The SSIM index is not a metric and it is badly evaluate the similarity of images

The article explored some properties of a very popular feature of image structural similarity, called the SSIM index. According to https://scholar.google.com, the article [3], where it was first described, has made more than 20,800 citations during the last 14 years. This indicator is actively used by the scientific community in imaging research. It acquired the status of an unofficial international standard for assessing image quality in the presence of a template, often referred to as the image quality metric. This article debunks some of the myths that have arisen around this index. A theorem is proved which states that the SSIM index and any of its linear transformations are not metric functions. In many publications and in the Matlab application software package in the description of the SSIM function, it is said that the SSIM index is used to measure the image quality. However, this index, as well as any comparison function with a reference image (such as full-reference), in principle, cannot assess the quality of the analyzed images. They estimate only a certain degree of similarity between the template image and its distorted copy. The article also shows that the SSIM index cannot always correctly determine the similarity of images of the same scene, while the Pearson linear correlation coefficient makes it much faster and more accurate.