perspective view of locus Jim Worthey,
                        Lighting and Color Research
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Jim Worthey • Lighting & Color Research • • 301-977-3551 • 11 Rye Court, Gaithersburg, MD 20878-1901, USA

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How White Light Works
LRO Lighting Research Symposium
Light and Color

Tuesday, 2006 Feb 7, 10:55 am
James A. Worthey, PE, PhD

Locus of  Unit Monochromats = The Eye's Vectorial Sensitivity to Color

Big Static Orthonormal
                                          Spectrum Locus

Talk Itself Starts Here

3 Main Ideas

Orthonormal Basis
Eye's Vectorial Sensitivity to Color
Any Vectors
Orthonormal Color Matching Functions.
What's "Orthonormal?"
Vectorial Sensitivity to narrow-band lights.
Tristimulus vectors
in Cohen's space.

Quick Example: Color Loss with Mercury Vapor Light.
Spectral comparison
                        daylight --> high pressure mercury
2 Lights That Match, with Dissimilar Spectra
Shifts of 64 Munsell Papers in Cohen's space when
Daylight --> High Pressure Mercury Vapor

Early Color-Matching Data of Guild or Wright.

Thoughts on Matching Data
The primary lights are not unique, and the same facts can be presented in alternate sets of graphs, an awkward situation. However,
  • Wright and Guild both used red, green, and blue primaries. The wavelengths are not really arbitrary.
  • The mixing of 3 primaries models technologies such as television.
  • Practical question: what wavelengths work best as primaries?

Same Color Matches, but the Functions Look Different:

Cone Sensitivities Color Matching Functions
CIE's x-bar, y-bar, z-bar

(x, y) Diagram
So, we think in terms of the chromaticity diagram... video colors,
                cone sens. peaks but those primary colors are still needed.
Television dots.

Which Wavelengths Act Most Strongly in Mixtures?

MacAdam, and later Thornton did calculations like this.

Narrow-band lights of constant power are mixed with equal energy light. Some wavelengths perturb the white chromaticity more than others.

Thornton coined the term "Prime Colors" for the 3 wavelengths that act most strongly.

Color Matching Functions Are in Fact Relatively Stable
When the Primary Wavelengths Are Changed.

Later Thornton found:
  • The color matching functions tend to peak at certain fixed wavelengths, despite perturbation of the primaries.
  • The peaks occur at the prime colors.
  • Prime color primaries run the experiment at minimum power.

Michael Brill's new theorem:

When one primary wavelength is changed (say the red wavelength only) then the associated (red) color matching function changes only in scale, not in shape.

(For a more leisurely discussion of shifting primary wavelengths, please click here.)

Red and Green Primaries Must Relate to the Overlap of Red and Green Sensitivities...
red, green cones
red cone
                sensitivity minus green cone sens.
ybar and an
                orthogonal function
Red and green cone functions are highly overlapping.
Subtracting green from red gives peaks to account for Prime Colors, but the scaling is arbitrary.
Achromatic sensitivity, y-bar, is a linear combination of red and green. Find a second combination that is orthogonal to it.

One Degree of Freedom Remains, to Re-mix ω1 and ω2,
But Keep the Mixtures Orthogonal.

We start with ω1 and ω2, which are linear combinations of red and green cone functions, and are orthonormal. Other orthonormal pairs of functions can be generated by a rotation matrix:

rotate by angle theta

Now Make a Parametric Plot of ω2 vs ω1.
Bingo, the Shape is Invariant.

Vectorial Sensitivity to Wavelength.
For each λ, the eye's sensitivity is a vector,
2, ω1) .

The spectrum locus is the eye's vectorial sensitivity to color. It is not a boundary.

The spectrum locus is alternatively the vectorial color of narrow-band lights, at unit power.

Vectorial addition of colors.
Prime colors the wavelengths where radius is a local maximum. The exact Prime Color is a little different.

Stalking Prime Colors in the 2 Dimensions of Red and Green.
Progress so far:
  • Computing red minus green gives tentative prime colors---the plus and minus peaks of an opponent-color graph.
  • But, the peaks depend on the scaling of red and green, an arbitrary element.
  • Since achromatic sensitivity (y-bar) is a linear combination of red and green, find an opponent-color function that is orthogonal to y-bar.
  • Now we have a rationalized vector space for colors.
  • The prime colors---the wavelengths that act most strongly in mixtures---are the wavelengths that map to the longest vectors, 536 and  604 nm.
  • If we extend this method to include blue cones, then the spectrum locus in 3 dimensions is what Jozef Cohen called The Locus of Unit Monochromats. It is the eye's Vectorial Sensitivity to Wavelength. Cohen used different steps to reach this point.

