Skip to main content
Log in

Manifold Based Low-Rank Regularization for Image Restoration and Semi-Supervised Learning

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Low-rank structures play important roles in recent advances of many problems in image science and data science. As a natural extension of low-rank structures for data with nonlinear structures, the concept of the low-dimensional manifold structure has been considered in many data processing problems. Inspired by this concept, we consider a manifold based low-rank regularization as a linear approximation of manifold dimension. This regularization is less restricted than the global low-rank regularization, and thus enjoy more flexibility to handle data with nonlinear structures. As applications, we demonstrate the proposed regularization to classical inverse problems in image sciences and data sciences including image inpainting, image super-resolution, X-ray computer tomography image reconstruction and semi-supervised learning. We conduct intensive numerical experiments in several image restoration problems and a semi-supervised learning problem of classifying handwritten digits using the MINST data. Our numerical tests demonstrate the effectiveness of the proposed methods and illustrate that the new regularization methods produce outstanding results by comparing with many existing methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Bennett, J., Lanning, S.: The Netflix prize. In: Proceedings of KDD cup and workshop, pp. 35. (2007)

  3. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 417–424. ACM Press/Addison-Wesley Publishing Co., (2000)

  4. Buades, A., Coll, B., Morel, J-M.: A non-local algorithm for image denoising. In: Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), vol. 2, pp. 60–65. IEEE, (2005)

  5. Buades, A., Coll, B., Morel, J-M.: Image enhancement by non-local reverse heat equation. Preprint CMLA, 22 (2006)

  6. Cai, J.F., Chan, R.H., Shen, Z.: Simultaneous cartoon and texture inpainting. Inverse Probl. Imaging 4(3), 379–395 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cai, J.F., Candès, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cai, J.F., Chan, R.H., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal. 24(2), 131–149 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chan, T.F., Kang, S.H.: Error analysis for image inpainting. J. Math. Imaging Vis. 26(1), 85–103 (2006)

    Article  MathSciNet  Google Scholar 

  11. Chan, T.F., Shen, J.: Nontexture inpainting by curvature-driven diffusions. J. Vis. Commun. Image Represent. 12(4), 436–449 (2001)

    Article  Google Scholar 

  12. Chan, T.F., Shen, J.: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  13. Chan, T.F., Shen, J., Zhou, H.M.: Total variation wavelet inpainting. J. Math. Imaging Vis. 25(1), 107–125 (2006)

    Article  MathSciNet  Google Scholar 

  14. Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising with block-matching and 3D filtering. In: Electronic Imaging. International Society for Optics and Photonics, (2006)

  15. Defrise, M., Clack, R.: A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection. IEEE Trans. Med. Imaging 13(1), 186–195 (1994)

    Article  Google Scholar 

  16. Dong, B., Ji, H., Li, J., Shen, Z., Xu, Y.: Wavelet frame based blind image inpainting. Appl. Comput. Harmon. Anal. 32(2), 268–279 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dong, B., Li, J., Shen, Z.: X-ray CT image reconstruction via wavelet frame based regularization and radon domain inpainting. J. Sci. Comput. 54(2–3), 333–349 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dong, B., Shen, Z.: Mra-based wavelet frames and applications. In: IAS Lecture Notes Series, Summer Program on “The Mathematics of Image Processing”, Park City Mathematics Institute, (2010)

  19. Dong, W., Shi, G., Li, X., Ma, Y., Huang, F.: Compressive sensing via nonlocal low-rank regularization. IEEE Trans. Image Process. 23(8), 3618–3632 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Feldkamp, L.A., Davis, L.C., Kress, J.W.: Practical cone-beam algorithm. J. Opt. Soc. Am. A 1(6), 612–619 (1984)

    Article  Google Scholar 

  21. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Sim. 7(3), 1005–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gilboa, G., Sochen, N., Zeevi, Y.Y.: Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Trans. Image Process. 11(7), 689–703 (2002)

    Article  Google Scholar 

  23. Gilboa, G., Sochen, N.A., Zeevi, Y.Y.: Image enhancement and denoising by complex diffusion processes. IEEE Trans. Pattern Anal. Machin. Intell. 26(8), 1020–1036 (2004)

    Article  Google Scholar 

  24. Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Gordon, R., Bender, R., Herman, G.T.: Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29(3), 471–481 (1970)

