Skip to main content
SpringerLink
Log in

Total Variation Wavelet Thresholding

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

We propose using Partial Differential Equation (PDE) techniques in wavelet based image processing to remove noise and reduce edge artifacts generated by wavelet thresholding. We employ a variational framework, in particular the minimization of total variation (TV), to select and modify the retained wavelet coefficients so that the reconstructed images have fewer oscillations near edges while noise is smoothed. Numerical experiments show that this approach improves the reconstructed image quality in wavelet compression and in denoising.

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

Similar content being viewed by others

References

  1. Alvarez L., Guichard F., Lions P.L., Morel J.M. (1993). Axioms and fundamental equations of image processing. Arch. Rational Mech. Anal. 16, 200–257

    MathSciNet  Google Scholar 

  2. Alvarez L., Morel J.M. (1994). Formalization and computational aspects of image analysis. Acta Numerica 3, 1–59

    Article  MathSciNet  Google Scholar 

  3. Bertalmio, M., Sapiro, G., Caselles, V., and Ballester, C. (1999). Image inpainting. Tech. Report, ECE-University of Minnesota.

  4. Blomgren, P., and Chan, T. F. (1996). Color TV: Total variation methods for restoration of vector valued images. CAM Report, No. 96-5, Dept. of Math., UCLA.

  5. Candès, E. J., and Guo, F. (2001). Edge-preserving image reconstruction from noisy radon data, in preparation (Invited Special Issue of the Journal of Signal Processing on Image and Video Coding Beyond Standards.).

  6. Chui C.K. (1997). Wavelet: A Mathematical Tool for Signal Analysis, SIAM Monogr. Math. Model. Comput. 1, SIAM, Philadelphia

    Google Scholar 

  7. Candès, E., and Donoho, D. (1999). Curvelets: A surprisingly effective nonadaptive of objects with edges. Tech. Report, Dept. of Stat., Stanford Univ..

  8. Chambolle, A., DeVore, R., Lee, N., and Lucier, B. (1998). Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Tran. Image Proc., Vol. 7, No. 3, Mar. 1998, pp. 319–333.

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Chan, T. F., Golub, G. H., and Mulet, P. (1996). A nonlinear primal-dual method for total variation-based image restoration. In ICAOS’96, 12th International Conference on Analysis and Optimization of Systems: Images, Wavelets, and PDEs, Paris, June 26–28, 1996, number 219 in Lecture Notes in Control and Information Sciences, pp. 241–252.

  11. Chan T.F., Mulet P. (1999). On the convergence of the lagged diffusivity fixed point method in total variation image restoration. SIAM J. Numer. Anal. 36, 354–367

    Article  MathSciNet  Google Scholar 

  12. Chan T.F., Osher S., Shen J. (2001). The digital TV filter and nonlinear denoising. IEEE Trans. Image Process. 10(2): 231–241

    Article  MATH  Google Scholar 

  13. Chan T.F., Shen J. (2002). Mathematical models for local non-texture inpainting. SIAM J. Appl. Math. 62(3): 1019–1043

    Article  MATH  MathSciNet  Google Scholar 

  14. Chan T.F., Vese L. (2001). Active contour without edges. IEEE Trans. Image Process. 10(2): 266–277

    Article  MATH  Google Scholar 

  15. Chan T.F., Wong CK. (1998). Total variation blind deconvolution. IEEE Trans. Image Process. 7, 370–375

    Article  Google Scholar 

  16. Chan, T. F., and Zhou, H. M. (1999). Adaptive ENO-wavelet transforms for discontinuous functions. CAM Report, No. 99-21, Dept. of Math., UCLA, June 1999. To appear in SIAM Numerical Analysis.

  17. Chan, T. F., and Zhou, H. M. (1998). Feature preserving lossy image compression using nonlinear PDEs. In Luk, F. T. (ed.), SPIE Proceedings on Advanced Signal Processing Algorithms, Architectures, and Implementations VIII, Vol. 3461. San Diego, California, July 1998, pp. 316–327.

  18. Chan, T. F., and Zhou, H. M. (2000). Optimal constructions of wavelet coefficients using total variation regularization in image compression. CAM Report, No. 00-27, Dept. of Math., UCLA, July 2000.

  19. Chan, T. F., and Zhou, H. M. (2000). Total variation improved wavelet thresholding in image compression. In Proceedings to the 2000 International Conference on Image Processing, Vancouver, BC, Canada, Sept. 10–13, 2000, pp. 391–394.

  20. Claypoole, P., Davis, G., Sweldens, W., and Baraniuk, R. (1999). Nonlinear wavelet transforms for image coding. Correspond. Author: Baraniuk, Dept. of Elec. and Comp. Sci., also Submit to IEEE Trans. on Image Proc., Preprint, 1999.

