. 39.
    T.F. Chan, C. Wong, Total variation blind deconvolution. IEEE Trans. Image Process. 7, 370–375 (1998)CrossRefGoogle Scholar
  24. 40.
    T.F. Chan, A. Marquina, P. Mulet, High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 41.
    T.F. Chan, S. Esedoglu, F.E. Park, A fourth order dual method for staircase reduction in texture extraction and image restoration problems. Technical report, UCLA CAM Report 05-28 (2005)Google Scholar
  26. 42.
    K. Chen, D.A. Lorenz, Image sequence interpolation using optimal control. J. Math. Imaging Vis. 41(3), 222–238 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 43.
    K. Chen, D.A. Lorenz, Image sequence interpolation based on optical flow, segmentation, and optimal control. IEEE Trans. Image Process. 21(3), 1020–1030 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 46.
    P.L. Combettes, V.R. Wajs, Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 49.
    G. David, Singular Sets of Minimizers for the Mumford-Shah Functional. Progress in Mathematics, vol. 233 (Birkhäuser, Basel, 2005)Google Scholar
  30. 55.
    V. Duval, J.-F. Aujol, Y. Gousseau, The TVL1 model: a geometric point of view. Multiscale Model. Simul. 8(1), 154–189 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 56.
    J. Eckstein, D.P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 57.
    I. Ekeland, R. Temam, Convex Analysis and Variational Problems. Studies in Mathematics and Its Applications, vol. 1 (North-Holland, Amsterdam, 1976)Google Scholar
  33. 58.
    H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems. Mathematics and Its Applications, vol. 375, 1st edn. (Kluwer Academic, Dordrecht, 1996)zbMATHCrossRefGoogle Scholar
  34. 60.
    L.C. Evans, A new proof of local \(\mathcal {C}^{1,\alpha }\) regularity for solutions of certain degenerate elliptic P.D.E. J. Differ. Equ. 45, 356–373 (1982)Google Scholar
  35. 61.
    L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions (CRC Press, Boca Raton, 1992)zbMATHGoogle Scholar
  36. 62.
    H. Federer, Geometric Measure Theory (Springer, Berlin, 1969)zbMATHGoogle Scholar
  37. 63.
    B. Fischer, J. Modersitzki, Ill-posed medicine — an introduction to image registration. Inverse Probl. 24(3), 034008 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 64.
    I. Galić, J. Weickert, M. Welk, A. Bruhn, A. Belyaev, H.-P. Seidel, Image compression with anisotropic diffusion. J. Math. Imaging Vis. 31, 255–269 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 65.
    E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80 (Birkhäuser, Boston, 1984)zbMATHCrossRefGoogle Scholar
  40. 66.
    G.H. Golub, C.F. Van Loan, Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, 4th edn. (Johns Hopkins University Press, Baltimore, 2013)Google Scholar
  41. 70.
    A. Haddad, Texture separation BV − G and BV − L 1 models. Multiscale Model. Simul. 6(1), 273–286 (electronic) (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 73.
    L. He, S.J. Osher, Solving the Chan-Vese model by a multiphase level set algorithm based on the topological derivative, in Scale Space and Variational Methods in Computer Vision, ed. by F. Sgallari, A. Murli, N. Paragios. Lecture Notes in Computer Science, vol. 4485 (Springer, Berlin, 2010), pp. 777–788Google Scholar
  43. 74.
    W. Hinterberger, O. Scherzer, Variational methods on the space of functions of bounded Hessian for convexification and denoising. Computing 76, 109–133 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 75.
    W. Hinterberger, O. Scherzer, C. Schnörr, J. Weickert, Analysis of optical flow models in the framework of the calculus of variations. Numer. Funct. Anal. Optim. 23(1), 69–89 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 76.
    M. Hintermüller, W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional. J. Math. Imaging Vis. 20(1–2), 19–42 (2004). Special issue on mathematics and image analysisMathSciNetzbMATHCrossRefGoogle Scholar
  46. 77.
    M. Holler, Theory and numerics for variational imaging — artifact-free JPEG decompression and DCT based zooming. Master’s thesis, Universität Graz (2010)Google Scholar
  47. 78.
    B.K.P. Horn, B.G. Schunck, Determining optical flow. Artif. Intell. 17, 185–203 (1981)CrossRefGoogle Scholar
  48. 80.
    J. Jost, Partial Differential Equations. Graduate Texts in Mathematics, vol. 214 (Springer, New York, 2002). Translated and revised from the 1998 German original by the authorGoogle Scholar
  49. 81.
    L.A. Justen, R. Ramlau, A non-iterative regularization approach to blind deconvolution. Inverse Probl. 22, 771–800 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 82.
