Abstract

We describe an innovative methodology for determining the quality of digital images. The method is based on measuring the variance of the expected entropy of a given image upon a set of predefined directions. Entropy can be calculated on a local basis by using a spatial/spatial-frequency distribution as an approximation for a probability density function. The generalized Rényi entropy and the normalized pseudo-Wigner distribution (PWD) have been selected for this purpose. As a consequence, a pixel-by-pixel entropy value can be calculated, and therefore entropy histograms can be generated as well. The variance of the expected entropy is measured as a function of the directionality, and it has been taken as an anisotropy indicator. For this purpose, directional selectivity can be attained by using an oriented 1-D PWD implementation. Our main purpose is to show how such an anisotropy measure can be used as a metric to assess both the fidelity and quality of images. Experimental results show that an index such as this presents some desirable features that resemble those from an ideal image quality function, constituting a suitable quality index for natural images. Namely, in-focus, noise-free natural images have shown a maximum of this metric in comparison with other degraded, blurred, or noisy versions. This result provides a way of identifying in-focus, noise-free images from other degraded versions, allowing an automatic and nonreference classification of images according to their relative quality. It is also shown that the new measure is well correlated with classical reference metrics such as the peak signal-to-noise ratio.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. Z. Wang and A. Bovik, "Why is image quality assessment so difficult?" IEEE International Conference on Acoustics Speech and Signal Processing (IEEE, 2002), pp. 3313-3316.
  2. Z. Zhang and R. S. Blum, "A categorization of multiscale-decomposition-based image fusion schemes with a performance study for a digital camera application," Proc. IEEE 87, 1315-1328 (1999).
    [CrossRef]
  3. Z. Wang and A. C. Bovik, "A universal image quality index," IEEE Signal Process. Lett. 9, 81-84 (2002).
    [CrossRef]
  4. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, "Image quality assessment: from error visibility to structural similarity," IEEE Trans. Image Process. 13, 600-612 (2004).
    [CrossRef]
  5. H. R. Sheikh, A. C. Bovik, and G. DeVeciana, "An information fidelity criterion for image quality assessment using natural scene statistics," IEEE Trans. Image Process. 14, 2117-2128 (2005).
    [CrossRef]
  6. H. R. Sheikh and A. C. Bovik, "Image information and visual quality," IEEE Trans. Image Process. 15, 430-444 (2006).
    [CrossRef]
  7. H. R. Sheikh, A. C. Bovik, and L. K. Cormack, "No-reference quality assessment using natural scene statistics: JPEG2000," IEEE Trans. Image Process. 14, 1918-1927 (2005).
    [CrossRef]
  8. C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, 1949).
  9. E. N. Kirsanova and M. G. Sadovsky, "Entropy approach in the analysis of anisotropy of digital images," Open Syst. Inf. Dyn. 9, 239-250 (2002).
    [CrossRef]
  10. W. J. Williams, M. L. Brown, and A. O. Hero, "Uncertainity, information and time-frequency distributions," Proc. SPIE 1566, 144-156 (1991).
    [CrossRef]
  11. P. Marziliano, F. Dufaux, S. Winkler, and T. Ebrahimi, "Perceptual blur and ringing metrics: application to JPEG2000," Signal Process. 19, 163-172 (2004).
  12. N. Cvejic, C. N. Canagarajah, and D. R. Bull, "Image fusion metric based on mutual information and Tsallis entropy," Electron. Lett. 42, 626-627 (2006).
    [CrossRef]
  13. C. S. Xydeas and V. Petkovic, "Objective image fusion performance measure," Electron. Lett. 36, 308-309 (2000).
    [CrossRef]
  14. G. Qu, D. Zhang, and P. Yang, "Information measure for performance of image fusion," Electron. Lett. 38, 313-315 (2002).
    [CrossRef]
  15. R. Danserau and W. Kinsner, "New relative multifractal dimension measures," in IEEE International Conference on Acoustics Speech and Signal Processing (IEEE, 2001), pp. 1741-1744.
