|Jim Worthey •
& Color Research • firstname.lastname@example.org
• 301-977-3551 • 11 Rye Court, Gaithersburg, MD 20878-1901, USA
Question: What is the proof that Jozef
Cohen’s Matrix R is invariant?
Proof that Projection Matrix R = A(A'A)−1A' is Invariant
To begin one may ask "With respect to
what is R invariant?" Consider
this miniature version of Fig. 1 in the Vectorial Color manuscript (or
In Eq. (2) and (3), matrices A,
B, C must be square and the inverses
must exist. In Eq. (1), A and B need not be square, but must be
conformable for multiplication.
|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.
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 of a matrix product: (AB)' = B' A'
- Inverse of a matrix product: (AB)−1
the idea of Eq. (2): (ABC)−1
Invariance of R: Now suppose
that the columns of A are a
set of color matching functions, such as the cone sensitivities. Let X be an invertible square matrix
that transforms A to a
different set of CMFs.
If A is the given set of CMFs, then
the CMFs are AX, then
Eq. (3) :
(8) is the same as Eq. (4), therefore R is invariant.
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).
Jim Worthey Home Page: http://www.jimworthey.com
2007 James A. Worthey, email: email@example.com
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