Planckian locus

In color theory, the Planckian locus is generally the path or "locus" that the color of a black body would take ina particular color space as the blackbody temperature changes. Generally, a color space is a set of three numbers ("X", "Y", and "Z", for example) which specify the color and brightness of a particular homogeneous visual stimulus. Sometimes we may only wish to deal with the chromaticity (color) of a visual stimulus. This is a two-dimensional space of two numbers ("x" and "y") which leave out the brightness information. In this case the Planckian locus is the path that the color of a black body takes in this chromaticity space as the blackbody temperature changes.

The Planckian locus in the XYZ color space

In the CIE XYZ color space, the three coordinates defining a color are given by "X", "Y", and "Z":cite book | author=Wyszecki, Günter and Stiles, Walter Stanley | title=Color Science: Concepts and Methods, Quantitative Data and Formulae |edition=2E| publisher=Wiley-Interscience | year=2000 | isbn=0-471-39918-3]

:$$X_T = int_0^infty X(lambda)I(lambda,T),dlambda

:$$Y_T = int_0^infty Y(lambda)I(lambda,T),dlambda

:$$Z_T = int_0^infty Z(lambda)I(lambda,T),dlambda

where "I"(λ,T) is the spectral radiance of the light being viewed, and "X"("λ"), "Y"("λ") and "Z"("λ") are the color matching functions of the CIE standard colorimetric observer, shown in the diagram on the right, and "λ" is the wavelength. The Planckian locus is determined by substituting into the above equations the black body spectral radiance, which is given by Planck's law:

:$$I(lambda,T) =frac{2pi hc^2}{lambda^5}frac{1}{expleft(frac{hc/lambda}{kT} ight)-1}

where::"I" is the black body spectral radiance (power per unit area per unit solid angle per unit wavelength):"T" is the temperature of the black body:"h" is Planck's constant:"c" is the speed of light:"k" is Boltzmann's constant

This will give the Planckian locus in CIE XYZ color space. If these coordinates are "X_T", "Y_T", "Z_T" where "T" is the temperature, then in the CIE chromaticity coordinates will be

:$$x_T = frac{X_T}{X_T+Y_T+Z_T}

:$$y_T = frac{Y_T}{X_T+Y_T+Z_T}

Approximation

The Planckian locus in "xy" space is depicted as a curve in the chromaticity diagram above. While it is possible to compute the CIE "xy" co-ordinates exactly given the above formulas, it is faster to use approximations. Since the mired scale changes more evenly along the locus than the temperature itself, it is common for such approximations to be functions of the reciprocal temperature. Kim "et al" uses a cubic spline: [US patent reference
number = 7024034
y = 2006
m = 04
d = 04
inventor = Kim "et al"
title = Color Temperature Conversion System and Method Using the Same] [cite journal|journal=Journal of the Korean Physical Society|volume=41|issue=6|month=December|year=2002|pages=865–871|title=Design of Advanced Color Temperature Control System for HDTV Applications| url=http://icpr.snu.ac.kr/resource/wop.pdf/J01/2002/041/R06/J012002041R060865.pdf|author=Bongsoon Kang, Ohak Moon, Changhee Hong, Honam Lee, Bonghwan Cho and Youngsun Kim]

$$x_c=egin{cases}-0.2661239 frac{10^9}{T^3} - 0.2343580 frac{10^6}{T^2} + 0.8776956 frac{10^3}{T} + 0.179910 & 1667 ext{K} leq T leq 4000 ext{K} \-3.0258469 frac{10^9}{T^3}+2.1070379 frac{10^6}{T^2} + 0.2226347 frac{10^3}{T} + 0.24039 & 4000 ext{K} leq T leq 25000 ext{K}end{cases}

$$y_c=egin{cases}-1.1063814 x_c^3 - 1.34811020 x_c^2 + 2.18555832 x_c - 0.20219683 & 1667 ext{K} leq T leq 2222 ext{K} \-0.9549476 x_c^3 - 1.37418593 x_c^2 + 2.09137015 x_c - 0.16748867 & 2222 ext{K} leq T leq 4000 ext{K} \+3.0817580 x_c^3 - 5.87338670 x_c^2 + 3.75112997 x_c - 0.37001483 & 4000 ext{K} leq T leq 25000 ext{K}end{cases}The Planckian locus can also be approximated in the CIE 1960 UCS, which is used to compute CCT and CRI, using the following expressions: [cite journal| journal=Color Research & Application|title=An algorithm to calculate correlated colour temperature|first=Michael P.|last=Krystek|volume=10|issue=1|month=January|year=1985|pages=38–40|doi=10.1002/col.5080100109|quote=A new algorithm to calculate correlated colour temperature is given. This algorithm is based on a rational Chebyshev approximation of the Planckian locus in the CIE 1960 UCS diagram and a bisection procedure. Thus time-consuming search procedures in tables or charts are no longer necessary.]

$$ar{u}(T)=frac{0.860117757+1.54118254 imes 10^{-4}T + 1.28641212 imes 10^{-7} T^2}{1+8.42420235 imes 10^{-4}T + 7.08145163 imes 10^{-7}T^2}

$$ar{v}(T)=frac{0.317398726+4.22806245 imes 10^{-5}T + 4.20481691 imes 10^{-8} T^2}{1-2.89741816 imes 10^{-5}T+1.61456053 imes 10^{-7}T^2}

