Suppose that an
achromatic target is viewed against a uniform background luminance B. For the target to be visible there must be sufficient luminance
contrast; i.e. the target must be brighter (or darker) than the background by some amount \Delta B. If the target is at
threshold (i.e. only just visible, or with some specified probability of detection) then the threshold contrast is defined as C = \Delta B / B . If B is in the range of
photopic vision, then as B varies we expect C = constant (
Weber's law). Suppose instead that B is in the
scotopic range, when the
quantum nature of light might be significant. Vision is initiated by a shower of (
visible spectrum)
photons coming from both target and background to the eye. The photon emission rate will be subject to some
probability distribution, so can be taken to lie in the range (N \pm \delta N) where N is the
mean of the distribution and \delta N is the
standard deviation. Luminance is directly proportional to shower rate over some sufficient time period, so we can write B = \phi N, \Delta B = \phi \Delta N for some fixed constant \phi and average rates N, \Delta N. Then C = \Delta N /N. Photons from the target and background are a visual
signal that the observer must discriminate from
noise. The likelihood of visibility will be related to the
signal to noise ratio. Imagine that the only noise is the variability of the photon shower, and that the eye is an ideal photon detector. Then the least amount of excess brightness required for the target to be visible will be directly proportional to the greatest accuracy with which the photon rate can be measured. So we can write \Delta N = \rho\delta N for some fixed constant \rho. If the photon shower is assumed to obey
Poisson statistics, then \delta N = \sqrt{N}. Then \Delta B = \phi \rho \sqrt{N} = \phi \rho \sqrt{B/\phi}, hence C \propto 1/\sqrt{B}. ==Empirical results==