The colour temperature noted $T_c$ is the temperature of a Planckian radiator whose radiation has the same chromaticity as that of a given stimulus. [1]
The correlated colour temperature noted $T_{cp}$ and shortened to $CCT$ is the temperature of the Planckian radiator having the chromaticity nearest the chromaticity associated with the given spectral distribution on a diagram where the (CIE 1931 2° Standard Observer based) $u^\prime, \cfrac{2}{3}v^\prime$ coordinates of the Planckian locus and the test stimulus are depicted. [2]
The CIE Standard Illuminant A, CIE Standard Illuminant D65 and CIE Illuminant E illuminants plotted in the CIE 1960 UCS Chromaticity Diagram:
In [1]:
%matplotlib inline
In [2]:
import colour
from colour.plotting import *
colour.utilities.filter_warnings(True, False)
colour_plotting_defaults()
planckian_locus_chromaticity_diagram_plot_CIE1960UCS(['A', 'D65', 'E'])
In [3]:
# Zooming into the *Planckian Locus*.
planckian_locus_chromaticity_diagram_plot_CIE1960UCS(
['A', 'D65', 'E'], bounding_box=[0.15, 0.35, 0.25, 0.45])
The concept of correlated colour temperature should not be used if the chromaticity of the test source differs more than $\Delta C=5\cdot10^{-2}$ from the Planckian radiator with: [2]
$$ \Delta C= \Biggl[ \Bigl(u_t^\prime-u_p^\prime\Bigr)^2+\cfrac{4}{9}\Bigl(v_t^\prime-v_p^\prime\Bigr)^2\Biggr]^{1/2} $$where $u_t^\prime$, $u_p^\prime$ refer to the test source, $v_t^\prime$, $v_p^\prime$ to the Planckian radiator.
Colour implements various methods for correlated colour temperature computation $T_{cp}$ from chromaticity coordinates $xy$ or $uv$ and chomaticity coordinates $xy$ or $uv$ computation from correlated colour temperature:
Robertson (1968) method is based on $T_{cp}$ computation by linear interpolation between two adjacent members of a defined set of 31 isotemperature lines. [3]
In the CIE 1960 UCS chromaticity diagram the distance $d_i$ of the chromaticity point of given source ($u_s$, $v_s$) from each of the chromaticity point ($u_i$, $v_i$) through which the $i$th isotemperature line of slope $t_i$ passes is calculated as follows: [3]
$$ \begin{equation} d_i=\cfrac{(v_s-v_i)-t_i(u_s-u_i)}{(1+t_i^2)^{1/2}} \end{equation} $$The chromaticity point ($u_s$, $v_s$) is located between the adjacent isotemperature lines $j$ and $j + 1$ if $d_j/d_{j+1} < 0$
$$ \begin{equation} T_c=\Biggl[\cfrac{1}{T_j}+\cfrac{\theta_1}{\theta_1+\theta_2}\biggl(\cfrac{1}{T_{j+1}}-\cfrac{1}{T_j}\biggr)\Biggr]^{-1} \end{equation} $$where $\theta_1$ and $\theta_2$ are respectively the angles between the isotemperature lines $T_j$ and $T_{j+1}$ and the line joining ($u_s$, $v_s$) to their intersection. Since the isotemperature lines are narrow spaced $\theta_1$ and $\theta_2$ are small enough that one can set $\theta_1/\theta_2 = \sin\theta_1/\sin\theta_2$. The above equation can then be written:
$$ \begin{equation} T_c=\Biggl[\cfrac{1}{T_j}+\cfrac{d_j}{d_j-d_{j+1}}\biggl(\cfrac{1}{T_{j+1}}-\cfrac{1}{T_j}\biggr)\Biggr]^{-1} \end{equation} $$The colour.uv_to_CCT_Robertson1968 definition is used to calculate the correlated colour temperature $T_{cp}$ and distance $D_{uv}$ ($d_i$):
