Color Theory Kurt Akeley CS248 Lecture 17 27 November 2007

transcript

Color Theory

Kurt Akeley

CS248 Lecture 17

27 November 2007

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

CS248 Lecture 17 Kurt Akeley, Fall 2007

Introduction

Difficulties:

I’m learning this too

Terminology is used carelessly

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

Caveats:

Not comprehensive Lots of CIE content, nothing on HSV or video

Units may be off (or missing all together)

Derived plots may differ slightly from standards

Human photoreceptors

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

Monochromatic scotopic vision

(low light levels)

Chromatic photopic vision

(high light levels)

There is a view direction (despite

what I told you). It doesn’t affect the projection, but it does affect the nature of the

receptor array.

Humans are trichromats

Spectral sensitivities of human cones

Stockman, MacLeod, and JohnsonJournal of the Optical Society of

AmericaVolume 10, Number 12, December

1993Table 8

Monochromatic

stimulus

Determined by chemistry and

retinal coverage

Monochromat: Horshoe crabs

Dichromat: dogsTrichromat: people, most

camerasTetrachromat: fancy

camerasPentachromat: Mallard

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

EvalSmjSRF(470);All three contributions are

non-zero (the plots overlap)!

Luminosity function

Sum of L, M, and S responses for

photopic vision ?(FvD pp. 576)

Comparison of luminosity functions

Derived from SMJ spectral-response

data by simple addition

From Wikipedia

Yes, the units are different. We are interested in the

shapes.

Luminous efficacy (and efficiency)

How much of the radiated energy (efficacy/efficiency) or wall-socket energy (overall efficacy/efficiency) of a light source is usable for vision?

Integrate the product of the luminosity function with the spectrum of the light source.

Source: Wikipedia

Metamers

Response is linear

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

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

Many different spectra will produce the same L,M,S response

These spectra are metamers

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

This aliasing is very convenient …

CIE 1931 RGB primaries

λB=435 nm

λG=546 nm

λR=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;

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 match a spectral color?

Primary amplitudes are adjusted to give

equal areas under all three color-matching

curves

Oops, what is a negative

primary contribution?

W = DomainSmjSRF(1);len = length(W);% compute M assuming equal-intensity primariesM = [ EvalSmjSRF(Rnm) * 1 EvalSmjSRF(Gnm) * 1 EvalSmjSRF(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) * Gs EvalSmjSRF(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]

Negative primaries

Can’t add a negative primary amount

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

Sensible but awkward

CIE 1931 XYZ primaries

Derived by linear transformation from the RGB primaries:

The matrix coefficients were (obviously) chosen carefully

We’ll see some of the desirable properties in later slides

The XYZ primaries are imaginary

They do not correspond directly to amounts of light

But they are very useful …

[ ] [ ]0.49 0.17697 0.00

10.31 0.81240 0.01

0.176970.20 0.01063 0.99

X Y Z R G B

é ùê úê ú= × ×ê úê úë û

CIE 1931 XYZ color-matching function

Intended property:

Y = luminosity

Intended property:

non-negative

A special metamer

400 nm

700 nm

Energy density

Dominant wavelength = wd

Excitation purity = (e2-e1) / (e2+e1)

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

These determine

chromaticity

XYZ chromaticity

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

Because the sum of the chromaticity values x, y, and z is always 1.0, a plot of any two of them loses no information

Such a plot is a chromaticity plot

x y zX Y Z X Y Z X Y Z

= = =+ + + + + +

xz chromaticity

You’ve never seen this plot, but it is

perfectly valid. The standard plots the x and y chromaticity

values.

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

CIE 1931 xy chromaticity (actual CIE standard)

Image from www.wikipedia.com

Pure (saturated) spectral colors around the edge of the

Less pure (desaturated) colors in the interior of

the plot

White at the centroid of the plot

(1/3, 1/3)

Are the colors correct ?

CIE 1931 rg chromaticity (derived from SMJ data)

Intended property:

rb chromaticity (derived from SMJ data)

Intended property:

Gamut is the chromaticities that can be generated by a set of primaries

Because everything we’ve done is linear, interpolation between chromaticities 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?

CIE 1931 RGB gamut

R = 700 nm

G = 546 nm

B = 438 nm

All dominant wavelengths can be

reproduced, but many cannot reach full

saturation.

No finite set of primaries can

reproduce the entire gamut. But more

primaries do a better job.

So the colors on the xy chromaticity diagram

cannot be correct, because no display

can accurately reproduce them!

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);

Short-persistence phosphor CRT gamut

Long-persistence phosphor CRT gamut

LCD projector gamut (hypothetical)

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 have wide

spectrums. Why?

Assumes projector light

has flat spectrum (wrong)

Subtractive color

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

This works for displays, but not for printing It could work for printing, if dye were reflective and

could be placed in very small, non-overlapping, regions

But (I guess) this doesn’t work well, so instead ink is designed to block light, and to be mixed on the surface of a white page

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

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

Subtractive color

Image from www.edmundoptics.com

Additive and subtractive primaries

The statement that “the additive primaries are red, green, and blue” is clearly incorrect

Any set of three non-colinear primaries yields a gamut

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

Additional (non-colinear) primaries are always better

Likewise, the statement that “the subtractive primaries are magenta, cyan, and yellow” is also incorrect, for the same reasons

Subtractive primaries must collectively block the entire visible spectrum, but many sets of blockers that do so are acceptable “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

Final thoughts

CIE RGB and CIE XYZ are color spaces

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

Refer to Marc Levoy’s 2006 CS248 lecture notes for many details: http://graphics.stanford.edu/courses/cs248-06/color/

color1.html

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 primaries

Color-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 reproduced

Subtractive colors

Are a convenient fiction (like ‘holes’ in semiconductor theory)

The true color arithmetic is unchanged

Assignments

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

Reading assignment: none (work on your projects)

Color Theory Kurt Akeley CS248 Lecture 17 27 November 2007

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