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Sensitivity and Variability Analysis for Image Denoising Using Maximum Likelihood Estimation of Exponential Distribution

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Abstract

In this paper, we have performed denoising when the pixel values of images are corrupted by Gaussian and Poisson noises. This paper introduces a new class exponential distribution which lies between Poisson and Gamma distributions. The proposed method combines the ion for denoising the pixels and later a minimization using log-likelihood estimation is performed. The characteristic equation is based on various image parameters like mean, variance, mean deviation, distortion index, shape and scale parameters for minimizing the noise and for maximizing image edge strength to enhance overall visual quality of the image. By utilizing the exponential distribution, we can adaptively control the distortion in the image by minimizing Gaussian and Poisson noises in accordance with the image feature. The simulation results indicate that the proposed algorithm is very efficient to strengthen edge information and remove noise. To provide a probabilistic model we have used statistical approximation of mean and variances. Later, we have evaluated sensitivity and variability effect as well on the image restoration. Experiments were conducted on different test images, which were corrupted by different noise levels in order to assess the performance of the proposed algorithm in comparison with standard and other related denoising methods.

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References

  1. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables, 3rd edn. (Dover Publications, New York, 1965)

    MATH  Google Scholar 

  2. S.K. Bar-Lev, P. Enis, Reproducibility and natural exponential families with power variance functions. Ann. Stat. 14(4), 1507–1522 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Besbeas, I. De Feis, T. Sapatinas, A comparative simulation study of wavelet shrinkage estimators for Poisson counts. Int. Stat. Rev. 72(2), 209–237 (2004)

    Article  MATH  Google Scholar 

  4. M.I.H. Bhuiyan, M.O. Ahmad, M.N.S. Swamy, Spatially adaptive thresholding in wavelet domain for despeckling of ultrasound images. IET Image Proc. 3(3), 147–162 (2009)

    Article  Google Scholar 

  5. G.G. Bhutada, R.S. Anand, S.C. Saxena, Image enhancement by wavelet-based thresholding neural network with adaptive learning rate. IET Image Proc. 5(7), 573–582 (2011)

    Article  MathSciNet  Google Scholar 

  6. J.M. Borwein, P.B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (Wiley, New York, 1998)

    MATH  Google Scholar 

  7. R.P. Brent, Multiple-Precision Zero-Finding Methods and the Complexity of Elementary Function Evaluation-Analytic Computational Complexity (Academic Press, New York, 1975), pp. 151–176

    Google Scholar 

  8. E. Chouzenoux, A. Jezierska, J.-C. Pesquet, H. Talbot, A convex approach for image restoration with exact Poisson–Gaussian likelihood. SIAM J. Imaging Sci. 8(4), 2662–2682 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Clevenson, J. Zidek, Simultaneous estimation of the means of independent Poisson laws. J. Am. Stat. Assoc. 70(351), 698–705 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. M.P. Eckert, A.P. Bradley, Perceptual quality metrics applied to still image compression. Signal Process. Spec. Issue Image Video Qual. Metr. 70(3), 177–200 (1998)

    MATH  Google Scholar 

  11. A.M. Eskicioglu, P.S. Fisher, Image quality measures and their performance. IEEE Trans. Commun. 43(12), 2959–2965 (1995)

    Article  Google Scholar 

  12. S.K.S. Fan, Y. Lin, A fast estimation method for the generalized Gaussian mixture distribution on complex images. Comput. Vis. Image Underst. 113(7), 839–853 (2009)

    Article  Google Scholar 

  13. M.A.T. Figueiredo, J.M. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization. IEEE Trans. Image Process. 19(12), 3133–3145 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Gai, G. Yang, M. Wan, L. Wang, Hidden Markov tree model of images using quaternion wavelet transform. Comput. Electr. Eng. 40(3), 819–832 (2014)

    Article  Google Scholar 

  15. M. Ghosh, J.T. Hwang, K.W. Tsui, Construction of improved estimators in multiparameter estimation for discrete exponential families. Ann. Stat. 11(2), 351–367 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. B. Girod, What’s Wrong with Mean-Squared Error-Digital Images and Human Vision (MIT Press, Cambridge, 1993), pp. 207–220

