Bilevel Image Denoising Using Gaussianity Tests

  • Jérôme Fehrenbach
  • Mila Nikolova
  • Gabriele Steidl
  • Pierre Weiss
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)

Abstract

We propose a new methodology based on bilevel programming to remove additive white Gaussian noise from images. The lower-level problem consists of a parameterized variational model to denoise images. The parameters are optimized in order to minimize a specific cost function that measures the residual Gaussianity. This model is justified using a statistical analysis. We propose an original numerical method based on the Gauss-Newton algorithm to minimize the outer cost function. We finally perform a few experiments that show the well-foundedness of the approach. We observe a significant improvement compared to standard TV-\(\ell ^2\) algorithms and show that the method automatically adapts to the signal regularity.

Keywords

Bilevel programming Image denoising Gaussianity tests Convex optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baus, F., Nikolova, M., Steidl, G.: Smooth objectives composed of asymptotically affine data-fidelity and regularization: Bounds for the minimizers and parameter choice. Journal of Mathematical Imaging and Vision 48(2), 295–307 (2013)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Candes, E., Demanet, L., Donoho, D., Ying, L.: Fast discrete curvelet transforms. Multiscale Modeling & Simulation 5(3), 861–899 (2006)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Dempe, S.: Foundations of Bilevel Programming. Springer (2002)Google Scholar
  4. 4.
    Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6, 721–741 (1984)CrossRefMATHGoogle Scholar
  5. 5.
    Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM Journal on Imaging Sciences 6(2), 938–983 (2013)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Labate, D., Lim, W.-Q., Kutyniok, G., Weiss, G.: Sparse multidimensional representation using shearlets. In: Papadakis, M., Laine, A.F., Unser, M.A. (eds.) Proceedings of Wavelets XI. Proc. SPIE, vol, 5914, San Diego (2005)Google Scholar
  7. 7.
    Luisier, F., Blu, T., Unser, M.: A new SURE approach to image denoising: Interscale orthonormal wavelet thresholding. IEEE Transactions on Image Processing 16(3), 593–606 (2007)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Morozov, V.A., Stessin, M.: Regularization Methods for Ill-posed Problems. CRC Press Boca Raton, FL (1993)MATHGoogle Scholar
  9. 9.
    Mumford, D., Desolneux, A., et al.: Pattern theory: The Stochastic Analysis of Real-world Signals (2010)Google Scholar
  10. 10.
    Nikolova, M.: Model distortions in Bayesian MAP reconstruction. Inverse Problems and Imaging 1(2), 399 (2007)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Rudin, L.I, Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena 60(1), 259–268 (1992)Google Scholar
  12. 12.
    Weiss, P., Blanc-Féraud, L., Aubert, G.: Efficient schemes for total variation minimization under constraints in image processing. SIAM Journal on Scientific Computing 31(3), 2047–2080 (2009)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jérôme Fehrenbach
    • 1
  • Mila Nikolova
    • 2
  • Gabriele Steidl
    • 3
  • Pierre Weiss
    • 4
  1. 1.CNRS, IMT (UMR5219) and ITAV (USR 3505)Université de ToulouseToulouseFrance
  2. 2.CNRS, CMLAENS CachanCachanFrance
  3. 3.University of KaiserslauternKaiserslauternGermany
  4. 4.CNRS, IMT (UMR5219) and ITAV (USR 3505)Université de ToulouseToulouseFrance

Personalised recommendations