A nonconvex and nonsmooth anisotropic total variation model for image noise and blur removal

  • Yanjun Ren
  • Liming TangEmail author


In this paper, a nonconvex and nonsmooth anisotropic total variation model is proposed, which can provide a very sparser representation of the derivatives of the function in horizontal and vertical directions. The new model can preserve sharp edges and alleviate the staircase effect often arising in total variation (TV) based models. We use graduated nonconvexity (GNC) algorithm to solve the proposed nonconvex and nonsmooth minimization problem. Starting with a convex initialization, it uses a family of nonconvex functional to gradually approach the original nonconvex functional. For each subproblem, we use function splitting technique to separately address the nonconvex and nonsmooth properties, and use augmented Lagrangian method (ALM) to solve it. Experiments are conducted for both synthetic and real images to demonstrate the effectiveness of the proposed model. In addition, we compare it with several state-of-the-art models in denoising and deblurring applications. The numerical results show that our model has the best performance in terms of PSNR and MSSIM indexes.


Total variation Nonconvex Nonsmooth Anisotropic total variation Graduated nonconvexity algorithm 



This work was supported in part by the Natural Science Foundation of China under Grant No. 61561019, and the Doctoral Scientific Fund Project of Hubei University for Nationalities under Grant No. MY2015B001.