In [4]:
colour.temperature.uv_to_CCT_Robertson1968((0.19783451566098664, 0.31221744678060825))
Out[4]:
colour.uv_to_CCT definition is implemented as a wrapper for various correlated colour temperature computation methods:
In [5]:
colour.uv_to_CCT((0.19783451566098664, 0.31221744678060825), 'Robertson 1968')
Out[5]:
Note: 'robertson1968' is defined as a convenient alias for 'Robertson 1968':
In [6]:
colour.uv_to_CCT((0.19783451566098664, 0.31221744678060825), 'robertson1968')
Out[6]:
Converting from correlated colour temperature $T_{cp}$ and distance $D_{uv}$ to chomaticity coordinates $uv$:
In [7]:
colour.CCT_to_uv(6503.03994225557, 'Robertson 1968', D_uv=0.0032556165414977167)
Out[7]:
In [8]:
colour.CCT_to_uv(6503.03994225557, 'robertson1968', D_uv=0.0032556165414977167)
Out[8]:
Ohno (2013) presented new practical accurate methods to calculate the correlated colour temperature $T_{cp}$ and distance $D_{uv}$ with an error of 1 $K$ in $T_{cp}$ range from 1000 to 20,000 and $\pm$0.03 in $D_{uv}$. [4]
The correlated colour temperature is calculated by searching the closest point on the Planckian locus on the CIE 1960 UCS chromaticity diagram but without the complexity of Roberston (1968) method.
A table of coordinates ($U_i$, $V_i$) of Planckian locus (Planckian ($u$, $v$) table) in the estimated range of correlated colour temperature needed is generated and then the distance $d_i$ from the chromaticity coordinates ($U_x$, $V_x$) of a test light source is calculated.
The point $i = m$ is the point where $d_i$ is the smallest in the table ensuring that the correlated colour temperature to be obtained lies between $T_{m-1}$ and $T_{m+1}$.
The previous computation is repeated $n$ times through cascade expansion in order to reduce errors.
A triangle is then formed by the chromaticity point ($U_x$, $V_x$) of the test light soure and the chromaticity points on Planckian locus at $T_{m-1}$ and $T_{m+1}$. The blackbody temperature $T_x$ for the closest point to the line between $T_{m-1}$ and $T_{m+1}$ is calculated as follows: [4]
$$ \begin{equation} T_x=T_{m-1}+(T_{m+1}-T_{m-1})\cdot\cfrac{x}{l} \end{equation} $$with
$$ \begin{equation} \begin{aligned} x&=\cfrac{d_{m-1}^2-d_{m+1}^2+l^2}{2l}\\ l&=\sqrt{(u_{m+1}-u_{m-1})^2+(v_{m+1}-v_{m-1})^2} \end{aligned} \end{equation} $$$D_{uv}$ is then calculated as follows:
$$ \begin{equation} D_{uv}=(d_{m-1}^2-x^2)^{1/2}\cdot sgn(v_x-v_{T_x}) \end{equation} $$with
$$ \begin{equation} \begin{aligned} v_{T_x}&=v_{m-1}+\{v_{m+1}-v_{m-1}\}\cdot x/l\\ SIGN(z)&=1\ for\ z \geq0\ and\ SIGN(z)=-1\ for\ z <0 \end{aligned} \end{equation} $$Errors due to the non linearity of the correlated colour temperature scale on ($u$, $v$) coordinates are reduced by applying the following correction:
$$ \begin{equation} T_{x,cor}=T_x\times 0.99991 \end{equation} $$This correction is not needed for Planckian ($u$, $v$) table with steps of 0.25% or smaller.
