# A Shift-Dependent Measure of Extended Cumulative Entropy and Its Applications in Blind Image Quality Assessment

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## Abstract

**:**

## 1. Introduction

## 2. Some Results on WECE and Its Dynamic Past Version

**Definition**

**1.**

**Remark**

**1.**

- i.
- If X is uniformly distributed in $[0,\theta ]$, then, ${\mathcal{CE}}_{n,k}^{w}\left(X\right)={\theta}^{2}{\left(\frac{k}{k+2}\right)}^{n+1}.$
- ii.
- If X has the $Fr\stackrel{\xb4}{e}chet$ distribution with $F\left(x\right)={e}^{\frac{-\theta}{x}}$, then for $n>2$ we have$${\mathcal{CE}}_{n,k}^{w}\left(X\right)=\frac{{k}^{3}{\theta}^{2}}{n(n-1)(n-2)}={k}^{3}{\mathcal{CE}}_{n,1}^{w}\left(X\right).$$
- iii.
- If X has an inverse Weibull distribution with $F\left(x\right)=exp(-{\left(\frac{\alpha}{x}\right)}^{\beta}),\phantom{\rule{0.277778em}{0ex}}\alpha ,\beta >0,$ then ${\mathcal{CE}}_{n,k}^{w}\left(X\right)=\frac{{\alpha}^{2}{k}^{\frac{\beta +2}{\beta}}}{\beta n!}\Gamma \left(\frac{n\beta -2}{\beta}\right).$
- iv.
- If $Y=aX+b$, with $a>0$ and $b\ge 0$, then ${\mathcal{CE}}_{n,k}^{w}\left(Y\right)={a}^{2}{\mathcal{CE}}_{n,k}^{w}\left(X\right)+ab{\mathcal{CE}}_{n,k}\left(X\right)$.

**Definition**

**2.**

- X is smaller than Y in the usual stochastic order (denoted by $X{\le}_{st}Y$) if $P(X\ge x)\le P(Y\ge x)$ for all x.
- X is smaller than Y in the likelihood ration ordering (denoted by $X{\le}_{lr}Y$) if $\frac{{f}_{Y}\left(x\right)}{{f}_{X}\left(x\right)}$ is increasing in x;
- X is smaller than Y in the reversed hazard rate order, denoted by $X{\le}_{rhr}Y$, if ${r}_{X}\left(x\right)\ge {r}_{Y}\left(x\right)$ for all x;
- X is smaller than Y in the decreasing convex order, denoted by $X{\le}_{dcx}Y$, if $\mathbb{E}\left(\varphi \right(X\left)\right)\le \mathbb{E}\left(\varphi \right(Y\left)\right)$ for all decreasing convex functions ϕ such that the expectations exist;
- X is smaller than Y in the dispersive order, denoted by $X{\le}_{disp}Y$, if ${F}^{-1}\left(v\right)-{F}^{-1}\left(u\right)\le {G}^{-1}\left(v\right)-{G}^{-1}\left(u\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}0<u\le v<1,$ where ${F}^{-1}$ and ${G}^{-1}$ are right continuous inverses of F and G, respectively;
- A non-negative random variable X is said to have a decreasing reversed hazard rate (DRHR) if ${r}_{X}\left(x\right)=\frac{f\left(x\right)}{F\left(x\right)}$ is decreasing in x;
- A non-negative random variable X is said to have a decreasing reversed hazard rate average (DRHRA) if $\frac{{r}_{X}\left(x\right)}{x}$ is a decreasing function in $x>0$. Note that DRHR classes of distributions are included in DRHRA classes of distributions.

