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.

Timothy D. Dixon, Eduardo Fernández Canga, Stavri G. Nikolov, Tom Troscianko, Jan M. Noyes, C. Nishan Canagarajah, and Dave R. Bull J. Opt. Soc. Am. A 24(12) B125-B135 (2007)

Di Wu, Xiudong Sun, Yuannan Xu, Yongyuan Jiang, and Chunfeng Hou Appl. Opt. 51(11) 1742-1748 (2012)

References

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“Buildings,” “Lighthouse,” “Statue,” and “Stream” taken from the LIVE database for BLUR degradation. The standard deviation of the Gaussian kernel is indicated below the reference number of each image (columns 1, 3, 5, and 7). The corresponding PSNR has been included in columns 2, 4, 6, and 8.

Table 3

Algorithm Evaluation Using the Four Images^{
a
}

Buildings

$\sigma \left(k\right)$/ (PSNR)

Lighthouse

$\sigma \left(k\right)$/ (PSNR)

Statue

$\sigma \left(k\right)$/ (PSNR)

Stream

$\sigma \left(k\right)$/ (PSNR)

#159 (0.000)

1

#164 (0.00)

1

#148 (0.000)

1

#151 (0.000)

1

#103 (0.031)

$\mathbf{0.99}$ (33.61)

#43 (0.019)

$\mathbf{0.99}$ (37.56)

#85 (0.015)

$\mathbf{0.96}$ (39.46)

#138 (0.031)

$\mathbf{0.98}$ (33.6)

#46 (0.058)

$\mathbf{0.94}$ (28.24)

#114 (0.039)

$\mathbf{0.97}$ (31.63)

#55 (0.046)

$\mathbf{0.82}$ (30.2)

#88 (0.062)

$\mathbf{0.93}$ (27.63)

#130 (0.2890)

$\mathbf{0.40}$ (15.49)

#40 (0.062)

$\mathbf{0.93}$ (27.58)

#25 (0.109)

$\mathbf{0.56}$ (23.12)

#2 (0.187)

$\mathbf{0.63}$ (18.48)

#61 (0.4062)

$\mathbf{0.26}$ (13.38)

#96 (0.171)

$\mathbf{0.69}$ (19.08)

#91 (0.203)

$\mathbf{0.38}$ (18.14)

#106 (0.312)

$\mathbf{0.36}$ (14.86)

#4 (1.9960)

$\mathbf{0.03}$ (8.65)

#66 (1.000)

$\mathbf{0.12}$ (10.19)

#145 (1.00)

$\mathbf{0.05}$ (9.63)

#131 (0.500)

$\mathbf{0.21}$ (12.28)

“Buildings,” “Lighthouse,” “Statue,” and “Stream” taken from the LIVE database for WHITE NOISE degradation. The standard deviation of the noise is indicated below the reference number of each image (columns 1, 3, 5, and 7). The corresponding PSNR has been included in columns 2, 4, 6, and 8.

Table 4

Algorithm Evaluation Using the Four Images^{
a
}

Buildings

$\sigma \left(k\right)$

Lighthouse

$\sigma \left(k\right)$

Statue

$\sigma \left(k\right)$

Stream

$\sigma \left(k\right)$

#157 (1.77)

1

#3 (0)

1

#13 (2.77)

1

#212 (0)

1

#227 (0)

$\mathbf{0.86}$

#57 (2.6)

$\mathbf{0.88}$

#70 (2.19)

$\mathbf{0.94}$

#137 (1.683)

$\mathbf{0.85}$

#163 (1.03)

$\mathbf{0.71}$

#231 (1.29)

$\mathbf{0.64}$

#130 (1.10)

$\mathbf{0.89}$

#185 (1.00)

$\mathbf{0.73}$

#43 (0.58)

$\mathbf{0.45}$

#44 (0.42)

$\mathbf{0.29}$

#208 (0)

$\mathbf{0.84}$

#16 (0.57)

$\mathbf{0.63}$

#162 (0.267)

$\mathbf{0.35}$

#86 (0.39)

$\mathbf{0.26}$

#221 (0.165)

$\mathbf{0.71}$

#85 (0.41)

$\mathbf{0.57}$

#204 (0.247)

$\mathbf{0.33}$

#161 (0.19)

$\mathbf{0.17}$

#11 (0.29)

$\mathbf{0.68}$

#100 (0.29)

$\mathbf{0.51}$

#131 (0.18)

$\mathbf{0.16}$

#217 (0.20)

$\mathbf{0.36}$

“Buildings,” “Lighthouse,” “Statue,” and “Stream” taken from the LIVE database for JPEG compression. The compression bitrate is indicated below the reference number of each image (columns 1, 3, 5, and 7).

