Abstract

Analogous to use of bullet scratches in forensic science, the authenticity of a digital image can be verified through the noise characteristics of an imaging sensor. In particular, photo-response non-uniformity noise (PRNU) has been used in source camera identification (SCI). However, this technique can be used maliciously to track or inculpate innocent people. To impede such tracking, PRNU noise should be suppressed significantly. Based on this motivation, we propose a counter forensic method to deceive SCI. Experimental results show that it is possible to impede PRNU-based camera identification for various imaging sensors while preserving the image quality.

© 2014 Optical Society of America

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References

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  1. J. Lukáš, J. Fridrich, M. Goljan, “Digital “bullet scratches” for images,” in “IEEE Int. Conf. on Image Processing 2005,” (IEEE, 2005), pp. III–65.
    [CrossRef]
  2. A. E. Dirik, H. Sencar, N. Memon, “Digital single lens reflex camera identification from traces of sensor dust,” IEEE T. Inf. Foren. Sec. 3, 539–552 (2008).
    [CrossRef]
  3. A. E. Dirik, H. Sencar, N. Memon, “Flatbed scanner identification based on dust and scratches over scanner platen,” in “Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP’09),” (IEEE, 2009), pp. 1385–1388.
    [CrossRef]
  4. K. S. Choi, E. Y. Lam, K. K. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 141, 11551–11565 (2006).
    [CrossRef]
  5. M. Goljan, J. Fridrich, “Camera identification from cropped and scaled images,” in “Proc. SPIE 6819, Security, Forensics, Steganography, and Watermarking of Multimedia Contents X,” (2008), pp. 68190E1–68190E13.
  6. K. Rosenfeld, H. T. Sencar, N. Memon, “A study of the robustness of prnu-based camera identification,” in “Proc. SPIE 7254, Media Forensics and Security,” (2009), p. 72540.
  7. M. Goljan, J. Fridrich, M. Chen, “Sensor noise camera identification: countering counter-forensics,” in “Proc. SPIE 7541, Media Forensics and Security II,” (2010), 607, pp. 75410S1–75410S12.
  8. M. Goljan, J. Fridrich, M. Chen, “Defending against fingerprint-copy attack in sensor-based camera identification,” IEEE Trans. Inf. Foren. Sec. 6, 227–236 (2011).
    [CrossRef]
  9. T. Gloe, M. Kirchner, A. Winkler, R. Böhme, “Can we trust digital image forensics?” in “Proc. ACM 15th Int. Conf. on Multimedia (MULTIMEDIA ’07),” (ACM Press, New York, USA, 2007), pp. 78–86.
  10. R. Böhme, M. Kirchner, “Counter-forensics: Attacking image forensics,” in Digital Image Forensics, H. T. Sencar, N. Memon, eds. (SpringerNew York, 2013), pp. 327–366.
    [CrossRef]
  11. M. Chen, J. Fridrich, M. Goljan, “Digital imaging sensor identification (further study),” in “Proc. SPIE 6505, Security, Steganography, and Watermarking of Multimedia Contents IX,” (2007), pp. 65050P1–65050P13.
  12. J. Lukáš, J. Fridrich, M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE T. Inf. Foren. Sec. 1, 205–214 (2006).
    [CrossRef]
  13. C.-T. Li, C.-Y. Chang, Y. Li, “On the repudiability of device identification and image integrity verification using sensor pattern noise,” in “Information Security and Digital Forensics,”, vol. 41 of Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, D. Weerasinghe, ed. (SpringerBerlin Heidelberg, 2010), pp. 19–25.
    [CrossRef]
  14. J. Lukáš, J. Fridrich, M. Goljan, “Detecting digital image forgeries using sensor pattern noise,” Proc. SPIE: Image and Video Communications and Processing5685, 249–260 (2005).
    [CrossRef]
  15. M. Goljan, “Digital camera identification from images - estimating false acceptance probability,” in “Digital Watermarking,”, vol. 5450 of Lecture Notes in Computer Science, H.-J. Kim, S. Katzenbeisser, A. Ho, eds. (SpringerBerlin Heidelberg, 2009), pp. 454–468.
    [CrossRef]
  16. J. Fridrich, “Sensor defects in digital image forensic,” in Digital Image Forensics, (SpringerNew York, 2012), chap. 5, pp. 179–219.
  17. M. Goljan, J. Fridrich, T. Filler, “Large scale test of sensor fingerprint camera identification,” in “SPIE Electronic Imaging,” (International Society for Optics and Photonics, 2009), pp. 72540I.
  18. J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice Hall, 1990).
  19. T. Gloe, R. Böhme, “The dresden image database for benchmarking digital image forensics,” J. Digital Forensic Practice 3, 150–159 (2010).
    [CrossRef]

