What is the Fundamental Metamer?

A^T |L₁〉 = A^T |L₂〉 . (1)

If Eq. (1) is true when A is one of the sets of functions, then it will also be true for A equal to any of the other three sets. Notice that A^T |L₁〉 is a 3-vector, so Eq. (1) is a comparison of 3-vectors. The actual 3-vectors will be different, depending on which functions are used for A. The set of CMFs is chosen arbitrarily, leading to arbitrary 3-vectors.

In short, the facts of color matching are well known, but are presented in a way that is subject to arbitrary transformation. Cohen sought a presentation that would be less arbitrary. We can't know what false starts he may have made, but what he found was a methodology in which any set of CMFs A can be used, but when the comparison is made, analogous to Eq. (1), the algebraic entities that are compared are not arbitrary. The fundamental metamer of L₁ is compared to the fundamental metamer of L₂ , and those are non-aribitrary. That is, they do not depend on which matrix A was used. Of course, Cohen was not saying that A for the 2-degree observer is equivalent to A for the 10-degree observer, or A for a Nikon camera. But the graphs above remind us that for a given observer, a representation A is subject to transformation. The fundamental metamer is not changed by a transformation of A.

It is not my purpose here to review all the algebra. One may recall briefly that for a light L₁, its fundamental metamer L*₁ can be found by

L*₁ = R L₁ , (2)

This is a mini version of Fig. 1 in the Vectorial Color manuscript. At upper left is a set of color-matching data as they might appear an experiment, and next comes human cone sensitivities. Then the familiar x-bar, y-bar, and z-bar are displayed, and the last graph shows a set of opponent-color functions.

The four graphs look dissimilar, but are related. If they are used as color-matching functions, all 4 sets of functions predict the same color matches. The fictitious experimental data, the cone sensitivities, and the opponent functions were all computed by adding and subtracting the CIE's three functions. Each set of functions is a linear combination of any other set.

In short, the facts of color matching have alternate representations. That is not a new idea, but part of our legacy of well-known color science from the 19th and 20th centuries. Cohen discovered something new, that projection Matrix R comes out the same, no matter which set of color matching functions is chosen as a starting point¹.

Matrix Theorems: We'll need a couple theorems about matrices. The prime symbol, ', denotes matrix transpose.

Transpose of a matrix product: (AB)' = B' A' . (1)
Inverse of a matrix product: (AB)⁻¹ = B⁻¹ A⁻¹ . (2)
Extending the idea of Eq. (2): (ABC)⁻¹ = C⁻¹ B⁻¹ A⁻¹ . (3)

R = AX[(AX)'(AX)]⁻¹(AX)' . (5)

R = AX[X'A'AX]⁻¹X'A' . (6)

Apply Eq. (3) :

R = AXX⁻¹(A'A)⁻¹X'⁻¹X'A' . (7)

R = A(A'A)⁻¹A' . (7)

Eq. (7) is the same as Eq. (4), therefore R is invariant.

1. Jozef B. Cohen and William E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psych. 95(4):537-564 (1982).