Abstract

Optimal denoising works at best on raw images (the image formed at the output of the focal plane, at the CCD or CMOS detector), which display a white signal-dependent noise. The noise model of the raw image is characterized by a function that given the intensity of a pixel in the noisy image returns the corresponding standard deviation; the plot of this function is the noise curve. This paper develops a nonparametric approach estimating the noise curve directly from a single raw image. An extensive cross-validation procedure is described to compare this new method with state-of-the-art parametric methods and with laboratory calibration methods giving a reliable ground truth, even for nonlinear detectors.

© 2014 Optical Society of America

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    [CrossRef]
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  3. J. Schmitt, J. L. Starck, J. M. Casandjian, J. Fadili, and I. Grenier, “Multichannel Poisson denoising and deconvolution on the sphere: application to the Fermi gamma ray space telescope,” Astron. Astrophys. 546, A114 (2012).
    [CrossRef]
  4. F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed Poisson–Gaussian noise,” IEEE Trans. Image Process. 20, 696–708 (2011).
    [CrossRef]
  5. F.-X. Dupé, J. M. Fadili, and J.-L. Starck, “A proximal iteration for deconvolving Poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321 (2009).
    [CrossRef]
  6. H. Rabbani, R. Nezafat, and S. Gazor, “Wavelet-domain medical image denoising using bivariate laplacian mixture model,” IEEE Trans. Biomed. Eng. 56, 2826–2837 (2009).
    [CrossRef]
  7. B. Zhang, J. M. Fadili, and J.-L. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
    [CrossRef]
  8. M. Lebrun, M. Colom, A. Buades, and J. M. Morel, “Secrets of image denoising cuisine,” Acta Numerica 21, 475–576 (2012).
  9. E. D. Kolaczyk, “Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds,” Statist. Sin. 9, 119–135 (1999).
  10. R. D. Nowak and R. G. Baraniuk, “Wavelet-domain filtering for photon imaging systems,” IEEE Trans. Image Process. 8, 666–678 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
    [CrossRef]
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  15. M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process. 20, 99–109 (2011).
    [CrossRef]
  16. A. Foi, “Noise estimation and removal in mr imaging: the variance-stabilization approach,” in 2011 IEEE International Symposium on Biomedical Imaging: From Nano to Macro (IEEE, 2011), pp. 1809–1814.
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  18. N. N. Ponomarenko, V. V. Lukin, S. K. Abramov, K. O. Egiazarian, and J. T. Astola, “Blind evaluation of additive noise variance in textured images by nonlinear processing of block DCT coefficients,” in Proceedings of the International Society for Optics and Photonics. Electronic Imaging, Image Processing: Algorithms and Systems II (2003), Vol. 5014, pp. 178–189.
  19. H. Rabbani and S. Gazor, “Local probability distribution of natural signals in sparse domains,” Int. J. Adapt. Control Signal Process. 28, 52–62 (2014).
  20. V. I. A. Katkovnik, V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006), Vol. PM157.
  21. S. Pyatykh, J. Hesser, and L. Zheng, “Image noise level estimation by principal component analysis,” IEEE Trans. Image Process. 22, 687–699 (2013).
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  23. D. L. Donoho and I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
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  24. D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).
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    [CrossRef]
  27. J. S. Lee and K. Hoppel, “Noise modelling and estimation of remotely-sensed images,” in Proceedings of the International Geoscience and Remote Sensing Symposium (1989), Vol. 2, pp. 1005–1008.
  28. G. A. Mastin, “Adaptive filters for digital image noise smoothing: an evaluation,” Comput. Vis. Graph. Image Process. 31, 103–121 (1985).
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    [CrossRef]
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    [CrossRef]
  36. M. Uss, B. Vozel, V. Lukin, S. Abramov, I. Baryshev, and K. Chehdi, “Image informative maps for estimating noise standard deviation and texture parameters,” EURASIP J. Advances Signal Process. 2011, 806516 (2011).
  37. C. Liu, R. Szeliski, S. B. Kang, C. L. Zitnick, and W. T. Freeman, “Automatic estimation and removal of noise from a single image,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 299–314 (2008).
    [CrossRef]
  38. P. L. Vora, J. E. Farrell, J. D. Tietz, and D. H. Brainard, “Linear models for digital cameras,” in IS&T Annual Conference (The Society for Imaging Science and Technology, 1997), pp. 377–382.
  39. M. Lebrun, A. Buades, and J. M. Morel, “A nonlocal Bayesian image denoising algorithm,” SIAM J. Imaging Sci. 6, 1665–1688 (2013).

2014

H. Rabbani and S. Gazor, “Local probability distribution of natural signals in sparse domains,” Int. J. Adapt. Control Signal Process. 28, 52–62 (2014).

2013

S. Pyatykh, J. Hesser, and L. Zheng, “Image noise level estimation by principal component analysis,” IEEE Trans. Image Process. 22, 687–699 (2013).
[CrossRef]

P. Milanfar, “A tour of modern image filtering: new insights and methods, both practical and theoretical,” IEEE Signal Process. Mag. 30(1), 106–128 (2013).
[CrossRef]

M. L. Uss, B. Vozel, V. V. Lukin, and K. Chehdi, “Image informative maps for component-wise estimating parameters of signal-dependent noise,” J. Electron. Imaging 22, 013019 (2013).
[CrossRef]

M. Lebrun, A. Buades, and J. M. Morel, “A nonlocal Bayesian image denoising algorithm,” SIAM J. Imaging Sci. 6, 1665–1688 (2013).