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proposition**

**5.**

**Proposition**

**6.**

- i.
- ${\mathcal{CE}}_{n,k}^{w}(X;\infty )={\mathcal{CE}}_{n,k}^{w}\left(X\right).$
- ii.
- $$\begin{array}{c}\hfill {\mathcal{CE}}_{n,k}^{w}(X;t)=\frac{{k}^{n+1}}{{\left[F\left(t\right)\right]}^{k}}\sum _{i=0}^{n}\frac{{(-1)}^{n-i}}{i!(n-i)!}{\left[\tilde{\Lambda}\left(t\right)\right]}^{n-i}{\int}_{0}^{t}x{\left[F\left(x\right)\right]}^{k}{\left[\tilde{\Lambda}\left(x\right)\right]}^{i}dx.\end{array}$$
- iii.
- $$\begin{array}{ccc}\hfill {\int}_{0}^{t}x{\left[F\left(x\right)\right]}^{k}{\left[\tilde{\Lambda}\left(x\right)\right]}^{n}dx& =& \frac{n!{\left[F\left(t\right)\right]}^{k}{\mathcal{CE}}_{n,k}^{w}(X;t)}{{k}^{n+1}}\hfill \\ & -& \sum _{i=0}^{n-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{i}}\right){(-1)}^{n-i}{\left[\tilde{\Lambda}\left(t\right)\right]}^{n-i}{\int}_{0}^{t}x{\left[F\left(x\right)\right]}^{k}{\left[\tilde{\Lambda}\left(x\right)\right]}^{i}dx.\hfill \end{array}$$

**Proposition**

**7.**

**Proof.**

**Remark**

**4.**

**Theorem**

**2.**

**Proof.**

**Proposition**

**8.**

**Proposition**

**9.**

**Definition**

**3.**

**Remark**

**5.**

**Proposition**

**10.**

**Proof.**

## 3. Properties of Conditional WECE

**Definition**

**4.**

**Lemma**

**1.**

**Proposition**

**11.**

**Proof.**

**Lemma**

**2.**

- i.
- ${\mathcal{CE}}_{n,k}^{w}\left(Z\right|Y,X)={\mathcal{CE}}_{n,k}^{w}\left(Z\right|Y),$
- ii.
- $\mathbb{E}\left[{\mathcal{CE}}_{n,k}^{w}\left(Z\right|Y)\right]\le \mathbb{E}\left[{\mathcal{CE}}_{n,k}^{w}\left(Z\right|X)\right].$

**Proof.**

- (i)
- By using the Markov property and definition of ${\mathcal{CE}}_{n,k}^{w}\left(Z\right|Y,X)$, the result follows.
- (ii)
- Let $\mathcal{G}=\sigma \left(X\right)$ and $\mathcal{F}=\sigma (X,Y)$, then from (20) we have$$\begin{array}{ccc}\hfill \mathbb{E}\left[{\mathcal{CE}}_{n,k}^{w}\left(Z\right|X)\right]& \ge & \mathbb{E}\left(\mathbb{E}\left[{\mathcal{CE}}_{n,k}^{w}\left(Z\right|X,Y)\right|X]\right)\hfill \\ & =& \mathbb{E}\left[{\mathcal{CE}}_{n,k}^{w}\left(Z\right|X,Y)\right]\hfill \\ & =& \mathbb{E}\left[{\mathcal{CE}}_{n,k}^{w}\left(Z\right|Y)\right],\hfill \end{array}$$

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Relationships with Other Reliability Functions

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Remark**

**6.**

## 5. Application of ${\mathcal{CE}}_{\mathbf{n},\mathbf{k}}^{\mathbf{w}}\left(\mathbf{X}\right)$ in Blind Image Quality Assessment