  16. L. Stankovic, "A measure of some time-frequency distributions concentration," Signal Process. 81, 621-631 (2001).
    [CrossRef]
  17. N. Wiener, Cybernetics (Wiley, 1948).
  18. A. Rényi, "Some fundamental questions of information theory," in Selected Papers of Alfréd Rényi, PálTurán, ed. (Akadémiai Kiadó, 1976), Vol. 3, pp. 526-552 A. Rényi,[Originally in Magy. Tud. Akad. III Oszt. Kózl. 10, 251-282 (1960)].
  19. T. H. Sang and W. J. Williams, "Rényi information and signal dependent optimal kernel design," in IEEE International Conference on Acoustics Speech and Signal Processing (IEEE, 1995), Vol. 2, pp. 997-1000.
  20. P. Flandrin, R. G. Baraniuk, and O. Michel, "Time-frequency complexity and information," in IEEE International Conference on Acoustics Speech and Signal Processing (IEEE, 1994), Vol. 3, pp. 329-332.
  21. R. Eisberg and R. Resnick, Quantum Physics (Wiley, 1974).
  22. L. D. Jacobson and H. Wechsler, "Joint spatial/spatial-frequency representation," Signal Process. 14, 37-68 (1988).
    [CrossRef]
  23. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  24. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, "The Wigner distribution--a tool for time frequency analysis, Parts I-III," Philips J. Res. 35, 217-250, 276-300, 372-389 (1980).
  25. K. H. Brenner, "A discrete version of the Wigner distribution function," in Proceedings of EURASIP, Signal Processing II: Theories and Applications (North Holland, 1983), pp. 307-309.
  26. B. Li, M. R. Peterson, and R. D. Freeman, "Oblique effect: a neural bias in the visual cortex," J. Neurophysiol. 90, 204-217 (2003).
    [CrossRef] [PubMed]
  27. E. Switkes, M. J. Mayer, and J. A. Sloan, "Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis," Vision Res. 18, 1393-1399 (1978).
    [CrossRef] [PubMed]
  28. R. J. Baddeley and P. J. B. Hancock, "A statistical analysis of natural images matches psychophysically derived orientation tuning curves," Proc. R. Soc. London, Ser. B 246, 219-223 (1991).
    [CrossRef]
  29. P. J. B. Hancock, R. J. Baddeley, and L. S. Smith, "The principal components of natural images," Network Comput. Neural Syst. 3, 61-70 (1992).
    [CrossRef]
  30. J. Haung and D. Mumford, "Statistics of natural images and models," in Proceedings of the ICCV, 1, 541-547 (1999).
  31. B. C. Hansen and E. A. Essock, "A horizontal bias in human visual processing of orientation and its correspondence to the structural components of natural scenes," J. Vision 4, 1044-1060 (2004).
    [CrossRef]
  32. M. S. Keil and G. Cristóbal, "Separating the chaff from the wheat: possible origins of the oblique effect," J. Opt. Soc. Am. A 17, 697-710 (2000).
    [CrossRef]
  33. R. Román, J. J. Quesada, and J. Martínez, "Multiresolution-information analysis for images," Signal Process. 24, 77-91 (1991).
    [CrossRef]
  34. Y. Qu, Z. Pu, H. Zhao, and Y. Zhao, "Comparison of different quality assessment functions in autoregulative illumination intensity algorithms," Opt. Eng. (Bellingham) 45, 117-201 (2006).
    [CrossRef]
  35. S. Fischer, F. Sroubek, L. Perrinet, R. Redondo, and G. Cristóbal, "Self-invertible 2D Gabor wavelets," Int. J. Comput. Vis. available at http://www.springerlink.com/content/07q411454q407047/fulltext.pdf.
  36. F. Sroubek, G. Cristóbal, and J. Flusser, "Combined superresolution and blind deconvolution," in Information Optics: 5th International Workshop (American Institute of Physics, 2006), paper CP860, pp. 15-26.
  37. H. R. Sheikh, Z. Wang, L. Cormack, and A. C. Bovik, "LIVE image quality assessment database," Release 2, 2005 [Online]. Available at http://live.ece.utexas.edu/research/quality.
  38. H. R. Sheikh, M. F. Sabir, and A. C. Bovik, "A statistical evaluation of recent full reference image quality assessment algorithms," IEEE Trans. Image Process. 15, 3440-3451 (2006).
    [CrossRef] [PubMed]
  39. M. Y. Shen and C. C. Jay Kuo, "Review of postprocessing techniques for compression artifact removal," J. Visual Commun. Image Represent 9, 2-14 (1998).
    [CrossRef]

