noise reduction - Does median blending / image stacking result in better or worse results than a long exposure?

For night photography, do long exposures work better or worse than taking a few photos and merging them in software (median blending / image stacking, like the excellent iOS app Cortex Camera)?

You can assume that the images are perfectly aligned -- assume the blending software corrects for both horizontal and vertical shifts and slight rotations. $2.99 iPhone apps do this, so it's not an unreasonable assumption. Alternatively, assume that a tripod + remote shutter release are being used, to eliminate movement between the photos. Long exposures use a tripod, of course.

So, does median blending result in better or worse image quality? Needless to say, assume other parameters (lens, aperture, ISO, camera...) are all the same.

(This is split from Can I merge multiple photos to reduce noise? but that question specifically assumes that I don't have a tripod, whereas this one allows for it. So they are different questions. I went through the links suggested there, but they didn't seem to clearly answer this question.)

Answer

Theoretically the mean of 10 one-second exposures should give the same amount of noise as one ten-second exposure.

The results in practice differ mainly on account of thermal noise. The longer the sensor is active during an exposure the warmer it gets which results in an increase in dark current noise. Multiple short exposures allow the sensor to cool in between. There is a threshold where thermal noise starts to become a real issue (I don't have the data to hand but it has been studied at length by astrophotographers).

Shorter exposures are also favored if one wants to avoid star trailing.

One advantage of multiple short exposures is that you have the choice of averaging methods, a long exposure can only combine photons in an additive way, whereas with multiple exposures you may capture the same amount of photons in total, you have extra information of when the photons arrived.

Certain types of noise will respond to certain methods, mean and median are usually pretty good and widely implemented. Better still would be a hybrid method like the alhpa-trimmed (truncated) mean. This discards outliers at either end that can skew the result and then computes the mean. You could do even better but that would involve profiling and calculating the parameters of the individual noise distributions.

However as the number of exposures increases the results will converge, i.e. if you shoot enough exposures it will cease to make a difference whether you use mean or median or some other method.