The luminous efficiency function is the ratio of radiant flux weighted according to $V(\lambda)$ to the corresponding radiant flux. [1] It characterizes the average spectral sensitivity of human visual perception of brightness. Brightness is defined as the attribute of a visual perception according to which an area appears to emit, or reflect, more or less light. [2]
Photometry is the measurement of quantities referring to radiation as evaluated according to a given spectral luminous efficiency function, e.g. $V(\lambda)$ or $V^\prime(\lambda)$. [3]
Radiometry is defined as the measurement of the quantities associated with optical radiation. [4]
Photometric quantities are weighted accordingly to human visual system spectral sensitivity within the wavelength range 360-780 nanometres ($nm$) while radiometric quantities represent unweighted absolute power within the wavelength range 0.01-1000 micrometres ($\mu m$).
Given the following terms and units table: [5]
| Term | Symbol | Defining Equation | Units | Units Name | Notes |
|---|---|---|---|---|---|
| Frequency | $v$ | $s^{-1}$ | Hertz | ||
| Wavelength | $\lambda$ | $\lambda=\cfrac{c}{v}$ | $m$ | Metre | $c$: Velocity of radiant energy in vacuum |
| Wavenumber | $m$ | $m=\cfrac{1}{\lambda}$ | $m^{-1}$ | Metre | |
| Solid angle | $\omega$ | $\omega=\cfrac{S}{r^2}$ | $sr$ | Steradian | $S$: Portion of sphere surface $r$: Radius of sphere, also distance between (1) of $dA_1$ and (2) of $dA_2$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ \ \ \ \ \ \ \ }$$\phantom{\ \ \ }$ |
Radiometry uses the following quantities, terms and units: [5]
| Quantity / Term | Symbol | Defining Equation | Units | Units Name | Notes |
|---|---|---|---|---|---|
| Radiant energy | $Q_e$ | $J$ | Joule | ||
| Radiant flux (or power) | $P_e$ | $W$ | Watt ($J\cdot sr^{-1}$) |
$d^2P_e=L_e\cfrac{dA_1\cos\varepsilon_1dA_2\cos\varepsilon_2}{r^2}$ | |
| Radiant exitance | $M_e$ | $M_e=\cfrac{dP_e}{dA_1}$ | $W\cdot m^{-2}$ | Watt per square metre | $dA_1$: Surface element of source |
| Irradiance | $E_e$ | $E_e=\cfrac{dP_e}{dA_2}$ | $W\cdot m^{-2}$ | Watt per square metre | $dA_1$: Surface element of receiver |
| Radiant intensity | $I_e$ | $I_e=\cfrac{dP_e}{d\omega_1}$ | $W\cdot sr^{-1}$ | Watt per steradian | $d\omega_1$: Element of solid angle with apex (2) at surface of source |
| Radiance | $L_e$ | $L_e=\cfrac{d^2P_e}{dA_1\cos\varepsilon_1d\omega_1}$ $L_e=\cfrac{d^2E_e}{dA_2\cos\varepsilon_2d\omega_2}$ $L_e=\cfrac{d(E_e)_n}{d\omega_2}$ |
$W\cdot m^{-2} \cdot sr^{-1}$ | Watt per square metre per steradian | $\varepsilon_1$: Angle between direction (1)-(2) and normal $n_1$ of $dA_1$ $\varepsilon_2$: Angle between direction (1)-(2) and normal $n_2$ of $dA_2$ $d\omega_2$: Element of solid angle with apex (2) at surface of receiver $dA_1\cos\varepsilon_1$: $dA_1$ orthogonally projected on plane perpendicular to direction (1)-(2) $dA_2\cos\varepsilon_2$: $dA_2$ orthogonally projected on plane perpendicular to direction (1)-(2) $d(E_e)_n=\cfrac{dE_e}{dA_2\cos\varepsilon_2}$ $d\omega_1=\cfrac{dA_1\cos\varepsilon_1}{r^2}$ $d\omega_2=\cfrac{dA_2\cos\varepsilon_2}{r^2}$ |
Photometry uses the following quantities, terms and units: [5]
| Quantity / Term | Symbol | Defining Equation | Units | Units Name | Notes |
|---|---|---|---|---|---|
| Luminous energy | $Q_v$ | $lm\cdot s$ | Lumen second | ||
| Luminous flux (or power) | $F_v$ (or $P_v$) | $F_v=KP_e$ $F_v=K_m\int_\lambda P_{e,\lambda}V(\lambda)d\lambda$ |
$lm$ | Lumen ($cd\cdot sr$) |
$P_e$: Radiant flux ($W$) $K$: Luminous efficacy ($lm\cdot W^{-1}$) |