  1. 1.
    Acar R, Vogel C (1997) Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems 10(6):1217–1229MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aubert G, Kornprobst P (2006) Mathematical problems in image processing: partial differential equations and the calculus of variations. Springer, BerlinzbMATHCrossRefGoogle Scholar
  3. 3.
    Blake A, Zisserman A (1987) The graduated non-convexity algorithm: visual reconstruction. MIT Press, Cambridge, pp 131–166CrossRefGoogle Scholar
  4. 4.
    Bredies K, Kunisch K, Pock T (2010) Total generalized variation. Siam Journal on Imaging Sciences 3(3):492–526MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Burrus C-S (2009) Iterative reweighted least squares. Commun Pure Appl Math 44(6):1–9Google Scholar
  6. 6.
    Callet PL, Autrusseau F (2005) Subjective quality assessment IRCCyN/IVC database.
  7. 7.
    Chambolle A, Pock T (2011) A first-order primal-dual algorithm for convex problems with applications to imaging. Journalof Mathematical Imaging and Vision 40(1):120–145MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chan TF, Esedoglu S, Park F-E (2007) Image decomposition combining staircase reduction and texture extraction. J Vis Commun Image Represent 18(6):464–486CrossRefGoogle Scholar
  9. 9.
    Charbonnier P (1997) Deterministic edge-preserving regularization in computed imaging. IEEE Trans Image Process 6(2):298–311MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen L, Bentley P, Mori K, Misawa K (2018) Drinet for medical image segmentation. IEEE Trans Med Imaging 37(11):2453–2462CrossRefGoogle Scholar
  11. 11.
    Chen X, Pan L (2018) A survey of graph cuts/graph search based medical image segmentation. IEEE Rev Biomed Eng:112–124CrossRefGoogle Scholar
  12. 12.
    Chen H, Wang C, Song Y, Li Z (2015) Split Bregmanized anisotropic total variation model for image deblurring. J Vis Commun Image Represent:282–293CrossRefGoogle Scholar
  13. 13.
    Chen B, Yang Z, Huang S, Cui Z (2017) Cyber-physical system enabled nearby traffic flow modelling for autonomous vehicles. IEEE 36th international performance computing and communications conference (IPCCC). IEEE, pp. 1–6Google Scholar
  14. 14.
    Daubechies I, Devore R, Fornasier M (2010) Iteratively reweighted least squares minimization for sparse recovery. Commun Pure Appl Math 63(1):1–38MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Esedoḡlu S, Osher S-J (2004) Decomposition of images by the anisotropic Rudin-Osher-Fatemi model. Commun Pure Appl Math 57(12):1609–1626MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Fornasier M, Rauhut H, Ward R (2010) Low-rank matrix recovery via iteratively reweighted least squares minimization. SIAM J Optim 21(4):1614–1640MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Geman S, Geman D (1984) Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell (6, 6):721–741zbMATHCrossRefGoogle Scholar
  18. 18.
    Gilboa G, Osher S (2008) Nonlocal operators with applications to image processing. SIAM Journal on Multiscale Modeling & Simulation 7(3):1005–1028MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Grasmair M, Lenzen F (2010) Anisotropic total variation filtering. Appl Math Optim 62(3):2323–2339MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hintermüller M, Wu T (2013) Nonconvex TVq-models in image restoration: analysis and a trust-region regularization based superlinearly convergent solver. SIAM Journal on Imaging Sciences 6(3):1385–1415MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Jalalzai K (2016) Some remarks on the staircasing phenomenon in total variation-based image denoising. Journal of Mathematical Imaging and Vision 54(2):256–268MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Jidesh P (2014) A convex regularization model for image restoration. Comput Electr Eng 40(8):66–78CrossRefGoogle Scholar
  23. 23.
    Jing L, Liu S, Li Z, Meng S (2009) An image reconstruction algorithm based on the extended Tikhonov regularization method for electrical capacitance tomography. Measurement 42(3):368–376CrossRefGoogle Scholar
  24. 24.
    Jung M, Kang M (2015) Efficient nonsmooth nonconvex optimization for image restoration and segmentation. J Sci Comput 62(2):336–370MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Knoll F, Bredies K, Pock T, Stollberger R (2011) Second order total generalized variation (TGV) for MRI. Magn Reson Med 65(2):480–491CrossRefGoogle Scholar
  26. 26.
    Liu X, Huang L (2014) A new nonlocal total variation regularization algorithm for image denoising. Math Comput Simul 97(1):224–233MathSciNetCrossRefGoogle Scholar
  27. 27.
    Liu J, Huang T-Z, Selesnick I-W, Lv X-G, Chen P-Y (2013) Image restoration using total variation with overlapping group sparsity. Inf Sci:232–246MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Lou Y, Zeng Y, Osher S, Xin J (2015) A weighted difference of anisotropic and isotropic total variation model for image processing. SIAM Journal on Imaging Sciences 8(3):1798–1823MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Lu C, Tang J, Yan S, Lin Z (2016) Nonconvex nonsmooth low-rank minimization via iteratively reweighted nuclear norm. IEEE Trans Image Process 25(2):829–839MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lysaker M, Lundervold A, Tai XC (2003) 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):79–90zbMATHCrossRefGoogle Scholar