2011 (1)

M. Goljan, J. Fridrich, M. Chen, “Defending against fingerprint-copy attack in sensor-based camera identification,” IEEE Trans. Inf. Foren. Sec. 6, 227–236 (2011).
[CrossRef]

2010 (1)

T. Gloe, R. Böhme, “The dresden image database for benchmarking digital image forensics,” J. Digital Forensic Practice 3, 150–159 (2010).
[CrossRef]

2008 (1)

A. E. Dirik, H. Sencar, N. Memon, “Digital single lens reflex camera identification from traces of sensor dust,” IEEE T. Inf. Foren. Sec. 3, 539–552 (2008).
[CrossRef]

2006 (2)

K. S. Choi, E. Y. Lam, K. K. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 141, 11551–11565 (2006).
[CrossRef]

J. Lukáš, J. Fridrich, M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE T. Inf. Foren. Sec. 1, 205–214 (2006).
[CrossRef]

Böhme, R.

T. Gloe, R. Böhme, “The dresden image database for benchmarking digital image forensics,” J. Digital Forensic Practice 3, 150–159 (2010).
[CrossRef]

T. Gloe, M. Kirchner, A. Winkler, R. Böhme, “Can we trust digital image forensics?” in “Proc. ACM 15th Int. Conf. on Multimedia (MULTIMEDIA ’07),” (ACM Press, New York, USA, 2007), pp. 78–86.

R. Böhme, M. Kirchner, “Counter-forensics: Attacking image forensics,” in Digital Image Forensics, H. T. Sencar, N. Memon, eds. (SpringerNew York, 2013), pp. 327–366.
[CrossRef]

Chang, C.-Y.

C.-T. Li, C.-Y. Chang, Y. Li, “On the repudiability of device identification and image integrity verification using sensor pattern noise,” in “Information Security and Digital Forensics,”, vol. 41 of Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, D. Weerasinghe, ed. (SpringerBerlin Heidelberg, 2010), pp. 19–25.
[CrossRef]

Chen, M.

M. Goljan, J. Fridrich, M. Chen, “Defending against fingerprint-copy attack in sensor-based camera identification,” IEEE Trans. Inf. Foren. Sec. 6, 227–236 (2011).
[CrossRef]

M. Chen, J. Fridrich, M. Goljan, “Digital imaging sensor identification (further study),” in “Proc. SPIE 6505, Security, Steganography, and Watermarking of Multimedia Contents IX,” (2007), pp. 65050P1–65050P13.

M. Goljan, J. Fridrich, M. Chen, “Sensor noise camera identification: countering counter-forensics,” in “Proc. SPIE 7541, Media Forensics and Security II,” (2010), 607, pp. 75410S1–75410S12.

Choi, K. S.

K. S. Choi, E. Y. Lam, K. K. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 141, 11551–11565 (2006).
[CrossRef]

Dirik, A. E.

A. E. Dirik, H. Sencar, N. Memon, “Digital single lens reflex camera identification from traces of sensor dust,” IEEE T. Inf. Foren. Sec. 3, 539–552 (2008).
[CrossRef]

A. E. Dirik, H. Sencar, N. Memon, “Flatbed scanner identification based on dust and scratches over scanner platen,” in “Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP’09),” (IEEE, 2009), pp. 1385–1388.
[CrossRef]

Filler, T.