| Luminous exitance | $Mv$ | $M=\cfrac{dF_v}{dA_1}$ | $lx$ | Lumen per square metre (or lux) ($lm\cdot m^{-2}$) |
|
| Illuminance | $E_v$ | $E_v=\cfrac{dF_v}{dA_2}$ | $lx$ | Lumen per square metre (or lux) ($lm\cdot m^{-2}$) |
$dA_2$: Surface element of receiver |
| Luminous intensity | $I_v$ | $I_v=\cfrac{dF_v}{d\omega_I}$ | $cd$ | Candela ($lm\cdot sr^{-1}$) |
Luminous flux per unit solid angle $d\omega_I$: Element of solid angle with apex (1) at surface of source |
| Luminance | $L_v$ | $L_v=\cfrac{d^2F_v}{dA_1\cos\varepsilon_1d\omega_1}$ $L_v=\cfrac{dI_v}{dA_1\cos\varepsilon_1}$ $L_v=\cfrac{d^2E_v}{dA_2\cos\varepsilon_2d\omega_2}$ $L_v=\cfrac{dE_{v,n}}{d\omega_2}$ |
$cd\cdot m^{-2}$ | Candela per square metre (or nits) ($lm\cdot m^{-2}\cdot sr^{-1}$) |
$dA_1$: Surface element of source $\varepsilon_1$ Angle between direction (1)-(2) and normal $n_1$ of $dA_1$ $dA_1\cos\varepsilon_1$: $dA_1$ orthogonally projected on plane perpendicular to direction (1)-(2) $dE_{v, n}$: Illuminance on $dA_2$ normal to the direction (1)-(2) $d\omega_2$: Element of solid angle with apex (2) at surface of receiver |
| Luminous efficacy function | $K(\lambda)$ | $K(\lambda)=K_mV(\lambda)$ with $K_m$=683 |
$V(\lambda)$: Relative photopic luminous efficiency function |
The following figures illustrate the notes from the above tables: [5]
In [1]:
from IPython.core.display import Image
Image(filename='resources/images/Photometric_Quantities_001.png')
Out[1]:
In [2]:
from IPython.core.display import Image
Image(filename='resources/images/Solid_Angle_001.png')
Out[2]:
As per the above tables, a quantity in the photometric system has a similar quantity in the radiometric system:
The human eye has two different types of photoreceptor cells:
Colour provides the following photopic and scotopic luminous efficiency functions:
In [3]:
import colour
sorted(colour.LEFS.keys())
Out[3]:
Note:
'cie_2_1924','cie_10_1964'and'cie_1951'are convenient aliases for respectively'CIE 1924 Photopic Standard Observer','CIE 1964 Photopic 10 Degree Standard Observer'and'CIE 1951 Scotopic Standard Observer'.
The CIE 1924 Photopic Standard Observer luminous efficiency function $V(\lambda)$ was established by the CIE in 1924 for a 2° angular subtense viewing field. It was derived from several independent experiments whose results were not averaged but weight assembled by Gibson and Tyndall (1923) from very different sets of data: [8]
The angular subtense viewing field size varied among those experiments and the surround field was not always of constant luminance, if present at all. The assembly of data gathered from very different experiments has subsequently affected the effectiveness of the luminous efficiency function $V(\lambda)$. [9]
Photopic luminous flux $P_v$ and radiant flux $P_e$ are related by the following equation: [8]
$$ P_v=K_m\int_\lambda P_{e,\lambda}V(\lambda)d\lambda $$Accordingly, the photopic luminous efficacy $K$ defined as follows:
$$ K(\lambda)=\cfrac{P_{v,\lambda}}{P_{e,\lambda}} $$is equal to:
$$ K(\lambda)=K_mV(\lambda) $$with $K_m=683\ lm\cdot W^-1$ at $\lambda_d=555\ nm$.
The CIE 1924 Photopic Standard Observer luminous efficiency function $V(\lambda)$ has the following range and increments:
In [4]:
colour.PHOTOPIC_LEFS['CIE 1924 Photopic Standard Observer'].shape
Out[4]:
In [5]:
from colour.plotting import *
In [6]:
colour_style();
In [7]:
plot_multi_sds(
[colour.PHOTOPIC_LEFS['CIE 1924 Photopic Standard Observer']],
y_label='Luminous Efficiency');
The CIE 1951 Scotopic Standard Observer luminous efficiency function $V^\prime(\lambda)$ was established by the CIE in 1951 and is a relative function of wavelength with peak at $\lambda_d=507\ nm$.
