characteristic vector

Standard illuminant

A standard illuminant is a profile or spectrum of visible light which is published in order to allow images or colors recorded under different lighting to be compared.

CIE illuminants

The International Commission on Illumination (usually abbreviated CIE for its French name) is the body responsible for publishing all of the well-known standard illuminants. Each of these is known by a letter or by a letter-number combination.

Illuminants A, B, and C were introduced in 1931, with the intention of respectively representing average incandescent light, direct sunlight, and average daylight. Illuminants D represent phases of daylight, Illuminant E is the equal-energy illuminant, while Illuminants F represent fluorescent lamps of various composition.

There are instructions on how to experimentally produce light sources ("standard sources") corresponding to the older illuminants. For the relatively newer ones (such as series D), experimenters are left to measure to profiles of their sources and compare them to the published spectra:

Nevertheless, they do provide a measure, called the Metamerism Index, to assess the quality of daylight simulators. The Metamerism Index tests how well five sets of metameric samples match under the test and reference illuminant. In a manner similar to the Color Rendering Index, the average difference between the metamers is calculated.

Illuminant A

The CIE defines illuminant A in these terms:

The spectral radiance of a black body follows Planck's law:

M_{e,lambda}(lambda,T)=frac{c_1 lambda^{-5}}{expleft(frac{c_2}{lambda T}right)-1}

At the time of standardizing illuminant A, both c_1=2pi cdot h cdot c^2 (which does not affect the relative SPD) and c_2=h cdot c/k were different. In 1968, the estimate of c2 was revised from 0.001438 m·K to 0.014388 m·K (and before that, it was 0.001435 m·K when illuminant A was standardized). This difference shifted the Planckian locus, changing the color temperature of the illuminant from its nominal 2848 K to 2856 K:

T_{new}=T_{old} times frac{1.4388}{1.435} = 2848 text{K} times 1.002648 = 2855.54 text{K}

In order to avoid further possible changes in the color temperature, the CIE now specifies the SPD directly, based on the original (1931) value of c2:

S_{A}(lambda)=100 left(frac{560}{lambda}right)^5 frac{exp frac{1.435 times 10^7}{2848 times 560}-1}{expfrac{1.435 times 10^7}{2848 lambda}-1}

The coefficients have been selected to achieve a peak SPD of 100 at 560 nm. The tristimulus values are (X,Y,Z) = (109.85,100.00,35.58), and the chromaticity coordinates using the standard observer are (x,y)=(0.44758, 0.40745).

Illuminants B and C

Illuminants B and C are daylight simulators. They are derived from Illuminant A by using a liquid filters. B served as a representative of noon sunlight, with a correlated color temperature (CCT) of 4874 K, while C represented average day light with a CCT of 6774 K. They are poor approximations of any common light source and deprecated in favor of the D series:

The liquid filters, designed by Raymond Davis, Jr. and Kasson S. Gibson in 1931, have a relatively high absorbance at the red end of the spectrum, effectively increasing the CCT of the gas lamp to daylight levels. This is similar in function to a CTO color gel that photographers and cinematographers use today, albeit much less convenient.

Each filter uses a pair of solutions, comprising specific amounts of distilled water, copper sulfate, mannite, pyridine, sulfuric acid, cobalt and ammonium sulfate. The solutions are separated by a sheet of uncolored glass. The amounts of the ingredients are carefully chosen so that their combination yields a color temperature conversion filter; that is, the filtered light is still white.

Illuminant series D

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 Ottowa, 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 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:

y=2.870x - 3.000x^2 - 0.275.

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):

S(lambda) = S_0(lambda) + M_1 S_1(lambda) + M_2 S_2(lambda)

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.

Expressing the chromaticities x and y as:

x=frac{X_0+M_1 X_1+M_2 X_2}{S_0+M_1 S_1 + M_2 S_2}

y=frac{Y_0+M_1 Y_1+M_2 Y_2}{S_0+M_1 S_1 + M_2 S_2}

and making use of known tristimulus values for the mean vectors, they were able to express M1 and M2 as follows:



The only problem is that this left unsolved the computation of the coordinate (x,y) 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 interpolation of the 10nm data set down to 5nm. 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, 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) S_D (lambda) of a D series illuminant can be derived from its chromaticity coordinates in the CIE 1931 color space, (x_D,y_D):

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}

y_D = -3.000 x_D^2 + 2.870 x_D - 0.275

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:

S_D(lambda)=S_0(lambda)+M_1 S_1(lambda)+M_2 S_2(lambda)




where S_0(lambda), S_1(lambda), S_2(lambda) are the mean and first two eigenvector SPDs, depicted above. 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 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.