Table 5

Algorithm Evaluation Using the Four Images^{
a
}

Buildings

$\sigma \left(k\right)$

Lighthouse

$\sigma \left(k\right)$

Statue

$\sigma \left(k\right)$

Stream

$\sigma \left(k\right)$

#199 (1.666)

1

#174 (1.54)

1

#162 (0.74)

1

#217 (1.48)

1

#222 (0.84)

$\mathbf{0.96}$

#51 (0.6505)

$\mathbf{0.97}$

#102 (2.41)

$\mathbf{0.99}$

#17 (0.40)

$\mathbf{0.98}$

#13 (0.40)

$\mathbf{0.89}$

#149 (0.364)

$\mathbf{0.96}$

#116 (0.222)

$\mathbf{0.97}$

#8 (0.71)

$\mathbf{0.93}$

#33 (0.37)

$\mathbf{0.88}$

#106 (0.242)

$\mathbf{0.92}$

#227 (0.05)

$\mathbf{0.97}$

#71 (0.19)

$\mathbf{0.79}$

#29 (0.20)

$\mathbf{0.79}$

#91 (0.242)

$\mathbf{0.92}$

#92 (0.377)

$\mathbf{0.95}$

#175 (0.050)

$\mathbf{0.58}$

#156 (0.12)

$\mathbf{0.69}$

#202 (0.05)

$\mathbf{0.75}$

#169 (0.07)

$\mathbf{0.85}$

“Buildings,” “Lighthouse,” “Statue,” and “Stream” taken from the LIVE database for JPEG2000 compression. The compression bitrate is indicated below the reference number of each image (columns 1, 3, 5, and 7).

Tables (5)

Table 1

Comparison of Different Image Quality Measures

Lena

PSNR

SSIM

$\sigma \left(t\right)$

MIT

PSNR

SSIM

$\sigma \left(t\right)$

#1

—

1

1

#7

—

1

1

#2

26.01

0.7923

0.82

#8

21.77

0.6618

0.77

#3

25.51

0.7766

0.80

#9

21.24

0.6273

0.74

#4

24.99

0.7459

0.72

#10

20.57

0.5917

0.66

#5

24.36

0.7129

0.71

#11

20.00

0.5574

0.65

#6

20.34

0.5357

0.55

#12

15.56

0.4002

0.39

Table 2

Algorithm Evaluation Using the Four Images^{
a
}

Buildings

$\sigma \left(k\right)$/ (PSNR)

Lighthouse

$\sigma \left(k\right)$/ (PSNR)

Statue

$\sigma \left(k\right)$/ (PSNR)

Stream

$\sigma \left(k\right)$/ (PSNR)

#159 (0.000)

1

#164 (0.00)

1

#148 (0.000)

1

#151 (0.000)

1

#45 (0.5624)

$\mathbf{0.71}$ (29.41)

#4 (0.4478)

$\mathbf{0.82}$ (39.15)

#98 (0.8489)

$\mathbf{0.58}$ (31.38)

#71 (0.4192)

$\mathbf{0.90}$ (37.4)

#7 (0.8489)

$\mathbf{0.49}$ (24.29)

#102 (0.8220)

$\mathbf{0.46}$ (28.32)

#77 (1.3072)

$\mathbf{0.41}$ (28.6)

#18 (0.7629)

$\mathbf{0.55}$ (25.29)

#62 (0.9348)

$\mathbf{0.45}$ (23.52)

#15 (1.1353)

$\mathbf{0.32}$ (25.93)

#131 (1.8228)

$\mathbf{0.28}$ (26.98)

#126 (0.834)

$\mathbf{0.51}$ (24.55)

#134 (1.5364)

$\mathbf{0.24}$ (20.62)

#97 (1.4791)

$\mathbf{0.23}$ (24.5)

#54 (2.166)

$\mathbf{0.22}$ (26.24)

#50 (1.020)

$\mathbf{0.42}$ (23.23)

#73 (2.6249)

$\mathbf{0.07}$ (18.64)

#24 (14.999)

$\mathbf{0.003}$ (18.5)

#120 (3.999)

$\mathbf{0.05}$ (24.02)

#58 (3.0833)

$\mathbf{0.07}$ (19.57)

“Buildings,” “Lighthouse,” “Statue,” and “Stream” taken from the LIVE database for BLUR degradation. The standard deviation of the Gaussian kernel is indicated below the reference number of each image (columns 1, 3, 5, and 7). The corresponding PSNR has been included in columns 2, 4, 6, and 8.