2012

M. Lebrun, M. Colom, A. Buades, and J. M. Morel, “Secrets of image denoising cuisine,” Acta Numerica 21, 475–576 (2012).

J. Schmitt, J. L. Starck, J. M. Casandjian, J. Fadili, and I. Grenier, “Multichannel Poisson denoising and deconvolution on the sphere: application to the Fermi gamma ray space telescope,” Astron. Astrophys. 546, A114 (2012).
[CrossRef]

2011

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed Poisson–Gaussian noise,” IEEE Trans. Image Process. 20, 696–708 (2011).
[CrossRef]

H. Rabbani, M. Sonka, and M. D. Abramoff, “Optical coherence tomography noise reduction using anisotropic local bivariate,” Int. J. Biomed. Imag. 3, 417491 (2011).

C. A. Deledalle, L. Denis, and F. Tupin, “Nl-insar: nonlocal interferogram estimation,” IEEE Trans. Geosci. Remote Sens. 49, 1441–1452 (2011).
[CrossRef]

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process. 20, 99–109 (2011).
[CrossRef]

M. Uss, B. Vozel, V. Lukin, S. Abramov, I. Baryshev, and K. Chehdi, “Image informative maps for estimating noise standard deviation and texture parameters,” EURASIP J. Advances Signal Process. 2011, 806516 (2011).

2009

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: application to photo-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[CrossRef]

F.-X. Dupé, J. M. Fadili, and J.-L. Starck, “A proximal iteration for deconvolving Poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321 (2009).
[CrossRef]

H. Rabbani, R. Nezafat, and S. Gazor, “Wavelet-domain medical image denoising using bivariate laplacian mixture model,” IEEE Trans. Biomed. Eng. 56, 2826–2837 (2009).
[CrossRef]

2008

B. Zhang, J. M. Fadili, and J.-L. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[CrossRef]

C. Liu, R. Szeliski, S. B. Kang, C. L. Zitnick, and W. T. Freeman, “Automatic estimation and removal of noise from a single image,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 299–314 (2008).
[CrossRef]

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17, 1737–1754 (2008).
[CrossRef]

2005

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

1999

E. D. Kolaczyk, “Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds,” Statist. Sin. 9, 119–135 (1999).

1997

R. D. Nowak and R. G. Baraniuk, “Wavelet-domain filtering for photon imaging systems,” IEEE Trans. Image Process. 8, 666–678 (1997).
[CrossRef]

1996

J. Immerkaer, “Fast noise variance estimation,” Comput. Vis. Image Underst. 64, 300–302 (1996).
[CrossRef]

1995

D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).

1994

D. L. Donoho and I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

1993

S. I. Olsen, “Estimation of noise in images: an evaluation,” Graph. Models Image Proc. 55, 319–323 (1993).

1990

P. Meer, J. M. Jolion, and A. Rosenfeld, “A fast parallel algorithm for blind estimation of noise variance,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 216–223 (1990).
[CrossRef]

1985

G. A. Mastin, “Adaptive filters for digital image noise smoothing: an evaluation,” Comput. Vis. Graph. Image Process. 31, 103–121 (1985).

1981

J. S. Lee, “Refined filtering of image noise using local statistics,” Comp. Graph. Image Proc. 15, 380–389 (1981).
[CrossRef]

1948

F. J. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika 35, 246–254 (1948).

Abramoff, M. D.

H. Rabbani, M. Sonka, and M. D. Abramoff, “Optical coherence tomography noise reduction using anisotropic local bivariate,” Int. J. Biomed. Imag. 3, 417491 (2011).

Abramov, S.

M. Uss, B. Vozel, V. Lukin, S. Abramov, I. Baryshev, and K. Chehdi, “Image informative maps for estimating noise standard deviation and texture parameters,” EURASIP J. Advances Signal Process. 2011, 806516 (2011).

Abramov, S. K.

N. N. Ponomarenko, V. V. Lukin, S. K. Abramov, K. O. Egiazarian, and J. T. Astola, “Blind evaluation of additive noise variance in textured images by nonlinear processing of block DCT coefficients,” in Proceedings of the International Society for Optics and Photonics. Electronic Imaging, Image Processing: Algorithms and Systems II (2003), Vol. 5014, pp. 178–189.

Anscombe, F. J.

F. J. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika 35, 246–254 (1948).

Astola, J.

V. I. A. Katkovnik, V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006), Vol. PM157.

Astola, J. T.

N. N. Ponomarenko, V. V. Lukin, M. S. Zriakhov, A. Kaarna, and J. T. Astola, “An automatic approach to lossy compression of AVIRIS images,” in International Geoscience and Remote Sensing Symposium (IEEE, 2007), pp. 472–475.

