2.5 Image Quality Metrics
2.5.1 Full-reference Metrics
These metrics are calculated between the original ground truth clean images and the corresponding generated de-noised images.
• PSNR: Given a clean image x and restored version of that image y with size m×n. The Peak Signal to Noise Ratio (PSNR) between x and y is calculated as:
P SN R(x, y) = 10log10
peak2 M SE(x, y)
(2.12) wherepeakdenotes the maximum possible intensity (forb-bit image,peak = 2b−1) andM SE(x, y) = mn1 Pm
• SSIM: The Structural Similarity Index (SSIM)  is another metric for image quality assessment, which considers the structural information.
The SSIM models the image distortion as combination of three modules - distortion in luminance (l), contrast distortion (c), and loss of correlation (s). The SSIMbetween image x and y is defined as:
SSIM(x, y) =l(x, y).c(x, y).s(x, y), (2.13) where
l(x, y) = 2µxµy+C1 µ2x+µ2y +C1
c(x, y) = 2σxσy +C2
s(x, y) = σxy+C3
l(x, y) measures the similarity of mean luminance of two images. l(x, y) is maximum (= 1) when µx = µy. c(x, y) compares the contrast of two images. It is maximum (= 1) whenσx =σy. s(x, y) compares the structure using correlation coefficient between two imagesxandy. TheSSIM∈[0,1], where 0 implies no correlation between images, and 1 implies x = y. C1, C2, andC3 are positive constants used to avoid zero denominator.
L 2 L 2 L 2
L 2 L 2 L 2
Figure 2.9: MS-SSIM evaluation system. L: low pass filtering and 2 ↓: downsample by factor of 2.
• MS-SSIM: The Multi-scale Structural Similarity Index (MS-SSIM) 
takes the clean image and the restored image as input, and iteratively ap- plies low-pass filtering and downsamples the image by a factor of 2. The original image is indexed at scale 1, and the maximum scale M. Atithscale, the contrast comparison and structure comparison are denoted as ci(x, y) andsi(x, y), respectively. The luminance is compared only at scale M and is denoted by lM(x, y). TheMS-SSIM is computed by combining these terms as:
SSIM(x, y) = [lM(x, y)]αM.
[ci(x, y)]βi[si(x, y)]γj (2.14) The relative importance of different components is tuned byαM,βi, andγj. A graphical demonstration of its evaluation system is presented in Figure 2.9.
• UQI: The Universal-Image-Quality Index (UQI)  can be written as Q= 4σxyx¯y¯
(σ2x+σy2)[(¯x)2+ (¯y)2] (2.15) where x =xj|j = 1,2, ..., N y = yj|j = 1,2, ..., N are original and test im- ages, respectively. ¯x= N1 PN
j=1xj, ¯y= N1 PN
j=1yj,σx2 = N1−1PN
j=1(xj−x)¯ 2, σy2 = N1−1PN
j=1(yj−y)¯ 2, and σxy = N−11 PN
j=1(xj−x)(y¯ j−y). The range of¯ Q belongs to [1,1] . The best value of Q= 1 is achieved when yj =xj, ∀i. The UQI can also be considered as the product of three different factors:
decay in correlation, distortion in luminance, and distortion in contrast as
(¯x)2+ (¯y)2. 2σxσy
• VIF: The Visual Information Fidelity (VIF)  measures the informa- tion fidelity for the input image in a statistical model of the Human Visual System (HVS) . For this, the VIF utilizes two variables. The first computes the statistics between the initial and the final stage of the vi- sual channel without distortion. Whereas the second considers the mutual information between the input of the distortion block and the output of the visual block. The VIF is estimated for a collection of N ×M wavelet coefficients from each sub-band as follows.
V IF = P
I(CN,i;EN,i|sN,i) (2.17) whereI(CN,i;FN,i|sN,i) andI(CN,i;EN,i|sN,i) are the information that can ideally be computed by the brain from a specific wavelet sub-band in the reference and the test images, respectively.
• LPIPS: The Learned Perceptual Image Patch Similarity (LPIPS) 
demonstrated that the deep neural network activations can be used as a perceptual similarity metric. For this, the authors utilized SqueezeNet , AlexNet  and VGG . It can be used to measure the difference between two image patches, with the output value higher means more difference and lower means more similar.
• FSIM:The The Feature Similarity Index (FSIM)  measures the qual- ity score based on the fact that the HVS understands an image mainly by processing its low-level features. With this motivation,FSIMconsiders the phase congruency (PC) as a significance of a local structure, and gradient magnitude (GM) to incorporate the contrast change. The inputRGBimage is initially converted into YCbCr color space to separate out the luminance
channel of the image. To formally address, let two images are xand y and their phase congruency can be denoted by P C1 andP C2, respectively. The PC and GM (G1, G2) maps extracted from two images x and y. FSIM can be defined and calculated based on P C1, P C2, G1 and G2. Firstly, the similarity between PC maps can be computed as
SP C = 2.P C1.P C2+T1
P C12+P C22 +T1
where T1 is a positive constant. Similarly, the same between the GM maps can be computed as
SG = 2.G1.G2+T2
where T2 is a positive constant.
Now, SP C and SG are combined together to compute the FSIMas
SL= [SP C]α.[SG]β, (2.20) where, α and β are used to adjust the relative importance of PC and GM features. Originally, in the paper, α=β = 1.
• CIEDE 2000: The CIEDE 2000  measure the difference between the color channels of the original clean and restored images. It considers the five corrections, namely: (a) a hue rotation term, to deal with the problematic blue region,(b) compensation for neutral colors (the primed values in the L*C*h differences), (c) compensation for lightness, (d) compensation for chroma, and (e) compensation for hue.
• Haar PSI: Haar Wavelet-based Perceptual Similarity Index (Haar PSI)  estimates the perceptual similarity between two images using features obtained by first performing Haar-wavelet decomposition. It later applies an additional non-linear mapping to the local similarities obtained from high- frequency Haar wavelet filter responses using logistic function owing to the explanation that it greatly models the thresholding in biological neurons.
The Haar PSIvalue of two similar images will be exactly one and the same of two completely different images will be close to zero.
• GMSD: Let d, r be the distorted and reference images, respectively.
Gradient Magnitude Similarity Deviation (GMSD)  first calculates the horizontal and vertical directional gradients by convolving Prewitt filter along the two directions, denoted as Gx, Gy. Then the image gradient maps for r and d at pixel i can be calculated as
Gx,r(i)2+Gy,r(i)2 Gd(i) =q
It then computes the gradient magnitude similarity (GMS) as GM S(i) = 2.Gr(i).Gd(i) +c
Gr(i)2+Gd(i)2+c, (2.22) where c is a numerical stability constant. The GMSD map then can be computed as
GM SD = v u u t 1
(GM S(i)−GM Sµ)2, (2.23) where GM Sµ is the mean of GMS map.
• SpEED-QA: Spatial Efficient Entropic Differencing for Image and Video Quality (SpEED-QA) relies on local spatial operations on given noisy and noise-free image frames and frame differences to estimate the perceptually relevant image/video quality features in an efficient manner.