Mean Square Error (MSE)¶
In the discrete case, the MSE is defined by:
\begin{equation} \text{MSE}(x,y) = \frac{1}{N}\sum_{i=1}^N (x_i-y_i)^2 \end{equation}where:
- $x$ is the original image.
- $y$ is the reconstructed image.
- $N$ is the number of pixels.
MSE expresses the average squared error between the two images $x$ and $y$. The maximum MSE is $d^2$, where $d$ is the dynamic range of $x$. When both images are identical, the (minimum) MSE is $0$.
Root MSE (RMSE)¶
Defined as:
\begin{equation} \text{RMSE}(x,y) = \sqrt{\text{MSE}(x,y)} \end{equation}where:
- $x$ is the original image.
- $y$ is the reconstructed image.
RMSE is similar to MSE, except that the dynamic range of the metric adapted to the dynamic range of the images. So, the maximum MSE is $d$ and the minimum MSE is $0$.
Signal to Noise Ratio (SNR)¶
Defined by:
\begin{equation} \text{SNR}(x,y) = 10\log_{10}\frac{\sigma^2}{\text{MSE}(x,y)} \end{equation}where $\sigma$ is the variance of $x$.
The SNR is the ratio between an estimation of the power of $x$ and a estimation power of $y$. The minimum value for the SNR is obtained when the MSE is $\sigma^2=\sigma^2$, i.e., the SNR cannot be smaller than $\log_{10}1$ dB (decibels).
Peak Signal to Noise Ratio (PSNR)¶
Defined as:
\begin{equation} \text{PSNR}(x,y) = 10\log_{10}\frac{p^2}{\text{MSE}(x,y)} \end{equation}where $p$ is the peak value of the signal (typically $255$ or $65535$, depending on the number of bits per pixel of $x$ and $y$).
As in the SNR, the PSNR is the ratio between an estimation of the maximum power of $x$ and a estimation power of $y$. The minimum value for the PSNR is obtained when the MSE is $d^2=p^2$, i.e., the PSNR cannot be smaller than $\log_{10}1$ dB (decibels). In image and video coding, the PSNR is more common that the SNR. A PSNR smaller than $20$ dB implies a big distortion in $y$, and values over $40$ dB, that $y$ becomes visually indistinguishable from $x$.
Structure Similarity Index Method (SSIM)¶
Defined by:
\begin{equation} \hbox{SSIM}(x,y) = \frac{(2\mu_x\mu_y + c_1)(2\sigma_{xy} + c_2)}{(\mu_x^2 + \mu_y^2 + c_1)(\sigma_x^2 + \sigma_y^2 + c_2)} \end{equation}where:
- $\mu_x$ is the average of $x$.
- $\mu_y$ is the average of $y$.
- $\sigma_x^2$ is the variance of $x$.
- $\sigma_y^2$ is the variance of $y$.
- $\sigma_{xy}$ us the covariance of $x$ and $y$.
- $c_1 = (k_1L)^2$ and $c_2 = (k_2L)^2$ are two constants, being $L$ the dynamic range of the pixel-values (typically this is $2^b-1$ where $b$ is the number of bits per pixel), $k_1 = 0.01$ and $k_2 = 0.03$.
Structural Similarity Index Method is a perception based model. SSIM estimates the perceived quality of images and videos. The maximum value for the SSIM is $1$ (perfect similarity) and the minimum is $-1$ (perfect dissimilarity).
Pearson Correlation Coefficient (PCC)¶
Defined as:
\begin{equation} \hbox{PCC}(x,y) = \frac{\sigma_{xy}}{\sigma_x \sigma_y} \end{equation}where:
- $\sigma_x$ is the standard deviation of $x$.
- $\sigma_y$ is the standard deviation of $y$.
- $\sigma_{xy}$ us the covariance of $x$ and $y$.
The PCC is a measure of the linear correlation between $x$ and $y$. The maximum value is $1$ (total positive linear correlation), $0$ means no linear correlation, and the minimum is $−1$ (total negative linear correlation).
import urllib.request
import math
try:
import cv2
except:
!pip3 install opencv-python --user
try:
import numpy as np
except:
!pip3 install numpy --user
try:
from matplotlib import pyplot as plt
except:
!pip3 install matplotlib --user
%matplotlib inline
try:
import skimage.metrics
except:
!pip3 install scikit-image --user
try:
import scipy.stats
except:
!pip3 install scipy --user
HTTP_response = urllib.request.urlopen('http://www.hpca.ual.es/~vruiz/images/lena.png')
arr = np.asarray(bytearray(HTTP_response.read()), dtype=np.uint8)
BGR = cv2.imdecode(arr,-1)
x = cv2.cvtColor(BGR, cv2.COLOR_BGR2YCrCb)[:,:,0]
plt.imshow(x, cmap="gray")
cv2.imwrite("lena.png", x)
!convert lena.png -quality 75% lena.jpg
y = cv2.imread("lena.jpg", cv2.IMREAD_UNCHANGED)
plt.imshow(y, cmap="gray")
plt.imshow(x-y+128, cmap="gray")
MSE = skimage.metrics.mean_squared_error(x, y)
print(f"MSE={MSE}")
RMSE = math.sqrt(MSE)
print(f"RMSE={RMSE}")
SNR = skimage.metrics.peak_signal_noise_ratio(x, y, data_range=np.std(x))
print(f"SNR={SNR}dB")
PSNR = skimage.metrics.peak_signal_noise_ratio(x, y, data_range=255)
print(f"PSNR={PSNR}dB")
SSIM = skimage.metrics.structural_similarity(x,y, data_range=y.max() - y.min())
print(f"SSIM={SSIM}")
PCC = scipy.stats.pearsonr(x.flatten(),y.flatten())[0]
print(f"PCC={PCC}")