**Example**

**1**

**(Blind Image Quality Assessment).**In this example a modified anisotropic image quality (AIQ) measure based on the WECE is used as a blind image quality index, which we call WECE-AIQ. The old AIQ is based on the using of R$\stackrel{\xb4}{e}$nyi entropy and the normalized pseudo-Wigner distribution [17]. We call this measure R$\stackrel{\xb4}{e}$nyi-AIQ. Dataset [18,19] is used in this example for blind image quality assessment. The dataset contains distorted images of three grayscale reference images: a horse, a harbor and a baby (Figure 1). The size and pixel values of the images are $512\times 512$ and in the range 0–255, respectively. The reference images are distorted using “flat allocation”; quantization of the LH sub-bands of a 5-level DWT of the image with equal distortion contrast at each scale (FLT), baseline JPEG compression (JPG), baseline JPEG-2000 compression (JP2), JPEG-2000+DCQ compression (DCQ), Gaussian blur filter (BLR) and additive Gaussian white noise (AGWN). These distortions are utilized to reference images in three levels: low quality (LQ), mid quality (MQ) and good quality (GQ). In this example, WECE is used instead of R$\stackrel{\xb4}{e}$nyi entropy for the estimation of the AIQ metric, and k and n are selected as 2 and 4 for WECE, respectively. For the assessment of the R$\stackrel{\xb4}{e}$nyi-AIQ and WECE-AIQ metrics, some full-reference image quality metrics are needed: PSNR, WSNR, a weighted SNR [20], a universal quality index (UQI) [21], a noise quality measure (NQM) [22], a structural similarity metric (SSIM) [23], a visual information fidelity (VIF) metric [24] and a visual SNR (VSNR) [18]. A bigger value of each of these metrics indicates a better quality of an image. The values of these metrics are available for images of the database used in this example [19]. Note that only gray scale images are considered in this example. For color images, only spatial structures cannot properly demonstrate the quality of an image. Visual damage caused by distortion of the image’s color must be considered. Therefore, a criterion for color distortion must be used. The color image can be decomposed into different color spaces such as RGB, CIE, YCbCr, YIQ, HIS etc. [25]. LMN space, with the optimized weights that are suitable for the human visual system (HVS), can be a good choice [25]. L is the luminance channel for evaluating the structure distortions of the images, and M and N are two chrominance channels which are used to characterize the image quality degradation caused by color distortions. an image quality metric is applied on the L channel for structure distortions measurement and on the M and N channels for color distortions measurement. The values of R$\stackrel{\xb4}{e}$nyi-AIQ, WECE-AIQ and full-reference metrics are depicted in Table 1. The biggest value of R$\stackrel{\xb4}{e}$nyi-AIQ and WECE-AIQ metrics are shown using bold numbers for each image. The performance of WECE-AIQ and R$\stackrel{\xb4}{e}$nyi-AIQ is measured using the times in which a full-reference criterion of the selected image of each approach is larger than in the other approaches. It can be seen from Table 1 which WECE-AIQ displayed a better performance than R$\stackrel{\xb4}{e}$nyi-AIQ for the “Horse (GQ)”, “Horse (LQ)”, “Harbor (GQ)”, “Harbor (MQ)” and "Harbor (LQ)" images. This shows that the quality of the selected images using the WECE-AIQ metric is better than the ones which were selected using the R$\stackrel{\xb4}{e}$nyi-AIQ metric. For visual analysis of the results of Table 1, corresponding images with the biggest values of R$\stackrel{\xb4}{e}$nyi-AIQ and WECE-AIQ metrics are shown in Figure 2, Figure 3 and Figure 4. It can be seen that in most cases, the visual quality of images which were selected using the WECE-AIQ metric was higher than the ones which were chosen using R$\stackrel{\xb4}{e}$nyi-AIQ. For more analysis of the results of Table 1, Spearman’s rank correlation coefficient (SRCC) was used in this example [26]. The results of this measure are shown in Table 2. Table 2 shows the SRCC between full-reference and blind image quality metrics for each image. Bold numbers show the bigger SRCC value of each full-reference metrics. In general, the Spearman’s rank correlation coefficient range is $[-1,1]$. In this example, each blind image quality metric that has a bigger Spearman’s rank correlation coefficient value than others is more useful for image quality assessment. Table 2 shows that for all images, the performance of WECE-AIQ was better than R$\stackrel{\xb4}{e}$nyi-AIQ. Additionally, the performance of WECE-AIQ for the harbor image was better than for the horse and baby image. The corresponding SRCC values of WECE-AIQ for the harbor image were positive in most cases. This shows that the quality ranks of images, which are selected using WECE-AIQ, are very similar to the quality ranks of full-reference metrics. Hence it seems that WECE-AIQ has worked much more effectively than R$\stackrel{\xb4}{e}$nyi-AIQ on the harbor image. Indeed, none of the full-reference image quality metric had a high correlation with the HVS. The accuracy of each one depends on the distortion type, context and texture of the distorted image. Therefore in general, the quality of a distorted image is evaluated using some of the full-reference image quality criteria. For further investigation of this subject, the Spearman’s rank correlation coefficients (SRCCs) between each of the full-reference criterions of the horse image are illustrated at Table 3. Contrary to what was expected, it is seen that the correlation between the full-reference criteria was not high in most cases. Additionally, as can be seen in Table 2, the correlation of R$\stackrel{\xb4}{e}$nyi-AIQ and WECE-AIQ with the full-reference image quality criteria was not high. This is due to the fact that each criterion evaluates the distorted image from a different point of view compared with the others. For example, PSNR calculates the difference between the distorted and reference images, while SSIM is based on the structural similarity between them. Indeed, none of the full-reference image quality criteria consider all of the properties of HVS. Therefore, in this research the performance of R$\stackrel{\xb4}{e}$nyi-AIQ and WECE-AIQ have been evaluated using the correlation between them and all of the full-reference image quality criteria.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Best quality images that were selected using the R$\stackrel{\xb4}{e}$nyi-AIQ metric from GQ, MQ and LQ distorted horse images (

