La Nature se dévoilant devant la Science

La Nature se dévoilant devant la Science
Louis-Ernest Barrias (1899)

• My Interest is the study of Nature &
my Science probably reflects that.
• I am not after Solutions or Applications:
my business is generating Problems.
• I am a physicist studying Consciousness,
mainly by way of Visual Perception.

1
Color Space
Jan Koenderink

Introduction:
“colorimetry”

physics
(optics)
psychology
of perception
sensory
physiology
intrinsically boring topic:
3
colorimetry
location of the talk
in a landscape of
the sciences
Color means something
entirely different in any new
field of endeavor.
For most people the “color” of
colorimetry is meaningless
(certainly colorless).

“Colorimetry” is a 19 th c. construction due to Maxwell,
Graßmann and Helmholtz, essentially a generalization of
“photometry” (an 18 th c. construction due to Lambert
and Bouguer).
4

When a beam of radiation enters the eye a “patch of light” is
often seen, although:
• you sometimes see light in the absence of radiation,
as in dreams,
e.m. radiation
• radiation may enter your eye without resulting in a
patch of light
“quale”
patch of light, as in temporary blindness or inattentiveness.
Is a “Color Science” possible at all? Are colors “mental paint”?
5

1
0.5
0
700
600
500
2.0 2.5 3.0 eV
6
eye
sun
400nm
Human observers are sensitive to electromagnetic
disturbances in the “visual range”, roughly 400-700nm

Consider the space S of incoherent beams. They may be
scaled (e.g, via neutral density filters) and added (e.g., via
superposition of irradiance on a white, Lambertian screen).
There is a null element, the “empty beam”, which is
simply no radiation at all.
S would be a linear space were it not for the fact that
the radiance has to be non–negative.
Thus real (actual) beams fill the “positive part” S + of
the linear (infinitely dimensional) “space of beams” S.
“Colors” (to be explained!) are elements of a 3 D lin-
ear space C. (The “trichromacy” was first suggested by
Thomas Young.) 7

The observer sets “color coordinates” {π1, π2, π3}
such that
π1 p1 + π2 p2 + π3 p3,
and some arbitrary beam s (colored paper) are
indiscriminable. The the beam s has been assigned
a point {π1, π2, π3} ∈ C. One technical problem is
that the coordinates are not always positive.
8
Maxwell picks some
arbitrary 3 D subspace of
S via a basis {p1, p2, p3}
(red, blue and green
papers). He implements
linear combination via a
spinning disk.

The whole procedure turns out to be linear within the
experimental uncertainty (ca. 0.1%).
The reason is that discriminability is fully decided at
the level of absorption of photons in the retinal
photopigments.
Equal absorption implies equal input to the brain,
thus the brain is simply used as a “null-indicator”!
That is why colorimetry can be a science whereas
“Color Science” is not.
9

1.0
0.8
0.6
0.4
0.2
0.0
400nm 500nm 600nm 700nm
1.2
1.0
0.8
0.6
0.4
0.2
0.0
400nm 500nm 600nm 700nm
1.2
1.0
0.8
0.6
0.4
0.2
0.0
400nm 500nm 600nm 700nm
beam A beam B beam C
Empirically “A looks like B”, “B looks like C” and “C looks like A”.
(They all appear like .)
Remember Euclid’s “The Elements”, Common notion #1:
“Things which equal the same thing also equal one another”.
Euclid takes reflexivity for granted (then symmetry & transitivity follow)
The binary relation “looks like” turns out to be an equivalence relation
(say “~”).

Colorimetry is based upon an equivalence relation ∼ (s, t), for s, t ∈ S.
When ∼ (s, t) then “s and t are indiscriminable”. Typically
¬ ∼ (s, t), thus indiscriminability is very special.
“Color space” C is defined as C = S/ ∼. Its elements (“colors”)
are equivalence sets of ∞ cardinality.
∼ (s, t) doesn’t imply s = t, but s−t ∈ K or ∼ (s−t, ∅), where ∅
is the “empty beam”. The linear space K is the “black space”.
Thus “Color Space” is the Space of Beams with the Black Space
collapsed to zero: C = S/K.
Empirically one finds dim C = 3 for generic human observers. All
colors are projections from S + . They form a convex cone C + ⊂ C.
11

12
The dimensions of
the black space are
like the “depth”
dimension in spatial
vision: They are
causally ineffective.
Large gaps in the
world can be absent
in vision.
Depth is 1 out of 3
dimensions, the
blacks are ∞-3 out of
∞ dimensions.