N. N. Ponomarenko, V. V. Lukin, S. K. Abramov, K. O. Egiazarian, and J. T. Astola, “Blind evaluation of additive noise variance in textured images by nonlinear processing of block DCT coefficients,” in Proceedings of the International Society for Optics and Photonics. Electronic Imaging, Image Processing: Algorithms and Systems II (2003), Vol. 5014, pp. 178–189.

Baraniuk, R. G.

R. D. Nowak and R. G. Baraniuk, “Wavelet-domain filtering for photon imaging systems,” IEEE Trans. Image Process. 8, 666–678 (1997).
[CrossRef]

Baryshev, I.

M. Uss, B. Vozel, V. Lukin, S. Abramov, I. Baryshev, and K. Chehdi, “Image informative maps for estimating noise standard deviation and texture parameters,” EURASIP J. Advances Signal Process. 2011, 806516 (2011).

Blu, T.

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed Poisson–Gaussian noise,” IEEE Trans. Image Process. 20, 696–708 (2011).
[CrossRef]

Bracho, R.

R. Bracho and A. C. Sanderson, “Segmentation of images based on intensity gradient information,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 1985), pp. 19–23.

Brainard, D. H.

P. L. Vora, J. E. Farrell, J. D. Tietz, and D. H. Brainard, “Linear models for digital cameras,” in IS&T Annual Conference (The Society for Imaging Science and Technology, 1997), pp. 377–382.

Buades, A.

M. Lebrun, A. Buades, and J. M. Morel, “A nonlocal Bayesian image denoising algorithm,” SIAM J. Imaging Sci. 6, 1665–1688 (2013).

M. Lebrun, M. Colom, A. Buades, and J. M. Morel, “Secrets of image denoising cuisine,” Acta Numerica 21, 475–576 (2012).

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

Casandjian, J. M.

J. Schmitt, J. L. Starck, J. M. Casandjian, J. Fadili, and I. Grenier, “Multichannel Poisson denoising and deconvolution on the sphere: application to the Fermi gamma ray space telescope,” Astron. Astrophys. 546, A114 (2012).
[CrossRef]

Chehdi, K.

M. L. Uss, B. Vozel, V. V. Lukin, and K. Chehdi, “Image informative maps for component-wise estimating parameters of signal-dependent noise,” J. Electron. Imaging 22, 013019 (2013).
[CrossRef]

M. Uss, B. Vozel, V. Lukin, S. Abramov, I. Baryshev, and K. Chehdi, “Image informative maps for estimating noise standard deviation and texture parameters,” EURASIP J. Advances Signal Process. 2011, 806516 (2011).

Coll, B.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

Colom, M.

M. Lebrun, M. Colom, A. Buades, and J. M. Morel, “Secrets of image denoising cuisine,” Acta Numerica 21, 475–576 (2012).

Dalalyan, A.

J. Salmon, C.-A. Deledalle, and A. Dalalyan, “Image denoising with patch based PCA: local versus global,” in Proceedings of the British Machine Vision Conference (BMVA, 2011), pp. 25.1–25.10.

Deledalle, C. A.

C. A. Deledalle, L. Denis, and F. Tupin, “Nl-insar: nonlocal interferogram estimation,” IEEE Trans. Geosci. Remote Sens. 49, 1441–1452 (2011).
[CrossRef]

Deledalle, C.-A.

J. Salmon, C.-A. Deledalle, and A. Dalalyan, “Image denoising with patch based PCA: local versus global,” in Proceedings of the British Machine Vision Conference (BMVA, 2011), pp. 25.1–25.10.

Denis, L.

C. A. Deledalle, L. Denis, and F. Tupin, “Nl-insar: nonlocal interferogram estimation,” IEEE Trans. Geosci. Remote Sens. 49, 1441–1452 (2011).
[CrossRef]

Donoho, D. L.

D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).

D. L. Donoho and I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

Dupé, F.-X.

F.-X. Dupé, J. M. Fadili, and J.-L. Starck, “A proximal iteration for deconvolving Poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321 (2009).
[CrossRef]

Egiazarian, K.

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17, 1737–1754 (2008).
[CrossRef]

V. I. A. Katkovnik, V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006), Vol. PM157.

Egiazarian, K. O.

N. N. Ponomarenko, V. V. Lukin, S. K. Abramov, K. O. Egiazarian, and J. T. Astola, “Blind evaluation of additive noise variance in textured images by nonlinear processing of block DCT coefficients,” in Proceedings of the International Society for Optics and Photonics. Electronic Imaging, Image Processing: Algorithms and Systems II (2003), Vol. 5014, pp. 178–189.

Fadili, J.

J. Schmitt, J. L. Starck, J. M. Casandjian, J. Fadili, and I. Grenier, “Multichannel Poisson denoising and deconvolution on the sphere: application to the Fermi gamma ray space telescope,” Astron. Astrophys. 546, A114 (2012).
[CrossRef]

Fadili, J. M.

F.-X. Dupé, J. M. Fadili, and J.-L. Starck, “A proximal iteration for deconvolving Poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321 (2009).
[CrossRef]

B. Zhang, J. M. Fadili, and J.-L. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[CrossRef]

Farrell, J. E.