Illuminant E

Illuminant E is an equal-energy radiator; it has a constant SPD inside the visible spectrum. It is useful as a theoretical reference; an illuminant that gives equal weight to all wavelengths, presenting an even color. It also has equal CIE XYZ tristimulus values, thus its chromaticity coordinates are (x,y)=(1/3,1/3). This is by design; the XYZ color matching functions are normalized such that their integrals over the visible spectrum are the same.

Illuminant E is not a black body, so it does not have a color temperature, but it can be approximated by a D series illuminant with a CCT of 5455 K. (Of the canonical illuminants, D55 is the closest.) Manufacturers sometimes compare light sources against Illuminant E to calculate the excitation purity.

Illuminant series F

The F series of illuminants represent various types of fluorescent lighting.

F1–F6 "standard" fluorescent lamps consist of two semi-broadband emissions of antimony and manganese activations in calcium halophosphate phosphor. F4 is of particular interest since it was used for calibrating the CIE Color Rendering Index (the CRI formula was chosen such that F4 would have a CRI of 51). F7–F9 are "broadband" (full-spectrum light) fluorescent lamps with multiple phosphors, and higher CRIs. Finally, F10–F12 are narrow triband illuminants consisting of three "narrowband" emissions (caused by ternary compositions of rare-earth phosphors) in the R,G,B regions of the visible spectrum. The phosphor weights can be tuned to achieve the desired CCT.

The spectra of these illuminants are published in Publication 15:2004.

White point

The spectrum of a standard illuminant, like any other profile of light, can be converted into tristimulus values. The set of three tristimulus coordinates of an illuminant is called a white point. If the profile is normalised, then the white point can equivalently be expressed as a pair of chromaticity coordinates.

If an image is recorded in tristimulus coordinates (or in values which can be converted to and from them), then the white point of the illuminant used gives the maximum value of the tristimulus coordinates that will be recorded at any point in the image, in the absence of fluorescence. It is called the white point of the image.

The process of calculating the white point discards a great deal of information about the profile of the illuminant, and so although it is true that for every illuminant the exact white point can be calculated, it is not the case that knowing the white point of an image alone tells you a great deal about the illuminant that was used to record it.

White points of standard illuminants

A list of standardized illuminants, their CIE chromaticity coordinates (x,y) of a perfect reflecting (or transmitting) diffuser, and their correlated color temperatures (CCTs) are given below. The CIE chromaticity coordinates are given for both the 2 degree field of view (1931) and the 10 degree field of view (1964). The color swatches represent the hue of each white point, calculated with luminance Y=0.54 and the standard observer, assuming correct sRGB display calibration.

White points
Name CIE 1931 CIE 1964 CCT (K) Hue Note
x2 y2 x10 y10
A 0.44757 0.40745 0.45117 0.40594 2856 Incandescent / Tungsten
B 0.34842 0.35161 0.3498 0.3527 4874 {obsolete} Direct sunlight at noon
C 0.31006 0.31616 0.31039 0.31905 6774 {obsolete} Average / North sky Daylight
D50 0.34567 0.35850 0.34773 0.35952 5003 Horizon Light. ICC profile PCS
D55 0.33242 0.34743 0.33411 0.34877 5503 Mid-morning / Mid-afternoon Daylight
D65 0.31271 0.32902 0.31382 0.33100 6504 Noon Daylight: Television, sRGB color space
D75 0.29902 0.31485 0.29968 0.31740 7504
E 1/3 1/3 1/3 1/3 5454 Equal energy
F1 0.31310 0.33727 0.31811 0.33559 6430 Daylight Fluorescent
F2 0.37208 0.37529 0.37925 0.36733 4230 Cool White Fluorescent
F3 0.40910 0.39430 0.41761 0.38324 3450 White Fluorescent
F4 0.44018 0.40329 0.44920 0.39074 2940 Warm White Fluorescent
F5 0.31379 0.34531 0.31975 0.34246 6350 Daylight Fluorescent
F6 0.37790 0.38835 0.38660 0.37847 4150 Lite White Fluorescent
F7 0.31292 0.32933 0.31569 0.32960 6500 D65 simulator, Daylight simulator
F8 0.34588 0.35875 0.34902 0.35939 5000 D50 simulator, Sylvania F40 Design 50
F9 0.37417 0.37281 0.37829 0.37045 4150 Cool White Deluxe Fluorescent
F10 0.34609 0.35986 0.35090 0.35444 5000 Philips TL85, Ultralume 50
F11 0.38052 0.37713 0.38541 0.37123 4000 Philips TL84, Ultralume 40
F12 0.43695 0.40441 0.44256 0.39717 3000 Philips TL83, Ultralume 30


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