BIT Numerical Mathematics

, Volume 59, Issue 4, pp 903–928 | Cite as

Directional total generalized variation regularization

  • Rasmus Dalgas Kongskov
  • Yiqiu DongEmail author
  • Kim Knudsen


In inverse problems, prior information and a priori-based regularization techniques play important roles. In this paper, we focus on image restoration problems, especially on restoring images whose texture mainly follow one direction. In order to incorporate the directional information, we propose a new directional total generalized variation (DTGV) functional, which is based on total generalized variation (TGV) by Bredies et al. After studying the mathematical properties of DTGV, we utilize it as regularizer and propose the \(\hbox {L}^2\hbox {-}\mathrm {DTGV}\) variational model for solving image restoration problems. Due to the requirement of the directional information in DTGV, we give a direction estimation algorithm, and then apply a primal-dual algorithm to solve the minimization problem. Experimental results show the effectiveness of the proposed method for restoring the directional images. In comparison with isotropic regularizers like total variation and TGV, the improvement of texture preservation and noise removal is significant.


Directional total generalized variation Prior information Regularization Variational model Primal-dual algorithm Image restoration 

Mathematics Subject Classification

49M29 65K10 65J22 90C47 94A08 



  1. 1.
    Aubert, G., Kornprobst, P.: Mathematical problems in image processing: partial differential equations and the calculus of variations, 2nd edn, pp xxxii+377, vol. 147. Springer, New York (2006)CrossRefGoogle Scholar
  2. 2.
    Bayram, I., Kamasak, M.E.: A directional total variation. Eur. Signal Process. Conf. 19(12), 265–269 (2012). CrossRefGoogle Scholar
  3. 3.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berkels, B., Burger, M., Droske, M., Nemitz, O., Rumpf, M.: Cartoon extraction based on anisotropic image classification. In: Proceedings of the vision, modeling and visualization, pp 293–300 (2006)Google Scholar
  5. 5.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2010). CrossRefzbMATHGoogle Scholar
  6. 6.
    Bredies, K.: Recovering Piecewise Smooth Multichannel Images by Minimization of Convex Functionals with Total Generalized Variation Penalty, pp. 44–77. Springer, Berlin (2014). CrossRefGoogle Scholar
  7. 7.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bredies, K., Valkonen, T.: Inverse problems with second-order total generalized variation constraints. In: Proceedings of SampTA 2011, 9th. International Conference on Sampling Theory and Applications, Singapore (2001)Google Scholar
  9. 9.
    Calatroni, L., Lanza, A., Sgallari, F., Pragliola, M.: A flexible space-variant anisotropic regularisation for image restoration with automated parameter selection. Arxiv preprint. (2018).
  10. 10.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chan, T., Wong, C.K.: Total variation blind deconvolution. IEEE Trans. Image Process. 7(3), 370–5 (1998). CrossRefGoogle Scholar
  13. 13.
    Delaney, A.H., Bresler, Y.: Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography. IEEE Trans. Image Process. 7(2), 204–221 (1998). CrossRefGoogle Scholar
  14. 14.
    Dong, Y., Hintermüller, M., Neri, M.: An efficient primal-dual method for \({L}^1\)TV image restoration. SIAM J. Appl. Math. 2(4), 1168–1189 (2009)zbMATHGoogle Scholar
  15. 15.
    Dong, Y., Zeng, T.: A convex variational model for restoring blurred images with multiplicative noise. SIAM J. Appl. Math. 6(3), 1598–1625 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Easley, G.R., Labate, D., Colonna, F.: Shearlet-based total variation diffusion for denoising. IEEE Trans. Image Process. 18(2), 260–268 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Esedoglu, S., Osher, S.J.: Decomposition of images by the anisotropic Rudin–Osher–Fatemi model. Commun. Pure Appl. Math. 57(12), 1609–1626 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Estellers, V., Soatto, S., Bresson, X.: Adaptive regularization with the structure tensor. IEEE Trans. Image Process. 24(6), 1777–1790 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fei, X., Wei, Z., Xiao, L.: Iterative directional total variation refinement for compressive sensing image reconstruction. IEEE Signal Process. Lett. 20(11), 1070–1073 (2013). CrossRefGoogle Scholar
  20. 20.
    Fernandez-Granda, C., Candes, E.J.: Super-resolution via transform-invariant group-sparse regularization. In: 2013 IEEE International Conference on Computer Vision, pp. 3336–3343 (2013).
  21. 21.
    Ferstl, D., Reinbacher, C., Ranftl, R., Ruether, M., Bischof, H.: Image guided depth upsampling using anisotropic total generalized variation. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 993–1000 (2013).
  22. 22.
    Granlund, G.H., Knutsson, H.: Signal Processing for Computer Vision. Springer, Dordrecht (1995). CrossRefGoogle Scholar
  23. 23.
    Hafner, D., Schroers, C., Weickert, J.: Introducing maximal anisotropy into second order coupling models. Ger. Conf. Pattern Recognit. 9358, 79–90 (2015)MathSciNetGoogle Scholar