**a**,

**c**and

**e**, respectively), and best quality images that were selected using the WECE-AIQ metric from GQ, MQ and LQ distorted horse images (

**b**,

**d**and

**f**, respectively).

**Figure 3.**Best quality images that were selected using the R$\stackrel{\xb4}{e}$nyi-AIQ metric from GQ, MQ and LQ distorted harbor images (

**a**,

**c**and

**e**, respectively), and best quality images that were selected using the WECE-AIQ metric from GQ, MQ and LQ distorted harbor images (

**b**,

**d**and

**f**, respectively).

**Figure 4.**Best quality images that were selected using the R$\stackrel{\xb4}{e}$nyi-AIQ metric from GQ, MQ and LQ distorted baby images (

**a**,

**c**and

**e**, respectively), and best quality images that were selected using the WECE-AIQ metric from GQ, MQ and LQ distorted baby images (

**b**,

**d**and

**f**, respectively).

Image | Distortion | Full-Reference Image Quality Metric | Blind Image Quality Metric | ||||||
---|---|---|---|---|---|---|---|---|---|

SSIM | VIF | NQM | UQI | PSNR | VSNR | R$\stackrel{\xb4}{\mathit{e}}$nyi-AIQ | WECE-AIQ | ||

Horse (GQ) | FLT | 0.933 | 0.570 | 19.391 | 0.833 | 28.983 | 20.597 | 0.00503661 | 0.00128575 |

JPG | 0.970 | 0.572 | 33.125 | 0.694 | 29.003 | 30.095 | 0.00564478 | 0.00114628 | |

JP2 | 0.946 | 0.427 | 30.446 | 0.656 | 28.870 | 27.700 | 0.0064663 | 0.00124661 | |

DCQ | 0.962 | 0.508 | 31.849 | 0.685 | 28.891 | 36.342 | 0.00577385 | 0.00112959 | |

BLR | 0.974 | 0.637 | 38.456 | 0.816 | 29.056 | 26.884 | 0.00565214 | 0.0013568 | |

AGWN | 0.907 | 0.559 | 29.675 | 0.659 | 28.822 | 28.584 | 0.00399926 | 0.00072772 | |

Horse (MQ) | FLT | 0.903 | 0.513 | 17.146 | 0.799 | 26.734 | 17.934 | 0.0054712 | 0.0011785 |

JPG | 0.926 | 0.374 | 28.033 | 0.589 | 26.701 | 23.736 | 0.0069904 | 0.0014497 | |

JP2 | 0.895 | 0.289 | 26.124 | 0.558 | 26.545 | 23.230 | 0.0057527 | 0.0013887 | |

DCQ | 0.938 | 0.416 | 31.940 | 0.624 | 26.590 | 27.577 | 0.0052392 | 0.0013404 | |

BLR | 0.944 | 0.498 | 34.419 | 0.702 | 26.733 | 22.566 | 0.0037091 | 0.0005268 | |

AGWN | 0.861 | 0.473 | 27.612 | 0.603 | 26.496 | 25.518 | 0.0054970 | 0.0016277 | |

Horse (LQ) | FLT | 0.840 | 0.437 | 13.808 | 0.709 | 23.777 | 14.561 | 0.00652794 | 0.0017846 |

