How to compute luminous intensity in a photograph?

Question

I have a photograph in raw format taken by a Nikon D-90 camera of a white LED spotlight shining on a wall in an otherwise dark room. I need to compute the relative luminous intensity of the bright center of the spotlight which is much brighter than the diffuse area around it. The reason is that I am trying to compute the maximum candelas of this spotlight from its lumens rating for that bright center area. I would like to know what percentage of the total light emitted (530 lumens) is going to the bright spot in the center since from that I can compute the candelas by dividing the fraction of the lumens that go to the center by the steradians of the center bright spot. I have Photoshop CS5 extended to examine the photograph. How can I get an approximation of the luminosity ratio of the brightness level in the center to the brightness of the whole area illuminated to compute this?

Will the RGB values of the photograph be a linear scale or some kind or logarithmic scale? If I can figure the relative brightness at a particular location, I could use the area that has that brightness level

Answer

The raw RGB values are in a linear scale. Beware however that the RGB values of the rendered image are far from linear. I don't know whether Photoshop will let you access the raw RGBs, but you need them to do the computation.

To compute the intensity, you first have to integrate the whole spotlight, i.e. add all the pixel values. Divide this sum by the average pixel value at the bright center (averaged over an integer number of 2×2 blocks). You get something that I will call N which is equivalent to a number of pixels, i.e. you can say that the whole spot has as much light as a notional spot of uniform brightness (equal to the maximum brightness of the real spot) and extended across N pixels. From the geometry of your setup, you can convert this number into a surface area S = N × S_p on the wall, where S_p is the wall surface corresponding to one image pixel. From here you can get a solid angle Ω = S / d², where d is the distance between the LED and the wall. Ω is given in steradians.

Finally, your answer is I = Φ/Ω, where Φ is the luminous flux un lumens and I the luminous intensity in lumens per steradian (i.e. in candela).

All this assumes that the distance from the camera to the wall is significantly larger than the spot diameter. Otherwise you would have to throw some cosines in the integration.

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