View source for Luminosity function

{{For|the luminosity function in astronomy|Luminosity function (astronomy)}}
[[Image:Luminosity.svg|thumb|right|400px|[[Photopic vision|Photopic]] (black) and [[scotopic vision|scotopic]] (green) luminosity functions.<ref name="scvl" group=c/> The photopic includes the CIE 1931 standard<ref name="ciexyz31" group=c/> (solid), the Judd–Vos 1978 modified data<ref name="vljv" group=c/> (dashed), and the Sharpe, Stockman, Jagla & Jägle 2005 data<ref name="ssvl2" group=c/> (dotted). The horizontal axis is wavelength in [[Nanometre|nm]].]]
[[Image:Srgbspectrum.png|thumb|right|400px]]

A '''luminosity function''' or '''luminous efficiency function''' describes the average [[spectral sensitivity]] of human visual perception of brightness.  It is based on subjective judgements of which of a pair of different-colored lights is brighter, to describe relative sensitivity to light of different wavelengths.  It should not be considered perfectly accurate, but it is a good representation of visual sensitivity of the human eye and it is valuable as a baseline for experimental purposes.  Different luminosity functions apply under different lighting condition, varying from photopic in brightly lit conditions through mesotopic to scotopic under low lighting conditions.  Without qualification, ''the luminosity function'' generally refers to the photopic luminosity function.

The CIE photopic luminosity function {{overline|''y''}}(''λ'') or ''V''(''λ'') 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 [[CIE 1931 color space#Color matching functions|color matching function]] in the [[CIE 1931 color space]].

==Details==
There are two luminosity functions in common use. For everyday light levels, the [[photopic vision|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 vision|scotopic]] curve applies. The photopic curve is the CIE standard curve used in the CIE 1931 color space.

The luminous flux (or visible power) in a light source is defined by the photopic luminosity function. The following equation calculates the total luminous flux in a source of light:
: <math>\Phi_\mathrm{v} = 683.002\ \mathrm{lm/W} \cdot \int^\infin_0 \overline{y}(\lambda) \Phi_{\mathrm{e},\lambda}(\lambda)\, \mathrm{d}\lambda,</math>
where
* Φ<sub>v</sub> is the [[luminous flux]], in lumens;
* Φ<sub>e,λ</sub> is the [[Radiant flux|spectral radiant flux]], in watts per nanometre;
* {{overline|''y''}}(''λ''), also known as ''V''(''λ''), is the luminosity function, dimensionless;
* ''λ'' is the wavelength, in nanometres.

Formally, the integral is the [[inner product]] of the luminosity function with the [[spectral power distribution]].<ref name="Charles2003"/> In practice, the integral is replaced by a sum over discrete wavelengths for which tabulated values of the luminosity function are available. The [[International Commission on Illumination|CIE]] distributes standard tables with luminosity function values at {{nowrap|5 nm}} intervals from {{nowrap|380 nm}} to {{nowrap|780 nm}}.<ref name="CIEdocs" group=cie/>

The standard luminosity function is normalized to a peak value of unity at {{nowrap|555 nm}} (see [[luminous coefficient]]). The value of the constant in front of the integral is usually rounded off to {{val|683|u=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 {{nowrap|1/683 W}} at a frequency of {{nowrap|540 THz}}, which corresponds to a standard air wavelength of {{nowrap|555.016 nm}} rather than {{val|555|u=nm}}, which is the peak of the luminosity curve. The value of {{overline|''y''}}(''λ'') is {{val|0.999997}} at {{val|555.016|u=nm}}, so that a value of 683/{{val|0.999997}} = 683.002 is the multiplicative constant.<ref name="Wyszecki2000"/>

The number 683 is connected to the modern (1979) definition of the [[candela]], the unit of [[luminous intensity]].<ref name="candela" group=cie/> 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''(''λ'') luminosity function,<ref name="CIE1926" group=cie/> which is included in the CIE 1931 color-matching functions as the {{overline|''y''}}(''λ'') 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,<ref name="Judd1975"/> improved by Vos in 1978,<ref name="Vos1978"/> resulted in a function known as CIE ''V''<sub>M</sub>(''λ'').<ref name="Stiles1955"/> More recently, Sharpe, Stockman, Jagla & Jägle (2005) developed a function consistent with the [http://www.cvrl.org/cones.htm Stockman & Sharpe cone fundamentals];<ref name="Sharpe2005"/> their curves are plotted in the figure above.