Vectorial Sensitivity to Wavelength, Now in 3 Dimensions.
Orthonormal Basis
Spectrum Locus
                    in Orthonormal Space
Now include the blue cones to create a set of 3 orthonormal color matching functions.
Combining the orthonormal functions into a parametric plot yields Jozef Cohen's "Locus of Unit Monochromats," the eye's Vectorial Sensitivity to Wavelength.

 Features of the Orthonormal System
  1. Axes have intuitive meanings: Achromatic, Red-Green, and Blue-Yellow.
  2. The first two functions, ω1(λ) and ω2(λ), combine red and green cones only.
  3. ω1(λ) is a multiple of the familiar y-bar.
  4. ω1(λ), ω2(λ) and ω3(λ) combine to give the eye's vectorial sensitivity to color.
  5. The functions are easily calculated, or get them from this link.
  6. A light's tristimulus vector has the same magnitude as the light's "fundamental metamer."
  7. Vector amplitude is non-arbitrary and has the units of the stimulus, such as radiance units. (Further explanation?)
  8. We don't usually learn to graph tristimulus vectors, or compute their magnitudes. Even graphs of the vector (X Y Z) would give some insight, but graphs and magnitudes mean more here.
  9. Algebraic benefit: orthonormal functions simplify derivations and formulas.
  10. Beginning students can be told flatly "This is the eye's vectorial sensitivity to color." Details can follow as needed.

 Working Class Summary of the Orthonormal System

The discussion started by seeking the wavelengths where the cone sensitivities are "the most different," and that idea evolved to give an orthonormal opponent system. Putting aside the fancy reasoning, what is new or not new about the orthonormal basis?

  • ω1(λ) is proportional to the old y-bar, so it is not new at all.
  • ω3(λ) is a kind of opponent function, but for practical purposes, it is very similar to blue cone sensitivity, meaning it's similar to z-bar. So ω3(λ) is not really new.
  • In the XYZ system, a non-intuitive feature is x-bar, an arbitrary magenta primary. It is replaced in the new system by ω2(λ), a kind of red-green opponent function. The opponent function confronts the overlap of cone sensitivities by finding a difference of red and green.
Graphical comparison of old & new functions.

Oh, Yes, Calculating Tristimulus Vectors.
Inner product defined:
inner product defined or similar.
Let L be a light, that is a Spectral Power Distribution.
Calculate XYZ vector
Calculate tristimulus vector in ortho
Tristimulus vector found the old way.
Tristimulus vector found the new way.
The calculation is essentially the same, but the benefits of the orthonormal color matching functions are tremendous!

Rearrange the Sums to Find a Vector for Each Wavelength Band
Mercury Vapor, Daylight Spectra
Mercury vapor
                    light and daylight
A daylight phase has the same tristimulus vector as a certain high-pressure mercury vapor light.

Earlier we saw the effect of the mercury light on 64 paint chips. Now we see it decomposed into its component colors.

Color Rendering
  • Smooth chain is daylight decomposed into narrow bands.
  • The other chain---a commercial mercury light---takes a shortcut to white, because it is poor in reds and greens.

Pick 3 LEDs Whose Peaks Are Near the Prime Colors;
Combine Them to Match Blackbody 5500 K.

Color Composition
Transition, 64 Color Chips
spds of bb, leds #80 57 28
Compose bb55, leds # 80, 57, 28
                    bb55 to leds # 80, 57, 28

Pick a redder red LED, and a greener green One.
Again combine them to match Blackbody 5500 K.

Color Composition
Transition, 64 Color Chips
Spectral comparison, bb, leds 86, 56, 28
Compose bb55, leds # 86, 56, 28
                    bb55 to leds # 86, 56, 28

So, the 2nd LED combination above doesn't dull reds and greens, but it exaggerates some of them.
Recall Yoshi Ohno's concern that the red prime color (603 nm) is too orange.
Address that issue with a double red primary...

Color Composition
Transition, 64 Color Chips
Spectral comparison, bb, leds 86, 78, 53, 28
Compose bb55, leds # 86, 78, 53, 28
Transition bb55 to leds # 86, 78, 53, 28

The orthonormal basis makes it convenient to present color data as vectors in Jozef Cohen's linear color space. Vector operations make use of linearity to do simple things, such as showing the vector components of a light's color. The color rendering examples present facts with no hidden assumptions. Each component vector, for example, is calculated by traditional colorimetry, but using the orthonormal basis, rather than XYZ.

