SSIM

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SSIM
SSIM

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The Structural SIMilarity (SSIM) index is a method for measuring the similarity between two images. The SSIM index is a full reference metric, in other words, the measuring of image quality based on an initial uncompressed or distortion-free image as reference. SSIM is designed to improve on traditional methods like PSNR and MSE, which have proved to be inconsistent with human eye perception. SSIM is also commonly used as a method of testing the quality of various lossy video compression methods. For example, the popular open source H.264 encoder x264 is set by default to display an SSIM value at the end of an encoding operation.

The SSIM index is a decimal value between 0 and 1. A value of 0 would mean zero correlation with the original image, and 1 means the exact same image. 0.95 SSIM, for example, would imply half as much variation from the original image as 0.90 SSIM. Through this index, image and video compression methods can be effectively compared.

The SSIM metric is calculated on various windows of an image. The measure between two windows of th size NxN $x$ and $y$ is :

$SSIM(x,y) = \frac{(2\mu_x\mu_y + c_1)(2cov_{xy} + c_2)}{(\mu_x^2 + \mu_y^2 + c_1)(\sigma_x^2 + \sigma_y^2 + c_2)}$

with

$μ x$ the average of $x$ ;
$μ y$ the average of $y$ ;
$\sigma_x^2$ the variance of $x$ ;
$\sigma_y^2$ the variance of $y$ ;
$c o v x y$ the covariance of $y$ ;
$c 1 = (k 1 L) 2$ , $c 2 = (k 2 L) 2$ two variables to stabilize the division with weak denominator ;
$L$ the dynamic of the pixel-values ;
$k 1$ = 0,01 and $k 2$ = 0,03 by default.

In order to evaluate the image quality this formula is applied only on the luminance. Typically it is calculated on window-sizes of 8x8. The window can be displaced pixel-by-pixel on the image but the authors propose to use only a subgoup of the possible windows to reduce the complexity of the calculation.