M. Goljan, J. Fridrich, T. Filler, “Large scale test of sensor fingerprint camera identification,” in “SPIE Electronic Imaging,” (International Society for Optics and Photonics, 2009), pp. 72540I.

Fridrich, J.

M. Goljan, J. Fridrich, M. Chen, “Defending against fingerprint-copy attack in sensor-based camera identification,” IEEE Trans. Inf. Foren. Sec. 6, 227–236 (2011).
[CrossRef]

J. Lukáš, J. Fridrich, M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE T. Inf. Foren. Sec. 1, 205–214 (2006).
[CrossRef]

M. Goljan, J. Fridrich, “Camera identification from cropped and scaled images,” in “Proc. SPIE 6819, Security, Forensics, Steganography, and Watermarking of Multimedia Contents X,” (2008), pp. 68190E1–68190E13.

J. Lukáš, J. Fridrich, M. Goljan, “Digital “bullet scratches” for images,” in “IEEE Int. Conf. on Image Processing 2005,” (IEEE, 2005), pp. III–65.
[CrossRef]

J. Fridrich, “Sensor defects in digital image forensic,” in Digital Image Forensics, (SpringerNew York, 2012), chap. 5, pp. 179–219.

M. Goljan, J. Fridrich, M. Chen, “Sensor noise camera identification: countering counter-forensics,” in “Proc. SPIE 7541, Media Forensics and Security II,” (2010), 607, pp. 75410S1–75410S12.

M. Chen, J. Fridrich, M. Goljan, “Digital imaging sensor identification (further study),” in “Proc. SPIE 6505, Security, Steganography, and Watermarking of Multimedia Contents IX,” (2007), pp. 65050P1–65050P13.

M. Goljan, J. Fridrich, T. Filler, “Large scale test of sensor fingerprint camera identification,” in “SPIE Electronic Imaging,” (International Society for Optics and Photonics, 2009), pp. 72540I.

J. Lukáš, J. Fridrich, M. Goljan, “Detecting digital image forgeries using sensor pattern noise,” Proc. SPIE: Image and Video Communications and Processing5685, 249–260 (2005).
[CrossRef]

Gloe, T.

T. Gloe, R. Böhme, “The dresden image database for benchmarking digital image forensics,” J. Digital Forensic Practice 3, 150–159 (2010).
[CrossRef]

T. Gloe, M. Kirchner, A. Winkler, R. Böhme, “Can we trust digital image forensics?” in “Proc. ACM 15th Int. Conf. on Multimedia (MULTIMEDIA ’07),” (ACM Press, New York, USA, 2007), pp. 78–86.

Goljan, M.

M. Goljan, J. Fridrich, M. Chen, “Defending against fingerprint-copy attack in sensor-based camera identification,” IEEE Trans. Inf. Foren. Sec. 6, 227–236 (2011).
[CrossRef]

J. Lukáš, J. Fridrich, M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE T. Inf. Foren. Sec. 1, 205–214 (2006).
[CrossRef]

M. Goljan, J. Fridrich, “Camera identification from cropped and scaled images,” in “Proc. SPIE 6819, Security, Forensics, Steganography, and Watermarking of Multimedia Contents X,” (2008), pp. 68190E1–68190E13.

J. Lukáš, J. Fridrich, M. Goljan, “Digital “bullet scratches” for images,” in “IEEE Int. Conf. on Image Processing 2005,” (IEEE, 2005), pp. III–65.
[CrossRef]

M. Goljan, J. Fridrich, M. Chen, “Sensor noise camera identification: countering counter-forensics,” in “Proc. SPIE 7541, Media Forensics and Security II,” (2010), 607, pp. 75410S1–75410S12.

M. Chen, J. Fridrich, M. Goljan, “Digital imaging sensor identification (further study),” in “Proc. SPIE 6505, Security, Steganography, and Watermarking of Multimedia Contents IX,” (2007), pp. 65050P1–65050P13.