JPG | 0.786 | 0.176 | 19.448 | 0.400 | 23.622 | 17.092 | 0.00549665 | 0.0016539 | |

JP2 | 0.753 | 0.122 | 19.999 | 0.371 | 23.230 | 15.921 | 0.0070989 | 0.0014571 | |

DCQ | 0.781 | 0.137 | 23.099 | 0.396 | 23.213 | 15.997 | 0.00697058 | 0.0012112 | |

BLR | 0.835 | 0.262 | 25.978 | 0.487 | 23.725 | 16.456 | 0.00395791 | 0.0011317 | |

AGWN | 0.777 | 0.363 | 24.709 | 0.513 | 23.300 | 21.530 | 0.00318548 | 0.0003756 | |

Harbor (GQ) | FLT | 0.935 | 0.608 | 14.953 | 0.772 | 31.098 | 18.362 | 0.00317073 | 0.00113086 |

JPG | 0.984 | 0.735 | 28.190 | 0.672 | 31.149 | 31.659 | 0.00302575 | 0.00103601 | |

JP2 | 0.949 | 0.493 | 24.223 | 0.585 | 31.118 | 24.349 | 0.00303644 | 0.00113519 | |

DCQ | 0.975 | 0.649 | 26.711 | 0.663 | 31.202 | 35.532 | 0.00313986 | 0.00110450 | |

BLR | 0.989 | 0.769 | 36.825 | 0.880 | 31.211 | 29.284 | 0.00226195 | 0.00151543 | |

AGWN | 0.934 | 0.640 | 26.318 | 0.658 | 31.097 | 26.079 | 0.00287300 | 0.000939305 | |

Harbor (MQ) | FLT | 0.906 | 0.545 | 12.597 | 0.721 | 28.740 | 15.843 | 0.0031356 | 0.0010805 |

JPG | 0.968 | 0.589 | 26.207 | 0.608 | 28.909 | 26.549 | 0.00292496 | 0.00115297 | |

JP2 | 0.918 | 0.363 | 21.363 | 0.515 | 28.792 | 21.245 | 0.002908160 | 0.00111366 | |

DCQ | 0.959 | 0.540 | 26.106 | 0.592 | 28.858 | 30.429 | 0.0030507 | 0.00119234 | |

BLR | 0.979 | 0.684 | 35.112 | 0.784 | 28.908 | 25.740 | 0.00195757 | 0.001470153 | |

AGWN | 0.895 | 0.552 | 24.244 | 0.607 | 28.724 | 23.095 | 0.00275998 | 0.001070499 | |

Harbor (LQ) | FLT | 0.854 | 0.462 | 9.254 | 0.651 | 25.556 | 12.315 | 0.00277902 | 0.00110425 |

JPG | 0.896 | 0.302 | 18.306 | 0.439 | 25.502 | 18.411 | 0.002717076 | 0.0014915792 | |

JP2 | 0.843 | 0.204 | 16.939 | 0.384 | 25.569 | 16.309 | 0.00231397 | 0.0010248261 | |

DCQ | 0.931 | 0.395 | 24.526 | 0.490 | 25.610 | 27.498 | 0.002142166 | 0.0010449862 | |

BLR | 0.939 | 0.490 | 30.010 | 0.576 | 25.802 | 19.524 | 0.001351146 | 0.0014994589 | |

AGWN | 0.818 | 0.438 | 21.897 | 0.526 | 25.536 | 19.318 | 0.002607232 | 0.0005917831 | |

Baby (GQ) | FLT | 0.948 | 0.614 | 22.824 | 0.843 | 34.485 | 23.352 | 0.001806857 | 0.000464815 |

JPG | 0.955 | 0.504 | 29.818 | 0.718 | 34.528 | 27.700 | 0.001632599 | 0.000500301 | |

JP2 | 0.945 | 0.413 | 28.877 | 0.675 | 34.504 | 26.049 | 0.001734634 | 0.00034369 | |

DCQ | 0.968 | 0.547 | 30.761 | 0.751 | 34.522 | 28.767 | 0.001640073 | 0.000445138 | |

BLR | 0.979 | 0.637 | 34.323 | 0.824 | 34.636 | 26.431 | 0.001428632 | 0.000429976 | |

AGWN | 0.963 | 0.716 | 32.966 | 0.732 | 34.564 | 31.574 | 0.00170072 | 0.000383635 | |

Baby (MQ) | FLT | 0.932 | 0.573 | 21.512 | 0.802 | 32.828 | 21.422 | 0.001849179 | 0.000506972 |

