Jim Worthey Talk, 2006 February 7

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 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,
(ω₂, 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.

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.


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

Axes have intuitive meanings: Achromatic, Red-Green, and Blue-Yellow.
The first two functions, ω₁(λ) and ω₂(λ), combine red and green cones only.
ω₁(λ) is a multiple of the familiar y-bar.
ω₁(λ), ω₂(λ) and ω₃(λ) combine to give the eye's vectorial sensitivity to color.
The functions are easily calculated, or get them from this link.
A light's tristimulus vector has the same magnitude as the light's "fundamental metamer."
Vector amplitude is non-arbitrary and has the units of the stimulus, such as radiance units. (Further explanation?)
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.
Algebraic benefit: orthonormal functions simplify derivations and formulas.
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:

or similar.

Let L be a light, that is a Spectral Power Distribution.


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


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.

Spectra	Color Composition	Transition, 64 Color Chips

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

Spectra	Color Composition	Transition, 64 Color Chips

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...

Spectra	Color Composition	Transition, 64 Color Chips

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

Cool White Fluorescent Light, a classic example where the shortage of red and green is intuitive
A commercial LED light, nominally 5500 K. With its 2-peaked spectrum, it resembles a fluorescent light.
A commercial LED light, "Warm White." This LED is a good approximation to blackbody.

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.

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.

William A. Thornton

Jozef B. Cohen

Background

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.

See http://www.jimworthey.com !

Seldom Asked Questions (links)

1. What Does "Orthonormal" Mean?

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

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

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