M. Goljan, “Digital camera identification from images - estimating false acceptance probability,” in “Digital Watermarking,”, vol. 5450 of Lecture Notes in Computer Science, H.-J. Kim, S. Katzenbeisser, A. Ho, eds. (SpringerBerlin Heidelberg, 2009), pp. 454–468.
[CrossRef]

J. Lukáš, J. Fridrich, M. Goljan, “Detecting digital image forgeries using sensor pattern noise,” Proc. SPIE: Image and Video Communications and Processing5685, 249–260 (2005).
[CrossRef]

M. Goljan, J. Fridrich, T. Filler, “Large scale test of sensor fingerprint camera identification,” in “SPIE Electronic Imaging,” (International Society for Optics and Photonics, 2009), pp. 72540I.

Kirchner, M.

R. Böhme, M. Kirchner, “Counter-forensics: Attacking image forensics,” in Digital Image Forensics, H. T. Sencar, N. Memon, eds. (SpringerNew York, 2013), pp. 327–366.
[CrossRef]

T. Gloe, M. Kirchner, A. Winkler, R. Böhme, “Can we trust digital image forensics?” in “Proc. ACM 15th Int. Conf. on Multimedia (MULTIMEDIA ’07),” (ACM Press, New York, USA, 2007), pp. 78–86.

Lam, E. Y.

K. S. Choi, E. Y. Lam, K. K. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 141, 11551–11565 (2006).
[CrossRef]

Li, C.-T.

C.-T. Li, C.-Y. Chang, Y. Li, “On the repudiability of device identification and image integrity verification using sensor pattern noise,” in “Information Security and Digital Forensics,”, vol. 41 of Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, D. Weerasinghe, ed. (SpringerBerlin Heidelberg, 2010), pp. 19–25.
[CrossRef]

Li, Y.

C.-T. Li, C.-Y. Chang, Y. Li, “On the repudiability of device identification and image integrity verification using sensor pattern noise,” in “Information Security and Digital Forensics,”, vol. 41 of Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, D. Weerasinghe, ed. (SpringerBerlin Heidelberg, 2010), pp. 19–25.
[CrossRef]

Lim, J. S.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice Hall, 1990).

Lukáš, J.

J. Lukáš, J. Fridrich, M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE T. Inf. Foren. Sec. 1, 205–214 (2006).
[CrossRef]

J. Lukáš, J. Fridrich, M. Goljan, “Digital “bullet scratches” for images,” in “IEEE Int. Conf. on Image Processing 2005,” (IEEE, 2005), pp. III–65.
[CrossRef]

J. Lukáš, J. Fridrich, M. Goljan, “Detecting digital image forgeries using sensor pattern noise,” Proc. SPIE: Image and Video Communications and Processing5685, 249–260 (2005).
[CrossRef]

Memon, N.

A. E. Dirik, H. Sencar, N. Memon, “Digital single lens reflex camera identification from traces of sensor dust,” IEEE T. Inf. Foren. Sec. 3, 539–552 (2008).
[CrossRef]

A. E. Dirik, H. Sencar, N. Memon, “Flatbed scanner identification based on dust and scratches over scanner platen,” in “Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP’09),” (IEEE, 2009), pp. 1385–1388.
[CrossRef]

K. Rosenfeld, H. T. Sencar, N. Memon, “A study of the robustness of prnu-based camera identification,” in “Proc. SPIE 7254, Media Forensics and Security,” (2009), p. 72540.

Rosenfeld, K.

K. Rosenfeld, H. T. Sencar, N. Memon, “A study of the robustness of prnu-based camera identification,” in “Proc. SPIE 7254, Media Forensics and Security,” (2009), p. 72540.

Sencar, H.

A. E. Dirik, H. Sencar, N. Memon, “Digital single lens reflex camera identification from traces of sensor dust,” IEEE T. Inf. Foren. Sec. 3, 539–552 (2008).
[CrossRef]

A. E. Dirik, H. Sencar, N. Memon, “Flatbed scanner identification based on dust and scratches over scanner platen,” in “Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP’09),” (IEEE, 2009), pp. 1385–1388.
[CrossRef]

Sencar, H. T.