===ISO Standard===
The ISO standard is ISO 11664-1:2007, soon to be replaced by ISO/CIE FDIS 11664-1. The standard provides an incremental table by nm of each value in the visible range.<ref>{{cite web |url=https://www.iso.org/standard/74164.html |title=Colorimetry -- Part 1: CIE standard colorimetric observers|access-date=December 9, 2018}}</ref> <ref>{{cite web|url=http://www.kayelaby.npl.co.uk/general_physics/2_5/2_5_3.html|title=Kay & Laby;tables of physical & chemical constants;General physics;SubSection: 2.5.3 Photometry|publisher=National Physical Laboratory; UK|access-date=December 9, 2018}}</ref>

==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 (color)|violet]], peaking around {{nowrap|507 nm}} for young eyes; the sensitivity is equivalent to {{val|1699|u=lm/W}}<ref name="Kohei2004"/> or {{val|1700|u=lm/W}}<ref name="Casimer1998"/> at this peak.

The standard scotopic luminosity function or ''V''{{′}}(''λ'') was adopted by the CIE in 1951, based on measurements by Wald (1945) and by Crawford (1949).<ref>[http://www.cvrl.org/database/text/lum/scvl.htm]</ref>

== Color blindness ==
[[File:LuminosityCurve2.svg|right|thumb|400px|Protanopic (green) and deuteranopic (red) luminosity functions.<ref name="judd">{{cite book|last=Judd|first=Deane B.|title=Contributions to Color Science|year=1979|publisher=NBS|location=Washington D.C. 20234|url=https://books.google.com/books?id=jgz_iI8NAzYC&pg=PA316|page=316}}</ref> For comparison, the standard photopic curve is shown in yellow.]]
[[Color blindness]] changes the sensitivity of the eye as a function of wavelength. For people with [[protanopia]], the peak of the eye's response is shifted toward the short-wave part of the spectrum (approximately 540&nbsp;nm), while for people suffering [[deuteranopia]], there is a slight shift in the peak of the spectrum, to about 560&nbsp;nm.<ref name="judd" /> People with protanopia have essentially no sensitivity to light of wavelengths more than 670&nbsp;nm.

Most non-[[primate]] [[mammals]] have the same luminosity function as people with protanopia.  Their insensitivity to long-wavelength red light makes it possible to use such illumination while studying the nocturnal life of animals.<ref>{{cite journal|author1=I. S. McLennan |author2=J. Taylor-Jeffs |lastauthoramp=yes |title=The use of sodium lamps to brightly illuminate mouse houses during their dark phases|journal=Laboratory Animals|year=2004|volume=38|pages=384–392|pmid=15479553|doi= 10.1258/0023677041958927|url=http://la.rsmjournals.com/content/38/4/384.full.pdf}}</ref>

For older people with normal color vision, the [[crystalline lens]] may become slightly yellow due to [[cataract]]s, which moves the maximum of sensitivity to the red part of the spectrum and narrows the range of perceived wavelengths.{{Citation needed|date=May 2014}}

==See also==
* [[Apparent Magnitude]]
* [[Color vision]]
* [[Quantum efficiency]], the image sensor equivalent
* [[A-weighting]] and [[equal-loudness contour]], related sound concepts

