Illuminant Computation - Revision history

Patapom at 17:07, 20 November 2016

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Patapom at 17:03, 20 November 2016

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Patapom at 16:10, 20 November 2016

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Patapom at 16:08, 20 November 2016

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Patapom: Patapom moved page Colorimetry DIlluminant Computation to Illuminant Computation

2016-11-20T16:06:24Z

Patapom moved page Colorimetry DIlluminant Computation to Illuminant Computation

Patapom at 16:56, 29 December 2011

2011-12-29T16:56:14Z

Patapom: Created page with 'Source from http://en.wikipedia.org/wiki/Standard_illuminant#Illuminant_series_D [[Image:CIE_illuminants_D_and_blackbody_small.gif|right|frame|Relative spectral power distributi…'

2011-12-29T16:51:40Z

Created page with 'Source from http://en.wikipedia.org/wiki/Standard_illuminant#Illuminant_series_D [[Image:CIE_illuminants_D_and_blackbody_small.gif|right|frame|Relative spectral power distributi…'

New page

Source from http://en.wikipedia.org/wiki/Standard_illuminant#Illuminant_series_D

[[Image:CIE_illuminants_D_and_blackbody_small.gif|right|frame|Relative spectral power distribution of illuminant D and a black body of the same correlated color temperature (in red), normalized about 560nm.]]

Derived by Judd, MacAdam, and Wyszecki, the '''D''' series of illuminants are constructed to represent natural daylight. They are difficult to produce artificially, but are easy to characterize mathematically.

H. W. Budde of the National Research Council of Canada in Ottawa, H. R. Condit and F. Grum of the Eastman Kodak Company in Rochester, New York, and S. T. Henderson and D. Hodgkiss of Thorn Electrical Industries in Enfield had independently measured the spectral power distribution (SPD) of daylight from 330 to 700 nm, totaling among them 622 samples. Judd ''et al.'' analyzed these samples and found that the (x,y) chromaticity coordinates had a simple, quadratic relation:

:<math>y=2.870x - 3.000x^2 - 0.275</math>.

Simonds supervised the characteristic vector analysis of the SPDs. Application of his method revealed that the SPDs could be satisfactorily approximated by using the mean (S0) and first two characteristic vectors (S1 and S2):

:<math>S(\lambda) = S_0(\lambda) + M_1 S_1(\lambda) + M_2 S_2(\lambda)</math>

In simpler terms, the SPD of the studied daylight samples can be expressed as the linear combination of three, fixed SPDs. The first vector (S0) is the mean of all the SPD samples, which is the best reconstituted SPD that can be formed with only a fixed vector. The second vector (S1) corresponds to yellow–blue variation, accounting for changes in the correlated color temperature due to presence or absence of clouds or direct sunlight. The third vector (S2) corresponds to pink–green variation caused by the presence of water in the form of vapor and haze.

To construct a daylight simulator of a particular correlated color temperature one merely needs to know the coefficients M1 and M2 of the characteristic vectors S1 and S2.

[[Image:CIE_illuminants_D_components.png|right|thumb|250px|Characteristic vectors of illuminant D; component SPDs S0 (blue), S1 (green), S2 (red).]]

Expressing the chromaticities x and y as:

<math>x=\frac{X_0+M_1 X_1+M_2 X_2}{S_0+M_1 S_1 + M_2 S_2}</math>

<math>y=\frac{Y_0+M_1 Y_1+M_2 Y_2}{S_0+M_1 S_1 + M_2 S_2}</math>

and making use of known tristimulus values for the mean vectors, they were able to express M1 and M2 as follows:
[[Image:PlanckianLocusCloseUp.png|thumb|Kelly's figures depicted the lines of constant correlated color temperature on the CIE 1960 UCS, as shown here, as well as the familiar xy diagram.]]

<math>M_1=\frac{-1.3515-1.7703x+5.9114y}{0.0241+0.2562x-0.7341y}</math>

<math>M_2=\frac{0.0300-31.4424x+30.0717y}{0.0241+0.2562x-0.7341y}</math>

The only problem is that this left unsolved the computation of the coordinate <math>(x,y)</math> for a particular phase of daylight. Judd ''et al.'' simply tabulated the values of certain chromaticity coordinates, corresponding to commonly-used correlated color temperatures, such as 5500 K, 6500 K, and 7500 K. For other color temperatures, one could consult figures made by Kelly. This problem was addressed in the CIE report that formalized illuminant D, with an approximation of the x coordinate in terms of the reciprocal color temperature, valid from 4000 K to 25,000 K. The y coordinate trivially followed from Judd's quadratic relation.

