You can use SSIM to measure the image quality as it is highy correlated with visual perception rather just using PSNR in dB. Here is a link for SSIM tool download

https://ece.uwaterloo.ca/~z70wang/research/ssim/

Check this link, it is very useful

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4830760/

If I understand your question, then:

Point (1) is a form of target detection problem in imagery, where the "target" is abnormal tissue, and the background scene is normal tissue. This means that many of the metrics used in target detection/classification in military imagery may be relevant for you (see mine or object detection on the seafloor from side-scan sonar images, for instance).

In point (2), "better" necessarily means that the "probability of detection" (PD) increases, and/or the "probability of false alarm" (PFA) decreases, where:

PD = probability of correctly registering the presence of abnormal tissue, given that there is abnormal tissue present in the image.

PFA = probability of incorrectly registering the presence of abnormal tissue, given that there is no abnormal tissue present in the image.

"To register the presence of abnormal tissue" means that the physician is motivated to take action (expend further resources, or change the course of a patient's care, etc.) as a result of inspecting an image. 

Changes in PD and PFA therefore assess changes in image quality in the most practical way possible, by the impact that image quality will have on the quality of the diagnosis by the physician.

You must always assess both PD and PFA together at the same time. Assessing one independent of the other is generally meaningless, because anybody can always get a high PD at the cost of an intolerably high PFA. 

To assess changes (improvements)  in PD and PFA is not easy. You need many "ground-truth" images in which the presence and absence of abnormal tissue is known with high certainty. Then you must get a number of experts (or an expert system) to make diagnoses on these images using an objective double-blind test methodology. Each expert will then yield a pair (PD, PFA), and a set of experts will yield a set of pairs (PD, PFA). It is the change of the set of pairs, before and after image enhancement, that you would be after. the complete analysis is more than can be covered here. 

As a rough rule of thumb, you need N >> 1/dP images in order to see differences in probability dP, in order for the differences to be statistically significant.  In order to see a 1 % improvement in PD, for instance, you'll need notably more than 1/0.01=100 ground-truth images.

A search of target detection and classification performance assessment literature might be in order.

Ronald

UIQI, SSIM, FSIM, VIFP, GMSD, and many more.