Are there any good "conceptual distances" for comparing two images?
Ok, I read the question posted in your link, so forget about MSE or SSIM.
What you can do is define a specific distance for the particular type of data you are using.
For example, if you are using video data, depending on what you are most interested in, you can define distance measure based on the average shot length, keyframe colour histogram, average motion in a shot, and so on.
If you are working on a database of face images, you can use e.g. PCA+eigenfaces to project each image on a N-dimensional space and compute an euclidean distance on that space; if you are working with e.g. flower pictures you can define a distance based on edge + colour information... 

Here´s my previous answer:

It´s not really clear to me what you mean for robustness/transparency, to be honest.

If you want to check which one of I_2 and I_3 is more similar to I_1 a simple way is to use compute the mean square error, defined as the sum of the squares of all the pixels in the image, normalized for the total # of pixels