Total generalized variation and shearlet transform based Poissonian image deconvolution

  • Xinwu LiuEmail author


Integrating the advantages of total generalized variation and shearlet transform, this article introduces a hybrid regularizers scheme for deconvolving Poissonian image. Computationally, a highly efficient alternating minimization algorithm associated with variable splitting approach is described to obtain the optimal solution in detail. Illustrationally, in comparison with several current state-of-the-art numerical methods, numerical simulations consistently demonstrate the outstanding performance of our proposed approach to deblurring Poissonian image, in terms of both restoration accuracy and feature-preserving ability.


Image deconvolution Poisson noise Total generalized variation Shearlet transform Alternating minimization method 



The author would like to thank the editors and anonymous reviewers for their helpful comments and valuable suggestions.


  1. 1.
    Bonettini S, Ruggiero V (2011) An alternating extragradient method for total variation-based image restoration from Poisson data. Inverse Probl 27(27):095001MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bonettini S, Zanella R, Zanni L (2009) A scaled gradient projection method for constrained image deblurring. Inverse Probl 25(1):711–723MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bratsolis E, Sigelle M (2011) A spatial regularization method preserving local photometry for Richardson-Lucy restoration. Astron Astrophys 375(375):1120–1128Google Scholar
  4. 4.
    Bredies K (2012) Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty. SFB-Report, pp 2012–006Google Scholar
  5. 5.
    Bredies K, Holler M (2013) A TGV regularized wavelet based zooming model. Scale space and variational methods in computer vision. Springer, Berlin, pp 149–160CrossRefGoogle Scholar
  6. 6.
    Bredies K, Kunisch K, Pock T (2010) Total generalized variation. SIAM J Imaging Sci 3(3):492–526MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bredies K, Dong Y, Hintermüller M (2012) Spatially dependent regularization parameter selection in total generalized variation models for image restoration. Int J Comput Math 90(1):109–123MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brune C, Burger M, Sawatzky A, Kösters T, Wübbeling F (2010) Forward backward EM-TV methods for inverse problems with Poisson noise. Preprint, University of MünsterGoogle Scholar
  9. 9.
    Brune C, Sawatzky A, Burger M (2011) Primal and dual Bregman methods with application to optical nanoscopy. Int J Comput Vis 92(2):211–229MathSciNetCrossRefGoogle Scholar
  10. 10.
    Carlavan M, Blanc-féraud L (2012) Sparse Poisson noisy image deblurring. IEEE Trans Image Process 21(4):1834–1846MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chaux C, Pesquet J-C, Pustelnik N (2009) Nested iterative algorithms for convex constrained image recovery problems. SIAM J Imaging Sci 2(2):730–762MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen D-Q (2014) Regularized generalized inverse accelerating linearized alternating minimization algorithm for frame-based Poissonian image deblurring. SIAM J Imaging Sci 7(2):716–739MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen D-Q, Cheng L-Z (2012) Spatially adapted regularization parameter selection based on the local discrepancy function for Poissonian image deblurring. Inverse Probl 28(1):015004MathSciNetCrossRefGoogle Scholar
  14. 14.
    Day N, Blanc-Feraud L, Zimmer C, Roux P, Kam Z, Olivo-Marin J-C, Zerubia J (2006) Richardson-lucy algorithm with total variation regularization for 3D confocal microscope deconvolution. Microsc Res Techniq 69(4):260–266CrossRefGoogle Scholar
  15. 15.
    Easley G, Labate D, Lim W Q (2008) Sparse directional image representations using the discrete shearlet transform. Appl Comput Harmon Anal 25(1):25–46MathSciNetCrossRefGoogle Scholar
  16. 16.
