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Journal of Mathematical Imaging and Vision

, Volume 61, Issue 1, pp 106–121 | Cite as

Total Variation for Image Denoising Based on a Novel Smart Edge Detector: An Application to Medical Images

  • Ahmed Ben SaidEmail author
  • Rachid Hadjidj
  • Sebti Foufou
Article
  • 215 Downloads

Abstract

In medical imaging applications, diagnosis relies essentially on good quality images. Edges play a crucial role in identifying features useful to reach accurate conclusions. However, noise can compromise this task as it degrades image information by altering important features and adding new artifacts rendering images non-diagnosable. In this paper, we propose a novel denoising technique based on the total variation method with an emphasis on edge preservation. Image denoising techniques such as the Rudin–Osher–Fatemi model which are guided by gradient regularizer are generally accompanied with staircasing effect and loss of details. To overcome these issues, our technique incorporates in the model functional, a novel edge detector derived from fuzzy complement, non-local mean filter and structure tensor. This procedure offers more control over the regularization, allowing more denoising in smooth regions and less denoising when processing edge regions. Experimental results on synthetic images demonstrate the ability of the proposed edge detector to determine edges with high accuracy. Furthermore, denoising experiments conducted on CT scan images and comparison with other denoising methods show the outperformance of the proposed denoising method.

Keywords

Computer tomography Medical images Total variation Image denoising Edge detector 

Notes

Acknowledgements

This publication was made possible by NPRP Grant #4-1165- 2-453 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.