P. L. Vora, J. E. Farrell, J. D. Tietz, and D. H. Brainard, “Linear models for digital cameras,” in IS&T Annual Conference (The Society for Imaging Science and Technology, 1997), pp. 377–382.

Foi, A.

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process. 20, 99–109 (2011).
[CrossRef]

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17, 1737–1754 (2008).
[CrossRef]

A. Foi, “Noise estimation and removal in mr imaging: the variance-stabilization approach,” in 2011 IEEE International Symposium on Biomedical Imaging: From Nano to Macro (IEEE, 2011), pp. 1809–1814.

Freeman, W. T.

C. Liu, R. Szeliski, S. B. Kang, C. L. Zitnick, and W. T. Freeman, “Automatic estimation and removal of noise from a single image,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 299–314 (2008).
[CrossRef]

Gazor, S.

H. Rabbani and S. Gazor, “Local probability distribution of natural signals in sparse domains,” Int. J. Adapt. Control Signal Process. 28, 52–62 (2014).

H. Rabbani, R. Nezafat, and S. Gazor, “Wavelet-domain medical image denoising using bivariate laplacian mixture model,” IEEE Trans. Biomed. Eng. 56, 2826–2837 (2009).
[CrossRef]

Grenier, I.

J. Schmitt, J. L. Starck, J. M. Casandjian, J. Fadili, and I. Grenier, “Multichannel Poisson denoising and deconvolution on the sphere: application to the Fermi gamma ray space telescope,” Astron. Astrophys. 546, A114 (2012).
[CrossRef]

Hesser, J.

S. Pyatykh, J. Hesser, and L. Zheng, “Image noise level estimation by principal component analysis,” IEEE Trans. Image Process. 22, 687–699 (2013).
[CrossRef]

Hoppel, K.

J. S. Lee and K. Hoppel, “Noise modelling and estimation of remotely-sensed images,” in Proceedings of the International Geoscience and Remote Sensing Symposium (1989), Vol. 2, pp. 1005–1008.

Immerkaer, J.

J. Immerkaer, “Fast noise variance estimation,” Comput. Vis. Image Underst. 64, 300–302 (1996).
[CrossRef]

Johnstone, I.

D. L. Donoho and I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[CrossRef]

Johnstone, I. M.

D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).

Jolion, J. M.

P. Meer, J. M. Jolion, and A. Rosenfeld, “A fast parallel algorithm for blind estimation of noise variance,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 216–223 (1990).
[CrossRef]

Kaarna, A.

N. N. Ponomarenko, V. V. Lukin, M. S. Zriakhov, A. Kaarna, and J. T. Astola, “An automatic approach to lossy compression of AVIRIS images,” in International Geoscience and Remote Sensing Symposium (IEEE, 2007), pp. 472–475.

Kang, S. B.

C. Liu, R. Szeliski, S. B. Kang, C. L. Zitnick, and W. T. Freeman, “Automatic estimation and removal of noise from a single image,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 299–314 (2008).
[CrossRef]

Katkovnik, V.

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17, 1737–1754 (2008).
[CrossRef]

V. I. A. Katkovnik, V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006), Vol. PM157.

Katkovnik, V. I. A.

V. I. A. Katkovnik, V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006), Vol. PM157.

Kolaczyk, E. D.

E. D. Kolaczyk, “Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds,” Statist. Sin. 9, 119–135 (1999).

Lebrun, M.

M. Lebrun, A. Buades, and J. M. Morel, “A nonlocal Bayesian image denoising algorithm,” SIAM J. Imaging Sci. 6, 1665–1688 (2013).

M. Lebrun, M. Colom, A. Buades, and J. M. Morel, “Secrets of image denoising cuisine,” Acta Numerica 21, 475–576 (2012).

Lee, J. S.

J. S. Lee, “Refined filtering of image noise using local statistics,” Comp. Graph. Image Proc. 15, 380–389 (1981).
[CrossRef]

J. S. Lee and K. Hoppel, “Noise modelling and estimation of remotely-sensed images,” in Proceedings of the International Geoscience and Remote Sensing Symposium (1989), Vol. 2, pp. 1005–1008.

Lefkimmiatis, S.

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: application to photo-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[CrossRef]

Lendl, M.

K. Rank, M. Lendl, and R. Unbehauen, “Estimation of image noise variance,” in IEEE Proceedings on Vision, Image and Signal Processing (IET, 1999), Vol. 146, pp. 80–84.

Liu, C.

C. Liu, R. Szeliski, S. B. Kang, C. L. Zitnick, and W. T. Freeman, “Automatic estimation and removal of noise from a single image,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 299–314 (2008).
[CrossRef]

Luisier, F.

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed Poisson–Gaussian noise,” IEEE Trans. Image Process. 20, 696–708 (2011).
[CrossRef]

Lukin, V.

M. Uss, B. Vozel, V. Lukin, S. Abramov, I. Baryshev, and K. Chehdi, “Image informative maps for estimating noise standard deviation and texture parameters,” EURASIP J. Advances Signal Process. 2011, 806516 (2011).

Lukin, V. V.