Judd ''et al.'' then extended the reconstituted SPDs to 300–330 nm and 700–830 nm by using Moon's spectral absorbance data of the Earth's atmosphere.

The tabulated SPDs presented by the CIE today are derived by linear interpolatio] of the 10 nm data set down to 5 nm. The limited nature of the photometric data is not an impediment to the calculation of the CIEXYZ tristimulus values since the CIE standard colorimetric observer's color matching functions are only tabulated from 380 to 780 nm in increments of 5 nm.

Similar studies have been undertaken in other parts of the world, or repeating Judd ''et al.'''s analysis with modern computational methods. In several of these studies, the daylight locus is notably closer to the Planckian locus than in Judd ''et al.''<ref>Studies from the 1960s and 1970s include: {{cite journal|journal=[[JOSA]]|volume=56|issue=4|month=April|year=1966|title=Spectroradiometric and Colorimetric Characteristics of Daylight in the Southern Hemisphere: Pretoria, South Africa|author=G. T. Winch, M. C. Boshoff, C. J. Kok, and A. G. du Toit|pages=456–464| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-56-4-456|quote=The derived chromaticities were found to be much closer to the full radiator locus than those previously published, which had been obtained in the northern hemisphere.|doi=10.1364/JOSA.56.000456}} {{cite journal|title=Spectral Distribution and Color of Tropical Daylight|journal=[[JOSA]]|first=S.R.|last=Das|coauthors=Sastri, V.D.P.|volume=55|issue=3|month=March|year=1965|pages=319–323| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-55-3-319|doi=10.1364/JOSA.55.000319}} {{cite journal|title=Typical Spectral Distributions and Color for Tropical Daylight|journal=[[JOSA]]|first=V.D.P.|last=Sastri|coauthors=Das, S.R.|pages=391–398|volume=58|issue=3|month=March|year=1968| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-58-3-391|doi=10.1364/JOSA.58.000391}} {{cite journal|title=Locus of daylight chromaticities in relation to atmospheric conditions|last=Sastri|first=V.D.P.|volume=9|issue=1|year=1976|month=January 11|pages=L1–L3|journal=[[Journal of Physics D: Applied Physics]]|doi=10.1088/0022-3727/9/1/001}} {{cite journal|title=Spectral distribution of Australian daylight|last=Dixon|first=E.R.|journal=[[JOSA]]|volume=68|issue=4|month=April|year=1978|pages=437–450| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-68-4-437|doi=10.1364/JOSA.68.000437}} Analyses using the faster computation of the 1990s and 2000s include: {{cite journal|title=Testing Linear Models on Spectral Daylight Measurements|first=Javier|last=Hernández-Andrés|coauthors=Javier Romero, Antonio García-Beltrán, and Juan L. Nieves|journal=Applied Optics|volume=37|issue=6|pages=971–977|year=1998|month=February 20|url=http://www.opticsinfobase.org/abstract.cfm?URI=ao-37-6-971|doi=10.1364/AO.37.000971|pmid=18268673}} {{cite journal|title=Color and spectral analysis of daylight in southern Europe|volume=18|issue=6|month=June|year=2001|journal=[[JOSA A]]|pages=1325–1335|first=Javier|last=Hernández-Andrés|coauthors=Javier Romero, Juan L. Nieves, and Raymond L. Lee, Jr.|doi=10.1364/JOSAA.18.001325}} {{cite conference
|title=Group theoretical investigations of daylight spectra
|author=Thanh Hai Bui, Reiner Lenz, Tomas Landelius
|conference=CGIV (European Conference on Colour Graphics, Imaging and Vision)
|year=2004
|pages=437–442
|accessdate = 2008-05-13
|url=http://staffwww.itn.liu.se/~reile/csp-pages/publications/reprints/cgiv04-grouptheory.pdf}}
</ref>

; Computation:

The relative [[spectral power distribution]] (SPD) <math>S_D (\lambda)</math> of a D series illuminant can be derived from its [[chromaticity coordinate]]s in the [[CIE 1931 color space]], <math>(x_D,y_D)</math>:<ref name=bruce>The coefficients differ from those in the original paper due to the change in the constants in [[Planck's law]]. See [http://www.brucelindbloom.com/index.html?Eqn_DIlluminant.html Lindbloom] for the current version, and [[Planckian locus]] for details.</ref>