In summary: Methodologically/conceptually colorimetry deals
only with the discriminability of beams.
Thus:
• (colorimetric) colors are not qualia
• the brain is irrelevant to colorimetry (though not to
“Color Science”!)
• Colorimetry ≠ psychology.
Color vision is uniquely specified though the “black space”,
that is the kernel of the projection.
Formal apparatus is mainly linear algebra augmented with
linear programming and/or iterative projection on convex
sets. 13

The spectral colors

From Newton derives the myth that one “sees by
wavelength”. Newton actually came to believe that:
• white light is “composed” of homogeneous lights (spectrum)
• homogeneous lights are “atomic” (experimentum crucis)
• all colors (qualia, i.e., “redness’’, etc.) are deterministically
and uniquely related to the homogeneous lights
Newton was wrong on all three counts.
His notions remain popular though.
15

If an entity can be decomposed into other entities this
does in no way entail that it is composed of it!
I.e., a sausage can be decomposed into slices, but it not
composed of slices before the actual slicing.
16

In a linear space infinitely many bases are possible and
any basis vector of one can be decomposed into all of
the other bases.
There are no “atomic parts” in a linear space. Any
element can be decomposed in infinitely many ways,
yet is in no way to be considered a “composite”.
One selects a basis on pragmatic grounds. The spectral
basis is in no way the most convenient one for
colorimetric purposes.
17

The Newtonean “spectrum” is visually incomplete.
The spectral colors don’t exhaust
the set of color experiences,
whereas the “color circle” does
The relation spectrum-color
circle was finally cleared up by
Ostwald in the early 20 th c.
18

700nm
530nm
} equal mixture
578nm
A linear combination of “homogeneous lights” of
530nm and 700nm can be made to be indiscriminable
from a homogeneous light of 578nm.
Colors are not uniquely tied to homogeneous lights.
19

• spectral radiant power is non-negative
• the monochromatic beams form a complete basis
of the space of beams
thus all colors are convex combinations of the
colors of monochromatic beams
Conclusion:
Colors & Spectra I
Colors are restricted to a convex cone in color space.
The generators are due to the monochromatic beams.
Empirical finding (Helmholtz):
There exist generators that are NOT due to any
monochromatic beam (the “purples”).
20

The object colors

eflectance
1
• each wavelength is an independent dimension
• spectral reflectance is in the range [0,1]
Π 0
Conclusion:
0 Π
Π
Π
2
wavelength
object color spectra are restricted to an ∞D hypercuboid in
the space of spectra
Colors & Spectra II
22
2

All spectral reflectances fill an
∞–dimensional hypercube
The projection in color space
is the “color solid”
(Schrödinger, 1920)
23

More precisely, Schrödinger showed that:
Consider all beams with spectra 0 ≤ S(λ) ≤ A(λ) for a cer-
tain fiducial beam A(λ) > 0. Then the length of any color
is limited (even without metric!). The maximum length is
obtained for beams such that
S(λ) = χ(λ)A(λ), where χ(λ) is a characteristic function
with at most 2 transitions in the spectrum. (⇒ Lyapunov’s
“bang–bang principle”.)
These “optimal colors” come in 4 varieties:
• pass band colors (greenish)
• stop band colors (purplish)
• short pass colors (bluish)
• long pass colors (reddish)
24

In classical colorimetry one has no metric. Color space is
only defined up to arbitrary linear isomorphisms.
Defining a scalar product in S allows one to define an or-
thogonal projection operator Λ, with
kerΛ = K, Λ 2 = Λ and Λ † = Λ.
With s ∈ S + one has Λs = sf ∈ F, where S = F ⊕ K.
“Visual space” F is uniquely determined and its points are
in 1–1 relation with the colors. (sf need not be in S + .)
sf is the “causally effective part” of s.
F is a metric space. It is a “true image” of S because ∞ D
unit hyperspheres of S project on 3 D unit spheres of F.
I use F for the illustrations. 25

1
0.5
0
1
0.5
0
0.6
0.4
0.2
0.
0.2
average daylight 5700ºK Planck
400nm 500nm 600nm 700nm
400nm 500nm 600nm 700nm
400nm 500nm 600nm 700nm
1
0.5
0
1
0.5
0
0.8
0.6
0.4
0.2
0.
0.2
400nm 500nm 600nm 700nm
400nm 500nm 600nm 700nm
400nm 500nm 600nm 700nm
spectrum
fundamental
component
black
component

Aside:
I’m skipping numerous technical problems here!
E.g., random 3D projections of ∞D hypercubes do not
“display the spectrum” as a well ordered smooth
curve.
It is not hard to characterize the “nice” projections (all
3x3 subdeterminants of the matrix of the projection
need to be of the same sign).
Fortunately the projection of the generic human
observer is of this “nice” type.
27

Generic (random) projection and a special (“nice”)
projection of the “spectral vertices” (all coordinates
zero, except for one which equals one) of a 17 D
hypercube.
28

The “optimal colors” are the brightest for their chromaticity
They lie on the boundary of the color solid, their spectral
reflectances are uniquely determined
29

Structure of the space
of object colors

Goethe against Newton
In the “Polemical Part” of his “Farbenlehre” Goethe
violently attacks Newton’s experiments and
conceptual ideas concerning optical radiation and
colors
31

The Goethe “edge colors”
are an alternative to the
monochromatic basis
They have the advantage
over the monochromatic
basis that they can be
produced from daylight
Schrödinger’s optimal colors
are produced as simple
combinations of pairs of
edge colors
32