2006 (4)

H. R. Sheikh and A. C. Bovik, "Image information and visual quality," IEEE Trans. Image Process. 15, 430-444 (2006).
[CrossRef]

N. Cvejic, C. N. Canagarajah, and D. R. Bull, "Image fusion metric based on mutual information and Tsallis entropy," Electron. Lett. 42, 626-627 (2006).
[CrossRef]

Y. Qu, Z. Pu, H. Zhao, and Y. Zhao, "Comparison of different quality assessment functions in autoregulative illumination intensity algorithms," Opt. Eng. (Bellingham) 45, 117-201 (2006).
[CrossRef]

H. R. Sheikh, M. F. Sabir, and A. C. Bovik, "A statistical evaluation of recent full reference image quality assessment algorithms," IEEE Trans. Image Process. 15, 3440-3451 (2006).
[CrossRef] [PubMed]

2005 (2)

H. R. Sheikh, A. C. Bovik, and L. K. Cormack, "No-reference quality assessment using natural scene statistics: JPEG2000," IEEE Trans. Image Process. 14, 1918-1927 (2005).
[CrossRef]

H. R. Sheikh, A. C. Bovik, and G. DeVeciana, "An information fidelity criterion for image quality assessment using natural scene statistics," IEEE Trans. Image Process. 14, 2117-2128 (2005).
[CrossRef]

2004 (3)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, "Image quality assessment: from error visibility to structural similarity," IEEE Trans. Image Process. 13, 600-612 (2004).
[CrossRef]

P. Marziliano, F. Dufaux, S. Winkler, and T. Ebrahimi, "Perceptual blur and ringing metrics: application to JPEG2000," Signal Process. 19, 163-172 (2004).

B. C. Hansen and E. A. Essock, "A horizontal bias in human visual processing of orientation and its correspondence to the structural components of natural scenes," J. Vision 4, 1044-1060 (2004).
[CrossRef]

2003 (1)

B. Li, M. R. Peterson, and R. D. Freeman, "Oblique effect: a neural bias in the visual cortex," J. Neurophysiol. 90, 204-217 (2003).
[CrossRef] [PubMed]

2002 (3)

G. Qu, D. Zhang, and P. Yang, "Information measure for performance of image fusion," Electron. Lett. 38, 313-315 (2002).
[CrossRef]

Z. Wang and A. C. Bovik, "A universal image quality index," IEEE Signal Process. Lett. 9, 81-84 (2002).
[CrossRef]

E. N. Kirsanova and M. G. Sadovsky, "Entropy approach in the analysis of anisotropy of digital images," Open Syst. Inf. Dyn. 9, 239-250 (2002).
[CrossRef]

2001 (1)

L. Stankovic, "A measure of some time-frequency distributions concentration," Signal Process. 81, 621-631 (2001).
[CrossRef]

2000 (2)

1999 (1)

Z. Zhang and R. S. Blum, "A categorization of multiscale-decomposition-based image fusion schemes with a performance study for a digital camera application," Proc. IEEE 87, 1315-1328 (1999).
[CrossRef]

1998 (1)

M. Y. Shen and C. C. Jay Kuo, "Review of postprocessing techniques for compression artifact removal," J. Visual Commun. Image Represent 9, 2-14 (1998).
[CrossRef]

