A global image fidelity metric: Visual distance and its properties

The purpose of full reference image quality indices like Mean Square Error (MSE) or SSIM is to predict the judgement of human observers in subjective quality assessment tasks. More advanced indices like SSIM or VIF are, however, rarely metrics in the strict sense, i.e. they don\´t define a distance between pairs of images that would describe how far these images are related. It was shown in a recent work, however, that SSIM can be "integrated" to such a global metric, in the following called Visual Distance, which behaves locally like SSIM, but globally like a distance in a curved space. It was also seen that the Visual Distance between two images can be interpreted as the number of almost invisible image deformations to transform one image into another. In this work, properties of Visual Distances will be discussed; these results will allow to extend the result from SSIM to its multi-scale variant MS-SSIM. It will also seen that human judgement will typically not define a metric, but it is conjectured that scores and Visual Distances are related by a monotonie Judgement Function. If so, it will be seen that the underlying Visual Distance can always be reconstructed from the scores up to a proportionality factor defined by the scale of the score.