K. Rosenfeld, H. T. Sencar, N. Memon, “A study of the robustness of prnu-based camera identification,” in “Proc. SPIE 7254, Media Forensics and Security,” (2009), p. 72540.

Winkler, A.

T. Gloe, M. Kirchner, A. Winkler, R. Böhme, “Can we trust digital image forensics?” in “Proc. ACM 15th Int. Conf. on Multimedia (MULTIMEDIA ’07),” (ACM Press, New York, USA, 2007), pp. 78–86.

Wong, K. K.

K. S. Choi, E. Y. Lam, K. K. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 141, 11551–11565 (2006).
[CrossRef]

IEEE T. Inf. Foren. Sec. (2)

J. Lukáš, J. Fridrich, M. Goljan, “Digital camera identification from sensor pattern noise,” IEEE T. Inf. Foren. Sec. 1, 205–214 (2006).
[CrossRef]

A. E. Dirik, H. Sencar, N. Memon, “Digital single lens reflex camera identification from traces of sensor dust,” IEEE T. Inf. Foren. Sec. 3, 539–552 (2008).
[CrossRef]

IEEE Trans. Inf. Foren. Sec. (1)

M. Goljan, J. Fridrich, M. Chen, “Defending against fingerprint-copy attack in sensor-based camera identification,” IEEE Trans. Inf. Foren. Sec. 6, 227–236 (2011).
[CrossRef]

J. Digital Forensic Practice (1)

T. Gloe, R. Böhme, “The dresden image database for benchmarking digital image forensics,” J. Digital Forensic Practice 3, 150–159 (2010).
[CrossRef]

Opt. Express (1)

K. S. Choi, E. Y. Lam, K. K. Wong, “Automatic source camera identification using the intrinsic lens radial distortion,” Opt. Express 141, 11551–11565 (2006).
[CrossRef]

Other (14)

M. Goljan, J. Fridrich, “Camera identification from cropped and scaled images,” in “Proc. SPIE 6819, Security, Forensics, Steganography, and Watermarking of Multimedia Contents X,” (2008), pp. 68190E1–68190E13.

K. Rosenfeld, H. T. Sencar, N. Memon, “A study of the robustness of prnu-based camera identification,” in “Proc. SPIE 7254, Media Forensics and Security,” (2009), p. 72540.

M. Goljan, J. Fridrich, M. Chen, “Sensor noise camera identification: countering counter-forensics,” in “Proc. SPIE 7541, Media Forensics and Security II,” (2010), 607, pp. 75410S1–75410S12.

T. Gloe, M. Kirchner, A. Winkler, R. Böhme, “Can we trust digital image forensics?” in “Proc. ACM 15th Int. Conf. on Multimedia (MULTIMEDIA ’07),” (ACM Press, New York, USA, 2007), pp. 78–86.

R. Böhme, M. Kirchner, “Counter-forensics: Attacking image forensics,” in Digital Image Forensics, H. T. Sencar, N. Memon, eds. (SpringerNew York, 2013), pp. 327–366.
[CrossRef]

M. Chen, J. Fridrich, M. Goljan, “Digital imaging sensor identification (further study),” in “Proc. SPIE 6505, Security, Steganography, and Watermarking of Multimedia Contents IX,” (2007), pp. 65050P1–65050P13.

J. Lukáš, J. Fridrich, M. Goljan, “Digital “bullet scratches” for images,” in “IEEE Int. Conf. on Image Processing 2005,” (IEEE, 2005), pp. III–65.
[CrossRef]

A. E. Dirik, H. Sencar, N. Memon, “Flatbed scanner identification based on dust and scratches over scanner platen,” in “Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP’09),” (IEEE, 2009), pp. 1385–1388.
[CrossRef]

C.-T. Li, C.-Y. Chang, Y. Li, “On the repudiability of device identification and image integrity verification using sensor pattern noise,” in “Information Security and Digital Forensics,”, vol. 41 of Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, D. Weerasinghe, ed. (SpringerBerlin Heidelberg, 2010), pp. 19–25.
[CrossRef]

