Those random colorful pixels are mainly caused by two effects going on in the camera, one is called dark current, the other read noise. There is also something called shot noise.
The way a digital camera sensor works is by having a load of pixels, each of which have an area sensitive to light on them. When a photon of light hits the pixel it is absorbed and causes an electron to be excited up in energy. At the end of the exposure you measure how much excited charge there is in each pixel, this tells you how much light hit it, and therefore how bright it should be in the final image.
Dark current is where electrons become excited due to something other than light striking them, this could be due to thermal effects for example where the random jiggling of energy in the pixel gives an electron enough energy for it to jump up and be excited. This results in fixed ‘hot pixels’ and randomly brighter pixels throughout the image. This happens in pretty much any electronic circuit with a photosensitive element causing a flow of charge. Since this is a flow of charge (current) being induced without light (dark), we call it dark current.
Next up when the charge is being read off the pixel to be measured the circuitry involved will add some amount of noise, this noise is being added when we read off the values of each pixel and so we call it read noise.
The final thing is shot noise, this isn’t like the others discussed in that it doesn’t come from the electronics or way we detect light, but from the nature of light itself. Light comes in little quantized packets called photons, this means that it isn’t a continuous thing. In the dark you’ll get random pixels which happened to end up with more photons being detected than others around them due to the random and quantized nature of light, this results in a fine variation in brightness throughout the image.
You’re going to get a lot of answers talking about ISO and how having a high ISO makes the image more noisy. This isn’t strictly speaking correct. Turning up the ISO is basically turning up the ‘volume knob’ of the image, it amplifies the signal from the sensor before it gets converted to a digital image. Because the signal is amplified the image becomes brighter, as does any noise in it. This makes the image **look** more noisy because the noise is brighter, but that actual amount hasn’t increased. In fact turning up the ISO decreases the read noise talked about earlier so it actually ends up making the image less noisy.
tl;dr Noise is due to the way we detect light, the way we read off that detection and statistical effects.
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