# Stevens power law or How bright is it?

[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.

Interesting. So what is K?

I guess that depends on how many drinks you had last night!
Sesitive eyes or not, the absolute number can not be expressed as we have no unit to express it in. K is considered to be a constant so in comparison between two levels, k disappears.
An example:
A light has 100 lumens out with XP-G and 125 lumens with an XP-G2. That is 1.25 times the lumens. If we use exp=0.25 (for hotspot angle 15-20 degr.) we take the square root two times and get 1.057, that is the sensation is 5.7% higher with the XP-G2.

Interesting, so I guess Boaz was pretty close. Maybe he knew?

Boaz knows all..... :steve:

Yes, I was also surprised/happy when Boaz’ number came up again. I think this is useful.