J. Lukáš, J. Fridrich, M. Goljan, “Detecting digital image forgeries using sensor pattern noise,” Proc. SPIE: Image and Video Communications and Processing5685, 249–260 (2005).
[CrossRef]

M. Goljan, “Digital camera identification from images - estimating false acceptance probability,” in “Digital Watermarking,”, vol. 5450 of Lecture Notes in Computer Science, H.-J. Kim, S. Katzenbeisser, A. Ho, eds. (SpringerBerlin Heidelberg, 2009), pp. 454–468.
[CrossRef]

J. Fridrich, “Sensor defects in digital image forensic,” in Digital Image Forensics, (SpringerNew York, 2012), chap. 5, pp. 179–219.

M. Goljan, J. Fridrich, T. Filler, “Large scale test of sensor fingerprint camera identification,” in “SPIE Electronic Imaging,” (International Society for Optics and Photonics, 2009), pp. 72540I.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice Hall, 1990).

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Figures (3)

Fig. 1
Fig. 1

Comparison of the PRNU removal methods for Sony H50. The proposed method “anonymized” 98% of the images taken with Sony H50. (Decision Threshold=50)

Fig. 2
Fig. 2

Comparison of the PRNU removal methods for Nikon D200. The proposed method “anonymized” all of the images taken with Nikon D200. (Decision Threshold=50)

Fig. 3
Fig. 3

Comparison of the PRNU removal methods for Panasonic FZ50. The proposed method “anonymized” 84.4% of the images taken with Panasonic FZ50. (Decision Thresh-old=50)

Tables (10)

Tables Icon

Table 1 The camera models used in the experiments from the Dresden Image Database.

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Table 2 Grid search statistics (num. of iteration and ψa) per camera.

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Table 3 Average PCE values (Decision threshold = 50)

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Table 4 Anonymization rates

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Table 5 PSNR [dB] after anonymization

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Table 6 Average correlation cofficients (F-50, Decision Threshold=0.0100)

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Table 7 The camera models used in the experiment.

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Table 8 The average number of iterations of grid search and the statistics of ψa per camera.

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Table 9 PCE Values and Anonymization Rates (F-50)

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Table 10 PCE Values and Anonymization Rates (F-100)

Equations (17)

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

I x = I 0 + ( I 0 F x + Φ )
N x ( i ) = I x ( i ) WDF ( I x ( i ) )
F ^ x = i = 1 M N x ( i ) I x ( i ) i = 1 M ( I x ( i ) ) 2
ρ ( u , v ; N , F ^ x ) = k = 1 K l = 1 L ( N [ k , l ] N ¯ ) ( F ^ x [ k + u , l + v ] F ^ x ¯ ) N N ¯ F ^ x F ^ x ¯
PCE = ρ peak 2 1 | s | | ε | s ε ρ s 2
N x = b F x I 0 + Φ 2
I x = I x ψ N x
I x = I 0 + ( F x I 0 + Φ 1 ) ψ ( b F x I 0 + Φ 2 )
I x = I 0 + ( 1 ψ b ) F x I 0 + ( Φ 1 ψ Φ 2 )
ψ 0 = 1 / b
PSNR ( I x , I x ) = 10 log 10 ( 255 2 ) 10 log 10 ( var ( F x I 0 ) + ψ 2 var ( Φ 2 ) )
PSNR ( I x , I x ) 10 log 10 ( 255 2 ) 10 log 10 ( ψ 2 var ( Φ 2 ) )
f PCE ( ψ ) = PCE ( I x ( ψ ) , F ^ x )
ψ o = argmin ψ [ 1 , ) ( f PCE ( ψ ) )
f PCE ( ψ a ) ε a
I x a = I x ψ a ( I x W ( I x ) )
AR ( ε a ) = 100 M i = 1 M S ( i ; ε a ) ; S ( i ; ε a ) = { 1 if f PCE ( ψ a ( i ) ) ε a ; i = 1 , , M 0 otherwise

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