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Brightness as a Function of Luminance

Let us now consider the brightness perception. Brightness is the magnitude of the subjective sensation produced by visible light. The light intensity can easily be measured, but brightness as a subjective phenomena cannot be exactly measured. Nevertheless, brightness is often approximated as log luminance, or luminance powered to 1/2 to 1/3 depending on the authors. More precise studies showed that there is no one single formula, but rather the brightness-luminance relation depends on the adaptation level and the surrounding light. We will describe the work of Stevens et al. [StSt63] extensively used by Tumblin and Rushmeier in developing their tone mapping operator in this section.

Stevens et al. [StSt63] devised the ``brils'' units to measure the subjective value of brightness. According to Stevens 1 bril equals the sensation of brightness induced in a fully dark-adapted eye by a brief exposure to a 5 degree white target of tex2html_wrap_inline4801 (tex2html_wrap_inline4803) luminance.

Note that two images with different luminance values can have the same brightness values, and appear to be the same. The reason lies in the adaptation mechanism, and the inability of neural units to transfer high dynamic range signals from the retina to the brain. Actually we are very poor judges of absolute luminances, all that we can judge is the change in luminance, i.e. the brightness.

What did Stevens do? He measured brightness as a function of luminance and adaptation by using ``haploskopic matching''. That means he tried to match the brightness when one eye is dark adapted (standard condition for brightness measuring) and the other eye is adapted to a test value. Brightness comparison between two eyes was made quickly, before either could change adaptation level significantly. Measured brightness is then:
where B is brightness in brils, tex2html_wrap_inline4807 is radiance of target in millilamberts, tex2html_wrap_inline4811 is threshold of detectable radiance in millilamberts (this depends on the adaptation radiance), and n and K are constants, dependent on the strength of the adapting field. For full dark-adaptation tex2html_wrap_inline4817, n=0.33, and K=10.

Stevens proposed the next equation from his measurements:
where, assuming tex2html_wrap_inline4823 is adapting, white background luminance in lamberts and tex2html_wrap_inline4825 is target luminance in lamberts, tex2html_wrap_inline4827 in dB, where tex2html_wrap_inline4831
and R is the target luminance difference in dB
After substituting S and R expressions in equation 2.19 we can write the final equation:
where B is brightness in brils, L is viewed (target) radiance in lamberts, tex2html_wrap_inline4823 is luminance of white surround and


These complex formulas provided by Stevens are, unfortunately, neither valid nor accurate when applied to more complex images. They are valid for laboratory settings only. Bartelson and Breneman [BaBr67] have measured many test photographs in order to find appropriate brightness versus luminance function for more complex images. They have proposed an extended formula for complex scenes:
where tex2html_wrap_inline4847 and tex2html_wrap_inline4849 are parameters dependent on viewing conditions and are given graphically.

next up previous contents
Next: Brightness as a Function Up: Human Vision Previous: Just Noticeable Difference