    Google Scholar 

  17. K. Hirakawa, P.J. Wolfe, Skellam shrinkage: wavelet-based intensity estimation for inhomogeneous Poisson data. IEEE Trans. Inf. Theory 58(2), 1080–1093 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Y.M. Huang, M.K. Ng, Y.W. Wen, A new total variation method for multiplicative noise removal. SIAM J. Imaging Sci. 2(1), 20–40 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. H.M. Hudson, A natural identity for exponential families with applications in multiparameter estimation. Ann. Stat. 6(3), 473–484 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. X. Jin, K. Hirakawa, Approximations to camera sensor noise. in Proceedings SPIE-IS & T , Image Processing: Algorithms and Systems XI, p. 8655 86550H-1–86550H-7 (2013)

  21. J.L. Knight, The joint characteristic function of linear and quadratic forms of non-normal variables. Indian J. Stat. Ser. A 47(2), 231–238 (1985)

    MATH  Google Scholar 

  22. E.D. Kolaczyk, R.D. Nowak, Multiscale likelihood analysis and complexity penalized estimation. Ann. Stat. 32(2), 500–527 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. C.E. Lawrence, F.G. Ronald, Measure Theory and Fine Properties of Functions (Studies in Advanced Mathematics) (CRC Press, Boca Raton, 1991)

    Google Scholar 

  24. T. Le, R. Chartrand, T.J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise. J. Math. Imaging Vis. 27(3), 257–263 (2007)

    Article  MathSciNet  Google Scholar 

  25. S. Levine, Y. Chen, J. Stanich, Image restoration via nonstandard diffusion. Department of Mathematics and Computer Science, Duquesne University, Pittsburgh, PA, Technical Report 04-01 (2004)

  26. A. Li, D. Chen, K. Lin et al., Nonlocal joint regularizations framework with application to image denoising. Circuits Syst. Signal Process. 35(8), 2932–2942 (2016)

    Article  Google Scholar 

  27. F. Luisier, T. Blu, M. Unser, Image denoising in mixed Poisson–Gaussian noise. IEEE Trans. Image Process. 20(3), 696–708 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Makitalo, A. Foi, Optimal inversion of the generalized Anscombe transformation for Poisson–Gaussian noise. IEEE Trans. Image Process. 22(1), 91–103 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. J.A. Nelder, Y. Lee, Likelihood, quasi-likelihood and pseudolikelihood: some comparisons. J. R. Stat. Soc. Ser. B (Methodol.) 54(1), 273–284 (1992)

    MathSciNet  Google Scholar 

  30. J.A. Nelder, D. Pregibon, An extended quasi-likelihood function. Biometrika 74(2), 221–232 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. S. Paul, D.B. Steve, Wavelet denoising of multicomponent images using Gaussian scale mixture models and a noise-free image as priors. IEEE Trans. Image Process. 16(7), 1865–1872 (2007)

    Article  MathSciNet  Google Scholar 

  32. M. Protter, I. Yavneh, M. Elad, Closed-form MMSE estimation for signal denoising under sparse representation modelling over a unitary dictionary. IEEE Trans. Signal Process. 58(7), 3471–3484 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Setzer, G. Steidl, T. Teuber, Deblurring Poissonian images by split Bregman techniques. J. Vis. Commun. Image Represent. 21, 193–199 (2010)

    Article  Google Scholar 

  34. S.V. Stehman, R.L. Czaplewski, Introduction to special issue on map accuracy. Environ. Ecol. Stat. 10(3), 301–308 (2003)

    Article  MathSciNet  Google Scholar 

  35. G. Steidl, T. Teuber, Removing multiplicative noise by Douglas–Rachford splitting. J. Math. Imaging Vis. 36(2), 168–184 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. G.M. Tallis, R. Light, The use of fractional moments for estimating the parameters of a mixed exponential distribution. Technometrics 10(1), 161–175 (1968)