    S. Kakutani, Concrete representation of abstract (m)-spaces (a characterization of the space of continuous functions). Ann. Math. Second Ser. 42(4), 994–1024 (1941)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 84.
    S.L. Keeling, W. Ring, Medical image registration and interpolation by optical flow with maximal rigidity. J. Math. Imaging Vis. 23, 47–65 (2005)MathSciNetCrossRefGoogle Scholar
  52. 87.
    S. Kindermann, S.J. Osher, J. Xu, Denoising by BV-duality. J. Sci. Comput. 28(2–3), 411–444 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 89.
    G.M. Korpelevič, An extragradient method for finding saddle points and for other problems. Ékonomika i Matematicheskie Metody 12(4), 747–756 (1976)MathSciNetGoogle Scholar
  54. 92.
    L.H. Lieu, L. Vese, Image restoration and decomposition via bounded Total Variation and negative Hilbert-Sobolev spaces. Appl. Math. Optim. 58, 167–193 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 93.
    P.-L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 95.
    M. Lysaker, A. Lundervold, X.-C. Tai, 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)zbMATHCrossRefGoogle Scholar
  57. 99.
    Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol. 22 (American Mathematical Society, Providence, 2001). The fifteenth Dean Jacqueline B. Lewis memorial lecturesGoogle Scholar
  58. 100.
    J. Modersitzki, FAIR: Flexible Algorithms for Image Registration. Fundamentals of Algorithms, vol. 6 (Society for Industrial and Applied Mathematics, Philadelphia, 2009)Google Scholar
  59. 101.
    D. Mumford, J. Shah, Optimal approximations by piecewise smooth functions and variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 102.
    H.J. Muthsam, Lineare Algebra und ihre Anwendungen, 1st edn. (Spektrum Akademischer Verlag, Heidelberg, 2006)zbMATHCrossRefGoogle Scholar
  61. 105.
    S.J. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153 (Springer, Berlin, 2003)Google Scholar
  62. 107.
    S.J. Osher, A. Sole, L. Vese, Image decomposition and restoration using Total Variation minimization and the h −1 norm. Multiscale Model. Simul. 1(3), 349–370 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 112.
    T. Pock, D. Cremers, H. Bischof, A. Chambolle, An algorithm for minimizing the Mumford-Shah functional, in 2009 IEEE 12th International Conference on Computer Vision (2009), pp. 1133–1140Google Scholar
  64. 113.
    L.D. Popov, A modification of the Arrow-Hurwicz method for search of saddle points. Math. Notes 28, 845–848 (1980)zbMATHCrossRefGoogle Scholar
  65. 118.
    R.T. Rockafellar, Convex Analysis. Princeton Mathematical Series (Princeton University Press, Princeton, 1970)zbMATHCrossRefGoogle Scholar
  66. 121.
    L.I. Rudin, S.J. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 60(1–4), 259–268 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 124.
    G. Sapiro, Color snakes. Comput. Vis. Image Underst. 68(2), 247–253 (1997)CrossRefGoogle Scholar
  68. 125.
    O. Scherzer, Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing 60(1), 1–27 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 126.
    O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, F. Lenzen, Variational Methods in Imaging (Springer, New York, 2009)zbMATHGoogle Scholar
  70. 127.
    L. Schwartz, Théorie des distributions, vol. 1, 3rd edn. (Hermann, Paris, 1966)Google Scholar
  71. 128.
    J.A. Sethian, Level Set Methods and Fast Marching Methods, 2nd edn. (Cambridge University Press, Cambridge, 1999)zbMATHGoogle Scholar
  72. 129.
    S. Setzer, G. Steidl, Variational methods with higher order derivatives in image processing, in Approximation XII, ed. by M. Neamtu, L.L. Schumaker (Nashboro Press, Brentwood, 2008), pp. 360–386Google Scholar
  73. 130.
    R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs, vol. 49 (American Mathematical Society, Providence, 1997)Google Scholar
  74. 131.
    J. Simon, Régularité de la solution d’une équation non linéaire dans R N, in Journées d’Analyse Non Linéaire, ed. by P. Bénilan, J. Robert. Lecture Notes in Mathematics, vol. 665 (Springer, Heidelberg, 1978)Google Scholar
  75. 138.
    G.N. Watson, A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library, 2nd edn. (Cambridge University Press, Cambridge, 1995). Revised editionGoogle Scholar
  76. 148.
    E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A - Linear Monotone Operators (Springer, New York, 1990)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Kristian Bredies
    • 1
  • Dirk Lorenz
    • 2
  1. 1.Institute for Mathematics and ScientificUniversity of GrazGrazAustria
  2. 2.BraunschweigGermany

Personalised recommendations