References

  1. 1.
    Zhong, J., Ning, R., Conover, D.: Image denoising based on multiscale singularity detection for cone beam CT breast imaging. IEEE Trans. Med. Imaging 23(6), 696–703 (2004)CrossRefGoogle Scholar
  2. 2.
    Petrongolo, M., Zhu, L.: Noise suppression for dual-energy CT through entropy minimization. IEEE Trans. Med. Imaging 34(11), 2286–2297 (2015)CrossRefGoogle Scholar
  3. 3.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  5. 5.
    Getreuer, P.: Rudin–Osher–Fatemi total variation denoising using split Bregman. Image Process. 2, 74–95 (2012)CrossRefGoogle Scholar
  6. 6.
    Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deng, L.J., Guo, H., Huang, T.Z.: A fast image recovery algorithm based on splitting deblurring and denoising. J. Comput. Appl. Math. 287, 88–97 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chambolle, A., Lions, P.L.: Image recovery via total variational minimization and related problems. Numer. Math. 76, 167–188 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chan, T.F., Mulet, P.: On the convergence of the lagged diffusivity fixed point method in total variation image restoration. SIAM J. Numer. Anal. 36, 354–367 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for \(l_1\)-minimization with applications to compressed sensing. J. Imaging Sci. 1, 143–168 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1, 248–272 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yan, J., Lu, W.S.: Image denoising by generalized total variation regularization and least square fidelity. Multimed. Syst. Signal Process. 26, 243–266 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, H., Wang, C., Song, Y., Li, Z.: Split Bregmanized anisotropic total variation model for image deblurring. J. Vis. Commun. Image Represent. 31, 282–293 (2015)CrossRefGoogle Scholar
  15. 15.
    Skinner, D., Foo, S., Mayer-Bäse, A.: Split Bregman’s optimization method for image construction in compressive sensing. In: Proceeding of SPIE 9118, Independent Component Analyses, Compressive Sampling, Wavelets, Neural Net, Biosystems, and Nanoengineering XII (2015)Google Scholar
  16. 16.
    Blomgren, P., Chan, T.F., Mulet, P.: Extensions to total variation denoising, In: Proceedings of SPIE, San Diego, 3162 (1997)Google Scholar
  17. 17.
    Chen, Y.M., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bollt, E.M., Chartrand, R., Esedoglu, S., Schultz, P., Vixie, K.R.: Graduated adaptive image denoising: local compromise between total variation and isotropic diffusion. Adv. Comput. Math. 31, 61–85 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, F., Li, Z., Pi, L.: Variable exponent functionals in image restoration. J. Appl. Math. Comput. 216, 870–882 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
  21. 21.
    Chen, Q., Montesinos, P., Sun, Q.S., Heng, P.A., Xia, D.S.: Adaptive total variation denoising based on difference curvature. Image Vis. Comput. 28, 298–306 (2010)CrossRefGoogle Scholar
  22. 22.
    Zeng, W., Lu, X., Fei, S.: Image super-resolution employing a spatial adaptive prior model. Neurocomputing 162, 218–233 (2015)CrossRefGoogle Scholar
  23. 23.
    Lefkimmiatis, S., Roussos, A., Maragos, P., Unser, M.: Structure tensor total variation. SIAM J. Imaging Sci. 8, 1090–1122 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)zbMATHGoogle Scholar
  25. 25.
    Fernandez, J., Li, S.: An improved algorithm for anisotropic nonlinear diffusion for denoising cryotomograms. J. Struct. Biol. 144, 152–161 (2003)CrossRefGoogle Scholar
  26. 26.
    Yang, D.G., Zhao, Z.X., Liu, L.M.: Face recognition based tensor structure. In: Proceeding of SPIE 8335, 2012 International Workshop on Image Processing and Optical Engineering (2011)Google Scholar
  27. 27.
    Khne, G., Weickert, J., Schuster, O., Richter, S.: A tensor-driven active contour model for moving object segmentation. In: Proceeding of International Conference on Image Processing, pp. 73–76 (2001)Google Scholar
  28. 28.
    Candes, E.J., Donoho, D.: Curvelets: a surprisingly effective nonadaptive representation of objects with edges. In: Curve and Surface Fitting, pp. 105–120. Vanderbilt University Press (2000)Google Scholar
  29. 29.
    Bhadauria, H.S., Dewal, M.L.: Medical image denoising using adaptive fusion of curvelet transform and total variation. Comput. Electr. Eng. 39, 1451–1460 (2013)CrossRefGoogle Scholar
  30. 30.
    Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation (TGV). SIAM J. Imaging Sci. 3, 92–526 (2010)CrossRefzbMATHGoogle Scholar
  32. 32.
    Valkonen, T., Bredies, K., Knoll, F.: Total generalized variation in diffusion tensor imaging. SIAM J. Imaging Sci. 6, 487–525 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation (TGV) for MRI, vol. 65, pp. 480–491 (2010)Google Scholar
  34. 34.
    Liu, X.: Augmented Lagrangian method for total generalized variation based Poissonian image restoration. Comput. Math. Appl. 71, 1694–1705 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lv, X.-G., Song, Y.-Z., Wang, S.-X., Li, J.: Image restoration with a high-order total variation minimization method. Appl. Math. Model. 37, 8210–8224 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Holler, M., Kunisch, K.: On infimal convolution of TV-type functionals and applications to video and image reconstruction. SIAM J. Imaging Sci. 7, 2258–2300 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Schloegl, M., Holler, M., Schwarzl, A., Bredies, K., Stollberger, R.: Infimal convolution of total generalized variation functionals for dynamic MRI. Magn. Reson. Med. 78, 142–155 (2017)CrossRefGoogle Scholar
  38. 38.
    Estellers, V., Soatto, S., Bresson, X.: Adaptive regularization with the structure tensor. IEEE Trans. Image Process. 24, 1777–1790 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Chang, C.C., Lee, J.S., Le, T.H.N.: Hybrid wet paper coding mechanism for steganography emloying \(n\)-indicator and fuzzy edge detector. Digit. Signal Proc. 20, 1286–1307 (2010)CrossRefGoogle Scholar
  40. 40.
    Amarunnishad, T.M., Govindan, V.K., Abraham, T.M.: A fuzzy edge complement operator. In: International Conference on Advanced Computing and Communications, pp. 344–348 (2006)Google Scholar
  41. 41.
    Chu, S.C., Huang, H.C., Shi, Y., Wu, S.Y., Shieh, C.S.: Genetic watermarking for zerotree-based application. Circuits Syst. Signal Process. 27, 171–182 (2008)CrossRefGoogle Scholar
  42. 42.
    Amarunnishad, T.M., Govindan, V.K., Abraham, T.M.: Improving BTC image compression using a fuzzy complement edge operator. Sig. Process. 88, 2989–2997 (2008)CrossRefzbMATHGoogle Scholar
  43. 43.
    Mathews, J., Nair, M.S., Jo, L.: Improved BTC algorithm for gray scale images using \(K\)-means quad clustering. In: International Conference on Neural Information Processing, pp. 9–17 (2012)Google Scholar
  44. 44.
    Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 60–65 (2005)Google Scholar
  45. 45.
    Chatterjee, P., Milanfar, P.: A generalization of non-local means via kernel regression. In: Proceedings of SPIE—The International Society for Optical Engineering, vol. 6814, p. 68140 (2008)Google Scholar
  46. 46.
    Mairal, J., Bach, F., Ponce, J., Sapiro, G., Zisserman, A.: Non-local sparse models for image restoration. In: International Conference on Computer Vision, pp. 2272–2279 (2009)Google Scholar
  47. 47.
    Coup, P., Yger, P., Prima, S., Hellier, P., Kervrann, C., Barillot, C.: An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images. IEEE Trans. Med. Imaging 27, 425–441 (2008)CrossRefGoogle Scholar
  48. 48.
    Buades, A., Coll, B., Morel, J.-M.: Nonlocal image and movie denoising. Int. J. Comput. Vis. 76, 123–139 (2008)CrossRefGoogle Scholar
  49. 49.
    Aubert, G., Kornprobst, P.: Mathematical Problem in Image Processing: Partial Differential Equations and the Calculus of Variations, 2nd edn. Springer, Berlin (2010)zbMATHGoogle Scholar
  50. 50.
    Kiranyaz, S., Ince, T., Gabbouj, M.: Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. Springer, Berlin (2013)zbMATHGoogle Scholar
  51. 51.
    Wang, L.-Y., Wei, Z.-H.: Fast gradient-based algorithm for total variation regularized Tomography reconstruction. In: 4th International Congress on Image and Signal Processing, pp. 1572–1576 (2011)Google Scholar
  52. 52.
    Sidky, E.Y., Kao, C.M., Pan, X.: Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT. J. X-ray Sci. Technol. 14, 119–139 (2006)Google Scholar
  53. 53.
    Sidky, E.Y., Pan, X.: Accurate image reconstruction in circular cone-beam computed tomography by total variation minimization: a preliminary investigation. In: IEEE Nuclear Science Symposium Conference Record, pp. 2904–2907 (2006)Google Scholar
  54. 54.
    Zhu, M.: Fast numerical algorithms for total variation based image restoration. Ph.D. Thesis, University of California at Los Angeles (2008)Google Scholar
  55. 55.
    Lee, S.H., Seo, J.K.: Noise removal with Gauss curvature-driven diffusion. IEEE Trans. Image Process. 14(7), 904–909 (2005)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3(3), 253–276 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Dong, W., Li, X., Zhang, L., Shi, G.: Sparsity-based image denoising via dictionary learning and structural clustering. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 457–464 (2011)Google Scholar
  59. 59.
    Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13, 600–612 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CSE Department, College of EngineeringQatar UniversityDohaQatar
  2. 2.Concordia UniversityMontrealCanada
  3. 3.Lab. Le2iUniversit de BourgogneDijonFrance
  4. 4.New York University of Abu DhabiAbu DhabiUAE

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