Is SSIM a measure of mutual information?
...is the quality measure SSIM between two images (one base-line, one distorted) directly correlated to the mutual information between the two [?]


The short answer to this is "yes" but it tells you nothing. Because the question is a little bit ill posed. 

You have two signals, let them be signals or images, it doesn't matter, you have two observations. And you want to assess how much they are alike.

What does alike mean? You might have an idea about this already, if it is sound do they sound similar? If it is images, do they look similar?

So, things look alike when they co-vary...Sound pressure over time covaries between two sound clips, brightness (reflected light flux) over space covaries between two images.

One of the first metrics of assessing how much alike two signals (images) are that someone might encounter is cross correlation .

If you employ (normalised) cross correlation, on your baseline image (i.e calculate its autocorrelation), you will observe the highest value at the centre of the spatial cross correlation. A value of 1.0.

Now, notice here, we decoupled the concept of co-variance from the way by which we assess it. Co-variance is this notion that things might be going up and down at the same time/place. Cross correlation is a quantification of that notion.

And, there are others. Cross correlation, mutual information, transfer entropy, granger causality, partial correlation, ... and many others.

ALL of these are correlated with each other. Their ranges, sign and even interpretation might be different, but if you adjust for all of these, in general, all of these metrics go in one direction when the observations co-vary and in another when they don't co-vary.

Let's turn to the SSIM, the SSIM is the product of three factors: luminance, contrast and structure with the following expressions:



(Image captured on wikipedia)

Where the $\mu$'s denote means and the $\sigma$'s denote variances with the $\sigma_{xy}$ in particular denoting covariance. By the way, you might want to notice the similarity of the $l,c,s$ components with the covariance and even the correlation coefficient.

Luminance doesn't even count, if I remove the mean from two images, the luminance component goes to zero. It is a little bit more difficult to zero the Structure but I can do it, if the two images do not covary with each other, a condition that is met if the two images are pure white noise. This will leave the contrast with some value that would have to be equal to the separate variances.

Long story short: SSIM contains a covariance term, covariance and mutual information are quantifications of the same concept, therefore, SSIM and mutual information are correlated.

BUT, this tells you nothing. The point is how are these two correlated and the interpretation. So, there is a different mapping between cross correlation and mutual information because mutual information has that $\log$ factor in it and, if it is mutual information then would it assume memory of some length?

So now we are on the next level, WHAT are you trying to achieve? To then move on to WHICH aspect of your signal do you need to quantify (and how) in order to get there?

Hope this helps.