M. L. Uss, B. Vozel, V. V. Lukin, and K. Chehdi, “Image informative maps for component-wise estimating parameters of signal-dependent noise,” J. Electron. Imaging 22, 013019 (2013).
[CrossRef]

N. N. Ponomarenko, V. V. Lukin, S. K. Abramov, K. O. Egiazarian, and J. T. Astola, “Blind evaluation of additive noise variance in textured images by nonlinear processing of block DCT coefficients,” in Proceedings of the International Society for Optics and Photonics. Electronic Imaging, Image Processing: Algorithms and Systems II (2003), Vol. 5014, pp. 178–189.

N. N. Ponomarenko, V. V. Lukin, M. S. Zriakhov, A. Kaarna, and J. T. Astola, “An automatic approach to lossy compression of AVIRIS images,” in International Geoscience and Remote Sensing Symposium (IEEE, 2007), pp. 472–475.

Makitalo, M.

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process. 20, 99–109 (2011).
[CrossRef]

Maragos, P.

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: application to photo-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[CrossRef]

Mastin, G. A.

G. A. Mastin, “Adaptive filters for digital image noise smoothing: an evaluation,” Comput. Vis. Graph. Image Process. 31, 103–121 (1985).

Meer, P.

P. Meer, J. M. Jolion, and A. Rosenfeld, “A fast parallel algorithm for blind estimation of noise variance,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 216–223 (1990).
[CrossRef]

Milanfar, P.

P. Milanfar, “A tour of modern image filtering: new insights and methods, both practical and theoretical,” IEEE Signal Process. Mag. 30(1), 106–128 (2013).
[CrossRef]

Morel, J. M.

M. Lebrun, A. Buades, and J. M. Morel, “A nonlocal Bayesian image denoising algorithm,” SIAM J. Imaging Sci. 6, 1665–1688 (2013).

M. Lebrun, M. Colom, A. Buades, and J. M. Morel, “Secrets of image denoising cuisine,” Acta Numerica 21, 475–576 (2012).

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

Nezafat, R.

H. Rabbani, R. Nezafat, and S. Gazor, “Wavelet-domain medical image denoising using bivariate laplacian mixture model,” IEEE Trans. Biomed. Eng. 56, 2826–2837 (2009).
[CrossRef]

Nowak, R. D.

R. D. Nowak and R. G. Baraniuk, “Wavelet-domain filtering for photon imaging systems,” IEEE Trans. Image Process. 8, 666–678 (1997).
[CrossRef]

Olsen, S. I.

S. I. Olsen, “Estimation of noise in images: an evaluation,” Graph. Models Image Proc. 55, 319–323 (1993).

Papandreou, G.

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: application to photo-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[CrossRef]

Poggio, T.

H. Voorhees and T. Poggio, “Detecting textons and texture boundaries in natural image,” in Proceedings of the First International Conference on Computer Vision London (IEEE, 1987), pp. 250–258.

Ponomarenko, N. N.

N. N. Ponomarenko, V. V. Lukin, S. K. Abramov, K. O. Egiazarian, and J. T. Astola, “Blind evaluation of additive noise variance in textured images by nonlinear processing of block DCT coefficients,” in Proceedings of the International Society for Optics and Photonics. Electronic Imaging, Image Processing: Algorithms and Systems II (2003), Vol. 5014, pp. 178–189.

N. N. Ponomarenko, V. V. Lukin, M. S. Zriakhov, A. Kaarna, and J. T. Astola, “An automatic approach to lossy compression of AVIRIS images,” in International Geoscience and Remote Sensing Symposium (IEEE, 2007), pp. 472–475.

Pyatykh, S.

S. Pyatykh, J. Hesser, and L. Zheng, “Image noise level estimation by principal component analysis,” IEEE Trans. Image Process. 22, 687–699 (2013).
[CrossRef]

Rabbani, H.

H. Rabbani and S. Gazor, “Local probability distribution of natural signals in sparse domains,” Int. J. Adapt. Control Signal Process. 28, 52–62 (2014).

H. Rabbani, M. Sonka, and M. D. Abramoff, “Optical coherence tomography noise reduction using anisotropic local bivariate,” Int. J. Biomed. Imag. 3, 417491 (2011).

H. Rabbani, R. Nezafat, and S. Gazor, “Wavelet-domain medical image denoising using bivariate laplacian mixture model,” IEEE Trans. Biomed. Eng. 56, 2826–2837 (2009).
[CrossRef]

Rank, K.

K. Rank, M. Lendl, and R. Unbehauen, “Estimation of image noise variance,” in IEEE Proceedings on Vision, Image and Signal Processing (IET, 1999), Vol. 146, pp. 80–84.

Rosenfeld, A.

P. Meer, J. M. Jolion, and A. Rosenfeld, “A fast parallel algorithm for blind estimation of noise variance,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 216–223 (1990).
[CrossRef]

Salmon, J.

J. Salmon, C.-A. Deledalle, and A. Dalalyan, “Image denoising with patch based PCA: local versus global,” in Proceedings of the British Machine Vision Conference (BMVA, 2011), pp. 25.1–25.10.

Sanderson, A. C.

R. Bracho and A. C. Sanderson, “Segmentation of images based on intensity gradient information,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 1985), pp. 19–23.

Schmitt, J.