    Article  Google Scholar 

  27. Hsieh, Jiang: Computed Tomography: Principles, Design, Artifacts and Recent Advances, vol. 114. SPIE press, Bellingham (2003)

    Google Scholar 

  28. Jia, X., Dong, B., Lou, Y., Jiang, S.B.: GPU-based iterative cone beam CT reconstruction using tight frame regularization. Phys. Med. Biol. 56(13), 3787 (2011)

    Article  Google Scholar 

  29. Jia, X., Lou, Y., Lewis, J., Li, R., Gu, X., Men, C., Jiang, S.B.: GPU-based fast low-dose cone beam CT reconstruction via total variation. J. X-ray Sci. Technol. 19(2), 139–153 (2011)

    Google Scholar 

  30. Kuang, D., Shi, Z., Osher, S., Bertozzi, A.: A harmonic extension approach for collaborative ranking. arXiv:1602.05127, (2016)

  31. LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324 (1998)

    Article  Google Scholar 

  32. Li, L., Chen, Z., Xing, Y., Zhang, L., Kang, K., Wang, G.: A general exact method for synthesizing parallel-beam projections from cone-beam projections via filtered backprojection. Phys. Med. Biol. 51(21), 5643–5654 (2006)

    Article  Google Scholar 

  33. Li, Z., Shi, Z., Sun, J.: Point integral method for solving poisson-type equations on manifolds from point clouds with convergence guarantees. Commun. Comput. Phys. 22(1), 228–258 (2017)

    Article  MathSciNet  Google Scholar 

  34. Noo, F., Bernard, C., Litt, F.X., Marchot, P.: A comparison between filtered backprojection algorithm and direct algebraic method in fan beam CT. Signal Process. 51(3), 191–199 (1996)

    Article  Google Scholar 

  35. Osher, S., Shi, Z., Zhu, W.: Low dimensional manifold model for image processing. SIAM J. Imaging Sci. (to appear, 2017)

  36. Peyré, G.: Manifold models for signals and images. Comput. Vis. Image Underst. 113(2), 249–260 (2009)

    Article  Google Scholar 

  37. Quan, Y., Ji, H., Shen, Z.: Data-driven multi-scale non-local wavelet frame construction and image recovery. J. Sci. Comput. 63(2), 307–329 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte Sächsische Akademie der Wissenschaften 69, 262–267 (1917)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Shen, J., Chan, T.F.: Mathematical models for local nontexture inpaintings. SIAM J. Appl. Math. 62(3), 1019–1043 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. Shi, Z., Osher, S., Zhu, W.: Low dimensional manifold model with semi-local patches. UCLA CAM Report (16–63) (2016)

  42. Shi, Z., Osher, S., Zhu, W.: Weighted graph laplacian and image inpainting. J. Sci. Comput. 577 (to appear, 2017)

  43. Shi, Z., Sun, J., Tian, M.: Harmonic extension. arXiv:1509.06458, (2015)

  44. Siddon, R.L.: Fast calculation of the exact radiological path for a 3-dimensional CT array. Med. Phys. 12, 252–255 (1985)

    Article  Google Scholar 

  45. Zhang, X., Chan, T.F.: Wavelet inpainting by nonlocal total variation. Inverse Probl. Imaging 4(1), 191–210 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhu, X., Ghahramani, Z., Lafferty, J.: Semi-supervised learning using gaussian fields and harmonic functions. ICML 3, 912–919 (2003)

    Google Scholar 

  47. Zhu, X., Goldberg, A.B.: Introduction to semi-supervised learning. Synth. Lect. Artif. Intell. Machin. Learn. 3(1), 1–130 (2009)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We thank Prof. Stanley Osher, Prof. Zuoqiang Shi and Mr. Wei Zhu kindly share their valuable comments and code of both LDMM and LDMM+WGL for comparisons.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rongjie Lai.

Additional information

The research of Rongjie Lai is partially supported by NSF Grant DMS–1522645.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lai, R., Li, J. Manifold Based Low-Rank Regularization for Image Restoration and Semi-Supervised Learning. J Sci Comput 74, 1241–1263 (2018). https://doi.org/10.1007/s10915-017-0492-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0492-x

Keywords

Navigation