More Interesting Examples

So, You Want a Simple Formula?
For color rendering, the usual standard of what we expect is blackbody radiation, or daylight. Those lights contain both red and green, and graphing the vector composition of the light shows a swing in the green direction, then back in the red direction. Other lights may have greater or lesser swings to green and back. A simple measure is the swing towards green, plus the swing back towards red. If L1 is the spectrum of the light under test, and L0 is an appropriate reference light such as blackbody, those swings can be summed, and then the ratio taken. The result g will be 1.0 if L1 has normal color rendering, and >1 or <1 if it exaggerates or diminishes red-green contrasts.
formula for
                      color-rendering ratio g
This formula is not tested, except by the examples above.

Special Credit

William A. Thornton
Jozef B. Cohen

Michael H. Brill
Tom Cornsweet (1970 Book)
Ronald W. Everson (taught color fundamentals)
David MacAdam and Gershon Buchsbaum who mentioned orthogonal color matching functions.

(But MacAdam gets a demerit for disparaging Cohen's work.)

Calculations were done with O-matrix software.

Bill Thornton
William A. Thornton
Jozef B. Cohen
Jozef B. Cohen

A draft manuscript is available on my web site, which talks about the orthonormal cmf's, Cohen's space, and color rendering. The title is "Vectorial Color."

Also on my web site is a manuscript that clarifies what Prime Colors are: Michael H. Brill and James A. Worthey, "Color Matching Functions When One Primary Wavelength is Changed".

General Background, including Thornton's and Cohen's work is in
Render Asking: James A. Worthey, "Color rendering: asking the question," Color Research and Application 28(6):403-412, December 2003.

 An alternate approach to color rendering, without vector diagrams, is in
Render Calc:  James A. Worthey, "Color rendering: a calculation that estimates colorimetric shifts," Color Research and Application 29(1):43-56, February 2004.

Seldom Asked Questions (links)

1. What Does "Orthonormal" Mean?

Why is the Tristimulus Vector the Best Measure of Stimulus Amplitude?

3. How Does the Orthonormal Basis Relate to Cohen's Matrix R?


Scroll No Farther
Material Below Addresses Obscure Questions

2 colors, C1 and C2 are set to equal power. The mixture is m.

Mixing of two colors at equal

Cone Sensitivities, Approximately Smith-Pokorny Primaries
Cone sensitivity functions

Calculating the Guth Opponent Functions
define matrix M-zero
Columns of C are x-bar, etc
Compute matrix of
                                          opponent vectors

As a matrix of 3 columns,

G = [achromatic, red-green, blue-yellow]471 x 3

Guth's 1980 Model, Approximately

As a matrix of 3 columns,

G = [achromatic, red-green, blue-yellow]471 x 3

                                model of Guth, Massof & Benzschawel

Now Take Redundancy out of Guth's Model by Gram-Schmidt
big G
                                to big Omega by Gram-Schmidt

inner products are Kronecker delta
big Omega in terms of little omega

Orthonormalized Opponent Functions
                      Basis = cmf's

Summary and Some New Issues
1. The initial stage of vision is linear and invariant.
2.  TV phosphor and autumn leaf need to act in the independent dimensions of color vision. Autumn leaf
3. Orthonormal color matching functions are as indpendent as possible.
4. Most arbitrariness drops out. One axis is whiteness.
5. No dispute with Jozef Cohen or Bill Thornton.

6. Some departure from Cohen's preferred treatment.

CIE says 471 numbers --> 3
But Cohen preferred the fundamental metamer as an invariant "color vector."
471 numbers
                                              --> 471 numbers
With orthonormal basis, we can have it both ways. Equation
N* squared
471 numbers
                                              --> 3 numbers
Tristimulus Vector is a Proxy for the Fundamental Metamer.

7. Color matching functions similar to raw data give an interesting spectrum locus, but X, Y, Z do not. See below.

Spectrum Locus for 4 Different Sets of Color Matching Functions
Locus in Orthonormal Space
Orthonormal Basis Functions
(Graph as Cohen drew it.)

Locus based on
                                        narrow band primaries
Color Matching functions similar to
Raw Experimental Data

Locus based on
                                        cones r, g, b.
Cone sensitivities, r, g, b
Locus based on x,
                                        y, z
CIE's x-bar, y-bar, z-bar

"Boomerang Graph," Not a Chromaticity Plot

Supplementary Material, Relationship to Jozef Cohen's Work

>>  Click Here

Copyright © 2006 James A. Worthey, email:
Page last modified, 2013 April 25, 20:08