<math>x_D = \begin{cases}
0.244063 + 0.09911 \frac{10^3}{T} + 2.9678 \frac{10^6}{T^2} - 4.6070 \frac{10^9}{T^3} & 4000K \leq T \leq 7000K \\
0.237040 + 0.24748 \frac{10^3}{T} + 1.9018 \frac{10^6}{T^2} - 2.0064 \frac{10^9}{T^3} & 7000K < T \leq 25000K
\end{cases}</math>

<math>y_D = -3.000 x_D^2 + 2.870 x_D - 0.275</math>

[[Image:Daylight-locus-in-CIE-1960-UCS.png|right|thumb|250px|Daylight locus in the CIE 1960 UCS. The isotherms are perpendicular to the Planckian locus. The two sections of the daylight locus, from 4000–7000 K and 7000–25000 K, are color coded. Note that the two loci are separated by a fairly even distance, of around <math>\Delta_{uv}=0.003</math>.]]

where T is the illuminant's CCT. The chromaticity coordinates of the Illuminants D are said to form the ''CIE Daylight Locus''. The relative SPD is given by:

<math>S_D(\lambda)=S_0(\lambda)+M_1 S_1(\lambda)+M_2 S_2(\lambda)</math>

<math>M_1=(-1.3515-1.7703x_D+5.9114y_D)/M</math>

<math>M_2=(0.03000-31.4424x_D+30.0717y_D)/M</math>

<math>M=0.0241+0.2562x_D-0.7341y_D</math>

where <math>S_0(\lambda), S_1(\lambda), S_2(\lambda)</math> are the mean and first two [[eigenvector]] SPDs, depicted above.<ref name=bruce/> The characteristic vectors both have a zero at 560 nm, since all the relative SPDs have been normalized about this point.

The CCTs of the canonical illuminants, D50, D55, D65, and D75, differ slightly from what their names suggest. For example, D50 has a CCT of 5003 K ("horizon" light), while [[CIE Standard Illuminant D65|D65]] has a CCT of 6504 K (noon light). As explained in a previous section, this is because the value of the constants in Planck's law have been slightly changed since the definition of these canonical illuminants, whose SPDs are based on the original values in Planck's law.

@@ Line 72: / Line 72: @@
-== Custom Illuminant Computation ==
+== Custom Illuminant Computation by Spectral Integration ==
 By combining the spectral radiance given by [[Colorimetry#Black_Body|Planck's law]], and the spectral power distribution of the [[Colorimetry#Color_matching_functions|CIE Color Matching Functions]], it's relatively easy to obtain the XYZ values of any white point given its Correlated Color Temperature.
@@ Line 82: / Line 82: @@
 These 2 sources come from my personal library named [https://github.com/Patapom/GodComplex God Complex].
-As you can see in the picture below, I'm plotting the locii of various white points from 1500K to 12000K by 500K increments and the results match pretty well (red dots).
+As you can see in the picture below, I'm plotting the loci of various white points from 1500K to 12000K by 500K increments and the results match pretty well (red dots).
 [[Image:WhitePointsLocii.png]]

← Older revision		Revision as of 17:03, 20 November 2016
Line 73:		Line 73:

	== Custom Illuminant Computation ==		== Custom Illuminant Computation ==
		+
		+	By combining the spectral radiance given by [[Colorimetry#Black_Body\|Planck's law]], and the spectral power distribution of the [[Colorimetry#Color_matching_functions\|CIE Color Matching Functions]], it's relatively easy to obtain the XYZ values of any white point given its Correlated Color Temperature.
		+
		+	You can find an example code here:
		+	* [http://patapom.com/Temp/SourceCodes/IlluminantComputation/ComputeWhitePointChromaticities.cpp ComputeWhitePointChromaticities.cpp] <= Actual code
		+	* [http://patapom.com/Temp/SourceCodes/IlluminantComputation/ColorMatchingFunctions.cpp ColorMatchingFunctions.cpp] <= Giant array of spectral power distribution values
		+
		+	These 2 sources come from my personal library named [https://github.com/Patapom/GodComplex God Complex].
		+
		+	As you can see in the picture below, I'm plotting the locii of various white points from 1500K to 12000K by 500K increments and the results match pretty well (red dots).
		+
		+	[[Image:WhitePointsLocii.png]]

