Image quality evaluation index

Generally, there are four methods to evaluate image noise , Namely ：

• Signal-to-noise ratio (Signal to Noise Ratio,SNR)
• Peak signal to noise ratio (Peak Signal to Noise Ratio, PSNR)
• Mean square error (Mean Square Error, MSE)
• Structural similarity (Structural SIMilarity, SSIM).

These four evaluation indicators are introduced below .

Mean square error （MSE）

Mean square difference is used to compare two images $K$, $I$ The mean square difference of
$$MSE=\frac{1}{mn}\sum _{i=0}^{n-1}\sum _{j=0}^{m-1}\Vert K(i,j)-I(i,j){\Vert }^2$$

Signal-to-noise ratio （SNR）

SNR Used to describe the ratio of signal to noise
$$SNR(dB)=10{\log }_{10}\left[\frac{{\sum }_{x=0}^{m-1}{\sum }_{y=0}^{n-1}(f(x,y){)}^2}{{\sum }_{x=0}^{m-1}{\sum }_{y=0}^{n-1}(f(x,y)-\hat{f}(x,y){)}^2}\right]$$

SNR Code implementation

The following code is to add that the signal-to-noise ratio is not leveldB Noise code of ：

def add_noise(y, level):
        with torch.no_grad():
            sigma = 10 ** (- (1 / 20) * level)
            y = y + sigma * torch.randn(y.size()).to(self.device)
        return y

Peak signal to noise ratio （PSNR）

grayscale PSNR Calculation

Given a size of $m\times n$ Clean image of $I$ And noise images $K$, Mean square error ($MSE$) Defined as ：
$$MSE=\frac{1}{mn}\sum _{i=0}^{m-1}\sum _{j=0}^{n-1}[I(i,j)-K(i,j){]}^2$$
On this basis ,PSNR(dB) Is defined as ：
$$PSNR=10\cdot {\log }_{10}\left(\frac{MA{X}_I^2}{MSE}\right)$$
among $MA{X}_I^2$ Is the maximum possible pixel value of the image . If every pixel is made up of 8 Bit binary to represent , So it's 255. Usually , If the pixel value is determined by $B$ Bit binary to represent , that $MA{X}_I=2^B-1$.

In a general way , in the light of uint8 data , The maximum pixel value is 255; For floating point data , The maximum pixel value is 1.

RGB Images PSNR Calculation

If it's a color image , There are usually three ways to calculate .

• Separate calculation RGB Three channels PSNR, Then take the average .
• Calculation RGB Three channel MSE , Then divide by 3 .
• Convert picture to YCbCr Format , Then just calculate Y Component, that is, the brightness component PSNR.

among , The second and third methods are more common .

Peak signal to noise ratio PSNR An objective measure of image distortion or noise level .2 Between two images PSNR The bigger the value is. , The more similar . The general benchmark is 30dB,30dB The following image degradation is more obvious .

PSNR Code implementation

Method 1 ： utilize skimage Modular compare_psnr Function calculation .

# method 1
diff = im1 - im2
mse = np.mean(np.square(diff))
psnr = 10 * np.log10(255 * 255 / mse)

# method 2
psnr = skimage.measure.compare_psnr(im1, im2, 255)

Method 2 ： From the big guy LeCun Of PSNR function ：

#  Calculation PSNR
def PSNR(target, pred, R=1, dummy=1e-4, reduction='mean'):
    with torch.no_grad():
        dims = (1, 2, 3) if len(target.shape) == 4 else 1
        mean_sq_err = ((target - pred) ** 2).mean(dims)
        mean_sq_err = mean_sq_err + (mean_sq_err == 0).float() * dummy  # if 0, fill with dummy -> PSNR of 40 by default
        output = 10 * torch.log10(R ** 2 / mean_sq_err)
        if reduction == 'mean':
            return output.mean()
        elif reduction == 'none':
            return output

Structural similarity （SSIM）

SSIM Describe the similarity between two images , The formula is based on samples $x$ and $y$ Three comparisons between ： brightness (luminance)、 Contrast (contrast) And structure (structure).
$$l(x,y)=\frac{2{\mu }_x{\mu }_y+c_1}{{\mu }_x^2+{\mu }_y^2+c_1}c(x,y)=\frac{2{\sigma }_x{\sigma }_y+c_2}{{\sigma }_x^2+{\sigma }_y^2+c_2}s(x,y)=\frac{{\sigma }_{xy}+c_3}{{\sigma }_x{\sigma }_y+c_3}$$
Usually take $c_3=c_2/2$.

• ${\mu }_x$ by $x$ The average of ,${\mu }_y$ by $x$ The average of ;${\sigma }_x^2$ by $x$ The variance of ,${\sigma }_y^2$ by $y$ The variance of ;${\sigma }_{xy}^2$ by $xy$ The variance of ;
• $c_1={\left(k_{1}L\right)}^2,c_2={\left(k_{2}L\right)}^2$ Is two constants , Avoid division by zero ,$L$ Is the range of pixel values ;
• $k_1=0.01,k_2=0.03$ As the default value .

that ：
$$\mathrm{SSIM}(x,y)=\left[l(x,y{)}^{\alpha }\cdot c(x,y{)}^{\beta }\cdot s(x,y{)}^{\gamma }\right]$$
take $\alpha ,\beta ,\gamma$ Set to 1, You can get
$$\mathrm{SSIM}(x,y)=\frac{\left(2{\mu }_x{\mu }_y+c_1\right)\left(2{\sigma }_{xy}+c_2\right)}{\left({\mu }_x^2+{\mu }_y^2+c_1\right)\left({\sigma }_x^2+{\sigma }_y^2+c_2\right)}$$

Take one from the picture every time you calculate $N\times M$ The window of , And then slide the window continuously to calculate , Finally, take the average value as the global SSIM.

SSIM Return the image of MSSIM. It's also a floating-point number between zero and one （ The higher, the better ）

SSIM Code implementation

# im1  and  im2  All gray images ,uint8  type 
ssim = skimage.measure.compare_ssim(im1, im2, data_range=255)