Truncated Fractional-Order Total Variation Model for Image Restoration

Raymond Honfu CHAN, Hai Xia LIANG*

*Corresponding author for this work

Research output: Journal PublicationsJournal Article (refereed)peer-review

5 Citations (Scopus)


Fractional-order derivative is attracting more and more interest from researchers working on image processing because it helps to preserve more texture than total variation when noise is removed. In the existing works, the Grunwald–Letnikov fractional-order derivative is usually used, where the Dirichlet homogeneous boundary condition can only be considered and therefore the full lower triangular Toeplitz matrix is generated as the discrete partial fractional-order derivative operator. In this paper, a modified truncation is considered in generating the discrete fractional-order partial derivative operator and a truncated fractional-order total variation (tFoTV) model is proposed for image restoration. Hopefully, first any boundary condition can be used in the numerical experiments. Second, the accuracy of the reconstructed images by the tFoTV model can be improved. The alternating directional method of multiplier is applied to solve the tFoTV model. Its convergence is also analyzed briefly. In the numerical experiments, we apply the tFoTV model to recover images that are corrupted by blur and noise. The numerical results show that the tFoTV model provides better reconstruction in peak signal-to-noise ratio (PSNR) than the full fractional-order variation and total variation models. From the numerical results, we can also see that the tFoTV model is comparable with the total generalized variation (TGV) model in accuracy. In addition, we can roughly fix a fractional order according to the structure of the image, and therefore, there is only one parameter left to determine in the tFoTV model, while there are always two parameters to be fixed in TGV model.

Original languageEnglish
Pages (from-to)561-578
Number of pages18
JournalJournal of the Operations Research Society of China
Issue number4
Early online date19 Jun 2019
Publication statusPublished - Dec 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2019, Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature.


  • Alternating directional method of multiplier
  • Fractional-order derivative
  • Image restoration
  • Total generalized variation
  • Total variation
  • Truncated fractional-order total variation model


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