I think of noise as variance, or standard deviation, or some statistical measure of variation in image values (take your pick). The 1-pixel image has zero variation. Hence zero noise."The 1-pixel image has zero variation. Hence zero noise."
Same thing happens in the real world when you view something with texture (like a rock or brick wall) and then move away from it. Eventually you will not see the texture anymore and you'll just see a flat tone.
- Actually, it has noise.
A measurement does not have to have spatial extent (more than 1 pixel, i.e. sampling in image space) for there to be both noise and signal present. In fact, information theory says that there must
be noise present.
If I mask off all except 1 pixel on an imaging sensor (let's keep it simple and give it perfect 100% quantum efficiency), and then fire exactly 10,000 photons at that pixel, and it spits out a count of 10,003 electrons...is there still "zero noise"? No...readout noise has added a random extra component, 3 electrons in this case.
But you are right to "think of noise as variance, or standard deviation, or some statistical measure of variation in image values". Where is the variation here? It's temporal
, not spatial. If I keep repeating the 10,000 photons experiment, I will keep getting different values - in a statistical distribution, the standard deviation of which is the camera's readout noise.
More realistically, I won't be able to release precisely 10,000 photons every time. Any light source will emit photons per Poisson statistics, so there will also be variations (of the order of the square root of 10,000 = 100) in the photon count reaching the sensor each time. Now I will get output counts like 9913, 10045, 10024, 9935, 10000, 9989, ...a broader distribution, due to the two sources of noise, one external and one internal to the single pixel.
Noise simply means that you cannot exactly measure the correct amount of a signal. There are plenty of light detectors (like photomultiplier tubes) which give no spatial information - they are essentially single-pixel devices. If these gave measurements which were truly noiseless, then using one of these on a telescope would allow me to perfectly
measure the brightness of every source in the night sky, right out to the dimmest and furthest galaxies. Such magic would make doing astronomy trivial! Alas, physics says it ain't so...