Table 3

Algorithm Evaluation Using the Four Images^{
a
}

Buildings

$\sigma \left(k\right)$/ (PSNR)

Lighthouse

$\sigma \left(k\right)$/ (PSNR)

Statue

$\sigma \left(k\right)$/ (PSNR)

Stream

$\sigma \left(k\right)$/ (PSNR)

#159 (0.000)

1

#164 (0.00)

1

#148 (0.000)

1

#151 (0.000)

1

#103 (0.031)

$\mathbf{0.99}$ (33.61)

#43 (0.019)

$\mathbf{0.99}$ (37.56)

#85 (0.015)

$\mathbf{0.96}$ (39.46)

#138 (0.031)

$\mathbf{0.98}$ (33.6)

#46 (0.058)

$\mathbf{0.94}$ (28.24)

#114 (0.039)

$\mathbf{0.97}$ (31.63)

#55 (0.046)

$\mathbf{0.82}$ (30.2)

#88 (0.062)

$\mathbf{0.93}$ (27.63)

#130 (0.2890)

$\mathbf{0.40}$ (15.49)

#40 (0.062)

$\mathbf{0.93}$ (27.58)

#25 (0.109)

$\mathbf{0.56}$ (23.12)

#2 (0.187)

$\mathbf{0.63}$ (18.48)

#61 (0.4062)

$\mathbf{0.26}$ (13.38)

#96 (0.171)

$\mathbf{0.69}$ (19.08)

#91 (0.203)

$\mathbf{0.38}$ (18.14)

#106 (0.312)

$\mathbf{0.36}$ (14.86)

#4 (1.9960)

$\mathbf{0.03}$ (8.65)

#66 (1.000)

$\mathbf{0.12}$ (10.19)

#145 (1.00)

$\mathbf{0.05}$ (9.63)

#131 (0.500)

$\mathbf{0.21}$ (12.28)

“Buildings,” “Lighthouse,” “Statue,” and “Stream” taken from the LIVE database for WHITE NOISE degradation. The standard deviation of the noise is indicated below the reference number of each image (columns 1, 3, 5, and 7). The corresponding PSNR has been included in columns 2, 4, 6, and 8.

Table 4

Algorithm Evaluation Using the Four Images^{
a
}

Buildings

$\sigma \left(k\right)$

Lighthouse

$\sigma \left(k\right)$

Statue

$\sigma \left(k\right)$

Stream

$\sigma \left(k\right)$

#157 (1.77)

1

#3 (0)

1

#13 (2.77)

1

#212 (0)

1

#227 (0)

$\mathbf{0.86}$

#57 (2.6)

$\mathbf{0.88}$

#70 (2.19)

$\mathbf{0.94}$

#137 (1.683)

$\mathbf{0.85}$

#163 (1.03)

$\mathbf{0.71}$

#231 (1.29)

$\mathbf{0.64}$

#130 (1.10)

$\mathbf{0.89}$

#185 (1.00)

$\mathbf{0.73}$

#43 (0.58)

$\mathbf{0.45}$

#44 (0.42)

$\mathbf{0.29}$

#208 (0)

$\mathbf{0.84}$

#16 (0.57)

$\mathbf{0.63}$

#162 (0.267)

$\mathbf{0.35}$

#86 (0.39)

$\mathbf{0.26}$

#221 (0.165)

$\mathbf{0.71}$

#85 (0.41)

$\mathbf{0.57}$

#204 (0.247)

$\mathbf{0.33}$

#161 (0.19)

$\mathbf{0.17}$

#11 (0.29)

$\mathbf{0.68}$

#100 (0.29)

$\mathbf{0.51}$

#131 (0.18)

$\mathbf{0.16}$

#217 (0.20)

$\mathbf{0.36}$

“Buildings,” “Lighthouse,” “Statue,” and “Stream” taken from the LIVE database for JPEG compression. The compression bitrate is indicated below the reference number of each image (columns 1, 3, 5, and 7).

Table 5

Algorithm Evaluation Using the Four Images^{
a
}

Buildings

$\sigma \left(k\right)$

Lighthouse

$\sigma \left(k\right)$

Statue

$\sigma \left(k\right)$

Stream

$\sigma \left(k\right)$

#199 (1.666)

1

#174 (1.54)

1

#162 (0.74)

1

#217 (1.48)

1

#222 (0.84)

$\mathbf{0.96}$

#51 (0.6505)

$\mathbf{0.97}$

#102 (2.41)

$\mathbf{0.99}$

#17 (0.40)

$\mathbf{0.98}$

#13 (0.40)

$\mathbf{0.89}$

#149 (0.364)

$\mathbf{0.96}$

#116 (0.222)

$\mathbf{0.97}$

#8 (0.71)

$\mathbf{0.93}$

#33 (0.37)

$\mathbf{0.88}$

#106 (0.242)

$\mathbf{0.92}$

#227 (0.05)

$\mathbf{0.97}$

#71 (0.19)

$\mathbf{0.79}$

#29 (0.20)

$\mathbf{0.79}$

#91 (0.242)

$\mathbf{0.92}$

#92 (0.377)

$\mathbf{0.95}$

#175 (0.050)

$\mathbf{0.58}$

#156 (0.12)

$\mathbf{0.69}$

#202 (0.05)

$\mathbf{0.75}$

#169 (0.07)

$\mathbf{0.85}$

“Buildings,” “Lighthouse,” “Statue,” and “Stream” taken from the LIVE database for JPEG2000 compression. The compression bitrate is indicated below the reference number of each image (columns 1, 3, 5, and 7).