Planckian locus

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In color theory, the Planckian locus is generally the path or locus that the color of a black body would take in a 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.

1 The Planckian locus in the XYZ color space
- 1.1 Approximation
2 Correlated color temperature
- 2.1 International Temperature Scale
3 References
4 External links

[edit] The Planckian locus in the XYZ color space

CIE 1931 Standard Colorimetric Observer functions used to create the plot below

Planckian locus in the CIE 1931 chromaticity diagram

In the CIE XYZ color space, the three coordinates defining a color are given by X, Y, and Z:^[1]

$X_T = \int_0^\infty X(\lambda)I(\lambda,T)\,d\lambda$

$Y_T = \int_0^\infty Y(\lambda)I(\lambda,T)\,d\lambda$

$Z_T = \int_0^\infty Z(\lambda)I(\lambda,T)\,d\lambda$

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:

$I(\lambda,T) =\frac{2\pi hc^2}{\lambda^5}\frac{1}{\exp\left(\frac{hc/\lambda}{kT}\right)-1}$

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

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

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

[edit] 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:^[2]^[3]

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

$y_c=\begin{cases} -1.1063814 x_c^3 - 1.34811020 x_c^2 + 2.18555832 x_c - 0.20219683 & 1667\text{K} \leq T \leq 2222\text{K} \\ -0.9549476 x_c^3 - 1.37418593 x_c^2 + 2.09137015 x_c - 0.16748867 & 2222\text{K} \leq T \leq 4000\text{K} \\ +3.0817580 x_c^3 - 5.87338670 x_c^2 + 3.75112997 x_c - 0.37001483 & 4000\text{K} \leq T \leq 25000\text{K} \end{cases}$

Kim et al's approximation to the Planckian locus (shown in red). The notches demarcate the three splines (shown in blue).

The Planckian locus can also be approximated in the CIE 1960 UCS, which is used to compute CCT and CRI, using the following expressions:^[4]

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

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

This approximation is accurate to within $\left| u-\bar{u} \right| < 8\times10^{-5}$ and $\left|v-\bar{v}\right|<9\times10^{-5}$ for $1000 K < T < 15,000 K$

[edit] Correlated color temperature

The correlated color temperature (T_cp) is the temperature of the Planckian radiator whose perceived colour most closely resembles that of a given stimulus at the same brightness and under specified viewing conditions

— CIE/IEC 17.4:1987, International Lighting Vocabulary (ISBN 3900734070)^[5]

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.^[6] Today, the CIE 1960 color space is deprecated for other purposes:^[7]

The 1960 UCS diagram and 1964 Uniform Space are declared obsolete recommendation in CIE 15.2 (1986), but have been retained for the time being for calculating colour rendering indices and correlated colour temperature.

—CIE 13.3 (1995), Method of Measuring and Specifying Colour Rendering Properties of Light Sources

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.^[8]

Close up of the CIE 1960 UCS. The isotherms are perpendicular to the Planckian locus, and are drawn to indicate the maximum distance from the locus that the CIE considers the correlated color temperature to the meaningful: $\Delta_{uv}=\pm 0.05$

[edit] 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, $c 2 = h c / k$ . 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:^[9]^[10]

$c_2=1.432 \times 10^{-2} \text{m·K}$ (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^[11]^[12]
$c_2=1.4380 \times 10^{-2} \text{m·K}$ (IPTS-48). In effect for Illuminant series D (formalized in 1967).
$c_2=1.4388 \times 10^{-2} \text{m·K}$ (ITS-68), (ITS-90). Often used in recent papers.
$c_2=1.4387752(25) \times 10^{-2} \text{m·K}$ (CODATA, 2006). Current value, as of 2008.^[13]

[edit] References

^ Wyszecki, Günter and Stiles, Walter Stanley (2000). Color Science: Concepts and Methods, Quantitative Data and Formulae, 2E, Wiley-Interscience. ISBN 0-471-39918-3.
^ US7,024,034 (PDF version) (2006-04-04) Kim et al Color Temperature Conversion System and Method Using the Same
^ Bongsoon Kang, Ohak Moon, Changhee Hong, Honam Lee, Bonghwan Cho and Youngsun Kim (December 2002). "Design of Advanced Color Temperature Control System for HDTV Applications". Journal of the Korean Physical Society 41 (6): 865-871.
^ Krystek, Michael P. (January 1985). "An algorithm to calculate correlated colour temperature". Color Research & Application 10 (1): 38-40. doi:10.1002/col.5080100109. “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.”
^ Borbély, Ákos; Sámson,Árpád;Schanda, János (December 2001). "The concept of correlated colour temperature revisited". Color Research & Application 26 (6): 450-457. doi:10.1002/col.1065.
^ Kelly, Kenneth L. (August 1963). "Lines of constant correlated color temperature based on MacAdam's (u,v) Uniform chromaticity transformation of the CIE diagram" (abstract). JOSA 53 (8).
^ Simons, Ronald Harvey; Bean, Arthur Robert (2001). Lighting Engineering: Applied Calculations. Architectural Press. ISBN 0750650516.
^ Ohno, Yoshi; Jergens, Michael (June 19 1999). Results of the Intercomparison of Correlated Color Temperature Calculation. CORM.
^ Janos Schanda (2007). "3: CIE Colorimetry", Colorimetry: Understanding the CIE System. Wiley Interscience, 37-46. ISBN 978-0-470-04904-4.
^ The ITS-90 Resource Site
^ Hall, J.A. (January 1967). "The Early History of the International Practical Scale of Temperature". Metrologia 3: 25-28. doi:10.1088/0026-1394/3/1/006.
^ Moon, Parry (March 1948). "A table of Planckian radiation" (abstract). JOSA 38 (3): 291-294.
^ Mohr, Peter J.; Taylor, Barry N. and Newell, David B. (2007). CODATA Recommended Values of the Fundamental Physical Constants: 2006.