Image similarity/distance measures: what is really behind MSE and SSIM?

Abstract

Similarity/distance measures play an important role in various signal/image processing applications such as classification, clustering, change detection and matching. In most cases, maybe excluding visual perception, the distance measure should be amplitude/intensity translation invariant what means that it depends only on the relative difference of compared variables/parameters, but not on their absolute values. The two most popular measures: mean squared error (MSE) and structural similarity (SSIM) index used in image processing have been analysed theoretically and experimentally by showing their origin, similarities/differences and main properties. Both measures depend on the same parameters: sample means, standard deviations and correlation coefficient. It has been shown that SSIM originates from the two generalised Dice measures and thus inherit their main property scale invariance. Consequently, this property leads to the dependence of the measure on absolute mean and standard deviation values. Similarly, MSE depends on the absolute standard deviation values. A new composite similarity/distance measure based on means, standard deviations and correlation coefficient (CMSC) which has been proposed recently exhibits translation invariance property with respect to means and standard deviations. Experiments on simulated and real data corrupted with various types of distortions: mean shift, contrast stretching, noise (additive/multiplicative, impulsive) and blurring, supported theoretical results.