Skip to main content
SpringerLink
Log in

A General Selective Averaging Method for Piecewise Constant Signal and Image Processing

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

Abstract

Piecewise constant signals and images, which are sampled from piecewise constant functions, are an important kind of data. Typical examples include bar code signals, images of texts, hand-written signatures, Quick Response codes (QR codes), logos and cartoons. Selective averaging method is a powerful technique for this kind of signal and image denoising. In this paper, we propose a general selective averaging method (GSAM) to use more flexible weights compared to the previous one. Some convergence results and a probabilistic interpretation are provided for its iterated version. For the choice of the weight parameter, we discuss its influence on the asymptotic rate of convergence. We also study its influence on the denoising results with a moderate number of iterations. Then, our method is compared to the iterated neighborhood filter in signal denoising. In 2D case, we propose a novel extension called the alternating GSAM (AGSAM). We similarly introduce an alternating neighborhood filter. Experimental results demonstrate that our method is especially effective for Gaussian noise removal from noisy piecewise constant signals and images.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

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
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Andelić, M., Da Fonseca, C.: Sufficient conditions for positive definiteness of tridiagonal matrices revisited. Positivity 15(1), 155–159 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Springer, Berlin (2009)

    MATH  Google Scholar 

  3. Barash, D., Comaniciu, D.: A common framework for nonlinear diffusion, adaptive smoothing, bilateral filtering and mean shift. Image Vis. Comput. 22(1), 73–81 (2004)

    Article  Google Scholar 

  4. Boyd, J.E., Meloche, J.: Binary restoration of thin objects in multidimensional imagery. IEEE Trans. Pattern Anal. Mach. Intell. 20(6), 647–651 (1998)

    Article  Google Scholar 

  5. Boyd, S., Diaconis, P., Xiao, L.: Fastest mixing markov chain on a graph. SIAM Rev. 46(4), 667–689 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

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

  8. Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Buades, A., Coll, B., Morel, J.M.: The staircasing effect in neighborhood filters and its solution. IEEE Trans. Image Process. 15(6), 1499–1505 (2006)

    Article  Google Scholar 

  10. Cai, J.F., Dong, B., Shen, Z.: Image restoration: a wavelet frame based model for piecewise smooth functions and beyond. Appl. Comput. Harmon. Anal. 41(1), 94–138 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cai, J.F., Ji, H., Shen, Z., Ye, G.B.: Data-driven tight frame construction and image denoising. Appl. Comput. Harmon. Anal. 37(1), 89–105 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chan, R.H., Chan, T.F., Shen, L., Shen, Z.: Wavelet algorithms for high-resolution image reconstruction. SIAM J. Sci. Comput. 24(4), 1408–1432 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chan, T.F., Esedoglu, S., Nikolova, M.: Finding the global minimum for binary image restoration. In: IEEE International Conference on Image Processing (ICIP) (2005)

  14. Chan, T.F., Shen, J., Vese, L.: Variational PDE models in image processing. Not. Am. Math. Soc. 50(1), 14–26 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Chan, T.F., Zhou, H.M.: Total variation wavelet thresholding. J. Sci. Comput. 32(2), 315–341 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cheng, Y.: Mean shift, mode seeking, and clustering. IEEE Trans. Pattern Anal. Mach. Intell. 17(8), 790–799 (1995)

    Article  Google Scholar 

  17. Choksi, R., van Gennip, Y.: Deblurring of one dimensional bar codes via total variation energy minimization. SIAM J. Imaging Sci. 3(4), 735–764 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Choksi, R., van Gennip, Y., Oberman, A.: Anisotropic total variation regularized \( {L}_{1}\) approximation and denoising/deblurring of 2d bar codes. Inverse Probl. Imaging 5(3), 591–617 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Deng, G., Cahill, L.: An adaptive gaussian filter for noise reduction and edge detection. In: Nuclear Science Symposium and Medical Imaging Conference, IEEE Conference Record, pp. 1615–1619 (1993)

  21. Donoho, D.L.: De-noising by soft-thresholding. IEEE Trans. Inf. Theory 41(3), 613–627 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Donoho, D.L., Johnstone, J.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3), 425–455 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Durrett, R.: Essentials of Stochastic Processes. Springer, Berlin (1999)

    MATH  Google Scholar 

  24. Elad, M.: On the origin of the bilateral filter and ways to improve it. IEEE Trans. Image Process. 11(10), 1141–1151 (2002)