Scotopic luminous flux $P_v^\prime$ and radiant flux $P_e$ are related by the following equation: [8]
$$ \begin{equation} P_v^\prime=K_m\int_\lambda P_{e,\lambda}^\prime V(\lambda)d\lambda \end{equation} $$Accordingly, the scotopic luminous efficacy $K^\prime$ defined as follows:
$$ \begin{equation} K^\prime(\lambda)=\cfrac{P_{v,\lambda}^\prime}{P_{e,\lambda}} \end{equation} $$is equal to:
$$ \begin{equation} K^\prime(\lambda)=K_m^\prime V(\lambda) \end{equation} $$with $K_m=1700\ lm\cdot W^-1$ at $\lambda_d=507\ nm$.
The CIE 1951 Scotopic Standard Observer luminous efficiency function $V^\prime(\lambda)$ has the following range and increments:
In [8]:
colour.SCOTOPIC_LEFS['CIE 1951 Scotopic Standard Observer'].shape
Out[8]:
In [9]:
plot_multi_sds(
[colour.SCOTOPIC_LEFS['CIE 1951 Scotopic Standard Observer']],
y_label='Luminous Efficiency');
Mesopic vision is a combination of photopic vision and scotopic vision and is defined for for $L_v\pm 0.2\ cd\cdot m^{-2}$. Although there is no standard for mesopic vision, a weighting function $V_m(\lambda)$ is defined as follows: [10]
$$ \begin{equation} V_m(\lambda)=(1-x)V^\prime(\lambda)+xV(\lambda) \end{equation} $$where $x$ is function of photopic luminance $L_p$ and can have value of:
| $L_p$ | Blue-heavy (MOVE) |
Blue-heavy (LRC) |
Red-heavy (MOVE) |
Red-heavy (LRC) |
|---|---|---|---|---|
| 0.01 | 0.13 | 0.04 | 0.00 | 0.01 |
| 0.1 | 0.42 | 0.28 | 0.34 | 0.11 |
| 1.0 | 0.70 | 1.00 | 0.68 | 1.00 |
| 10 | 0.98 | 1.00 | 0.98 | 1.00 |
The MOVE is an european research consortium and LRC stands for Lighting Research Center.
The colour.mesopic_weighting_function implements support for mesopic vision following the above equation:
In [10]:
colour.colorimetry.mesopic_weighting_function(wavelength=500, Lp=0.2)
Out[10]:
We can use different parameters for $x$:
In [11]:
colour.colorimetry.mesopic_weighting_function(500, 0.2, source='Red Heavy', method='LRC')
Out[11]:
Conveniently the colour.sd_mesopic_luminous_efficiency_function definition implements the creation of a mesopic luminous efficiency function based on colour.mesopic_weighting_function definition:
In [12]:
# Plotting a mesopic luminous efficiency function with photopic luminance of 0.2.
sd_mesopic_luminous_efficiency_function = colour.sd_mesopic_luminous_efficiency_function(0.2)
plot_multi_sds(
[sd_mesopic_luminous_efficiency_function,
colour.PHOTOPIC_LEFS['CIE 1924 Photopic Standard Observer'],
colour.SCOTOPIC_LEFS['CIE 1951 Scotopic Standard Observer']],
y_label='Luminous Efficiency');
Judd (1951) proposed a modified CIE 1924 Photopic Standard Observer luminous efficiency function $V(\lambda)$ because of effectiveness issues in the blue end of the visible spectrum leading to inadequacies in the CIE 1931 2° Standard Observer $\bar{x}(\lambda)$,$\bar{y}(\lambda)$,$\bar{z}(\lambda)$ colour matching functions. [11]
The Judd Modified CIE 1951 Photopic Standard Observer luminous efficiency function $V(\lambda)$ has the following range and increments:
In [13]:
colour.PHOTOPIC_LEFS['Judd Modified CIE 1951 Photopic Standard Observer'].shape
Out[13]:
In [14]:
plot_multi_sds(
[colour.PHOTOPIC_LEFS['Judd Modified CIE 1951 Photopic Standard Observer']],
y_label='Luminous Efficiency');
The modification Judd (1951) proposed is in the wavelength region below 460 nm, where the blue sensitivity was slightly increased:
In [15]:
plot_multi_sds(
[colour.PHOTOPIC_LEFS['CIE 1924 Photopic Standard Observer'],
colour.PHOTOPIC_LEFS['Judd Modified CIE 1951 Photopic Standard Observer']],