==References==
{{reflist|3|refs=
<ref name="Charles2003">{{cite book
 | title   = Digital Video and HDTV: Algorithms and Interfaces
 | author  = Charles A. Poynton
 | publisher = Morgan Kaufmann
 | year   = 2003
 | isbn   = 1-55860-792-7
 | url    = https://books.google.com/books?id=ra1lcAwgvq4C&pg=RA1-PA205&dq=luminance+dot-product
 }}</ref>
<ref name="Wyszecki2000">{{cite book
 |author1=Wyszecki, Günter |author2=Stiles, W.S.
  |lastauthoramp=yes | title   = Color Science - Concepts and Methods, Quantitative Data and Formulae
 | edition  = 2nd
 | publisher = Wiley-Interscience
 | year   = 2000
 | isbn   = 0-471-39918-3
 }}</ref>
<ref name="Judd1975">{{cite book
 |author1=Judd, Deane B. |author2=Wyszecki, Günter
  |lastauthoramp=yes | title   = Color in Business, Science and Industry
 | edition  = 3rd
 | publisher = John Wiley
 | year   = 1975
 | isbn   = 0-471-45212-2
 }}</ref>
<ref name="Vos1978">{{cite journal
 | author = Vos, J. J.
 | title  = Colorimetric and photometric properties of a 2° fundamental observer
 | journal = Color Research and Application
 | year  = 1978
 | volume = 3
 | issue  = 3
 | pages  = 125–128
 | doi   = 10.1002/col.5080030309
 }}</ref>
<ref name="Stiles1955">{{cite journal
 | author1 = Stiles, W. S.
 | author2 = Burch, J. M.
 | title  = Interim report to the Commission Internationale de l'Eclairage Zurich 1955, on the National Physical Laboratory's investigation of colour-matching
 | journal = Optica Acta
 | year  = 1955
 | volume = 2
 | pages  = 168–181
 | bibcode = 1955AcOpt...2..168S
 | doi   = 10.1080/713821039
 | issue  = 4
 }}</ref>
<ref name="Sharpe2005">{{cite journal
 | author1 = Sharpe, L. T.
 | author2 = Stockman, A.
 | author3 = Jagla, W.
 | author4 = Jägle, H.
 | year  = 2005
 | title  = A luminous efficiency function, V*(λ), for daylight adaptation
 | url   = http://calendar.arvo.org/5/11/3/Sharpe-2005-jov-5-11-3.pdf
 | archiveurl=https://web.archive.org/web/20120426041536/http://calendar.arvo.org/5/11/3/Sharpe-2005-jov-5-11-3.pdf
 | archivedate=Apr 26, 2012
 | journal = Journal of Vision
 | volume = 5
 | number = 11
 | pages  = 948–968
 | doi   = 10.1167/5.11.3
 }}</ref>
<ref name="Kohei2004">{{cite book
 | title   = Light Pollution Handbook
 | author1  = Kohei Narisada
 | author2  = Duco Schreuder
 | publisher = Springer
 | year   = 2004
 | isbn   = 1-4020-2665-X
 }}</ref>
<ref name="Casimer1998">{{cite book
 | title   = Handbook of Applied Photometry
 | author  = Casimer DeCusatis
 | publisher = Springer
 | year   = 1998
 | isbn   = 1-56396-416-3
 }}</ref>
}}

===CIE documents===
{{reflist|group=cie|refs=
<ref name="CIEdocs" group=cie>{{cite web
 | url  = http://www.cie.co.at/index.php/LEFTMENUE/index.php?i_ca_id=298
 | title = CIE Selected Colorimetric Tables
 | archiveurl = https://web.archive.org/web/20170131100357/http://files.cie.co.at/204.xls
 | archivedate = 2017-01-31
 }}</ref>
<ref name="candela" group=cie>16th [[General Conference on Weights and Measures|Conférence générale des poids et mesures]] Resolution 3, CR, 100 (1979), and ''Metrologia'', '''16''', 56 (1980).</ref>
<ref name="CIE1926" group=cie>{{cite book
 | author  = CIE
 | title   = Commission internationale de l'Eclairage proceedings, 1924
 | publisher = Cambridge University Press, Cambridge
 | year   = 1926
 }}</ref>
}}

===Curve data===
{{reflist|group=c|refs=
<ref name="scvl">{{cite web
 | title    = CIE Scotopic luminosity curve (1951)
 | url     = http://www.cvrl.org/database/text/lum/scvl.htm
 | archiveurl = https://web.archive.org/web/20081228115119/http://www.cvrl.org/database/text/lum/scvl.htm
 | archivedate = 2008-12-28
 }}</ref>
<ref name="ciexyz31">{{cite web
 | title    = CIE (1931) 2-deg color matching functions
 | url     = http://www.cvrl.org/database/text/cmfs/ciexyz31.htm
 | archiveurl = https://web.archive.org/web/20081228084047/http://www.cvrl.org/database/text/cmfs/ciexyz31.htm
 | archivedate = 2008-12-28
 }}</ref>
<ref name="vljv">{{cite web
 | title    = Judd–Vos modified CIE 2-deg photopic luminosity curve (1978)
 | url     = http://www.cvrl.org/database/text/lum/vljv.htm
 | archiveurl = https://web.archive.org/web/20081228083025/http://www.cvrl.org/database/text/lum/vljv.htm
 | archivedate = 2008-12-28
 }}</ref>
<ref name="ssvl2">{{cite web
 | title    = Sharpe, Stockman, Jagla & Jägle (2005) 2-deg V*(l) luminous efficiency function
 | url     = http://www.cvrl.org/database/text/lum/ssvl2.htm
 | archiveurl = https://web.archive.org/web/20070927222337/http://www.cvrl.org/database/text/lum/ssvl2.htm
 | archivedate = 2007-09-27
 }}</ref>
}}

==External links==
* [http://www.cvrl.org/lumindex.htm Color and Research Vision Laboratory - luminous efficiency data tables]

[[Category:Physical quantities]]
[[Category:Photometry]]