# Luminance¶

The Luminance $L_v$ is the quantity defined by the formula: [1]

$$$$L_v=\cfrac{d\Phi_v}{dAcos\theta d\Omega}$$$$

where $d\Phi_v$ is the luminous flux transmitted by an elementary beam passing through the given point and propagating in the solid angle, $d\Omega$, containing the given direction. $dA$ is the area of a section of that beam containing the given point. $\theta$ is the angle between the normal to that section and the direction of the beam.

$L_v$ unit is candela per square metre (or nits) $cd\cdot m^{-2}=lm\cdot m^{-2}\cdot sr^{-1}$.

Colour defines the following luminance computation methods:

In [1]:
import colour

sorted(colour.LUMINANCE_METHODS.keys())

Out[1]:
['ASTM D1535',
'CIE 1976',
'Fairchild 2010',
'Fairchild 2011',
'Newhall 1943',
'astm2008',
'cie1976']

Note: 'astm2008' and 'cie1976' are convenient aliases for respectively 'ASTM D1535' and 'CIE 1976'.

## Newhall, Nickerson, and Judd (1943) Method¶

Newhall, Nickerson, and Judd (1943) fitted a quintic-parabola function to the adjusted Munsell-Sloan-Godlove reflectances, the resulting equation computing luminance $R_Y$ as function of Munsell value $V$ is expressed as follows: [2]

$$$$R_Y=1.2219V-0.23111V^2+0.23951V^3-0.021009V^4+0.0008404V^5$$$$

The colour.luminance_Newhall1943 definition is used to compute luminance $R_Y$:

In [2]:
colour.colorimetry.luminance_Newhall1943(3.74629715382)

Out[2]:
10.408987457743208

Note: Input Munsell value $V$ is in domain [0, 10], output luminance $R_Y$ is in domain [0, 100].

The colour.luminance definition is implemented as a wrapper for various luminance computation methods:

In [3]:
colour.colorimetry.luminance(3.74629715382, method='Newhall 1943')

Out[3]:
0.43750976730517127

## ASTM D1535-08$^{\epsilon 1}$ (2008) Method¶

Since 1943, the reference white used for the Munsell Renotation System has changed.

As a result the quintic-parabola function from Newhall, Nickerson, and Judd (1943) has been adjusted: Each coefficient of the function has been multiplied by 0.975, the reflectance factor of magnesium oxide with respect to the perfect reflecting diffuser and then rounded to five digits.

The updated equation for computing luminance $Y$ as function of the Munsell value $V$ is expressed as follows: [3]

$$$$Y=1.1914V-0.22533V^2+0.23352V^3-0.020484V^4+0.00081939V^5$$$$

The colour.luminance_ASTMD153508 definition is used to compute luminance $Y$:

In [4]:
colour.colorimetry.luminance_ASTMD153508(3.74629715382)

/Users/kelsolaar/Documents/Development/colour-science/colour/colour/utilities/verbose.py:235: ColourUsageWarning: "colour.colorimetry.luminance_ASTMD153508" object has been renamed to "colour.colorimetry.luminance_ASTMD1535".
warn(*args, **kwargs)

Out[4]:
10.148809678226682

Note: Input Munsell value $V$ is in domain [0, 10], output luminance $Y$ is in domain [0, 100].

Using the colour.luminance wrapper definition:

In [5]:
colour.luminance(3.74629715382, method='ASTM D1535')

Out[5]:
0.42659001108260142
In [6]:
colour.luminance(3.74629715382, method='astm2008')

Out[6]:
0.42659001108260142

## CIE 1976 Method¶

The CIE $L^a^b^$* approximately uniform colourspace defined in 1976 computes the *luminance* $Y$ quantity as follows: [4]

$$$$Y=\begin{cases}Y_n*\biggl(\cfrac{L^*+16}{116}\biggr)^3 & for\ L^*>\kappa*\epsilon\\ Y_n*\biggl(\cfrac{L^*}{\kappa}\biggr) & for\ L^*<=\kappa*\epsilon \end{cases}$$$$

where $Y_n$ is the reference white luminance. with \begin{aligned} \epsilon&\ =\begin{cases}0.008856 & Actual\ CIE\ Standard\\ 216\ /\ 24389 & Intent\ of\ the\ CIE\ Standard \end{cases}\\ \kappa&\ =\begin{cases}903.3 & Actual\ CIE\ Standard\\ 24389\ /\ 27 & Intent\ of\ the\ CIE\ Standard \end{cases} \end{aligned}

The original $\epsilon$ and $\kappa$ constants values have been shown to exhibit discontinuity at the junction point of the two functions grafted together to create the Lightness $L^*$ function. [5]

Colour uses the rational values instead of the decimal values for these constants.

See Also: The CIE $L^*a^*b^*$ Colourspace notebook for in-depth information about the CIE $L^a^b^$* colourspace.

The colour.luminance_CIE1976 definition is used to compute Luminance $Y$:

In [7]:
colour.colorimetry.luminance_CIE1976(37.9856290977)

Out[7]:
10.080000000026304

Note: Input Lightness $L^*$ and and $Y_n$ are in domain [0, 100], output luminance $Y$ is in domain [0, 100].

Using the colour.luminance wrapper definition:

In [8]:
colour.luminance(37.9856290977, method='CIE 1976')

Out[8]:
10.080000000026304
In [9]:
colour.luminance(37.9856290977, method='cie1976')

Out[9]:
10.080000000026304

## Fairchild and Wyble (2010) Method¶

In [10]:
colour.colorimetry.luminance_Fairchild2010(24.902290269546651, 1.836)

Out[10]:
0.10079999999999999
In [11]:
colour.luminance(24.902290269546651, method='Fairchild 2010', epsilon=1.836)

Out[11]:
10.079999999999998

## Fairchild and Chen (2011) Method¶

In [12]:
colour.colorimetry.luminance_Fairchild2011(26.459509817572265, 0.710)

Out[12]:
0.10079999999999999
In [13]:
colour.luminance(26.459509817572265, method='Fairchild 2011', epsilon=0.710)

Out[13]:
10.079999999999998

## Bibliography¶

1. ^ CIE. (n.d.). 17-711 luminance (in a given direction, at a given point of a real or imaginary surface) [Lv; L]. Retrieved July 09, 2014, from http://eilv.cie.co.at/term/711
2. ^ Newhall, S. M., Nickerson, D., & Judd, D. B. (1943). Final report of the OSA subcommittee on the spacing of the munsell colors. JOSA, 33(7), 385. doi:10.1364/JOSA.33.000385
3. ^ ASTM International. (n.d.). ASTM D1535-08e1 Standard Practice for Specifying Color by the Munsell System. doi:10.1520/D1535-08E01
4. ^ Wyszecki, G., & Stiles, W. S. (2000). CIE 1976 (Luv)-Space and Color-Difference Formula. In Color Science: Concepts and Methods, Quantitative Data and Formulae* (p. 167). Wiley. ISBN:978-0471399186
5. ^ Lindbloom, B. (2003). A Continuity Study of the CIE L* Function. Retrieved February 24, 2014, from http://brucelindbloom.com/LContinuity.html