← Older revision		Revision as of 16:10, 20 November 2016
Line 70:		Line 70:

	The CCTs of the canonical illuminants, D<sub>50</sub>, D<sub>55</sub>, D<sub>65</sub>, and D<sub>75</sub>, differ slightly from what their names suggest. For example, D50 has a CCT of 5003 K ("horizon" light), while D65 has a CCT of 6504 K (noon light). As explained in a previous section, this is because the value of the constants in Planck's law have been slightly changed since the definition of these canonical illuminants, whose SPDs are based on the original values in Planck's law.		The CCTs of the canonical illuminants, D<sub>50</sub>, D<sub>55</sub>, D<sub>65</sub>, and D<sub>75</sub>, differ slightly from what their names suggest. For example, D50 has a CCT of 5003 K ("horizon" light), while D65 has a CCT of 6504 K (noon light). As explained in a previous section, this is because the value of the constants in Planck's law have been slightly changed since the definition of these canonical illuminants, whose SPDs are based on the original values in Planck's law.
		+
		+
		+	== Custom Illuminant Computation ==

← Older revision		Revision as of 16:08, 20 November 2016
Line 1:		Line 1:
		+
		+	== D-Illuminant Computation ==
		+
	Source from http://en.wikipedia.org/wiki/Standard_illuminant#Illuminant_series_D		Source from http://en.wikipedia.org/wiki/Standard_illuminant#Illuminant_series_D

