SSIM as global quality metric: A differential geometry view

While traditional image quality metrics like MSE are mathematically well understood and tractable, they are known to correlate weakly to image distortion as observed by human observers. To address this situation, many full reference quality indices have been suggested over the years that correlate better to human perception, one of them being the well-known Structural Similarity Index by Wang and Bovik. However, while these expressions show higher correlations, they are often not very tractable mathematically, and - in specific - are rarely metrics in the strict mathematical sense. Specifically, the triangle inequality is often not satisfied, which could either be seen as an effect of the human visual system being unable to compare images that are visually too different, or as a defect of the index capturing the global situation correctly. In this article, the latter position is taken, and it is shown how the SSIM can be understood as a local approximation of a global metric, namely the geodesic distance on a manifold. While the metric cannot be computed explicitly in most cases, it is nevertheless shown that in specific cases its expression is identical to Weber´s Law of luminance sensitivity of the human eye.