But as you say "The precise gain depends on the nature of the noise" - if the fex noise is defined as grain larger than 1 pixel in the photo, the noise reduction will not fall linearly with photo size reduction - but rather fall rather softly at first and then accelerate with later photo resizing. The actual mathematical algoritm one uses also plays a part. Am I right here?
Partly. When speaking of the nature of the noise, I wasn't really referring to the size of grains or something - in a digital camera noise quite naturally is spatially limited to the pixel size. I was referring to things like the frequency distribution of the noise.
However, there may be some spatial aspects to digital camera noise, too, such as when you have a very bright pixel, noise in the surrounding pixels might be affected. Those things are difficult to model, but when we are talking about downscaling an image 20 times we can probably mostly ignore them, because I'm not sure how a bright pixel would affect a photosite 20 pixels away (and if it did, there would be major things to worry about regarding image quality of the sensor!)
If you assume something like film grain, which (when talking about digital images) consists of lumps larger than the resolution of the image, the mathematics breaks down and works differently.
Niels said:
On perception of images you claim that "I'm saying "both" (i.e. not only mathematical, but also practical)" - and then explains scanning and downsizing tecniques, as examples of perception of images.
Actually I wasn't referring to the perception of images, but to the mathematics of the scanning and downsizing process. It's not like we can retreat into subjectivity, into "judging" things and into and "perception is different" just because it's pictures we're talking about. The fact that in scanning you gain 1 bit (= 1 stop) worth of useful dynamic range above the noise threshold for every increase in the number of scanning passes by a factor of two, for example, is a very simple mathematical result.
This kind of mathematical result applies even if there is no human looking at the picture at all. You can measure the dynamic range above the noise threshold, or you can take the histogram, do a Fourier transformation of it and see if it's jagged or not, without ever looking at the picture. With downsizing it's basically the same, admittedly with some pitfalls such as those mentioned above, but even those largely aren't perception-related.
I claim that if you do a blinded trial with experienced people you would - to a certain extent - be able to "see through" the compressed image in order to estimate the original noise of the picture. (of course you can fool these people if you wantet to :=)
No, it's not that easy to take people and give them two pictures downscaled 20 times and ask them "which one of them was the noisy one". Once you mash 20 pixels into one, they are gone. There is basically no way of reconstructing them, the "fine microstructures" are gone forever, and the large structures become the new fine microstructures. Otherwise it would be easy to do CSI-style image manipulations where you have a ten-pixel blob through a magical process you arrive at a clear image of a human face.