y_label='Luminous Efficiency');
Vos (1978) slightly refined Judd Modified CIE 1951 Photopic Standard Observer luminous efficiency function $V(\lambda)$. [11]
The Judd-Vos Modified CIE 1978 Photopic Standard Observer luminous efficiency function $V(\lambda)$ has the following range and increments:
In [16]:
colour.PHOTOPIC_LEFS['Judd-Vos Modified CIE 1978 Photopic Standard Observer'].shape
Out[16]:
In [17]:
plot_multi_sds(
[colour.PHOTOPIC_LEFS['Judd-Vos Modified CIE 1978 Photopic Standard Observer']],
y_label='Luminous Efficiency');
In [18]:
plot_multi_sds(
[colour.PHOTOPIC_LEFS['Judd Modified CIE 1951 Photopic Standard Observer'],
colour.PHOTOPIC_LEFS['Judd-Vos Modified CIE 1978 Photopic Standard Observer']],
y_label='Luminous Efficiency');
The CIE 1964 Photopic 10° Standard Observer luminous efficiency function $V_{10}(\lambda)$ is the the CIE 1964 10° Standard Observer $\bar{y}_{10}(\lambda)$ colour matching function and is considered to be the most representative for large angular subtense viewing fields. [11]
The CIE 1964 Photopic 10° Standard Observer luminous efficiency function $V_{10}(\lambda)$ has the following range and increments:
In [19]:
colour.PHOTOPIC_LEFS['CIE 1964 Photopic 10 Degree Standard Observer'].shape
Out[19]:
In [20]:
plot_multi_sds(
[colour.PHOTOPIC_LEFS['CIE 1964 Photopic 10 Degree Standard Observer']],
y_label='Luminous Efficiency');
Stockman, Jagle, Pirzer and Sharpe (2005) proposed a new luminous efficiency function noted $V^*(\lambda)$, improving on the CIE 1924 Photopic Standard Observer luminous efficiency function $V(\lambda)$ and based on the linear combination of the Stockman and Sharpe (2000) long-wave sensitive ($L-$), medium-wave sensitive ($M-$)cones spectral sensitivities functions. [9][12]
In 2008, the luminous efficiency function $V^*(\lambda)$ has been corrected to account for luminous efficiency dependence on chromatic adaptation. [13]
If the photopic sensitivity curve and the cone fundamentals are defined on energy basis, then luminous efficiency function $V_F(\lambda)$ given in terms of the energy-based cone fundamentals $\bar{l}(\lambda)$, $\bar{m}(\lambda)$ and renormalised to unity peak sensitivities can be expressed as follows for a 2° angular subtense viewing field: [14]
$$ \begin{equation} V_{Fe}(\lambda)=\cfrac{1.980647+\bar{l}(\lambda)+\bar{m}(\lambda)}{2.87090767} \end{equation} $$The CIE 2008 2° Physiologically Relevant Luminous Efficiency Function $V_F(\lambda)$ has respectively the following shape and increments:
In [21]:
colour.PHOTOPIC_LEFS['CIE 2008 2 Degree Physiologically Relevant LEF'].shape
Out[21]:
Similarly, for a 10° angular subtense viewing field, the luminous efficiency function $V_{F,10}(\lambda)$ given in terms of the energy-based cone fundamentals $\bar{l_{10e}}(\lambda)$, $\bar{m_{10e}}(\lambda)$ and renormalised to unity peak sensitivities can be expressed as follows: [15]
$$ \begin{equation} V_{F,10e}(\lambda)=\cfrac{1.981377+\bar{l_{10e}}(\lambda)+\bar{m_{10e}}(\lambda)}{2.85979294} \end{equation} $$The CIE 2008 10° Physiologically Relevant Luminous Efficiency Function $V_{F,10e}(\lambda)$ has respectively the following shape and increments:
In [22]:
colour.PHOTOPIC_LEFS['CIE 2008 10 Degree Physiologically Relevant LEF'].shape
Out[22]:
In [23]:
plot_multi_sds(
[colour.PHOTOPIC_LEFS['CIE 2008 2 Degree Physiologically Relevant LEF'],
colour.PHOTOPIC_LEFS['CIE 2008 10 Degree Physiologically Relevant LEF']],
y_label='Luminous Efficiency');