|Jim Worthey •
& Color Research • firstname.lastname@example.org
• 301-977-3551 • 11 Rye Court, Gaithersburg, MD
|What Does "Orthonormal" Mean?
In the example above, the
vectors have 2 components, (x,
y) . The inner-product
generalizes to any number of dimensions, and even to continuous
functions. As an alternative to the dot, •, an inner product can
indicated with angle brackets, 〈〉: If
vectors are perpendicular, then
their dot product is zero:
a•b = a1*b1
+ a2*b2 = −2 + 2
Orthogonal means perpendicular, so vectors a and b are orthogonal. A dot
product is also
called an inner product.
For an example with
continuous functions, sin(x)
orthogonal to cos(x).
|sine and cosine
|When sin(x) and cos(x)
the graph has equal + and − areas. 〈sin(x)|cos(x)〉 = 0, see below.
other hand, 〈sin(x)|sin(x)〉 = π .
So, sin(x) and cos(x) are orthogonal, but they
normalized. A function f(x) is normalized if 〈f(x)|f(x)〉
= 1. On the other hand, 〈sin(x)|sin(x)〉
= π, see above. If we define a new function s(x) = π−0.5sin(x), then 〈s(x)|s(x)〉
= 1. Function s(x) is normalized.
If a set of functions, such as ω1, ω2, ω3,
are orthogonal to each other, but each of them is normalized,
are an orthonormal set.
of orthonormal functions is a powerful tool for deriving needed
formulas in the simplest form.
|Why the Tristimulus Vector is the
Measure of Stimulus Amplitude.
1. Because the basis functions
orthogonal, they are sensitivities for independent features of the
stimulus. The achromatic, or whiteness function, ω1, is
all-positive, so the first component of the tristimulus vector
a positive number. The second tristimulus value measures redness
greenness. The third tristimulus value
measures blueness versus yellowness, although no light goes very
"yellow" by this measure.
|The orthonormal color matching
functions map lights into a color space, as pictured at
Monochromatic lights of unit power map to the Locus of
Monochromats. Jozef B. Cohen said that this Locus
"true structure of color space."
Without challenging Cohen's insight, it is interesting
to seek a more
step-by-step explanation. Why are tristimulus vectors
orthonormal basis the best measure of color stimuli,
amplitude? Some of the answers are verbal and intuitive;
to linear algebra and what Cohen called "the fundamental
The term "basis functions" is used here and elsewhere as
a synonym for
"orthonormal color matching functions."
A person may well object that red, green and so forth are
dimensions that should be decided by an observer, whereas this
space was cooked up using mathematical ideas. The objection is
We could go further and point out that the Prime Colors, which
large in this space, are not synonymous with such important
psychological landmarks as unique red, unique green, etc. The
color-mixing stage of vision plays its own unique role. As
explained it, the color-mixing, or transduction stage acts to lose information. If a
mixture of narrow-band red and narrow-band green makes a match to
narrow-band yellow, then the eye has lost the information that the
mixture stimulus is physically different from the single
The color-mixing stage of vision is about matches and mismatches.
Prime Colors, which technically are Prime Wavelengths, are
wavelengths that map to the longest vectors. If we are inventing a
technology such as televison, we
to get a signal through the
information-losing stage, and we need to know about
space and Prime Colors. Other
issues can then be handled as they arise in later stages of the
visual system, and later stages of the discussion.
Anyway, the basis functions themselves map to the axes. Functions
are independent mathematically, and in their intuitive meaning,
perpendicular directions in the color space. This "makes sense"
convenient in further analysis. By contrast, X and Y in the CIE
are not independent. They come from highly overlapping
functions, x-bar and y-bar.
2. If independent stimulus properties project onto the
axes, then the components of the tristimulus vector can be added
vector components normally add. That is, the vector's total length
the square root of the sum of the squares of the components. We
call this The Simple Algebraic Explanation.
The simple explanation is valid, but open to some objections. One
say, well, OK, red versus green is independent of whiteness or
blueness, but what about scaling? How do we know that we are not
too much weight, or too little weight, to redness in relation to
whiteness? The non-mathematical counter-argument, for persons who
not have color deficiency, is that the 3D graph "looks about
has about the same reach in each independent direction. If you
that a red bell pepper is "really red," and snow is "really
so forth, then the scaling is reasonable.
3. The more algebraic answer is essentially the explanation of why
the 3D graphic of the locus of unit monochromats indeed looks so
reasonable. Recall that there are no adjustable
parameters in the development of the color space. One starts with
certain functions such as cone sensitivities, but any
their scaling is removed in the steps that generate the
The remainder of the more algebraic discussion, for now, is spelled out on a separate page. So click here.
2004 James A. Worthey, email: email@example.com
Page last modified, 2013 April