J. Schmitt, J. L. Starck, J. M. Casandjian, J. Fadili, and I. Grenier, “Multichannel Poisson denoising and deconvolution on the sphere: application to the Fermi gamma ray space telescope,” Astron. Astrophys. 546, A114 (2012).
[CrossRef]

Sonka, M.

H. Rabbani, M. Sonka, and M. D. Abramoff, “Optical coherence tomography noise reduction using anisotropic local bivariate,” Int. J. Biomed. Imag. 3, 417491 (2011).

Starck, J. L.

J. Schmitt, J. L. Starck, J. M. Casandjian, J. Fadili, and I. Grenier, “Multichannel Poisson denoising and deconvolution on the sphere: application to the Fermi gamma ray space telescope,” Astron. Astrophys. 546, A114 (2012).
[CrossRef]

Starck, J.-L.

F.-X. Dupé, J. M. Fadili, and J.-L. Starck, “A proximal iteration for deconvolving Poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321 (2009).
[CrossRef]

B. Zhang, J. M. Fadili, and J.-L. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[CrossRef]

Szeliski, R.

C. Liu, R. Szeliski, S. B. Kang, C. L. Zitnick, and W. T. Freeman, “Automatic estimation and removal of noise from a single image,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 299–314 (2008).
[CrossRef]

Tietz, J. D.

P. L. Vora, J. E. Farrell, J. D. Tietz, and D. H. Brainard, “Linear models for digital cameras,” in IS&T Annual Conference (The Society for Imaging Science and Technology, 1997), pp. 377–382.

Trimeche, M.

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17, 1737–1754 (2008).
[CrossRef]

Tupin, F.

C. A. Deledalle, L. Denis, and F. Tupin, “Nl-insar: nonlocal interferogram estimation,” IEEE Trans. Geosci. Remote Sens. 49, 1441–1452 (2011).
[CrossRef]

Unbehauen, R.

K. Rank, M. Lendl, and R. Unbehauen, “Estimation of image noise variance,” in IEEE Proceedings on Vision, Image and Signal Processing (IET, 1999), Vol. 146, pp. 80–84.

Unser, M.

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed Poisson–Gaussian noise,” IEEE Trans. Image Process. 20, 696–708 (2011).
[CrossRef]

Uss, M.

M. Uss, B. Vozel, V. Lukin, S. Abramov, I. Baryshev, and K. Chehdi, “Image informative maps for estimating noise standard deviation and texture parameters,” EURASIP J. Advances Signal Process. 2011, 806516 (2011).

Uss, M. L.

M. L. Uss, B. Vozel, V. V. Lukin, and K. Chehdi, “Image informative maps for component-wise estimating parameters of signal-dependent noise,” J. Electron. Imaging 22, 013019 (2013).
[CrossRef]

Voorhees, H.

H. Voorhees and T. Poggio, “Detecting textons and texture boundaries in natural image,” in Proceedings of the First International Conference on Computer Vision London (IEEE, 1987), pp. 250–258.

Vora, P. L.

P. L. Vora, J. E. Farrell, J. D. Tietz, and D. H. Brainard, “Linear models for digital cameras,” in IS&T Annual Conference (The Society for Imaging Science and Technology, 1997), pp. 377–382.

Vozel, B.

M. L. Uss, B. Vozel, V. V. Lukin, and K. Chehdi, “Image informative maps for component-wise estimating parameters of signal-dependent noise,” J. Electron. Imaging 22, 013019 (2013).
[CrossRef]

M. Uss, B. Vozel, V. Lukin, S. Abramov, I. Baryshev, and K. Chehdi, “Image informative maps for estimating noise standard deviation and texture parameters,” EURASIP J. Advances Signal Process. 2011, 806516 (2011).

Zhang, B.

B. Zhang, J. M. Fadili, and J.-L. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[CrossRef]

Zheng, L.

S. Pyatykh, J. Hesser, and L. Zheng, “Image noise level estimation by principal component analysis,” IEEE Trans. Image Process. 22, 687–699 (2013).
[CrossRef]

Zitnick, C. L.

C. Liu, R. Szeliski, S. B. Kang, C. L. Zitnick, and W. T. Freeman, “Automatic estimation and removal of noise from a single image,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 299–314 (2008).
[CrossRef]

Zriakhov, M. S.

N. N. Ponomarenko, V. V. Lukin, M. S. Zriakhov, A. Kaarna, and J. T. Astola, “An automatic approach to lossy compression of AVIRIS images,” in International Geoscience and Remote Sensing Symposium (IEEE, 2007), pp. 472–475.

Acta Numerica

M. Lebrun, M. Colom, A. Buades, and J. M. Morel, “Secrets of image denoising cuisine,” Acta Numerica 21, 475–576 (2012).

Astron. Astrophys.

J. Schmitt, J. L. Starck, J. M. Casandjian, J. Fadili, and I. Grenier, “Multichannel Poisson denoising and deconvolution on the sphere: application to the Fermi gamma ray space telescope,” Astron. Astrophys. 546, A114 (2012).
[CrossRef]

Biometrika

F. J. Anscombe, “The transformation of Poisson, binomial and negative-binomial data,” Biometrika 35, 246–254 (1948).

D. L. Donoho and I. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
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J. S. Lee, “Refined filtering of image noise using local statistics,” Comp. Graph. Image Proc. 15, 380–389 (1981).
[CrossRef]

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G. A. Mastin, “Adaptive filters for digital image noise smoothing: an evaluation,” Comput. Vis. Graph. Image Process. 31, 103–121 (1985).