JPG | 0.919 | 0.376 | 26.591 | 0.630 | 32.738 | 24.065 | 0.001550735 | 0.000520883 | |

JP2 | 0.918 | 0.307 | 26.283 | 0.611 | 32.759 | 23.314 | 0.001649043 | 0.000329263 | |

DCQ | 0.938 | 0.365 | 27.239 | 0.661 | 32.846 | 24.059 | 0.00159274 | 0.0003811 | |

BLR | 0.964 | 0.530 | 30.984 | 0.769 | 32.931 | 23.131 | 0.001232089 | 0.00038633 | |

AGWN | 0.946 | 0.647 | 31.594 | 0.665 | 32.740 | 29.220 | 0.001609792 | 0.000381473 | |

Baby (LQ) | FLT | 0.907 | 0.523 | 19.740 | 0.735 | 30.772 | 19.064 | 0.001757395 | 0.000498084 |

JPG | 0.859 | 0.264 | 22.941 | 0.507 | 30.722 | 20.583 | 0.001516522 | 0.001729756 | |

JP2 | 0.877 | 0.222 | 23.077 | 0.529 | 30.794 | 19.833 | 0.001483539 | 0.0046381869 | |

DCQ | 0.915 | 0.283 | 26.433 | 0.613 | 30.803 | 19.935 | 0.001341795 | 0.0042625907 | |

BLR | 0.935 | 0.402 | 26.743 | 0.688 | 31.012 | 19.711 | 0.000895241 | 0.003490746 | |

AGWN | 0.916 | 0.561 | 30.352 | 0.581 | 30.740 | 26.674 | 0.001570303 | 0.000022357 |

**Table 2.**Spearman’s rank correlation coefficient (SRCC) between full-reference and blind image quality metrics for each image.

Image | Blind Image Quality Index | Full-Reference Image Quality Metric | |||||
---|---|---|---|---|---|---|---|

SSIM | VIF | NQM | UQI | PSNR | VSNR | ||

Horse (GQ) | R$\stackrel{\xb4}{e}$nyi-AIQ | 0.485 | −0.428 | 0.42 | −0.371 | 0.085 | 0.2 |

WECE - AIQ | 0.485 | 0.485 | 0.257 | 0.6 | 0.714 | −0.771 | |

Horse (MQ) | R$\stackrel{\xb4}{e}$nyi-AIQ | 0.028 | −0.485 | −0.314 | −0.257 | 0.028 | −0.028 |

WECE -AIQ | 0.085 | 0.142 | −0.485 | 0.314 | 0.485 | −0.542 | |

Horse (LQ) | R$\stackrel{\xb4}{e}$nyi-AIQ | −0.257 | −0.657 | −0.485 | −0.657 | −0.428 | −0.771 |

WECE -AIQ | 0.428 | 0.028 | −0.942 | 0.028 | 0.371 | −0.657 | |

Harbor (GQ) | R$\stackrel{\xb4}{e}$nyi-AIQ | −0.371 | −0.714 | −0.714 | −0.314 | −0.257 | −0.257 |

WECE -AIQ | 0.485 | −0.028 | 0.085 | 0.257 | 0.257 | −0.257 | |

Harbor (MQ) | R$\stackrel{\xb4}{e}$nyi-AIQ | −0.257 | −0.485 | −0.542 | −0.142 | −0.085 | −0.085 |

WECE -AIQ | 0.942 | 0.314 | 0.771 | 0.257 | 0.828 | 0.657 | |

Harbor (LQ) | R$\stackrel{\xb4}{e}$nyi-AIQ | −0.714 | −0.2 | −0.828 | 0.085 | −0.257 | −0.714 |

