The following errata correct the April 2022 Sky & Telescope Article, Image Stacking Demystified, by Richard S. Wright, Jr.
- The cumulative image noise is proportional to the square root of the number of frames being stacked.
- Mathematically, the shot noise from the imager and associated electronics can be considered to be added to the signal.
- The horizontal axis in both graphs on p.56 should be labeled “number of stacked frames,” and the vertical axis should be labeled “quantity.”
EXPLANATION
1. The S&T author writes, “Shot noise is also quantifiable — it’s simply the square root of the signal value.” This is incorrect, as are the numerical examples that follow the statement.1 The noise and signal components are separate entities, and one cannot say that one of them is a function (square root) of the other. (Actually, some systems do have an interaction between the two, but those are usually 2nd-order, minor effects, and are not relevant in the image stacking situation.) The noise value is completely determined by the physics of the imaging device and the transistors in the related electronics, and is independent of the signal – it’s even there when there is no signal (e.g., a dark frame). The incorrect statement would imply that a dark frame has zero noise, which is not true: in addition to fixed pattern noise (which we can reduce by subtracting a dark frame when doing advanced image processing), the dark frame will have its own random noise, too.
What is really happening is the following. When we stack multiple images, we are literally adding the images together, pixel-by-pixel. That means that the signal components get added together, and so do the noise components. When the images are properly aligned (registered), the signal components at each pixel from one frame to the next add together as correlated data, since they are part of the same image. This combination is literally a simple addition, so image stacking increases the signal component in proportion to the number of frames being stacked.
However, the noise component at each pixel from frame to frame is uncorrelated, because it is a random process. The noise components add together as orthogonal vectors, which means that the noise value increases by the square root of the number of frames being added together. (The stacked images are then re-scaled, so that the resulting image doesn’t get progressively brighter everywhere – but this, of course, scales the noise by the same amount.) The signal-to-noise ratio improvement is therefore proportional to the square-root of the number of frames that are stacked.
2. The S&T author writes, “It’s important to bear in mind that this noise is not something that gets added so much as something that’s missing.” This is incorrect. The mathematical modelling and analysis of signals with noise accounts for each of these elements as an added component; there is nothing “missing” from the original signal, which still exists in the image capture. In practice, this can be readily seen by using a spectrum analyzer, which will show that the signal and noise are separate components. The shot noise from the imager and associated electronics should be considered to be added to the signal.
3. The graphs on p.56 are labeled incorrectly. The horizontal axis in both graphs should be labeled “number of stacked frames,” and the vertical axis should be labeled “quantity,” as it represents either signal or noise in the left-hand graph, and signal-to-noise ratio in the right-hand graph.
—agc
FOOTNOTE (1):
There is a different quantity, known as photon noise, which is characterized as the square root of the photon signal, but this is not the dominant factor in our calculation of signal-to-noise ratio, because we are considering the net effect over a set of stacked frames.
REFERENCES:
Wikipedia: Shot Noise – note the discussion regarding “square root of the expected number of events.”
Wikipedia: Gaussian Noise – “values at any pair of times are identically distributed and statistically independent (and hence uncorrelated).”
Philippe Cattin, Image Restoration: Introduction to Signal and Image Processing.
Robert Fisher, et al, Image Synthesis — Noise Generation.