After finding $T_{m-1}$ and $T_{m+1}$ as described in the triangular solution method above, $d_{m−1}$, $d_m$, $d_{m+1}$ are fitted to a parabolic function. The polynomial is derived from $d_{m−1}$, $d_m$, $d_{m+1}$ and $T_{m−1}$, $T_m$, $T_{m+1}$ as: [4]
$$ \begin{equation} d(T)=aT^2+bT+c \end{equation} $$where
$$ \begin{equation} \begin{aligned} a&\ =[T_{m-1}(d_{m+1}-d_m)+T_m(d_{m-1}-d_{m+1})+T_{m+1}(d_m-d_{m-1})]\cdot X^{-1}\\ b&\ =-[T_{m-1}^2(d_{m+1}-d_m)+T_m^2(d_{m-1}-d_{m+1})+T_{m+1}^2(d_m-d_{m-1})]\cdot X^{-1}\\ c&\ =-[d_{m-1}(T_{m+1}-T_m)\cdot T_m\cdot T_{m+1}+d_m(T_{m-1}-T_{m+1})\cdot T_{m-1}\cdot T_{m+1}+d_{m+1}(T_m-T_{m-1})\cdot T_{m-1}\cdot T_m]\cdot X^{-1} \end{aligned} \end{equation} $$with
$$ \begin{equation} X=(T_{m+1}-T_m)(T_{m-1}-T_{m+1})(T_m-T_{m-1}) \end{equation} $$The correlated colour temperature $T=T_x$ is then obtained as follows:
$$ \begin{equation} T_X=-\cfrac{b}{2a}\qquad\because d^\prime(T)=2aT_x+b=0 \end{equation} $$The correction factor $T_{x,cor}$ for nonlinearity is applied as described in the triangular solution method.
$D_{uv}$ is then calculated as follows:
$$ \begin{equation} D_{uv}=SIGN(v_x-v_{T_x})\cdot(aT_{x,cor}^2+bT_{x,cor}+c) \end{equation} $$with
$$ \begin{equation} SIGN(z)=1\ for\ z \geq0\ and\ SIGN(z)=-1\ for\ z <0 \end{equation} $$The parabolic solution works accurately except in on or near the Planckian locus. Taking triangular solution results for $|D_{uv}| < 0.002$ and the parabolic solution results for other regions solves that problem. [4]
The colour.uv_to_CCT_Ohno2013 definition is used to calculate the correlated colour temperature $T_{cp}$ and distance $D_{uv}$ ($d_i$):
In [9]:
colour.temperature.uv_to_CCT_Ohno2013((0.19783451566098664, 0.31221744678060825))
Out[9]:
Precision can be changed by passing a value to the iterations argument:
In [10]:
for i in range(10):
print(colour.temperature.uv_to_CCT_Ohno2013(
(0.19783451566098664, 0.31221744678060825), iterations=i + 1))
Using the colour.uv_to_CCT wrapper definition:
In [11]:
colour.uv_to_CCT((0.19783451566098664, 0.31221744678060825), 'Ohno 2013')
Out[11]:
Note: 'ohno2013' is defined as a convenient alias for 'Ohno 2013':
In [12]:
colour.uv_to_CCT((0.19783451566098664, 0.31221744678060825), 'ohno2013')
Out[12]:
Converting from correlated colour temperature $T_{cp}$ and distance $D_{uv}$ to chomaticity coordinates $uv$:
In [13]:
colour.CCT_to_uv(6503.03994225557, 'Ohno 2013', D_uv=0.0032556165414977167, )
Out[13]:
In [14]:
colour.CCT_to_uv(6503.03994225557, 'Ohno 2013', D_uv=0.0032556165414977167)
Out[14]:
McCamy (1992) proposed an equation to compute correlated colour temperature $T_{cp}$ from CIE 1931 2° Standard Observer chromaticity coordinates $x$, $y$ by using a chromaticity epicenter ($x_e$, $y_e$) where the isotemperature lines in some of the correlated colour temperature range converge and the inverse slope of the line $n$ that connects it to $x$, $y$. [5]
The cubic approximation equation is defined as follows: [5]