  21. Dobson D., Vogel C.R. (1997). Convergence of an iterative method for total variation denoising. SIAM J. Num. Anal. 34, 1779–1971

    Article  MATH  MathSciNet  Google Scholar 

  22. Donoho D. (1995). De-noising by soft thresholding. IEEE Trans. Inf. Th. 41, 613–627

    Article  MATH  MathSciNet  Google Scholar 

  23. Donoho, D. (1997). Wedgelets: Nearly-Minimax Estimation of Edges. Tech. Report, Dept. of Stat., Stanford Univ.

  24. Donoho, D. (1998). Orthonormal Ridgelets and Linear Singularities, Tech. Report, Dept. of Stat., Stanford Univ.

  25. Donoho, D. (1999). Sparse Components of Images and Optimal Atomic Decompositions. Tech. Report, Dept. of Stat., Stanford Univ.

  26. Durand, S., and Froment, J. (2001). Artifact free signal denoising with wavelets. In Proceedings of ICASSP’01, Vol. 6, 2001, pp. 3685–3688.

  27. Special Issue on partial differential equations and geometry-driven diffusion in image processing and analysis. IEEE Trans. Image Proc., Vol. 7, No. 3, Mar. 1998.

  28. Hansen, P. C. (1999). The L-curve and its use in the numerical treatment of inverse problems. Tech. Report, IMM-REP 99-15, Dept. of Math. Model., Tech. Univ. of Denmark.

  29. Harten A. (1993). Discrete multi-resolution analysis and generalized wavelet. Appl. Numer. Math. 12, 153–192

    Article  MATH  MathSciNet  Google Scholar 

  30. Harten, A. (1994). Multiresolution representation of data, II. general framework. CAM Report 94-10, UCLA, Los Angeles, CA.

  31. Malgouyres, F. (2000). Increase in the resolution of digital images: variational theory and applications, Ph.D. thesis, Ecole Normale Supérieure de Cachan, 2000, Cachan, France.

  32. Malgouyres, F. (2002). Mathematical analysis of a model which combines total variation and wavelet for image restoration. Journal of information processes, 2:1, 2002, pp. 1–10.

    Google Scholar 

  33. Malgouyres F., Guichard F. (2001). Edge Direction preserving image zooming: a mathematical and numerical analysis. SIAM, J. Num. Anal. 39(1): 1–37

    Article  MATH  MathSciNet  Google Scholar 

  34. Mallat S. (1998). A Wavelet Tour of Signal Processing. Academic Press, San Diego, CA

    MATH  Google Scholar 

  35. Meyer, Y. (2001). Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, volume 22 of University Lecture Series. AMS, Providence.

  36. Morel, J. M., and Solimini, S. (1994). Variational Methods in Image Segmentation, Birkhauser.

  37. Mumford D., Shah J. (1989). Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685

    Article  MATH  MathSciNet  Google Scholar 

  38. Olshausen, B. A., and Field, D. J. (1979). Emergence of simple-cell receptive field properties by learning a sparse code for natural images, Nature 281, pp. 607–609.

    Google Scholar 

  39. Marquina, A., and Osher, S. (1999). Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal. CAM Report, No.~99-5, Dept. of Math., UCLA.

  40. Le Pennec, E., and Mallat, S. (2000). Image compression with geometrical wavelets. In IEEE Conference on Image Processing(ICIP), Vancouver, September, 2000.

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  43. Sapiro G., Tannenbaum A. (1993). Affine invariant scale-space. Internet. J. Comput. Vision 11, 25–44

    Article  Google Scholar 

  44. Steidl G., Weickert J., Brox T., Mrzek P., Welk M. (2004). On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs. SIAM J. Num. Ana. 42(2): 686–713

    Article  MATH  Google Scholar 

  45. Strang G., Nguyen T. (1996). Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley, MA

    Google Scholar 

  46. Strong, D. M., Blomgren, P., and Chan, T. F. (1997). Spatially adaptive local feature-driven total variation minimizing image restoration. Proceedings of SPIE, Vol. 3167, 1997, pp. 222–233 .

  47. Vogel C., Oman M. (1996). Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17, 227–238

    Article  MATH  MathSciNet  Google Scholar 

  48. Vogel C., Oman M. (1998). Fast, robust total variation–based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7, 813–824

    Article  MATH  MathSciNet  Google Scholar 

  49. Weickert J. (1998). Anisotropic Diffusion in Image Processing. ECMI Series, Teubner, Stuttgart

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of California, Los Angeles, CA, 90095-1555, USA

    Tony F. Chan

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Hao-Min Zhou

Authors
  1. Tony F. Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hao-Min Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hao-Min Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chan, T.F., Zhou, HM. Total Variation Wavelet Thresholding. J Sci Comput 32, 315–341 (2007). https://doi.org/10.1007/s10915-007-9133-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-007-9133-0

Keywords

Search

Navigation