Comput. Vis. Image Underst.

J. Immerkaer, “Fast noise variance estimation,” Comput. Vis. Image Underst. 64, 300–302 (1996).
[CrossRef]

EURASIP J. Advances Signal Process.

M. Uss, B. Vozel, V. Lukin, S. Abramov, I. Baryshev, and K. Chehdi, “Image informative maps for estimating noise standard deviation and texture parameters,” EURASIP J. Advances Signal Process. 2011, 806516 (2011).

Graph. Models Image Proc.

S. I. Olsen, “Estimation of noise in images: an evaluation,” Graph. Models Image Proc. 55, 319–323 (1993).

IEEE Signal Process. Mag.

P. Milanfar, “A tour of modern image filtering: new insights and methods, both practical and theoretical,” IEEE Signal Process. Mag. 30(1), 106–128 (2013).
[CrossRef]

IEEE Trans. Biomed. Eng.

H. Rabbani, R. Nezafat, and S. Gazor, “Wavelet-domain medical image denoising using bivariate laplacian mixture model,” IEEE Trans. Biomed. Eng. 56, 2826–2837 (2009).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

C. A. Deledalle, L. Denis, and F. Tupin, “Nl-insar: nonlocal interferogram estimation,” IEEE Trans. Geosci. Remote Sens. 49, 1441–1452 (2011).
[CrossRef]

IEEE Trans. Image Process.

R. D. Nowak and R. G. Baraniuk, “Wavelet-domain filtering for photon imaging systems,” IEEE Trans. Image Process. 8, 666–678 (1997).
[CrossRef]

S. Lefkimmiatis, P. Maragos, and G. Papandreou, “Bayesian inference on multiscale models for Poisson intensity estimation: application to photo-limited image denoising,” IEEE Trans. Image Process. 18, 1724–1741 (2009).
[CrossRef]

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process. 20, 99–109 (2011).
[CrossRef]

B. Zhang, J. M. Fadili, and J.-L. Starck, “Wavelets, ridgelets, and curvelets for poisson noise removal,” IEEE Trans. Image Process. 17, 1093–1108 (2008).
[CrossRef]

F. Luisier, T. Blu, and M. Unser, “Image denoising in mixed Poisson–Gaussian noise,” IEEE Trans. Image Process. 20, 696–708 (2011).
[CrossRef]

F.-X. Dupé, J. M. Fadili, and J.-L. Starck, “A proximal iteration for deconvolving Poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321 (2009).
[CrossRef]

A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. 17, 1737–1754 (2008).
[CrossRef]

S. Pyatykh, J. Hesser, and L. Zheng, “Image noise level estimation by principal component analysis,” IEEE Trans. Image Process. 22, 687–699 (2013).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

P. Meer, J. M. Jolion, and A. Rosenfeld, “A fast parallel algorithm for blind estimation of noise variance,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 216–223 (1990).
[CrossRef]

C. Liu, R. Szeliski, S. B. Kang, C. L. Zitnick, and W. T. Freeman, “Automatic estimation and removal of noise from a single image,” IEEE Trans. Pattern Anal. Mach. Intell. 30, 299–314 (2008).
[CrossRef]

Int. J. Adapt. Control Signal Process.

H. Rabbani and S. Gazor, “Local probability distribution of natural signals in sparse domains,” Int. J. Adapt. Control Signal Process. 28, 52–62 (2014).

Int. J. Biomed. Imag.

H. Rabbani, M. Sonka, and M. D. Abramoff, “Optical coherence tomography noise reduction using anisotropic local bivariate,” Int. J. Biomed. Imag. 3, 417491 (2011).

J. Am. Stat. Assoc.

D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).

J. Electron. Imaging

M. L. Uss, B. Vozel, V. V. Lukin, and K. Chehdi, “Image informative maps for component-wise estimating parameters of signal-dependent noise,” J. Electron. Imaging 22, 013019 (2013).
[CrossRef]

Multiscale Model. Simul.

A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Simul. 4, 490–530 (2005).
[CrossRef]

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M. Lebrun, A. Buades, and J. M. Morel, “A nonlocal Bayesian image denoising algorithm,” SIAM J. Imaging Sci. 6, 1665–1688 (2013).

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E. D. Kolaczyk, “Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds,” Statist. Sin. 9, 119–135 (1999).

Other

A. Foi, “Noise estimation and removal in mr imaging: the variance-stabilization approach,” in 2011 IEEE International Symposium on Biomedical Imaging: From Nano to Macro (IEEE, 2011), pp. 1809–1814.

J. Salmon, C.-A. Deledalle, and A. Dalalyan, “Image denoising with patch based PCA: local versus global,” in Proceedings of the British Machine Vision Conference (BMVA, 2011), pp. 25.1–25.10.

