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A novel image denoising approach based on a curvature-based regularization

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Abstract

The classical total variation (TV) model has made great successes in image denoising due to the edge-preserving property of the TV regularization. However, it is well known that the TV model suffers from the staircase effects and the loss of image details. In order to overcome these problems, high-order variational models have been widely used to yield better quality of denoised images. In this paper, we first propose a new high-order image denoising model based on the sum of squared principal curvatures of the image surface of a given image. As a result, the associated Euler–Lagrange equation is highly nonlinear and of fourth order so standard numerical techniques such as gradient descent methods are not appropriate. We therefore propose an efficient numerical solution using the split Bregman (SB) method. Numerical tests not only show that the proposed curvature model is more robust in removing noise and preserving structural information for a wide range of applications than some existing high-order variational models, but also that the proposed numerical solution is accurate in delivering visually pleasing results.

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References

  1. You, Y.L., Kaveh, M.: Fourth-order partial differential equations for noise removal. IEEE Trans. Image Process. 9(10), 1723–1730 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Scherzer, O.: Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing 60(1), 1–27 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lysaker, M., Lundervold, A., Tai, X.-C.: 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)

    Article  MATH  Google Scholar 

  4. Hinterberger, W., Scherzer, O.: Variational methods on the space of functions of bounded hessian for convexification and denoising. Computing 76(1), 109–133 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergounioux, M., Piffet, L.: A second-order model for image denoising. Set-Valued Var. Anal. 18(3–4), 277–306 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lai, R.J., Tai, X.-C., Chan, T.F.: A ridge and corner preserving model for surface restoration. SIAM J. Sci. Comput. 35(2), 675–695 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chan, T.F., Esedoglu, S., Park, F.: A fourth order dual method for staircase reduction in texture extraction and image restoration problems. In: 2010 17th IEEE International Conference on Image Processing (ICIP), Hongkong, China, pp. 4137–4140 (2010)

  8. Zheng, S.X., Pan, Z.K., Jiang, C.X., Wang, G.D.: A new fast algorithm for image denoising. In: 3rd International Conference on Multimedia Technology, pp. 682–689 (2013)

  9. Wang, G.D., Xu, J., Dong, Q., Pan, Z.L.: Active contour model coupling with higher order diffusion for medical image segmentation. Int. J. Biomed. Imaging 2014, 1–8 (2014)

    Article  Google Scholar 

  10. Chan, R.H., Liang, H.X., Wei, S.H., Nikolova, M., Tai, X.-C.: High-order total variation regularization approach for axially symmetric object tomography from a single radiograph. Inverse Probl. Imagin. 9(1), 55–77 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Papafitsoros, K., Schönlieb, C.B.: A combined first and second order variational approach for image reconstruction. J. Math. Imagin. Vis. 48(2), 308–338 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bredies, K., Kunisch, K., Pock, T.: Color TV total variation methods for restoration of vector-valued images. SIAM J. Imagin. Sci. 3(3), 492–526 (2010)

    Article  MATH  Google Scholar 

  14. Zhu, W., Chan, T.F.: Image denoising using mean curvature of image surface. SIAM J. Imagin. Sci. 5(1), 1–32 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brito-Loeza, C., Chen, K., Uc-Cetina, V.: Image denoising using the Gaussian curvature of the image surface. Numer. Meth. Part. Differ. Equ. 32(3), 1066–1089 (2016)

  16. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lu, W., Duan, J., Qiu, Z., Pan, Z., Lid, R.W., Bai, L.: Implementation of high-order variational models made easy for image processing. Math. Methods Appl. Sci. 39, 4208–4233 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hue, N.M., Thanh, D.N.H., Thanh, L.T., Hien, N.N., Prasath, V.B.S.: Image denoising with overlapping group sparsity and second order total variation regularization. In: 6th NAFOSTED Conference on Information and Computer Science (NICS), pp. 370–374 (2019)

  19. Hien, N.N., Thanh, D.N.H., Erkan, U., Tavares, J.M.R.S.: Image noise removal method based on thresholding and regularization techniques. IEEE Access 10, 71584–71597 (2022)

  20. Thanh, D.N.H., Prasath, V.B.S., Hieu, L.M., Dvoenko, S.: An adaptive method for image restoration based on high-order total variation and inverse gradient. SIViP 14, 1189–1197 (2020)

    Article  Google Scholar 

  21. Goldstein, T., Osher, S.: The split bregman method for l1-regularized problems. SIAM J. Imag. Sci. 2(2), 323–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Brito-Loeza, C., Chen, K.: On fast iterative algorithms for solving the minimisation of curvature-related functionals in surface fairing. Int. J. Comput. Math. 90(1), 92–110 (2013)

  23. Chambolle, A., Pock, T.: Total roto-translational variation. Numer. Math. 142, 611–666 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chan, T.F., Kang, S.H., Shen, J.: Euler’s elastica and curvature based inpaintings. SIAM J. Appl. Math. 63(2), 564–592 (2002)

  25. Sroisangwan, P., Chumchob, N.: A New Numerical Method for \(\text{G}\)aussian Curvatuve Based Image Restoration. In: Proceedings of Annual Pure and Applied Mathematics Conference 2018, Chulalongkorn University, Bangkok, Thailand, pp. 54-66 (2018)

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Acknowledgements

The authors would like to thank the anonymous reviewers for their useful suggestions to improve the quality of the paper.

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This research is not financially supported from a research funding agency.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand

    Siriwan Chankan, Noppadol Chumchob & Pornpimon Sroisangwan

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  1. Siriwan Chankan
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  2. Noppadol Chumchob
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  3. Pornpimon Sroisangwan
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Contributions

NC is the main investigator and wrote the manuscript. SC and PS did all numerical experiments and helped by discussing theoretical and numerical work.

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Correspondence to Noppadol Chumchob.

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Chankan, S., Chumchob, N. & Sroisangwan, P. A novel image denoising approach based on a curvature-based regularization. SIViP 17, 2129–2136 (2023). https://doi.org/10.1007/s11760-022-02427-5

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