I don't understand how the camera can work out the white-balance to use in a given scene.
I could see it working if there is an obvious colour-cast (for example: under fluorescent lights). Does it compare the histograms from the different colour channels and try to make them match up to some degree? Even then I can only imagine it working reliably in very well-defined circumstances.
Can someone explain how it is implemented in today's cameras, and how well it typically works?
Answer
The original assumption is that the average scene should be color neutral and therefore by computing the average color in the scene and then applying the same correction to every pixel you would get a scene whose average color is neutral which should have the correct white-balance. This will fail when there is a dominant color and the scene.
Algorithms got more sophisticated over the years with lots of technical papers and patents written on the subject. They added more intelligence like clamping to the set of known illuminants.
The exact algorithm differs between cameras and it seems to work extremely well outdoors during the day, where there is little variation. Under artificial light there is much more variance and it is rather hit or miss. Older digital cameras were particularly bad but it has been improving on average.
The very best white-balance performance I've ever seen was on the HP Photosmart R967. DC Resource noticed this and commented that they should win the Nobel prize! Several recent compact cameras also do an excellent job. The advantage of a mirrorless camera over a DSLR for this is that it can read data from all over the sensor. DSLRs can now do that in Live-View mode.
Some DSLRs use an entirely different approach which is to measure white-balance instead. This is the case for the Olympus E-5. It has a dedicated 'external' sensor which measures the light falling on the camera. You can turn this off for cases when you are shooting from a different lighting than your subject.
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