This approximation is accurate to within $$left| u-ar{u} ight| < 8 imes10^{-5} and $$left|v-ar{v} ight|<9 imes10^{-5} for $$1000K

Correlated color temperature

quote|The correlated color temperature (T_cp) is the temperatureof the Planckian radiator whose perceived colour most closely resembles that of a given stimulus at the same brightness and under specified viewing conditions| [http://www.cie.co.at/publ/abst/17-4-89.html CIE/IEC 17.4:1987] |International Lighting Vocabulary (ISBN 3900734070) [cite journal|title=The concept of correlated colour temperature revisited|first=Ákos|last=Borbély|coauthors=Sámson,Árpád;Schanda, János|volume=26|issue=6|pages=450–457|month=December|year=2001|doi=10.1002/col.1065|journal=Color Research & Application| url=http://www.knt.vein.hu/staff/schandaj/SJCV-Publ-2005/462.doc]

The mathematical procedure for determining the correlated color temperature involves finding the closest point to the light source's white point on the Planckian locus. Since the CIE's 1959 meeting in Brussels, the Planckian locus has been computed using the CIE 1960 color space, also known as MacAdam's (u,v) diagram. [cite journal|title=Lines of constant correlated color temperature based on MacAdam's (u,v) Uniform chromaticity transformation of the CIE diagram|first=Kenneth L.|last=Kelly|journal=JOSA|volume=53|issue=8|month=August|year=1963| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-53-8-999|format=abstract] Today, the CIE 1960 color space is deprecated for other purposes: [cite book|title=Lighting Engineering: Applied Calculations|first=Ronald Harvey|last=Simons|coauthors=Bean, Arthur Robert|publisher=Architectural Press|isbn=0750650516|year=2001| url=http://books.google.com/books?id=SWzBKDGHxOUC&pg=PA289&dq=%22correlated+colour+temperature%22+CIE+macadam&ei=vFq7R5foOZy8zASDp42PAw&sig=6c_-HQ14a6yS2t5iUKMCkTGdFok]

Owing to the perceptual inaccuracy inherent to the concept, it suffices to calculate to within 2K at lower CCTs and 10K at higher CCTs to reach the threshold of imperceptibility. [cite web|title=Results of the Intercomparison of Correlated Color Temperature Calculation| url=http://cie2.nist.gov/CR3/Documents/Results_CCTcomparison.pdf|month=June 19|year=1999|publisher=CORM|first=Yoshi|last=Ohno|coauthors=Jergens, Michael]

International Temperature Scale

The Planckian locus is derived by the determining the chromaticity values of a Planckian radiator using the standard colorimetric observer. The relative SPD of Planckian radiator follows Planck's law, and depends on the second radiation constant, $$c_2=hc/k. As measuring techniques have improved, the General Conference on Weights and Measures has revised its estimate of this constant, with the International Temperature Scale (and briefly, the "International Practical Temperature Scale"). These successive revisions caused a shift in the Planckian locus and, as a result, the correlated color temperature scale. Before ceasing publication of standard illuminants, the CIE worked around this problem by explicitly specifying the form of the SPD, rather than making references to black bodies and a color temperature. Nevertheless, it is useful to be aware of previous revisions in order to be able to verify calculations made in older texts: [cite book|title=Colorimetry: Understanding the CIE System|author=Janos Schanda|publisher=Wiley Interscience|year=2007|chapter=3: CIE Colorimetry|page=37-46|isbn=978-0-470-04904-4] [ [http://www.its-90.com/its-90p4.html The ITS-90 Resource Site] ]
* $$c_2=1.432 imes 10^{-2} ext{m·K} (ITS-27). Note: Was in effect during the standardization of Illuminants A, B, C (1931), however the CIE used the value recommended by the U.S. National Bureau of Standards, 1.435 × 10^-2 [cite journal|title=The Early History of the International Practical Scale of Temperature|first=J.A.|last=Hall|doi=10.1088/0026-1394/3/1/006|journal=Metrologia|volume=3|pages=25–28|month=January|year=1967] [cite journal|title=A table of Planckian radiation|first=Parry|last=Moon|month=March|year=1948|journal=JOSA|volume=38|issue=3|pages=291–294|url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-38-3-291|format=abstract]
* $$c_2=1.4380 imes 10^{-2} ext{m·K} (IPTS-48). In effect for Illuminant series D (formalized in 1967).
* $$c_2=1.4388 imes 10^{-2} ext{m·K} (ITS-68), (ITS-90). Often used in recent papers.
* $$c_2=1.4387752(25) imes 10^{-2} ext{m·K} (CODATA, 2006). Current value, as of 2008. [cite web|title=CODATA Recommended Values of the Fundamental Physical Constants: 2006|year=2007|first=Peter J.|last=Mohr|coauthors=Taylor, Barry N. and Newell, David B.|url=http://physics.nist.gov/cuu/Constants/codata.pdf]

References

External links

* [http://www.vendian.org/mncharity/dir3/blackbody/UnstableURLs/bbr_color.html Numerical table of color temperature and the corresponding xy and sRGB coordinates for both the 1931 and 1964 CMFs] , by Mitchell Charity.
* [http://www.aim-dtp.net/aim/technology/cie_xyz/k2xy.txt Planckian xy locus for 1000K-25000K using the 2° CMF, in 1K increments] , by Timo Autiokari.