[Edited 23.sep-2012]

I recently had a brief discussion with Scaru about using a formula for perceived brightness in order to compare outputs from two lights. Searching for a formula for perceived brightness as a function of measured brightness I found nothing on the Internet - until now.

A useful formula appears to be ‘Stevens power law’.

A more understandable interpretation of the formula can be found here.

The table in the first article mentions an exponent ‘a’ in the formula:

Perceived brightness = k * (measured brightness)^a

‘k’ is a constant that is not interesting as we only use the formula to compare two levels of light, then ‘k’ will be divided out.

A very important factor that influences the perceived level of light is the angle of sight that the light is observed under, e.g. a point source (0 degr.), the hotspot from a thrower (about 5 degr.) or from a ‘standard’ light (10-20 degr).

For a point source (e.g. looking into a flashlight from a distance) the ‘a’ is 0.5 so the perceived brightness is proportional to the square root of the light output in Candela.

For a 5 degr. target i.e. viewing a hotspot from a thrower pointed to a wall the ‘a’ is 0.33 so the perceived brightness is proportional to the cubic root of the light output in Candela (in the hotspot).

For a hotspot from an XM-L in a P60 (which has a 15-20 degr. viewing angle) one has to guess the exponent, perhaps it is more like a=0.25 (‘quadruple root’).

A ‘funny’ thing is that for a point source briefly flashed, the perceived brightness is directly proportional to the light output (a=1).

I find this article very interesting and usable to calculate e.g. mode level spacing.