    Article  MathSciNet  Google Scholar 

  37. K.E. Timmermann, R.D. Nowak, Multiscale modeling and estimation of Poisson processes with application to photon-limited imaging. IEEE Trans. Inf. Theory 45(3), 846–862 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Turek, I. Yavneh, M. Protter, M. Elad, On MMSE and MAP denoising under sparse representation modeling over a unitary dictionary. IEEE Trans. Signal Process. 59(8), 3526–3535 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. J.S. Uebersax, A generalized Kappa coefficient. Educ. Psychol. Meas. 42, 181–183 (1982)

    Article  Google Scholar 

  40. A. Veevers, M.C.K. Tweedie, Variance-stabilizing transformation of a Poisson variate by a beta function. J. Roy. Stat. Soc. Ser. C (Appl. Stat.) 20(3), 304–308 (1971)

    Google Scholar 

  41. Z. Wang, A.C. Bovik, A universal image quality index. IEEE Signal Process. Lett. 9(3), 81–84 (2002)

    Article  Google Scholar 

  42. Z. Wang, A.C. Bovik, L. Lu, Why is image quality assessment so difficult. IEEE Int. Conf. Acoust. Speech Signal Proces. Orlando FL. USA 4, 3313–3316 (2002)

    Google Scholar 

  43. T.J. Watson, D.V. Chudnovsky, G.V. Chudnovsky, Approximations and Complex Multiplication According to Ramanujan. IBM T.J, Watson Research Center (1987)

  44. R.M. Willett, R.D. Nowak, Multiscale Poisson intensity and density estimation. IEEE Trans. Inf. Theory 53(9), 3171–3187 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Winkler, Perceptual distortion metric for digital color video. Proc. SPIE Hum. Vis. Electron. Imaging IV 3644, 175–184 (1999)

    Article  Google Scholar 

  46. M.V. Wüthrich, Claims reserving using Tweedie’s compound Poisson model. ASTIN Bull. 33(2), 331–346 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. H.Y. Yang, X.Y. Wang, T.X. Qua, Image denoising using bilateral filter and Gaussian scale mixtures in shiftable complex directional pyramid domain. Comput. Electr. Eng. 37(5), 655–667 (2011)

    Google Scholar 

  48. S. Yun, H. Woo, A new multiplicative denoising variational model based on \(m\)th root transformation. IEEE Trans. Image Process. 21(5), 2523–2533 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. B. Zhang, J.M. Fadili, J.-L. Starck, Wavelets, ridgelets, and curvelets for Poisson noise removal. IEEE Trans. Image Process. 17(7), 1093–1108 (2008)

    Article  MathSciNet  Google Scholar 

  50. B. Zhang, J. Fadili, J.-L. Starck, J.-C. Olivo-Marin, Multiscale variance-stabilizing transform for mixed-Poisson–Gaussian processes and its applications in bioimaging. in IEEE International Conference on Image Processing (ICIP’07), vol. 6, p. VI-233–VI-236 (2007)

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Authors and Affiliations

  1. Universty of Information Science and Technology, Ohrid, Macedonia

    Amita Nandal & Ninoslav Marina

  2. National Institute of Technology, Hamirpur, India

    Arvind Dhaka

  3. Faculty of Electrical Engineering, Autonomous University of Zacatecas, Zacatecas, Mexico

    Hamurabi Gamboa-Rosales, Jorge I. Galvan-Tejada, Carlos E. Galvan-Tejada, Arturo Moreno-Baez & Huizilopoztli Luna-Garcia

  4. CONACYT - Autonomous University of Zacatecas, Zacatecas, Mexico

    Jose M. Celaya-Padilla

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  1. Amita Nandal
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  2. Arvind Dhaka
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Correspondence to Amita Nandal.

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Nandal, A., Dhaka, A., Gamboa-Rosales, H. et al. Sensitivity and Variability Analysis for Image Denoising Using Maximum Likelihood Estimation of Exponential Distribution. Circuits Syst Signal Process 37, 3903–3926 (2018). https://doi.org/10.1007/s00034-018-0746-3

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  • DOI: https://doi.org/10.1007/s00034-018-0746-3

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