Appendix B: Calculating color rendering metrics | Light Sources and Color | Lighting Answers

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Volume 8 Issue 1	October 2004

color rendering index (CRI) - A rating index commonly used to represent how well a light source renders the colors of objects that it illuminates. For a CRI value of 100, the maximum value, the colors of objects can be expected to be seen as they would appear under an incandescent or daylight spectrum of the same correlated color temperature (CCT). Sources with CRI values less than 50 are generally regarded as rendering colors poorly, that is, colors may appear unnatural. correlated color temperature (CCT) - A specification for white light sources used to describe the dominant color tone along the dimension from warm (yellows and reds) to cool (blue). Lamps with a CCT rating below 3200 K are usually considered warm sources, whereas those with a CCT above 4000 K usually considered cool in appearance. Temperatures in between are considered neutral in appearance. Technically, CCT extends the practice of using temperature, in kelvins (K), for specifying the spectrum of light sources other than blackbody radiators. Incandescent lamps and daylight closely approximate the spectra of black body radiators at different temperatures and can be designated by the corresponding temperature of a blackbody radiator. The spectra of fluorescent and LED sources, however, differ substantially from black body radiators yet they can have a color appearance similar to a blackbody radiator of a particular temperature as given by CCT. efficacy - The ratio of the light output of a lamp (lumens) to its active power (watts), expressed as lumens per watt. spectral power distribution (SPD) - A representation of the radiant power emitted by a light source as a function of wavelength. blackbody radiator - A temperature radiator of uniform temperature whose radiant output in all parts of the spectrum is the maximum obtainable from any temperature radiator at the same temperature. Such a radiator is called a blackbody because it absorbs all the radiant energy that falls upon it. All other temperature radiators can be classed as non-blackbodies. Non-blackbodies radiate less in some or all wavelength intervals than a blackbody of the same size and the same temperature. chromaticity - The dominant or complementary wavelength and purity aspects of the color taken together, or of the aspects specified by the chromaticity coordinates of the color taken together. It describes the properties of light related to hue and saturation, but not luminance (brightness). color appearance - The resultant color perception that includes the effects of spectrum, background contrast, chromatic adaptation, color constancy, brightness, size and saturation. color consistency - The measure of how close in color appearance random samples of a lamp or source tend to be. color matching - The action of making a color appear the same as a given color. Often used as a method of evaluating the ability of a light source to render colors faithfully. color stability - The ability of a lamp or light source to maintain its color rendering and color appearance properties over its life. The color properties of some discharge light sources may tend to shift over the life of the lamp. full-spectrum index (FSI) - A mathematical measure of how much a light source's spectrum deviates from an equal energy spectrum, based on the slope of its cumulative spectrum. full-spectrum color index (FSCI) - A mathematical transformation of full-spectrum index into a zero to 100 scale, where the resulting values are directly comparable to color rendering index. An equal energy spectrum is defined as having an FSCI value of 100, a “standard warm white” fluorescent lamp has an FSCI value of 50, and a monochromatic light source (e.g., low pressure sodium) has an FSCI value of 0. gamut area - A measure of color rendering based upon volume in color space. It is the range of colors achievable on a given color reproduction medium (or present in an image on that medium) under a given set of viewing conditions. hue - The attribute of a light source or illuminated object that determines whether it is red, yellow, green, blue, or the like. isotemperature - A set of coordinates within which all points have the same temperature. In a color space diagram, isotemperature lines represent lights with identical correlated color temperatures. metamers - Lights of the same color but of different spectral power distribution. photopic - Vision mediated essentially or exclusively by the cones. It is generally associated with adaptation to a luminance of at least 3.4 cd/m². primary - Any one of three lights in terms of which a color is specified by giving the amount of each required to match it by additive combination.

Appendix B: MATLAB script for calculating measures of light source color: CCT, CRI, GA, and FSI

% Second, calculate Correlated Color Temperature (CCT), Tc.

load ('isoTempLinesNewestFine.mat', 'T', 'ut', 'vt', 'tt');

% Find adjacent lines to (us, vs)

n = length (T);

index = 0;

d1 = ((v-vt(1)) - tt(1)*(u-ut(1)))/sqrt(1+tt(1)*tt(1));

for i=2:n

d2 = ((v-vt(i)) - tt(i)*(u-ut(i)))/sqrt(1+tt(i)*tt(i));

if (d1/d2 < 0)

index = i;

break;

else

d1 = d2;

end

end

if index == 0

Tc = -1; % Not able to calculate CCT, u, v coordinates outside range.

return

end

% Calculate CCT by interpolation between isotemperature lines
Tc = 1/(1/T(index-1)+d1/(d1-d2)*(1/T(index)-1/T(index-1)));

% Third, calculate the Color Rendering Indices (CRI and its 14 indices)

% Calculate Reference Source Spectrum, spdref.

if (Tc < 5000)

c1 = 3.7418e-16;

c2 = 1.4388e-2;

spdref = c1 * (1e-9*wavelength_spd).^-5 ./ (exp(c2./(Tc.* 1e-9*wavelength_spd)) - 1);

else

if (Tc <= 25000)

load('CIEDaySn','wavelength','S0','S1','S2');

if (Tc <= 7000)

xd = -4.6070e9 / Tc.^3 + 2.9678e6 / Tc.^2 + 0.09911e3 / Tc + 0.244063;

else

xd = -2.0064e9 / Tc.^3 + 1.9018e6 / Tc.^2 + 0.24748e3 / Tc + 0.237040;

end

yd = -3.000*xd*xd + 2.870*xd - 0.275;

M1 = (-1.3515 - 1.7703*xd + 5.9114*yd) / (0.0241 + 0.2562*xd - 0.7341*yd);

M2 = (0.0300 - 31.4424*xd + 30.0717*yd) / (0.0241 + 0.2562*xd - 0.7341*yd);

spdref = S0 + M1*S1 + M2*S2;

spdref = interp1(wavelength,spdref,wavelength_spd);

spdref(isnan(spdref)) = 0.0;

else

R = -1;

return

else

end

% Load data for the spectral reflectance data of 14 color samples
TCS = load ('Tcs.txt');
TCS = TCS/1000;

% Interpolate TCS values from 5 nm to spd nm increments
TCS_1 = zeros (14,length(wavelength_spd));
wavelength_5 = 380:5:750;

for i = 1:14

TCS_1(i,:) = interp1(wavelength_5,TCS(i,:),wavelength_spd');

TCS_1(i,isnan(TCS_1(i,:))) = 0.0; % remove NaN from vector.

end

% Calculate u, v chromaticity coordinates of samples under test illuminant, uk, vk and
% reference illuminant, ur, vr.
uki = zeros(1,14);
vki = zeros(1,14);
uri = zeros(1,14);
vri = zeros(1,14);
Yknormal = 100 / Y;
Yk = Y*Yknormal;
uk = 4*X/(X+15*Y+3*Z);
vk = 6*Y/(X+15*Y+3*Z);
X = sum(spdref .* xbar);
Y = sum(spdref .* ybar);
Z = sum(spdref .* zbar);
Yrnormal = 100 / Y;
Yr = Y*Yrnormal;
ur = 4*X/(X+15*Y+3*Z);
vr = 6*Y/(X+15*Y+3*Z);

for i = 1:14

X = sum(spd .* TCS_1(i,:)' .* xbar);

Y = sum(spd .* TCS_1(i,:)' .* ybar);

Z = sum(spd .* TCS_1(i,:)' .* zbar);

Yki(i) = Y*Yknormal;

uki(i) = 4*X/(X+15*Y+3*Z);

vki(i) = 6*Y/(X+15*Y+3*Z);

X = sum(spdref .* TCS_1(i,:)' .* xbar);

Y = sum(spdref .* TCS_1(i,:)' .* ybar);

Z = sum(spdref .* TCS_1(i,:)' .* zbar);

Yri(i) = Y*Yrnormal;

uri(i) = 4*X/(X+15*Y+3*Z);

vri(i) = 6*Y/(X+15*Y+3*Z);

end

% Check tolerance for reference illuminant
DC = sqrt((uk-ur).^2 + (vk-vr).^2);
if DC>0.0054
return
end

% Apply adaptive (perceived) color shift.
ck = (4 - uk - 10*vk) / vk;
dk = (1.708*vk + 0.404 - 1.481*uk) / vk;
cr = (4 - ur - 10*vr) / vr;
dr = (1.708*vr + 0.404 - 1.481*ur) / vr;

for i = 1:14

cki = (4 - uki(i) - 10*vki(i)) / vki(i);

dki = (1.708*vki(i) + 0.404 - 1.481*uki(i)) / vki(i);

ukip(i) = (10.872 + 0.404*cr/ck*cki - 4*dr/dk*dki) / (16.518 + 1.481*cr/ck*cki - dr/dk*dki);

vkip(i) = 5.520 / (16.518 + 1.481*cr/ck*cki - dr/dk*dki);

end

% Transformation into 1964 Uniform space coordinates.

for i = 1:14

Wstarr(i) = 25*Yri(i).^.333333 - 17;

Ustarr(i) = 13*Wstarr(i)*(uri(i) - ur);

Vstarr(i) = 13*Wstarr(i)*(vri(i) - vr);

Wstark(i) = 25*Yki(i).^.333333 - 17;

Ustark(i) = 13*Wstark(i)*(ukip(i) - ur); % after applying the adaptive color shift, u'k = ur

Vstark(i) = 13*Wstark(i)*(vkip(i) - vr); % after applying the adaptive color shift, v'k = vr

end

% Determination of resultant color shift, delta E.
deltaE = zeros(1,14);

for i = 1:14

deltaE(i) = sqrt((Ustarr(i) - Ustark(i)).^2 + (Vstarr(i) - Vstark(i)).^2 + (Wstarr(i) - Wstark(i)).^2);

Ra14(i) = 100 - 4.6*deltaE(i);

end

Ra = sum(Ra14(1:8))/8;

% fourth, calculate the gamut area formed by the 8 CIE standard color samples

ukii=[uki(:,1:8),uki(1)];
vkii=1.5*[vki(:,1:8),vki(1)];
Ga=polyarea(ukii,vkii);
% Normalize gamut area to equal energy source
Ga=Ga/0.00728468*100;

% fifth, calculate the FSI (full spectrum index)

% Calculates the Full-spectrum Index

% FSI = fsi(spd,startw,endw,incrementw)
% spd is a single column or row vector and startw, endw and incrementw specify the
% starting, ending and increment of wavelength in nm.

% FSI = fsi(spd)
% spd is a two column matrix with wavelength values in column 1 and spd values in
% column 2. Wavelength values are assumed to be in units of nm.

if length(varargin)==0

[rows columns] = size(spd);

if columns > 2

error('Not column oriented data. Try transposing spd');

end

wavelength_spd = spd(:,1);

spd = spd(:,2);

else

startw = varargin{1}

endw = varargin{2}

incrementw = varargin{3}

wavelength_spd = (startw:incrementw:endw)';

[rows columns] = size(spd);

if columns > 1

error('Detected multiple columns of data. Try transposing spd');

end

end

% Interpolate to wavelength interval of 1nm from 380nm to 730nm
numWave = 351;
t=(380:1:730)';
spd=interp1(wavelength_spd,spd,t,'spline');
spd(isnan(spd)) = 0.0;
spd = spd/sum(spd); % Normalize the relative spd so that the total power equals 1
%Equal energy cumulative spd
EEcum=(1/numWave:1/numWave:1)';
%Calculate FSI

for j=1:numWave

cum = cumsum(spd); % A MatLab function for cumulative sums

sqrDiff = (cum-EEcum).^2;

sumSqrDiff(j)=sum(sqrDiff);

spd=circshift(spd,1);

end

FSI=mean(sumSqrDiff);

% Note: To make FSI more directly comparable to CRI, FSI values in Figure 14 have been converted to a 0-100 scale, with an equal energy spectrum defined as having an FSI value of 100, and all practical light sources having FSI values lower than 100; a monochromatic light source (e.g., low pressure sodium) has a value of 0.