1992 (1)

P. J. B. Hancock, R. J. Baddeley, and L. S. Smith, "The principal components of natural images," Network Comput. Neural Syst. 3, 61-70 (1992).
[CrossRef]

1991 (3)

R. J. Baddeley and P. J. B. Hancock, "A statistical analysis of natural images matches psychophysically derived orientation tuning curves," Proc. R. Soc. London, Ser. B 246, 219-223 (1991).
[CrossRef]

R. Román, J. J. Quesada, and J. Martínez, "Multiresolution-information analysis for images," Signal Process. 24, 77-91 (1991).
[CrossRef]

W. J. Williams, M. L. Brown, and A. O. Hero, "Uncertainity, information and time-frequency distributions," Proc. SPIE 1566, 144-156 (1991).
[CrossRef]

1988 (1)

L. D. Jacobson and H. Wechsler, "Joint spatial/spatial-frequency representation," Signal Process. 14, 37-68 (1988).
[CrossRef]

1980 (1)

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, "The Wigner distribution--a tool for time frequency analysis, Parts I-III," Philips J. Res. 35, 217-250, 276-300, 372-389 (1980).

1978 (1)

E. Switkes, M. J. Mayer, and J. A. Sloan, "Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis," Vision Res. 18, 1393-1399 (1978).
[CrossRef] [PubMed]

1932 (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Electron. Lett. (3)

N. Cvejic, C. N. Canagarajah, and D. R. Bull, "Image fusion metric based on mutual information and Tsallis entropy," Electron. Lett. 42, 626-627 (2006).
[CrossRef]

C. S. Xydeas and V. Petkovic, "Objective image fusion performance measure," Electron. Lett. 36, 308-309 (2000).
[CrossRef]

G. Qu, D. Zhang, and P. Yang, "Information measure for performance of image fusion," Electron. Lett. 38, 313-315 (2002).
[CrossRef]

IEEE Signal Process. Lett. (1)

Z. Wang and A. C. Bovik, "A universal image quality index," IEEE Signal Process. Lett. 9, 81-84 (2002).
[CrossRef]

IEEE Trans. Image Process. (5)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, "Image quality assessment: from error visibility to structural similarity," IEEE Trans. Image Process. 13, 600-612 (2004).
[CrossRef]

H. R. Sheikh, A. C. Bovik, and G. DeVeciana, "An information fidelity criterion for image quality assessment using natural scene statistics," IEEE Trans. Image Process. 14, 2117-2128 (2005).
[CrossRef]

H. R. Sheikh and A. C. Bovik, "Image information and visual quality," IEEE Trans. Image Process. 15, 430-444 (2006).
[CrossRef]

H. R. Sheikh, A. C. Bovik, and L. K. Cormack, "No-reference quality assessment using natural scene statistics: JPEG2000," IEEE Trans. Image Process. 14, 1918-1927 (2005).
[CrossRef]

H. R. Sheikh, M. F. Sabir, and A. C. Bovik, "A statistical evaluation of recent full reference image quality assessment algorithms," IEEE Trans. Image Process. 15, 3440-3451 (2006).
[CrossRef] [PubMed]

J. Neurophysiol. (1)

B. Li, M. R. Peterson, and R. D. Freeman, "Oblique effect: a neural bias in the visual cortex," J. Neurophysiol. 90, 204-217 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

J. Vision (1)

B. C. Hansen and E. A. Essock, "A horizontal bias in human visual processing of orientation and its correspondence to the structural components of natural scenes," J. Vision 4, 1044-1060 (2004).
[CrossRef]

J. Visual Commun. Image Represent (1)

M. Y. Shen and C. C. Jay Kuo, "Review of postprocessing techniques for compression artifact removal," J. Visual Commun. Image Represent 9, 2-14 (1998).
[CrossRef]

Network Comput. Neural Syst. (1)

