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# Luminosity function

The luminosity function or luminous efficiency function describes the average sensitivity of the human eye to light of different wavelengths. It should not be considered perfectly accurate in every case, but it is a very good representation of human eye sensitivity and it is valuable as a baseline for experimental purposes. It is a standard function established by the Commission Internationale de l'Éclairage (CIE) and may be used to convert radiant energy into luminous (i.e., visible) energy. It also forms the central color matching function in the CIE 1931 color space.

## Details

There are actually two luminosity functions in common use. For everyday light levels, the photopic luminosity function best approximates the response of the human eye. For low light levels, the response of the human eye changes, and the scotopic curve applies. The photopic curve is the CIE standard curve used in the CIE 1931 color space.

The luminous flux (or visible energy) in a light source is defined by the photopic luminosity function. The following equation calculates the total luminous flux in a source of light.

$F=683.002 mathrm\left\{lm/W\right\}cdot int^infin_0 overline\left\{y\right\}\left(lambda\right) J\left(lambda\right) dlambda$

where

$F,$ is the luminous flux in lumens,
$J\left(lambda\right),$ is the power spectral density of the radiation (power per unit wavelength), in watts per metre.
$overline\left\{y\right\}\left(lambda\right)$ (also known as $V\left(lambda\right),$) is the standard luminosity function (which is dimensionless).
$lambda,$ is wavelength in metres.

Formally, the integral is the inner product of the luminosity function with the light spectrum. In practice, the functions of wavelength are discretized, and the inner product is computed as an ordinary vector dot product; the CIE distributes standard tables discretized into 5 nm samples from 380 nm to 780 nm.

The standard luminosity function is normalized to a peak value of unity at 555 nm (see luminous coefficient). The value of the constant in front of the integral is usually rounded off to 683 lm/W. The small excess fractional value comes from the slight mismatch between the definition of the lumen and the peak of the luminosity function. The lumen is defined to be unity for a radiant energy of 1/683 watt at a frequency of 540 THz, which corresponds to a standard air wavelength of 555.016 nm rather than 555 nm, which is the peak of the luminosity curve. The value of $overline\left\{y\right\}\left(lambda\right)$ is 0.999997 at 555.016 nm, so that a value of 683/0.999997 = 683.002 is the multiplicative constant. The number 683 is connected to the modern (1979) definition of the candela, the unit of luminous intensity. This arbitrary number made the new definition give numbers equivalent to those from the old definition of the candela.

## Improvements to the standard

The CIE 1924 photopic $V\left(lambda\right)$ luminosity function, which is included in the CIE 1931 color-matching functions as the y function, has long been acknowledged to underestimate the contribution of the blue end of the spectrum to perceived luminance. There have been numerous attempts to improve the standard function, to make it more representative of human vision. Judd in 1951, improved by Vos in 1978, resulted in a function known as CIE $V_M\left(lambda\right)$. More recently, Sharpe, Stockman, Jagla & Jägle (2005) developed a function consistent with the Stockman & Sharpe cone fundamentals; their curves are plotted in the figure above.

## Scotopic luminosity

For very low levels of intensity (scotopic vision), the sensitivity of the eye is mediated by rods, not cones, and shifts toward the violet, peaking around 507 nm for young eyes; the sensitivity is equivalent to 1699 lm/W or 1700 lm/W at this peak.

The standard scotopic luminosity function or $V^prime\left(lambda\right)$ was adopted by the CIE in 1951, based on measurements by Wald (1945) and by Crawford (1949).