Finding largest level of imperceptible noise in an image Suppose that we want to find the "largest level" of a random noise that can safely be added to an image such that the resultant noisy image cannot be distinguishable from the original noise-free image. For this purpose, we design the following subjective experiment:

Let I be the reference (noise-free) image. We first add a high level of noise to I to get a noisy image J. We then display I and J side by side, and ask a user to reduce the level of the noise in J gradually using a slider until he/she cannot distinguish between the reference image and the noisy image. The resultant level is then considered as the largest level of imperceptible noise for I.

But do you think that this experiment is technically sound? Also, can we use an adaptive procedure like QUEST to find the desired threshold/noise level? What do you think?
You are describing two classical psychophysical techniques. The first one is called matching: you try to perceptually match stimulus J to stimulus I. The problem I can see here is that under the level of noise for which the two images are indiscriminable (called absolute threshold), any value would be a match, so that's an ambiguous task. There is another technique where you would ask your observer to press a button when she first notices a difference between the two images while the noise is slowly increasing. You alternate with trials where the observer reports when she cannot perceive a difference between the two images while the noise is slowly decreasing (these values will be different because of perceptual hysteresis). Then you take the average of these values.

The second technique is called a 2-alternative forced choice task. It's the technique used by most psychophysical studies today. There are different ways you could implement it but the simplest would be to ask your observer which of 2 images displayed simultaneously (or successively) contains noise. Then you can indeed use an adaptive technique such as QUEST, a fixed-step staircase or any of the hundreds of staircases that exist to find the threshold at which your observer is N% correct (N being an arbitrary value you fix in advance).

A last technique could be useful. If you want to find not only the threshold at which your observer can perceive noise but the entire function that links noise level to perceived "noisiness", then you could use Maximum Likelihood Difference Scaling. See the paper below.

Sources

Any psychophysics textbook would do, this a very good one:
Kingdom, F. A., &amp; Prins, N. (2010). Psychophysics: A practical introduction.

MLDS technique:
https://jov.arvojournals.org/article.aspx?articleid=2192635