P. J. B. Hancock, R. J. Baddeley, and L. S. Smith, "The principal components of natural images," Network Comput. Neural Syst. 3, 61-70 (1992).
[CrossRef]

Open Syst. Inf. Dyn. (1)

E. N. Kirsanova and M. G. Sadovsky, "Entropy approach in the analysis of anisotropy of digital images," Open Syst. Inf. Dyn. 9, 239-250 (2002).
[CrossRef]

Opt. Eng. (Bellingham) (1)

Y. Qu, Z. Pu, H. Zhao, and Y. Zhao, "Comparison of different quality assessment functions in autoregulative illumination intensity algorithms," Opt. Eng. (Bellingham) 45, 117-201 (2006).
[CrossRef]

Philips J. Res. (1)

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, "The Wigner distribution--a tool for time frequency analysis, Parts I-III," Philips J. Res. 35, 217-250, 276-300, 372-389 (1980).

Phys. Rev. (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Proc. IEEE (1)

Z. Zhang and R. S. Blum, "A categorization of multiscale-decomposition-based image fusion schemes with a performance study for a digital camera application," Proc. IEEE 87, 1315-1328 (1999).
[CrossRef]

Proc. R. Soc. London, Ser. B (1)

R. J. Baddeley and P. J. B. Hancock, "A statistical analysis of natural images matches psychophysically derived orientation tuning curves," Proc. R. Soc. London, Ser. B 246, 219-223 (1991).
[CrossRef]

Proc. SPIE (1)

W. J. Williams, M. L. Brown, and A. O. Hero, "Uncertainity, information and time-frequency distributions," Proc. SPIE 1566, 144-156 (1991).
[CrossRef]

Signal Process. (4)

P. Marziliano, F. Dufaux, S. Winkler, and T. Ebrahimi, "Perceptual blur and ringing metrics: application to JPEG2000," Signal Process. 19, 163-172 (2004).

L. Stankovic, "A measure of some time-frequency distributions concentration," Signal Process. 81, 621-631 (2001).
[CrossRef]

R. Román, J. J. Quesada, and J. Martínez, "Multiresolution-information analysis for images," Signal Process. 24, 77-91 (1991).
[CrossRef]

L. D. Jacobson and H. Wechsler, "Joint spatial/spatial-frequency representation," Signal Process. 14, 37-68 (1988).
[CrossRef]

Vision Res. (1)

E. Switkes, M. J. Mayer, and J. A. Sloan, "Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis," Vision Res. 18, 1393-1399 (1978).
[CrossRef] [PubMed]

Other (13)

J. Haung and D. Mumford, "Statistics of natural images and models," in Proceedings of the ICCV, 1, 541-547 (1999).

K. H. Brenner, "A discrete version of the Wigner distribution function," in Proceedings of EURASIP, Signal Processing II: Theories and Applications (North Holland, 1983), pp. 307-309.

S. Fischer, F. Sroubek, L. Perrinet, R. Redondo, and G. Cristóbal, "Self-invertible 2D Gabor wavelets," Int. J. Comput. Vis. available at http://www.springerlink.com/content/07q411454q407047/fulltext.pdf.

F. Sroubek, G. Cristóbal, and J. Flusser, "Combined superresolution and blind deconvolution," in Information Optics: 5th International Workshop (American Institute of Physics, 2006), paper CP860, pp. 15-26.

H. R. Sheikh, Z. Wang, L. Cormack, and A. C. Bovik, "LIVE image quality assessment database," Release 2, 2005 [Online]. Available at http://live.ece.utexas.edu/research/quality.

N. Wiener, Cybernetics (Wiley, 1948).

A. Rényi, "Some fundamental questions of information theory," in Selected Papers of Alfréd Rényi, PálTurán, ed. (Akadémiai Kiadó, 1976), Vol. 3, pp. 526-552 A. Rényi,[Originally in Magy. Tud. Akad. III Oszt. Kózl. 10, 251-282 (1960)].