    Figueiredo M, Bioucas-Dias J (2010) Restoration of Poissonian images using alternating direction optimization. IEEE Trans Image Process 19(12):3133–3145MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gilboa G, Osher S (2007) Nonlocal operators with applications to image processing. Multiscale Model Simul 7(3):1005–1028MathSciNetCrossRefGoogle Scholar
  18. 18.
    Guo W, Qin J, Yin W (2014) A new detail-preserving regularity scheme. SIAM J Imaging Sci 7(2):1309–1334MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hajiaboli MR (2011) An anisotropic fourth-order diffusion filter for image noise removal. Int J Comput Vis 92(2):177–191MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hajiaboli MR (2010) A self-governing fourth-order nonlinear diffusion filter for image noise removal. IPSJ Trans Comput Vision Appl 2:94–103CrossRefGoogle Scholar
  21. 21.
    Häuser S, Steidl G (2014) Fast finite shearlet transform: a tutorial. arXiv:1202.1773
  22. 22.
    Jiang H, Zhang Y, Ma L, Yang X, Liu Y (2014) A Shearlet-based filter for low-dose mammography. In: Fujita H, Hara T, Muramatsu C (eds) Breast imaging. IWDM 2014, Lect Notes Comput Sci, vol 8539, pp 707–714Google Scholar
  23. 23.
    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
  24. 24.
    Kryvanos A, Hesser J, Steidl G (2005) Nonlinear image restoration methods for marker extraction in 3D fluorescent microscopy. Comput Imaging III Proc SPIE 5674:432–443CrossRefGoogle Scholar
  25. 25.
    Kutyniok G, Labate D (2012) Shearlets: multiscale analysis for multivariate data. Birkhäuser BaselGoogle Scholar
  26. 26.
    Lan X, Ma AJ, Yuen PC (2014) Multi-cue visual tracking using robust feature-level fusion based on joint sparse representation. In: Proc CVPR, pp 1194–1201Google Scholar
  27. 27.
    Lan X, Ma AJ, Yuen PC, Chellappa R (2015) Joint sparse representation and robust feature-level fusion for multi-cue visual tracking. IEEE Trans Image Process 24(12):5826–5841MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lan X, Zhang S, Yuen PC, Chellappa R (2018) Learning common and feature-specific patterns: a novel multiple-sparse-representation-based tracker. IEEE Trans Image Process 27(4):2022–2037MathSciNetCrossRefGoogle Scholar
  29. 29.
    Liu X (2016) Augmented Lagrangian method for total generalized variation based Poissonian image restoration. Comput Math Appl 71(8):1694–1705MathSciNetCrossRefGoogle Scholar
  30. 30.
    Liu X, Huang L (2013) Poissonian image reconstruction using alternating direction algorithm. J Electron Imaging 22(3):033007 1–9Google Scholar
  31. 31.
    Liu X, Huang L (2014) A new nonlocal total variation regularization algorithm for image denoising. Math Comput Simulat 97:224–233MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lysaker M, Lundervold A, Tai X-C (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):1579–1590CrossRefGoogle Scholar
  33. 33.
    Sarder P, Nehorai A (2006) Deconvolution method for 3-D fluorescence microscopy images. IEEE Signal Proc Mag 23(3):32–45CrossRefGoogle Scholar
  34. 34.
    Sawatzky A, Brune C, Wübbeling F, Kösters T, Schäfers K, Burger M (2008) Accurate EM-TV algorithm in PET with low SNR. IEEE Nucl Sci Symp Conf Rec, pp 5133–5137Google Scholar
  35. 35.
    Setzer S, Steidl G, Teuber T (2010) Deblurring Poissonian images by split Bregman techniques. J Vis Commun Image R 21(3):193–199CrossRefGoogle Scholar
  36. 36.
    Shepp LA, Vardi Y (1982) Maximum likelihood reconstruction for emission tomography. IEEE Trans Med Imaging 1(2):113–122CrossRefGoogle Scholar
  37. 37.
    You Y-L, Kaveh M (2000) Fourth-order partial differential equations for noise removal. IEEE Trans Image Process 9(10):1723–1730MathSciNetCrossRefGoogle Scholar
  38. 38.
    Valkonen T, Bredies K, Knoll F (2013) Total generalized variation in diffusion tensor imaging. SIAM J Imaging Sci 6(1):487–525MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zhang L, Zhang L, Mou X, Zhang D (2011) FSIM: a feature similarity index for image qualtiy assessment. IEEE Trans Image Process 20(8):2378–2386MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematics and Computational ScienceHunan University of Science and TechnologyXiangtanChina

Personalised recommendations