    Article  MathSciNet  Google Scholar 

  25. Elad, M., Aharon, M.: Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736–3745 (2006)

    Article  MathSciNet  Google Scholar 

  26. Esedoglu, S.: Blind deconvolution of bar code signals. Inverse Probl. 20(1), 121 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Guidotti, P., Longo, K.: Two enhanced fourth order diffusion models for image denoising. J. Math. Imaging Vis. 40(2), 188–198 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Härdle, W.: Applied Nonparametric Regression. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  29. Ji, H., Luo, Y., Shen, Z.: Image recovery via geometrically structured approximation. Appl. Comput. Harmon. Anal. 41(1), 75–93 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kornprobst, P., Deriche, R., Aubert, G.: Nonlinear operators in image restoration. In: IEEE Conference on ComputerVision and Pattern Recognition (CVPR), pp. 325–330 (1997)

  31. Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  32. Liu, N., Zheng, X., Sun, H., Tan, X.: Two-dimensional bar code out-of-focus deblurring via the increment constrained least squares filter. Pattern Recognit. Lett. 34(2), 124–130 (2013)

    Article  Google Scholar 

  33. 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 

  34. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  35. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom. 60(1), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  36. Seneta, E.: Non-negative matrices and Markov chains. Springer, Berlin (2006)

    MATH  Google Scholar 

  37. Shen, Y., Lam, E.Y., Wong, N.: A signomial programming approach for binary image restoration by penalized least squares. IEEE Trans. Circuits Sys. II Exp. Briefs 55(1), 41–45 (2008)

    Article  Google Scholar 

  38. Singer, A., Shkolnisky, Y., Nadler, B.: Diffusion interpretation of nonlocal neighborhood filters for signal denoising. SIAM J. Imaging Sci. 2(1), 118–139 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Smith, S.M., Brady, J.M.: Susana new approach to low level image processing. Int. J. Comput. Vis. 23(1), 45–78 (1997)

    Article  Google Scholar 

  40. Strang, G.: The discrete cosine transform. SIAM Rev. 41(1), 135–147 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Taylor, M.E.: Partial Differential Equations I: Basic Theory, 2nd edn. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  42. Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: IEEE International Conference on Computer Vision (ICCV), pp. 839–846 (1998)

  43. Van Gennip, Y., Athavale, P., Gilles, J., Choksi, R.: A regularization approach to blind deblurring and denoising of QR barcodes. IEEE Trans. Image Process. 24(9), 2864–2873 (2015)

    Article  MathSciNet  Google Scholar 

  44. Wand, M.P., Jones, M.C.: Kernel Smoothing. CRC Press, Boca Raton (1994)

    MATH  Google Scholar 

  45. Wang, W., Wen, S., Wu, C., Deng, J.: Denoising piecewise constant images with selective averaging and outlier removal (submitted) (2017)

  46. Weickert, J., Romeny, B., Viergever, M.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7(3), 398–410 (1998)

    Article  Google Scholar 

  47. Wu, C., Tai, X.C.: Augmented lagrangian method, dual methods, and split bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu, L., Lu, C., Xu, Y., Jia, J.: Image smoothing via \(\text{ L }_{0}\) gradient minimization. ACM Trans. Graph. 30(6), 174:1–174:12 (2011)

    Google Scholar 

  49. Yaroslavsky, L.P.: Digital Picture Processing. Springer, Berlin (1985)

    Book  Google Scholar 

  50. 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 

  51. Zhang, J.: An alternating minimization algorithm for binary image restoration. IEEE Trans. Image Process. 21(2), 883–888 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zhang, Y., Dong, B., Lu, Z.: \(\ell _{0}\) minimization for wavelet frame based image restoration. Math. Comput. 82(282), 995–1015 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 11301289, 11531013, 11371341 and 11426236). We thank the authors of Ji et al. [29] for providing us their code. We would like to also thank the anonymous referees for valuable comments and suggestions, which significantly improved the content of the paper.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    Weina Wang & Jiansong Deng

  2. School of Mathematical Sciences, Nankai University, Tianjin, 300071, China

    Chunlin Wu

Authors
  1. Weina Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunlin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiansong Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunlin Wu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, W., Wu, C. & Deng, J. A General Selective Averaging Method for Piecewise Constant Signal and Image Processing. J Sci Comput 76, 1078–1104 (2018). https://doi.org/10.1007/s10915-018-0650-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0650-9

Keywords

Search

Navigation