|Jim Worthey • Lighting & Color
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Question: Is Jozef Cohen’s Matrix R
related to the Moore-Penrose pseudoinverse?
Projection Matrix R = A(A'A)−1A' , Derived from the
Pseudoinverse Formula, A+
= (A'A)−1A' .
Jozef Cohen derived the formula for the projection Matrix R
without reference to the concept of pseudoinverse. However,
projection Matrix R can be derived from the Moore-Penrose
pseudoinverse, which is also familiar in color science1.
The Wikipedia article can be our reference on pseudoinverse: http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse
Matrix Transpose and Hermitian:
The notation in the Wikipedia article clashes slightly with
Cohen and Kappauf. The pseudoinverse formulas apply to matrices
having complex numbers as elements. Then an asterisk, *, to the
right of a matrix indicates the Hermitian operation. Given a
matrix A, its
Hermitian, A*, is
found by taking the transpose of A, and then the complex conjugate of each
element. For color vectors and related matrices, complex numbers
are not used. Therefore, for our purposes,
A' (below or in
Cohen's articles) = A*
(in Wikipedia's pseudoinverse discussion)
Recall the use and meaning of projection matrix R:
Let L be any
vector, such as the spectrum of a light. Let A be a matrix whose
columns are linearly independent vectors, such as 3
color-matching functions. Then L* is defined to be the projection of L into the subspace of
the columns of A.
An equivalent statement is that L* is that linear combination of the
columns of A which
is a least-squares best fit to L. Matrix R is the one-step solution:
notation such as A* appears
in the Wikipedia article, it can be read as meaning A-transpose,
written as A'. The
clash occurs because Cohen used the asterisk suffix, *, to
indicate the fundamental metamer, which has nothing to do
with transpose, or with complex numbers. In everything
below, Cohen's notation is followed. The prime symbol, ',
denotes matrix transpose, and the asterisk, *, denotes a
vector projected into a subspace.
L* = R L ,
Restatement of the original question: A
vector L is given,
and we want to approximate it by a linear combination of the
columns of A. The
linear combination is specified by a coefficient vector u. Then
where R is known to be:
L ≈ Au
1. Brian A. Wandell and
Joyce E. Farrell, “Water into Wine: Converting Scanner RGB
to Tristimulus XYZ,” Proceedings
of SPIE Vol.1909,
pp. 92-100, (1993).
task is to solve for u.
For concreteness, suppose that A has 31 rows and 3 columns, then Eq. (4) is
a set of 31 equations in 3 unknowns, the unknowns being the
components of u. The
unknowns are overdetermined, so we seek a best fit solution
for u. Cohen solved the
least squares problem directly, but now we use the
Special case applies:
The pseudoinverse of A
is indicated as A+.
In the special case that the columns of A are linearly
independent, an explicit formula can be written:
A+ = (A' A)−1 A' .
problems, the columns of A will be linearly independent,
therefore Eq. (5) applies. Using A+
as an inverse in Eq. (4), we find
u = A+ L
Then from Eq. (5), substitute the explicit
formula for A+
= (A' A)−1 A' L
The approximation to L in Eq. (4) is our L*,
so substituting Eq. (7) into Eq. (4) gives
= Au = A (A' A)−1 A' L
Comparing Eq. (8)
to Eq. (2), we see
R = A(A'A)−1A'
as Eq. (3), but now deduced by Eq. (5), (6),
(7) and (8).
In short, Cohen's formula for R can be
deduced from the formula for the
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