  31. 31.
    Moll J-S (2005) The anisotropic total variation flow. Math Ann 332(1):177–218MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Needell D (2009) Noisy signal recovery via iterative reweighted L1-minimization. Asilomar Conference on Signals, Systems and Computers IEEE Press, pp. 113–117Google Scholar
  33. 33.
    Nguyen D, Jeon J (2015) Multiple-constraint variational framework and image restoration problems. IET Image Process 9(6):435–449CrossRefGoogle Scholar
  34. 34.
    Nikolova M (1999) Markovian reconstruction using a GNC approach. IEEE Trans Image Process 8(9):1204–1220CrossRefGoogle Scholar
  35. 35.
    Nikolova M, Ng M-K, Tam C-P (2010) Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Trans Image Process 19(12):3073–3088MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Nikolova M, Ng M-K, Zhang S, Ching W-K (2008) Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization. SIAM Journal on Imaging Sciences 1(1):2–25MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Papafitsoros K, Schönlieb C-B (2014) A combined first and second order variational approach for image reconstruction. Journal of Mathematical Imaging and Vision 48(2):308–338MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Perona P, Malik J (1990) Scale-space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Mach Intell 12(7):629–639CrossRefGoogle Scholar
  39. 39.
    Phillips D (1962) A technique for the numerical solution of certain integral equations of the first kind. J ACM 9(1):84–97MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Ponomarenko P, Lukin V, Zelensky A, Egiazarian K, Carli M, Battisti F (2009) Tid2008- a database for evaluation of full-reference visual quality assessment metrics. Adv Mod Radioelectron 10(4):30–45Google Scholar
  41. 41.
    Rudin L-I, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. Physica D Nonlinear Phenomena 60(1–4):259–268MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Rusu C, Dumitrescu B (2012) Iterative reweighted l1 design of sparse FIR filters. Signal Process 92(4):905–911CrossRefGoogle Scholar
  43. 43.
    Shi M, Han T, Liu S (2015) Total variation image restoration using hyper-Laplacian prior with overlapping group sparsity. Signal Process:65–76CrossRefGoogle Scholar
  44. 44.
    Shi M, Xu T, Feng L, Liang J, Zhang K (2013) Single image deblurring using novel image prior constraints. Optik 124(20):4429–4434CrossRefGoogle Scholar
  45. 45.
    Tang L, Fang Z (2016) Edge and contrast preserving in total variation image denoising. EURASIP Journal on Advances in Signal Processing 2016(1):1–13MathSciNetCrossRefGoogle Scholar
  46. 46.
    Tihonov A (1963) On the solution of ill-posed problems and the method of regularization. Dokl Akad Nauk SSSR:501–504Google Scholar
  47. 47.
    Tikhonov A-N, Arsenin V-Y (1977) Solutions of ill-posed problems. Wiley, New YorkzbMATHGoogle Scholar
  48. 48.
    Wang Z, Bovik A-C, Sheikh H-R (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13(4):600–612CrossRefGoogle Scholar
  49. 49.
    Wang L, Zhou S, Kari A (2016) Super-resolution image reconstruction method using homotopy regularization. Multimed Tools Appl 75(23):15993–16016CrossRefGoogle Scholar
  50. 50.
    Xu L, Zheng S, Jia J (2013) Unnatural L0 sparse representation for natural image deblurring. IEEE Conference on Computer Vision and Pattern Recognition:1107–1114Google Scholar
  51. 51.
    Yan C, Li L, Zhang C, Liu B, Zhang Y, Dai Q (2019) Cross-modality bridging and knowledge transferring for image understanding. IEEE Transactions on MultimediaGoogle Scholar
  52. 52.
    Yan C, Tu Y, Wang X, Zhang Y, Hao X, Zhang Y, Dai Q (2019) STAT: spatial-temporal attention mechanism for video captioning. IEEE Transactions on MultimediaGoogle Scholar
  53. 53.
    Yan C, Xie H, Chen J, Zha Z, Zhang Y, Dai Q (2018) A fast Uyghur text detector for complex background images. IEEE Transactions on Multimedia 20(12):3389–3398CrossRefGoogle Scholar
  54. 54.
    Yang Z, Jia D, Loannidis S, Sheng B (2018) Intermediate data caching optimization for multi-stage and parallel big data frameworks. IEEE 11th international conference on CLOUD computing (CLOUD), pp. 277–284Google Scholar
  55. 55.
    Yang J, Wright J, Huang T-S, Ma Y (2010) Image super-resolution via sparse representation. IEEE Trans Image Process 19(11):2861–2873MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Zhang H, Tang L, Fang Z, Xiang C, Li C (2018) Nonconvex and nonsmooth total generalized variation model for image restoration. Signal Process:69–85CrossRefGoogle Scholar
  57. 57.
    Zuo Z, Yang W, Lan X, Liu L, Hu J, Yan L (2014) Adaptive nonconvex nonsmooth regularization for image restoration based on spatial information. Circuits, Systems, and Signal Processing 33(8):2549–2564MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of ScienceHubei Minzu UniversityEnshiPeople’s Republic of China

Personalised recommendations