T. H. Sang and W. J. Williams, "Rényi information and signal dependent optimal kernel design," in IEEE International Conference on Acoustics Speech and Signal Processing (IEEE, 1995), Vol. 2, pp. 997-1000.

P. Flandrin, R. G. Baraniuk, and O. Michel, "Time-frequency complexity and information," in IEEE International Conference on Acoustics Speech and Signal Processing (IEEE, 1994), Vol. 3, pp. 329-332.

R. Eisberg and R. Resnick, Quantum Physics (Wiley, 1974).

R. Danserau and W. Kinsner, "New relative multifractal dimension measures," in IEEE International Conference on Acoustics Speech and Signal Processing (IEEE, 2001), pp. 1741-1744.

Z. Wang and A. Bovik, "Why is image quality assessment so difficult?" IEEE International Conference on Acoustics Speech and Signal Processing (IEEE, 2002), pp. 3313-3316.

C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, 1949).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Thirty-six images used for empirically determining the directional entropy in natural images. Framed image processing is described in Fig. 2 and in the text.

Fig. 2
Fig. 2

Test scheme consisting in 21 degraded images. Blur decreases from −10 to 0 and noise increases from 0 to 10. The central image is the original source image.

Fig. 3
Fig. 3

Expected value of the pixelwise Rényi entropy of the 21 images of the test set presented in Fig. 2.

Fig. 4
Fig. 4

A. Standard deviation of the expected values of the Rényi directional entropy for the images shown in Fig. 2. B. Range of the expected values of the Rényi directional entropy for the images in Fig. 2. The variability refers to six different equally spaced orientations of the entropy in the image. The maximum variability corresponds to the original image, as an in-focus, noise-free version of the test set defined in Fig. 2.

Fig. 5
Fig. 5

Upper row (from left to right): original Lena image (#1) and progressively degraded blurred and noisy versions. Bottom row (from left to right): original MIT image (#7) and progressively degraded blurred and noisy versions. Images are courtesy of Sylvain Fischer [35].

Fig. 6
Fig. 6

Classification obtained with two sets of test images in a superresolution scenario. From left to right, image quality decreases and a quantitative figure of merit (standard deviation of directional entropy per pixel) is given at the bottom of each image. Images are courtesy of Filip Šroubek.

Tables (5)

Tables Icon

Table 1 Comparison of Different Image Quality Measures

Tables Icon

Table 2 Algorithm Evaluation Using the Four Images a

Tables Icon

Table 3 Algorithm Evaluation Using the Four Images a

Tables Icon

Table 4 Algorithm Evaluation Using the Four Images a

Tables Icon

Table 5 Algorithm Evaluation Using the Four Images a

Equations (13)

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

R α = 1 1 α log 2 ( n k P α [ n , k ] ) .
H = n k P x [ n , k ] log 2 ( P x [ n , k ] )
R E α = 1 1 α log 2 ( n k P α [ n , k ] n k P [ n , k ] ) with α 2 .
R V 3 = 1 2 log 2 ( n k P 3 [ n , k ] n k P [ n , k ] ) .
R ̆ 3 = 1 2 log 2 ( n k P ̆ 3 [ n , k ] ) .
R ̆ 3 [ n ] = 1 2 log 2 ( k P ̆ 3 [ n , k ] ) .
W z [ n , k ] = 2 m = N 2 N 2 1 z [ n + m ] z * [ n m ] e 2 i ( 2 π m N ) k .
R 3 [ n ] = 1 2 log 2 ( k = 1 N P ̆ n 3 [ k ] ) .
R ¯ [ t , θ s ] = n R 3 [ n , θ s ] M ,
σ ( t ) = s = 1 S ( μ t R ¯ ( t , θ s ) ) 2 S ,
μ t = s = 1 S R ¯ ( t , θ s ) S ,
r g ( t ) = max { R ¯ ( t , θ s ) } min { R ¯ ( t , θ s ) } .
σ ̃ = σ ( 1 ( K S L ) β )

Metrics