$$ \begin{equation} T_{cp}=-449n^3+3525n^2-6823.3n+5520.33 \end{equation} $$where
$$ \begin{equation} n=\cfrac{x-x_e}{y-ye}\\ x_e=0.3320\qquad y_e=0.1858 \end{equation} $$The colour.xy_to_CCT_mccamy definition is used to calculate the correlated colour temperature $T_{cp}$ from CIE 1931 2° Standard Observer chromaticity coordinates $x$, $y$:
In [15]:
colour.temperature.xy_to_CCT_McCamy1992((0.31271, 0.32902))
Out[15]:
The colour.xy_to_CCT definition is implemented as a wrapper for various correlated colour temperature computation methods from CIE 1931 2° Standard Observer chromaticity coordinates $x$, $y$:
In [16]:
colour.xy_to_CCT((0.31271, 0.32902), 'McCamy 1992')
Out[16]:
Note: 'mccamy1992' is defined as a convenient alias for 'McCamy 1992':
In [17]:
colour.xy_to_CCT((0.31271, 0.32902), 'mccamy1992')
Out[17]:
Hernandez-Andres, Lee and Romero (1999) extended McCamy (1992) work by using a second epicenter to extend the accuracy over a wider correlated colour temperature and chromaticity coordinates range ($3000$–$10^6K$). [6]
The new extended equation to calculate the correlated colour temperature $T_{cp}$ is defined as follows: [6]
$$ \begin{equation} T_{cp}=A_0+A_1exp(-n/t_1)+A_2exp(-n/t_2)+A_3exp(-n/t_3) \end{equation} $$where
$$ \begin{equation} n=\cfrac{x-x_e}{y-ye}\\ \end{equation} $$with
| Constants | $T_{cp}$ Range ($K$) $3000$-$50,000$ | $T_{cp}$ Range ($K$) $50,000$-$8\times10^5$ |
|---|---|---|
| $A_0$ | $-949.86315$ | $36284.48953$ |
| $A_1$ | $6253.80338$ | $0.00228$ |
| $t_1$ | $0.92159$ | $0.07861$ |
| $A_2$ | $28.70599$ | $5.4535\times10^{-36}$ |
| $t_2$ | $0.20039$ | $0.01543$ |
| $A_3$ | $0.00004$ | |
| $t_3$ | $0.07125$ | |
| $x_e$ | $0.3366$ | $0.3356$ |
| $y_e$ | $0.1735$ | $0.1691$ |
The colour.xy_to_CCT_hernandez definition is used to calculate the correlated colour temperature $T_{cp}$ from CIE 1931 2° Standard Observer chromaticity coordinates $x$, $y$:
In [18]:
colour.temperature.xy_to_CCT_Hernandez1999((0.31271, 0.32902))
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Using the colour.xy_to_CCT wrapper definition:
In [19]:
colour.xy_to_CCT((0.31271, 0.32902), 'Hernandez 1999')
Out[19]:
Note: 'hernandez1999' is defined as a convenient alias for 'Hernandez 1999':
In [20]:
colour.xy_to_CCT((0.31271, 0.32902), 'hernandez1999')
Out[20]:
Krystek (1985) proposed a polynomial approximation valid from $1000K$ to $15000K$. [7]
The CIE UCS colourspace chromaticity coordinates $u$, $v$ are given by the following equations: [7]
$$ \begin{equation} \begin{aligned} u&\ =\cfrac{0.860117757 + 1.54118254 \times 10^{-4} T + 1.28641212 \times 10^{-7} T^2}{1 + 8.42420235 \times 10^{-4} T + 7.08145163 \times 10^{-7} T^2}\\ v&\ =\cfrac{0.317398726 + 4.22806245 \times 10^{-5} T + 4.20481691 \times 10^{-8} T^2}{1 - 2.89741816 \times 10^{-5} T + 1.61456053 \times 10^{-7} T^2} \end{aligned} \end{equation} $$The colour.CCT_to_uv_Krystek1985 definition is used to calculate the CIE UCS colourspace chromaticity coordinates $u$, $v$ from correlated colour temperature $T_{cp}$:
In [21]:
colour.temperature.CCT_to_uv_Krystek1985(6504.389383048972)
Out[21]:
Using the colour.CCT_to_uv wrapper definition:
In [22]:
colour.CCT_to_uv(6504.389383048972, 'Krystek 1985')
Out[22]:
Kang et al. (2002) proposed an advanced colour-temperature control system for HDTV applications in the range from $1667K$ to $25000K$. [8]
The CIE 1931 2° Standard Observer chromaticity coordinates $x$, $y$ are given by the following equations: [8]
$$ \begin{equation} \begin{aligned} x&\ =\begin{cases}-0.2661239\cfrac{10^9}{T_{cp}^3}-0.2343589\cfrac{10^6}{T_{cp}^2}+0.8776956\cfrac{10^3}{T_{cp}}+0.179910 & for\ 1667K\leq T_{cp}\leq4000k\\ -3.0258469\cfrac{10^9}{T_{cp}^3}+2.1070379\cfrac{10^6}{T_{cp}^2}+0.2226347\cfrac{10^3}{T_{cp}}+0.24039 & for\ 4000K\leq T_{cp}\leq25000k\end{cases}\\ y&\ =\begin{cases}-1.1063814x^3-1.34811020x^2+2.18555832x-0.20219683 & for\ 1667K\leq T_{cp}\leq2222k\\ -0.9549476x^3-1.37418593x^2+2.09137015x-0.16748867 & for\ 2222K\leq T_{cp}\leq4000k\\ 3.0817580x^3-5.8733867x^2+3.75112997x-0.37001483 & for\ 4000K\leq T_{cp}\leq25000k\end{cases} \end{aligned} \end{equation} $$The colour.CCT_to_xy_kang definition is used to calculate the CIE 1931 2° Standard Observer chromaticity coordinates $x$, $y$ from correlated colour temperature $T_{cp}$:
In [23]:
colour.temperature.CCT_to_xy_Kang2002(6504.389383048972)
Out[23]:
The colour.CCT_to_xy definition is implemented as a wrapper for various CIE 1931 2° Standard Observer chromaticity coordinates $x$, $y$ computation from correlated colour temperature:
In [24]:
colour.CCT_to_xy(6504.389383048972, 'Kang 2002')
Out[24]:
Note: 'kang2002' is defined as a convenient alias for 'Kang 2002':
In [25]:
colour.CCT_to_xy(6504.389383048972, 'kang2002')
Out[25]:
Judd et al. (1964) defined the following equations to calculate the CIE 1931 2° Standard Observer chromaticity coordinates $x_D$, $y_D$ of a CIE Illuminant D Series: [9]
$$ \begin{equation} \begin{aligned} x_D&\ =\begin{cases}-4,6070\cfrac{10^9}{T_{cp}^3}+2.9678\cfrac{10^6}{T_{cp}^2}+0.09911\cfrac{10^3}{T_{cp}}+0.244063 & for\ 4000K\leq T_{cp}\leq7000k\\ -2.0064\cfrac{10^9}{T_{cp}^3}+1.9018\cfrac{10^6}{T_{cp}^2}+0.24748\cfrac{10^3}{T_{cp}}+0.237040 & for\ 7000K\leq T_{cp}\leq25000k\end{cases}\\ y_D&\ =-3.000x_D^2+0.2.870x_D-0.275 \end{aligned} \end{equation} $$The colour.CCT_to_xy_CIE_D definition is used to calculate the CIE 1931 2° Standard Observer chromaticity coordinates $x$, $y$ of a CIE Illuminant D Series from correlated colour temperature $T_{cp}$:
In [26]:
colour.temperature.CCT_to_xy_CIE_D(6504.389383048972)
Out[26]:
Using the colour.CCT_to_xy wrapper definition:
In [27]:
colour.CCT_to_xy(6504.389383048972, 'CIE Illuminant D Series')
Out[27]:
Note: 'cie_d' is defined as a convenient alias for 'CIE Illuminant D Series':
In [28]:
colour.CCT_to_xy(6504.389383048972, 'cie_d')
Out[28]: