2007.11.27 CS248-17 Color Theory

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Color Theory

Kurt Akeley

CS248 Lecture 17

27 November 2007

http://graphics.stanford.edu/courses/cs248-07/

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CS248 Lecture 17 Kurt Akeley, Fall 2007

Introduction

Difficulties:

Im learning this too

Terminology is used carelessly

There are lots of standardsCIE 1931, CIE 1964, Vos 1976, SMJ 1993,

Caveats:

Not comprehensive

Lots of CIE content, nothing on HSV or videoUnits may be off (or missing all together)

Derived plots may differ slightly from standards

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Human photoreceptors

www.stat.auckland.ac.nz/~ihaka/787/lectures-vision.pdf

Monochromaticscotopic vision

(low light levels)Chromaticphotopic vision

(high light levels)

There is a view direction(despite what I told you).

It doesnt affect theprojection, but it does

affect the nature of thereceptor array.

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Humans are trichromats

Spectral sensitivities of human conesStockman, MacLeod, and Johnson

Journal of the Optical Society of AmericaVolume 10, Number 12, December 1993

Table 8

Monochromaticstimulus

Determined by chemistryand retinal coverage

LM

S

Monochromat: Horshoe crabsDichromat: dogs

Trichromat: people, most camerasTetrachromat: fancy camerasPentachromat: Mallard ducks

[L M S] = EvalSmjSRF(lambda);[0.0197 0.0372 0.0104] =EvalSmjSRF(470);

All three contributions arenon-zero (the plots overlap)!

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Luminosity function

Sum of L, M, and Sresponses for photopic

vision ?(FvD pp. 576)

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Comparison of luminosity functions

Derived from SMJspectral-response data by

simple addition

From Wikipedia

Yes, the units are

different. We areinterested in the shapes.

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Luminous efficacy (and efficiency)

How much of the

radiated energy(efficacy/efficiency)or wall-socketenergy (overallefficacy/efficiency)of a light source isusable for vision?

Integrate the product ofthe luminosityfunction with thespectrum of the lightsource.

Source: Wikipedia

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Metamers

Response is linear

So response to an energy spectrum is the sum of the responses tothe individual spectral components

Same integration as for luminous efficiency, but doneseparately for L, M, and S, rather than for their sum

Many different spectra will produce the same L,M,S responseThese spectra are metamers

Metamers are form of aliasing : undersampling (only three conetypes) causes different signals to appear equivalent

This aliasing is very convenient

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CIE 1931 RGB primaries

B=435 nm

G=546 nm

R =700 nm[L M S] = EvalSmjSRF(700)*Sr +EvalSmjSRF(546)*Sg +EvalSmjSRF(435)*Sb);

M = [ EvalSmjSRF(700)EvalSmjSRF(546)EvalSmjSRF(435) ];

R = 1;G = 1;B = 1;

[L M S] = [R G B] * M;

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RGB color-matching function

M = [ EvalSmjSRF(700)EvalSmjSRF(546)EvalSmjSRF(435) ];

R = 1;G = 1;B = 1;

[L M S] = [R G B] * M;

[R G B] = [L M S] * Minv;

CMF(lambda) = EvalSmjSRF(lambda) * Minv;

What proportion of the R, G, and B primaries is required to matcha spectral color?

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Primary amplitudesare adjusted to give

equal areas under allthree color-matching

curves

Oops, what is anegative primary

contribution?

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W = DomainSmjSRF(1);len = length(W);% compute M assuming equal-intensity primaries

M = [ EvalSmjSRF(Rnm) * 1EvalSmjSRF(Gnm) * 1EvalSmjSRF(Bnm) * 1];

Minv = inv(M);% integrate the color-matching functionCMFsum = [0 0 0];for n = 1 : len

CMFsum = CMFsum + EvalSmjSRF(W(n)) * Minv;end% recompute M such that the color-matching curves have equal integrals% this amounts to scaling the primary intensities by Rs, Gs, and BsRs = CMFsum(1) / CMFsum(3);Gs = CMFsum(2) / CMFsum(3);Bs = 1;

M = [ EvalSmjSRF(Rnm) * Rs

EvalSmjSRF(Gnm) * GsEvalSmjSRF(Bnm) * Bs]; Minv = inv(M);

% compute the color-matching functionCMF = zeros(len, 3);for n = 1 : len

CMF(n,:) = EvalSmjSRF(W(n)) * Minv;end

[390 391 392 729 730]

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Negative primaries

Cant add a negative primary amount

So a negative match adds the (opposite of) the negative amount tothe source being matched, rather than to the matching light

Sensible but awkward

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CIE 1931 XYZ primaries

Derived by linear transformation from the RGB primaries:

The matrix coefficients were (obviously) chosen carefully

Well see some of the desirable properties in later slides

The XYZ primaries are imaginaryThey do not correspond directly to amounts of light

But they are very useful

[ ] [ ]

0.49 0.17697 0.001

0.31 0.81240 0.010.17697

0.20 0.01063 0.99

X Y Z R G B

=

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CIE 1931 XYZ color-matching function

Intended property:Y = luminosity

Intended property:non-negative

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A special metamer

400 nm 700 nm

Energy density

wd

e1

e2

Dominant wavelength = wd

Excitation purity = (e 2-e 1) / (e 2+e 1)

Luminance is related to the integration of thespectrum (as we saw before)

These determinechromaticity

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XYZ chromaticity

Rescale X, Y, and Z to remove luminance, leaving chromaticity:

Because the sum of the chromaticity values x, y, and z is always1.0, a plot of any two of them loses no information

Such a plot is a chromaticity plot

X Y Z x y z

X Y Z X Y Z X Y Z = = =

+ + + + + +

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xz chromaticity

Youve never seen thisplot, but it is perfectly

valid. The standardplots the x and y

chromaticity values.

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CIE 1931 xy chromaticity (derived from SMJ data)

Intended property:non-negative

Intended property:white point at(1/3, 1/3, 1/3)

Intended property:fitted to edge of right

triangle

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CIE 1931 rg chromaticity (derived from SMJ data)


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rb chromaticity (derived from SMJ data)


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Gamut

Gamut is the chromaticities that can be generated by a set ofprimaries

Because everything weve done is linear, interpolation betweenchromaticities on a chromaticity plot is also linear

Thus the gamut is the convex hull of the primary chromaticities

What is the gamut of the CIE 1931 primaries?

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CIE 1931 RGB gamut

R = 700 nm

G = 546 nm

B = 438 nm

All dominant wavelengthscan be reproduced, butmany cannot reach full

saturation.

No finite set of primariescan reproduce the entire

gamut. But more primariesdo a better job.

So the colors on the xychromaticity diagram cannot

be correct, because nodisplay can accurately

reproduce them!

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MATLAB code

RGBr = EvalSmjSRF(Rnm) * Minv;RGBg = EvalSmjSRF(Gnm) * Minv;RGBb = EvalSmjSRF(Bnm) * Minv;XYZr = RGBr * CieM;XYZg = RGBg * CieM;

XYZb = RGBb * CieM;xyzr = XYZr / (XYZr(1) + XYZr(2) + XYZr(3));xyzg = XYZg / (XYZg(1) + XYZg(2) + XYZg(3));xyzb = XYZb / (XYZb(1) + XYZb(2) + XYZb(3));

plot([xyzr(1) xyzg(1) xyzb(1) xyzr(1)], ...[xyzr(2) xyzg(2) xyzb(2) xyzr(2)], ...

c. -', 'linewidth', lw);

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Short-persistence phosphor CRT gamut

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Long-persistence phosphor CRT gamut

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LCD projector gamut (hypothetical)

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MATLAB and dichroic filter source

RGBr = [0 0 0];RGBg = [0 0 0];RGBb = [0 0 0];for n = 1 : len

TF = EvalDichroicRgbTF(W(n));RGBr = RGBr + (TF(1) .* CMF(n,:));RGBg = RGBg + (TF(2) .* CMF(n,:));RGBb = RGBb + (TF(3) .* CMF(n,:));

endXYZr = RGBr * CieM;XYZg = RGBg * CieM;XYZb = RGBb * CieM;xyzr = XYZr / (XYZr(1) + XYZr(2) + XYZr(3));xyzg = XYZg / (XYZg(1) + XYZg(2) + XYZg(3));xyzb = XYZb / (XYZb(1) + XYZb(2) + XYZb(3));

plot([xyzr(1) xyzg(1) xyzb(1) xyzr(1)], ...[xyzr(2) xyzg(2) xyzb(2) xyzr(2)], ...

c. -', 'linewidth', lw);

Image from www.edmundoptics.com

Realistic primaries havewide spectrums. Why?

Assumes projectorlight has flat

spectrum (wrong)

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Subtractive color

We have been adding primaries with spectra that are (almost)

mutually exclusiveThis works for displays, but not for printing

It could work for printing, if dye were reflective and couldbe placed in very small, non-overlapping, regions

But (I guess) this doesnt work well, so instead ink isdesigned to block light, and to be mixed on the surface of awhite page

Typical (additive) RGB filters have transmission functions like these:

What would you expect the transmission functions of subtractivefilters to look like?

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Subtractive color

Image from www.edmundoptics.com

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Additive and subtractive primaries

The statement that the additive primaries are red, green, andblue is clearly incorrect

Any set of three non-colinear primaries yields a gamut

Primaries that appear red, green, and blue are a goodchoice, but not the only choice

Additional (non-colinear) primaries are always betterLikewise, the statement that the subtractive primaries are

magenta, cyan, and yellow is also incorrect, for the samereasons

Subtractive primaries must collectively block the entirevisible spectrum, but many sets of blockers that do so areacceptable primaries

The use of black ink (the k in cmyk) is a good example

Modern ink-jet printers often have 6 or more ink colors

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Final thoughts

CIE RGB and CIE XYZ are color spaces

Many other color spaces, with different strengths and weaknesses,are possible

Refer to Marc Levoys 2006 CS248 lecture notes for many details: http://graphics.stanford.edu/courses/cs248-06/color/color1.html

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Summary

Perceived color is the ratio of (linear) L, M, and S stimulation

It is not possible to distinguish between metamers

Three degrees of freedom need for (at least) three primaries

Everything derives from the human spectral-response function

Luminosity function (I think)

Reasonable choices for primariesColor-matching functions (RGB and XYZ)

Chromaticity diagrams (rg and xy)

Three primaries cannot reproduce the entire human gamut

Use chromaticity diagram to understand what can be reproducedSubtractive colors

Are a convenient fiction (like holes in semiconductor theory)

The true color arithmetic is unchanged

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Assignments

Next lecture: Some combination of lighting theory and z-buffertheory, TBD (Thursday 29 November)

Reading assignment: none (work on your projects)

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2007.11.27 CS248-17 Color Theory

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