N. N. Ponomarenko, V. V. Lukin, S. K. Abramov, K. O. Egiazarian, and J. T. Astola, “Blind evaluation of additive noise variance in textured images by nonlinear processing of block DCT coefficients,” in Proceedings of the International Society for Optics and Photonics. Electronic Imaging, Image Processing: Algorithms and Systems II (2003), Vol. 5014, pp. 178–189.

N. N. Ponomarenko, V. V. Lukin, M. S. Zriakhov, A. Kaarna, and J. T. Astola, “An automatic approach to lossy compression of AVIRIS images,” in International Geoscience and Remote Sensing Symposium (IEEE, 2007), pp. 472–475.

K. Rank, M. Lendl, and R. Unbehauen, “Estimation of image noise variance,” in IEEE Proceedings on Vision, Image and Signal Processing (IET, 1999), Vol. 146, pp. 80–84.

H. Voorhees and T. Poggio, “Detecting textons and texture boundaries in natural image,” in Proceedings of the First International Conference on Computer Vision London (IEEE, 1987), pp. 250–258.

R. Bracho and A. C. Sanderson, “Segmentation of images based on intensity gradient information,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 1985), pp. 19–23.

J. S. Lee and K. Hoppel, “Noise modelling and estimation of remotely-sensed images,” in Proceedings of the International Geoscience and Remote Sensing Symposium (1989), Vol. 2, pp. 1005–1008.

V. I. A. Katkovnik, V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006), Vol. PM157.

P. L. Vora, J. E. Farrell, J. D. Tietz, and D. H. Brainard, “Linear models for digital cameras,” in IS&T Annual Conference (The Society for Imaging Science and Technology, 1997), pp. 377–382.

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Figures (7)

Fig. 1.
Fig. 1.

Noise ground-truth curves obtained for a Nikon D80 camera with fixed ISOs of (a) 1250 and (c) 1600 and four exposure times, t{1/30s,1/250s,1/400s,1/640s}, using laboratory calibration. Channels G1 and G2 give the same STD. The obtained curves overlap perfectly. Since they cover different color intervals, their fusion yields a complete noise curve (b), (d).

Fig. 2.
Fig. 2.

Linear approximation of the variance ground truth in Fig. 1 with the Nikon D80 with ISO 1600 (solid lines, original values; dashed lines, approximation). Exposure time (a) 1/30s and (b) 1/640s. Because of the saturation at the darkest zones, the estimated noise in the dark gray level does not follow a linear model.

Fig. 3.
Fig. 3.

Noise curve obtained (b) when the saturated pixels are avoided in the noise estimation and (c) when they are taken into account by using the Ponomarenko et al. method [31] with 49 bins in an image with saturated pixels (a).

Fig. 4.
Fig. 4.

RMSE between the methods and the ground truth for all 20 images in our dataset. In general, the RMSE of the modified Ponomarenko et al. (red curve) method is close to zero, which means that indeed it can be considered a nonparametric ground-truth curve. The estimations given by Foi et al. (green curve) and Uss et al. (blue curve) are really close to the nonparametric ground truth, and therefore they are also validated by our approach. (a) Obtained RMSEs/image, (b) detailed view.

Fig. 5.
Fig. 5.

Highly textured images that caused small oscillations in the noise curves with the proposed method and wrong results with Foi et al.: images (a) no. 19 and (b) no. 20 (see the obtained RMSEs in Fig. 4). (c), (d) Ground truth obtained with the series (green), the nonparametric ground truth (darker blue), the Uss et al. method (brighter blue), and the Foi et al. method (red).

Fig. 6.
Fig. 6.

Examples of images [(a) no. 8 and (b) no. 12 in our dataset] in which all algorithms estimated the noise correctly. (c), (d) Their noise curves along all the intensity range. (e), (f) Detail of the noise curves only within the range of the estimation given by the modified Ponomarenko et al. method (nonparametric ground truth, green curve). Note that the Foi et al. method (red curve) matches accurately the ground-truth curves (green and blue), since it is designed to predict the shape of the curve under saturation conditions, whereas the Uss et al. estimation is overall correct, except in the saturation zone.

Fig. 7.
Fig. 7.

Details of the denoising results with the NL-Bayes algorithm using the noise curves obtained from the noisy images (a), (e) with (b), (f) modified Ponomarenko et al., (c), (g) Uss et al., and (d), (h) Foi et al. methods. Image (a) is test image no. 3 of our dataset (it is very dark, so we increased the brightness for visualization purposes), where the Foi et al. method was unable to give a reliable estimation and thus the noise is not removed at the dark zones and remains visible (the bag over the table). Image (e) is a detail from test image no. 20 of our dataset, where the Uss et al. and the modified Ponomarenko et al. methods give a valid estimation, whereas the Foi method overestimates.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

δ(i,j){1,(i+j<T)and(i+j0)low freq;0,(i+jT)or(i+j=0)med./high freq.
VmL1θi=0w1j=0w1[Dm(i,j)]2δ(i,j),
θ=i=0w1j=0w1δ(i,j)
VH(i,j)1Kk=0K1[D(k)(i,j)]2,
σ^[mediani,j(VH(i,j)|i+jT)]1/2.

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