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A Blind Poisson–Gaussian Noise Separation Using Learning Copula Densities

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Abstract

This paper develops a novel copula denoising optimization solution for separating Poisson noise from images, or removing mixtures of Poisson and Gaussian noise. The proposed approach is elaborated in two steps: first, a spatial bilateral total variation (BTV) regularization is used to reduce Gaussian noise; second, a learning copula procedure is employed to separate the Poisson noise from the ideal image. This leads to capture different image features while significantly reducing the noise. Analytically, we include results on the approximation of the Poisson component as well as the resolution of the proposed optimization model. In addition, to resolve the BTV minimization problem, we proposed an alternating direction method of multipliers algorithm. Finally, numerical results are provided to remove noise while preserving important details and features, along with convincing comparisons to demonstrate the performance of the proposed approach. We show, in particular, that using a large database can improve the robustness of the proposed algorithm.

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Data availability

The datasets generated during the current real study are available in https://www.oasis-brains.org/#data.

Notes

  1. https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/BSDS300/html/dataset/images.html.

  2. http://www.oasis-brains.org.

References

  1. L. Afraites, A. Hadri, A. Laghrib, A denoising model adapted for impulse and gaussian noises using a constrained-PDE. Inverse Probl. 36(2), 025006 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. M.M. Ali, N.N. Mikhail, M.S. Haq, A class of bivariate distributions including the bivariate logistic. J. Multiv. Anal. 8(3), 405–412 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. M.R. Banham, A.K. Katsaggelos, Digital image restoration. IEEE Signal Process. Mag. 14(2), 24–41 (1997)

    Article  Google Scholar 

  4. L. Calatroni, J.C. De Los Reyes, C.-B. Schonlieb, Infimal convolution of data discrepancies for mixed noise removal. SIAM J. Imaging Sci. 10(3), 1196–1233 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. D.G. Clayton, A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65(1), 141–151 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. B.R. Corner. Information content analysis and noise characterization in remote sensing image interpretation (The University of Nebraska-Lincoln, 2004)

  7. M.N. Do, M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans. image Process. 14(12), 2091–2106 (2005)

    Article  Google Scholar 

  8. S. Durand, J. Fadili, M. Nikolova, Multiplicative noise removal using l1 fidelity on frame coefficients. J. Math. Imaging Vis. 36(3), 201–226 (2010)

    Article  Google Scholar 

  9. M. El Helou, S. Süsstrunk, Blind universal Bayesian image denoising with gaussian noise level learning. IEEE Trans. Image Process. 29, 4885–4897 (2020)

    Article  MATH  Google Scholar 

  10. I. El Mourabit, M. El Rhabi, A. Hakim, A. Laghrib, E. Moreau, A new denoising model for multi-frame super-resolution image reconstruction. Signal Process. 132, 51–65 (2017)

    Article  Google Scholar 

  11. M.J. Frank, On the simultaneous associativity of \(F(x,\, y)\) and \(x+y-F(x,\, y)\). Aequationes Math. 19(2–3), 194–226 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Ghazdali, M. El Rhabi, H. Fenniri, A. Hakim, A. Keziou, Blind noisy mixture separation for independent/dependent sources through a regularized criterion on copulas. Signal Process. 131, 502–513 (2017)

    Article  Google Scholar 

  13. J.J. Goldberger, J. Ng, Practical signal and image processing in clinical cardiology (Springer, 2010)

  14. B. Goyal, A. Dogra, S. Agrawal, B.S. Sohi, A. Sharma, Image denoising review: from classical to state-of-the-art approaches. Inf. Fusion 55, 220–244 (2020)

    Article  Google Scholar 

  15. K.H. Jin, M.T. McCann, E. Froustey, M. Unser, Deep convolutional neural network for inverse problems in imaging. IEEE Trans. Image Process. 26(9), 4509–4522 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Keziou, H. Fenniri, A. Ghazdali, E. Moreau, New blind source separation method of independent/dependent sources. Signal Process. 104, 319–324 (2014)

    Article  Google Scholar 

  17. D.S. Marcus, T.H. Wang, J. Parker, J.G. Csernansky, J.C. Morris, R.L. Buckner, Open access series of imaging studies (oasis): cross-sectional MRI data in young, middle aged, nondemented, and demented older adults. J. Cogn. Neurosci. 19(9), 1498–1507 (2007)

    Article  Google Scholar 

  18. G.R. Naik, W. Wang et al., Blind source separation (Springer, Berlin, 2014), pp.978–3

    Book  Google Scholar 

  19. R.B. Nelsen, An introduction to copulas, 2nd edn. (Springer Series in Statistics. Springer, New York, 2006)

    MATH  Google Scholar 

  20. B. Saboury, M.A. Morris, M. Nikpanah, T.J. Werner, E.C. Jones, A. Alavi, Reinventing molecular imaging with total-body pet, part ii: clinical applications. Pet Clin. 15(4), 463 (2020)

    Article  Google Scholar 

  21. H.R. Shahdoosti, Z. Rahemi, Edge-preserving image denoising using a deep convolutional neural network. Signal Process. 159, 20–32 (2019)

    Article  Google Scholar 

  22. W. Shi, Q. Ling, K. Yuan, W. Gang, W. Yin, On the linear convergence of the ADMM in decentralized consensus optimization. IEEE Trans. Signal Process. 62(7), 1750–1761 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Sklar, Fonctions de répartition à \(n\) dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231 (1959)

    MathSciNet  MATH  Google Scholar 

  24. M.K. Tripathi, D.D. Maktedar, A role of computer vision in fruits and vegetables among various horticulture products of agriculture fields: A survey. Inf. Process. Agric. 7(2), 183–203 (2020)

    Google Scholar 

  25. B. Xu, D. Kocyigit, R. Grimm, B.P. Griffin, F. Cheng. Applications of artificial intelligence in multimodality cardiovascular imaging: a state-of-the-art review. Progress in cardiovascular diseases, 2020

  26. X. Zhang, M.K. Ng, M. Bai, A fast algorithm for deconvolution and Poisson noise removal. J. Sci. Comput. 75(3), 1535–1554 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Zhao, Y.-W. Wen, M. Ng, H. Li, A nonlocal low rank model for Poisson noise removal. Inverse Probl. Imaging 15(3), 519 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are grateful to the anonymous referees for the corrections and useful suggestions that have improved this article.

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Authors and Affiliations

  1. LIPIM, ENSA Khouribga, Université Sultan Moulay Slimane, Beni Mellal, Morocco

    Abdelghani Ghazdali

  2. Lab SIV, Université Ibn Zohr, Agadir, Morocco

    Aissam Hadri

  3. EMI FST Béni-Mellal, Université Sultan Moulay Slimane, Beni Mellal, Morocco

    Amine Laghrib & Mourad Nachaoui

Authors
  1. Abdelghani Ghazdali
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  2. Aissam Hadri
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  3. Amine Laghrib
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  4. Mourad Nachaoui
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Correspondence to Amine Laghrib.

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Ghazdali, A., Hadri, A., Laghrib, A. et al. A Blind Poisson–Gaussian Noise Separation Using Learning Copula Densities. Circuits Syst Signal Process 42, 6564–6590 (2023). https://doi.org/10.1007/s00034-023-02326-1

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  • DOI: https://doi.org/10.1007/s00034-023-02326-1

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