  24. 24.
    Holler, M., Kunisch, K.: On infimal convolution of total variation type functionals and applications. SIAM J. Imaging Sci. 7(4), 2258–2300 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jespersen, K.M., Zangenberg, J., Lowe, T., Withers, P.J., Mikkelsen, L.P.: Fatigue damage assessment of uni-directional non-crimp fabric reinforced polyester composite using X-ray computed tomography. Compos. Sci. Technol. 136, 94–103 (2016). CrossRefGoogle Scholar
  26. 26.
    Jespersen, K.M., Zangenberg, J., Lowe, T., Withers, P.J., Mikkelsen, L.P.: X-ray CT Data: Fatigue Damage in Glass Fibre/Polyester Composite Used for Wind Turbine Blades [Data-set] (2016).
  27. 27.
    Jonsson, E., Chan, T., Huang, S.C.: Total variation regularization in positron emission tomography. Tech. rep., Dept. Mathematics, University of California, Los Angeles (1998)Google Scholar
  28. 28.
    Kongskov, R., Dong, Y.: Tomographic reconstruction methods for decomposing directional components. Inverse Probl. Imaging 12, 1429–1442 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kongskov, R., Dong, Y.: Directional total generalized variation regularization for impulse noise removal. In: Proceedings of Scale Space and Variational Methods in Computer Vision 2017, LNCS 10302, pp. 221–231 (2017)Google Scholar
  30. 30.
    Lefkimmiatis, S., Roussos, A., Maragos, P., Unser, M.: Structure tensor total variation. SIAM J. Imaging Sci. 8(2), 1090–1122 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Nesterov, Y.: A method of solving a convex programming problem with convergence rate O (1/k2). Sov. Math. Dokl. 27(2), 372–376 (1983)zbMATHGoogle Scholar
  32. 32.
    Nikolova, M.: Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math. 61(2), 633–658 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ranftl, R., Gehrig, S., Pock, T., Bischof, H.: Pushing the limits of stereo using variational stereo estimation. IEEE Intell. Veh. Symp. Proc. 1, 401–407 (2012). CrossRefGoogle Scholar
  34. 34.
    Ring, W.: Structural properties of solutions to total variation regularization problems. ESAIM: M2AN 34(4), 799–810 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 60(1–4), 259–268 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sandoghchi, S.R., Jasion, G.T., Wheeler, N.V., Jain, S., Lian, Z., Wooler, J.P., Boardman, R.P., Baddela, N.K., Chen, Y., Hayes, J.R., Fokoua, E.N., Bradley, T., Gray, D.R., Mousavi, S.M., Petrovich, M.N., Poletti, F., Richardson, D.J.: X-ray tomography for structural analysis of microstructured and multimaterial optical fibers and preforms. Opt. Express 22(21), 26181 (2014). CrossRefGoogle Scholar
  37. 37.
    Scherzer, O.: Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing 60(1), 1–27 (1998). MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sciacchitano, F., Dong, Y., Zeng, T.: Variational approach for restoring blurred images with Cauchy noise. SIAM J. Appl. Math. 8(3), 1894–1922 (2015)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Setzer, S., Steidl, G.: Variational methods with higher order derivatives in image processing. Approx. Theory XII San Antonio 2007, 360–385 (2008)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Setzer, S., Steidl, G., Teuber, T.: Restoration of images with rotated shapes. Numer. Algorithms 48(1–3), 49–66 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Setzer, S., Steidl, G., Teuber, T.: Infimal convolution regularizations with discrete ll-type functionals. Commun. Math. Sci. 9(3), 797–827 (2011)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Steidl, G., Teuber, T.: Anisotropic smoothing using double orientation. In: Tai, X.-C., et al. (Eds.) Proceedings of the Scale Space and Variational Methods in Computer Vision, Second International Conference, pp. 477–489 (2009)Google Scholar
  43. 43.
    Steidl, G., Teuber, T.: Diffusion tensors for denoising sheared and rotated rectangles. IEEE Trans. Image Process. 12, 2640–2648 (2009)CrossRefGoogle Scholar
  44. 44.
    Temam, R.: Mathematical Problems in Plasticity. Gaulthier-Villars (1985)Google Scholar
  45. 45.
    Turgay, E., Akar, G.B.: Directionally Adaptive Super-resolution. In: 2009 16th IEEE International Conference Image Processing, (1), pp. 1201–1204 (2009).
  46. 46.
    Vogel, C.R., Oman, M.E.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17(1), 227–238 (1996). MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Weickert, J.: Anisotropic diffusion in image processing. In: European Consortium for Mathematics in Industry, p. xii+170. B. G. Teubner, Stuttgart (1998)zbMATHGoogle Scholar
  48. 48.
    Weickert, J.: Coherence-enhancing diffusion filtering. Int. J. Comput. Vis. 31(2), 111–127 (1999). MathSciNetCrossRefGoogle Scholar
  49. 49.
    Weickert, J., Romeny, B.M.T.H., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7, 398–410 (1998)CrossRefGoogle Scholar
  50. 50.
    Zhang, H., Wang, Y.: Edge adaptive directional total variation. J. Eng. (2013). CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKgs. LyngbyDenmark

Personalised recommendations