Metrics for Noisy Images
How would you expect the effect of a denoising model? Do you want it to exceed your ability to observe the noise pattern and remove them, or to replace you and remove the known pattern which you define as noise?

I think most denoising algorithms target the 2nd option. That's the reason why it makes more sense to quantify and compare different algorithms through simulations. The paper assumes the main source of noise comes from speckle, which can be further assumed as Gaussian noise. So they added normal distribution values onto the 'clean' image, and PSNR, SNR, and MSE all get involved with the variation of the image. Such variation is just due to the adding values. They are actually trying to find out which algorithm can reduce such variation to the closest level of 'clean' image. So the comparisons have to be implemented with the known 'clean' image.

Strictly speaking, the 'clean' image in the paper is not clean at all. It is experimental data from ultrasonic B mode scan, and includes the unknown noise. I remember you asked a question in SO very long ago, claiming that you failed to reproduce some of the comparison results from the paper. That's very possible, since as I mentioned above, you can only evaluate your algorithms via simulations, and the 'clean' image should have been generated as a computational phantom. Otherwise you don't even know which part of the image from experiment, other than your added value,  is treated as noise in your algorithm.

As far as I know, SNR can be used as the quantification of experimental data, but it is not calculated that way in your paper. The aim of your algorithm is to enhance the region of interest (organ, tissues in the ultrasound image), making it clearer to the diagnosis. As a result, you'd better choose such regions by yourself, and select other background regions as the  'noise' part in SNR calculation. If you have more than one regions not viewed as background, contrast to noise ratio (CNR) is also suited in your contrast quantification (this is more used in MRI image analysis). Besides, edge spread function (ESF) along with modulation transfer function are often used in optical imaging.

You may also find some books on image analysis useful in image quality evaluation.