WECE -AIQ | 0.714 | 0.2 | 0.085 | 0.142 | −0.028 | −0.028 | |

Baby (GQ) | R$\stackrel{\xb4}{e}$nyi-AIQ | −0.771 | −0.142 | −0.7714 | 0.028 | −0.828 | −0.428 |

WECE -AIQ | −0.142 | −0.2 | −0.028 | −0.6 | 0.085 | 0.657 | |

Baby (MQ) | R$\stackrel{\xb4}{e}$nyi-AIQ | −0.485 | 0.085 | −0.6 | 0.085 | −0.2 | −0.257 |

WECE -AIQ | −0.085 | −0.485 | −0.028 | −0.314 | 0.085 | −0.028 | |

Baby (LQ) | R$\stackrel{\xb4}{e}$nyi-AIQ | −0.371 | 0.428 | −0.428 | 0.028 | −0.771 | 0.085 |

WECE -AIQ | 0.314 | 0.314 | −0.2 | 0.6 | 0.257 | −0.542 |

Image | Image Quality Index | Full-Reference Image Quality Index | |||||
---|---|---|---|---|---|---|---|

SSIM | VIF | NQM | UQI | PSNR | VSNR | ||

Horse (GQ) | SSIM | 1 | 0.542857 | 0.942857 | 0.314286 | 0.828571 | 0.142857 |

VIF | 0.542857 | 1 | 0.485714 | 0.771429 | 0.828571 | −0.31429 | |

NQM | 0.942857 | 0.485714 | 1 | 0.085714 | 0.657143 | 0.314286 | |

UQI | 0.314286 | 0.771429 | 0.085714 | 1 | 0.771429 | −0.48571 | |

PSNR | 0.828571 | 0.828571 | 0.657143 | 0.771429 | 1 | −0.25714 | |

VSNR | 0.142857 | −0.31429 | 0.314286 | −0.48571 | −0.25714 | 1 | |

Horse (MQ) | SSIM | 1 | 0.2 | 0.771429 | 0.428571 | 0.6 | −0.08571 |

VIF | 0.2 | 1 | −0.02857 | 0.942857 | 0.6 | −0.48571 | |

NQM | 0.771429 | −0.02857 | 1 | 0.085714 | 0.028571 | 0.371429 | |

UQI | 0.428571 | 0.942857 | 0.085714 | 1 | 0.714286 | −0.42857 | |

PSNR | 0.6 | 0.6 | 0.028571 | 0.714286 | 1 | −0.71429 | |

VSNR | −0.08571 | −0.48571 | 0.371429 | −0.42857 | −0.71429 | 1 | |

Horse (LQ) | SSIM | 1 | 0.657143 | −0.25714 | 0.657143 | 0.828571 | −0.25714 |

VIF | 0.657143 | 1 | −0.08571 | 1 | 0.771429 | 0.085714 | |

NQM | −0.25714 | −0.08571 | 1 | −0.08571 | −0.25714 | 0.542857 | |

UQI | 0.657143 | 1 | −0.08571 | 1 | 0.771429 | 0.085714 | |

PSNR | 0.828571 | 0.771429 | −0.25714 | 0.771429 | 1 | −0.14286 | |

VSNR | −0.25714 | 0.085714 | 0.542857 | 0.085714 | −0.14286 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Tahmasebi, S.; Keshavarz, A.; Longobardi, M.; Mohammadi, R.
A Shift-Dependent Measure of Extended Cumulative Entropy and Its Applications in Blind Image Quality Assessment. *Symmetry* **2020**, *12*, 316.
https://doi.org/10.3390/sym12020316

**AMA Style**

Tahmasebi S, Keshavarz A, Longobardi M, Mohammadi R.
A Shift-Dependent Measure of Extended Cumulative Entropy and Its Applications in Blind Image Quality Assessment. *Symmetry*. 2020; 12(2):316.
https://doi.org/10.3390/sym12020316

**Chicago/Turabian Style**

Tahmasebi, Saeid, Ahmad Keshavarz, Maria Longobardi, and Reza Mohammadi.
2020. "A Shift-Dependent Measure of Extended Cumulative Entropy and Its Applications in Blind Image Quality Assessment" *Symmetry* 12, no. 2: 316.
https://doi.org/10.3390/sym12020316