  Assessing the similarity of two images is a complex task that attracts
significant efforts in the image processing community. The widely used
Structural Similarity Index Measure (SSIM) addresses this problem by
quantifying a perceptual structural similarity. In this paper we consider a
recently introduced continuous SSIM (cSSIM), which allows one to analyze
sequences of images of increasingly fine resolutions, and further extend the
definition of the index to encompass the locally weighted version that is used
in practice. For both the local and the global versions, we prove that the
continuous index includes the classical SSIM as a special case, and we provide
a precise connection between image similarity measured by the cSSIM and by the
$L_2$ norm. Using this connection, we derive bounds on the cSSIM by means of
bounds on the $L_2$ error, and we even prove that the two error measures are
equivalent in certain circumstances. We exploit these results to obtain precise
rates of convergence with respect to the cSSIM for several concrete image
interpolation methods, and we further validate these findings by different
numerical experiments. This newly established connection paves the way to
obtain novel insights into the features and limitations of the SSIM, including
on the effect of the local weighted window on the index performances.