@@ Line 1: / Line 1: @@
 Source from http://en.wikipedia.org/wiki/Standard_illuminant#Illuminant_series_D
 [[Image:CIE_illuminants_D_and_blackbody_small.gif|right|frame|Relative spectral power distribution of illuminant D and a black body of the same correlated color temperature (in red), normalized about 560nm.]]
@@ Line 26: / Line 27: @@
 and making use of known tristimulus values for the mean vectors, they were able to express M<sub>1</sub> and M<sub>2</sub> as follows:
-[[Image:PlanckianLocusCloseUp.png|thumb|Kelly's figures depicted the lines of constant correlated color temperature on the CIE 1960 UCS, as shown here, as well as the familiar xy diagram.]]
+[[Image:Planckian-Locus.png|thumb|Kelly's figures depicted the lines of constant correlated color temperature on the CIE 1960 UCS, as shown here, as well as the familiar xy diagram.]]
 <math>M_1=\frac{-1.3515-1.7703x+5.9114y}{0.0241+0.2562x-0.7341y}</math>
@@ Line 38: / Line 39: @@
 The tabulated SPDs presented by the CIE today are derived by linear interpolatio] of the 10&nbsp;nm data set down to 5&nbsp;nm. The limited nature of the photometric data is not an impediment to the calculation of the CIEXYZ tristimulus values since the CIE standard colorimetric observer's color matching functions are only tabulated from 380 to 780&nbsp;nm in increments of 5&nbsp;nm.
-Similar studies have been undertaken in other parts of the world, or repeating Judd ''et al.'''s analysis with modern computational methods.  In several of these studies, the daylight locus is notably closer to the Planckian locus than in Judd ''et al.''<ref>Studies from the 1960s and 1970s include:<br>{{cite journal|journal=[[JOSA]]|volume=56|issue=4|month=April|year=1966|title=Spectroradiometric and Colorimetric Characteristics of Daylight in the Southern Hemisphere: Pretoria, South Africa|author=G. T. Winch, M. C. Boshoff, C. J. Kok, and A. G. du Toit|pages=456–464| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-56-4-456|quote=The  derived chromaticities  were  found  to  be much  closer  to  the  full  radiator  locus  than  those  previously  published, which  had  been obtained  in  the  northern  hemisphere.|doi=10.1364/JOSA.56.000456}}<br>{{cite journal|title=Spectral Distribution and Color of Tropical Daylight|journal=[[JOSA]]|first=S.R.|last=Das|coauthors=Sastri, V.D.P.|volume=55|issue=3|month=March|year=1965|pages=319–323| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-55-3-319|doi=10.1364/JOSA.55.000319}}<br>{{cite journal|title=Typical Spectral Distributions and Color for Tropical Daylight|journal=[[JOSA]]|first=V.D.P.|last=Sastri|coauthors=Das, S.R.|pages=391–398|volume=58|issue=3|month=March|year=1968| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-58-3-391|doi=10.1364/JOSA.58.000391}}<br>{{cite journal|title=Locus of daylight chromaticities in relation to atmospheric conditions|last=Sastri|first=V.D.P.|volume=9|issue=1|year=1976|month=January 11|pages=L1–L3|journal=[[Journal of Physics D: Applied Physics]]|doi=10.1088/0022-3727/9/1/001}}<br>{{cite journal|title=Spectral distribution of Australian daylight|last=Dixon|first=E.R.|journal=[[JOSA]]|volume=68|issue=4|month=April|year=1978|pages=437–450| url=http://www.opticsinfobase.org/abstract.cfm?URI=josa-68-4-437|doi=10.1364/JOSA.68.000437}}<br>Analyses using the faster computation of the 1990s and 2000s include:<br>{{cite journal|title=Testing Linear Models on Spectral Daylight Measurements|first=Javier|last=Hernández-Andrés|coauthors=Javier Romero, Antonio García-Beltrán, and Juan L. Nieves|journal=Applied Optics|volume=37|issue=6|pages=971&ndash;977|year=1998|month=February 20|url=http://www.opticsinfobase.org/abstract.cfm?URI=ao-37-6-971|doi=10.1364/AO.37.000971|pmid=18268673}}<br>{{cite journal|title=Color and spectral analysis of daylight in southern Europe|volume=18|issue=6|month=June|year=2001|journal=[[JOSA A]]|pages=1325–1335|first=Javier|last=Hernández-Andrés|coauthors=Javier Romero, Juan L. Nieves, and Raymond L. Lee, Jr.|doi=10.1364/JOSAA.18.001325}}<br>{{cite conference
+Similar studies have been undertaken in other parts of the world, or repeating Judd et al.'s analysis with modern computational methods. In several of these studies, the daylight locus is notably closer to the Planckian locus than in Judd et al.
 ; Computation:
-The relative [[spectral power distribution]] (SPD) <math>S_D (\lambda)</math> of a D series illuminant can be derived from its [[chromaticity coordinate]]s in the [[CIE 1931 color space]], <math>(x_D,y_D)</math>:<ref name=bruce>The coefficients differ from those in the original paper due to the change in the constants in [[Planck's law]]. See [http://www.brucelindbloom.com/index.html?Eqn_DIlluminant.html Lindbloom] for the current version, and [[Planckian locus]] for details.</ref>
+The relative spectral power distribution (SPD) <math>S_D (\lambda)</math> of a D series illuminant can be derived from its chromaticity coordinates in the CIE 1931 color space, <math>(x_D,y_D)</math>:
 <math>x_D = \begin{cases}
@@ Line 71: / Line 64: @@
 <math>M=0.0241+0.2562x_D-0.7341y_D</math>
-where <math>S_0(\lambda), S_1(\lambda), S_2(\lambda)</math> are the mean and first two [[eigenvector]] SPDs, depicted above.<ref name=bruce/> The characteristic vectors both have a zero at 560&nbsp;nm, since all the relative SPDs have been normalized about this point.
+where <math>S_0(\lambda), S_1(\lambda), S_2(\lambda)</math> are the mean and first two eigenvector SPDs, depicted above. The characteristic vectors both have a zero at 560&nbsp;nm, since all the relative SPDs have been normalized about this point.
-The CCTs of the canonical illuminants, D<sub>50</sub>, D<sub>55</sub>, D<sub>65</sub>, and D<sub>75</sub>, differ slightly from what their names suggest. For example, D50 has a CCT of 5003&nbsp;K ("horizon" light), while [[CIE Standard Illuminant D65|D65]] has a CCT of 6504&nbsp;K (noon light). As explained in a previous section, this is because the value of the constants in Planck's law have been slightly changed since the definition of these canonical illuminants, whose SPDs are based on the original values in Planck's law.
+The CCTs of the canonical illuminants, D<sub>50</sub>, D<sub>55</sub>, D<sub>65</sub>, and D<sub>75</sub>, differ slightly from what their names suggest. For example, D50 has a CCT of 5003&nbsp;K ("horizon" light), while D65 has a CCT of 6504&nbsp;K (noon light). As explained in a previous section, this is because the value of the constants in Planck's law have been slightly changed since the definition of these canonical illuminants, whose SPDs are based on the original values in Planck's law.

← Older revision	Revision as of 16:06, 20 November 2016
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