Goethe’s “edge colors” are seen as you look at a
black/white edge. Different edge colors appear for
edges like this or this
Goethe considered Newton’s spectrocope slit
as a combination of edges, thus Newton’s spectrum
as a mixture of edge colors
In this mindset it is natural to explore complementary
slits ! In this way Goethe discovered the
“inverted spectrum”
All Newton’s experiments can be repeated with the
inverted spectrum, but you don’t need a dark room for
that. This was important to Goethe
33

The spectrum contains no purples, whereas the
inverted spectrum contains no greens
There is an optimum slit width of the spectroscope.
Either too narrow or too wide a slit width yields black or
white, but no color
widen slit ➠
34

spectrum
&
inverted spectrum
35
“color circle”

700 nm There exist no
600 nm
500 nm
400 nm
400 nm 500 nm 600 nm 700 nm
36
complementary
wavelengths for the
medium wavelength
range (greens).
This runs counter
Newton’s claims and was
discovered by Helmholtz
in the 1850’s.

Like the optimal colors, exactly one half of the RGB
colors is “spectral” or “Newtonean” the other half is
non-Newtonean and derives from the inverted
spectrum
37

“Babinet’s Principle”: complementary objects
yield complementary optical images
For a “complementary slit” you
obtain the “inverted spectrum”
Newton’s experimentum crucis is
guaranteed to work for the
“inverse homogeneous lights’’ too 38

All of Newton’s experiments (also the famous
experimentum crucis) can be repeated for the inverted
spectrum – and with analogous results!
39

Newton’s “spectral
resolution” was
something like
50-100nm
40

“Bad” spectral resolution
can be a good thing
(Goethe, Schopenhauer)
The more “monochromatic”
a beam is, the worse its
color becomes!
Spectrum colors go black,
inverted spectrum colors go
white
41

0.75
slit width
1
0.5
0.25
0
400 500 600 700
The optimum resolution for seeing vivid, bright colors
is surprisingly bad
best
fix the
dominant
wavelength
nm
42

Aside:
“Color is seeing by wavelength” is a misleading concept.
A much more apt thought model is:
43
The human observer
detects:
• the overall slope
(subjectively the yellow-
blue dimension)
• and the curvature
(subjectively the cyanmagenta
dimension)
of the spectrum.

spectrum
locus
Parameterization of
chromaticity by
spectral slope and
curvature in the
RGB triangle.
44

The Goethe “edge colors”
are an alternative to the
monochromatic basis.
They have the advantage
over the monochromatic
basis that they can be
produced from daylight.
Schrödinger’s optimal colors
are produced as simple
combinations of pairs of
edge colors.
45

There exist two families of edge colors, they are each
other’s complementaries (or rather “supplementaries”
since they add to white) 46

“warm” edge color series
“cool”edge color series
The edge colors lack the greens and the magentas
Goethe noticed they can be obtained from combinations
of two edge colors, thus Goethe came to see Newton’s
spectral colors as “composite” (similar conceptual error)
47

The edge color loci are spirals on
the surface of the color solid 48

cool
loop
warm
loop
The edge color locus in color space as seen
from the direction of the achromatic axis
The edge colors divide into two spirals of opposite
chirality.
The boundary of Schrödinger’s color solid can easily be
constructed geometrically through translated edge
color arcs.
49

The Newtonian spectrum, Goethe’s Kantenfarben and
the painter’s color circle have different topologies and
metrics, a source of endless confusion in the literature.

The edge color loci
appear as you plot the
colors of random
spectra.

Schopenhauer’s
“Parts of White”

Arthur Schopenhauer
Arthur Schopenhauer was a pupil of Goethe
He wrote a book on color in which he treated colors as
“parts of daylight”
53

Two complementary edge colors can be regarded as “two halves of daylight”
Schopenhauer noticed that there exist two ways to “cut daylight into two halves”
such that the parts are most strongly colored
54

“Best” colors occur when the cut locus
is complementary to a spectrum limit
55

1. Aguilonius's color sequence [17]: from white, through yellow, red and blue to bl
~d}luar~, ~iotctt, ~Slau, ~tiiu, ~at~, ~3tanae, Oelb, ~r
0 '/, '/, y, '/2 2/, '/, 1
2. Schopenhauer's color sequence [3]: from black, through violet, blue, green, ora
llow to white. Goethe's three pairs of contrasting colors, including their white conte
The Schopenhauer “Parts of White”
Notice his usage of: “Violett” (= blue)
e light and the 'shadowy' are not always so thoroughly, as it we
“Blau” (= cyan)
ically', mixed: alongside the 'qualitative' color-creating activity of t
there is also quantitative 'mechanical' blending. This produces the gr
“Roth” (= magenta)
from white through various degrees of gray to black. Light is thus dete
“Orange” (= red)
by two separate variables: the qualitative and the quantitative retin
ties. Schopenhauer's theory of light and colors has two dimensions.

The spectral region left over between the two optimal cuts
yields green, its complement is magenta
Thus Schopenhauer obtained the cardinal colors as “parts”
57

Schopenhauer also considered tripartitions
One way to obtain an “optimal” tripartition of daylight is to find two cuts such
that the three parts span a “crate” that exhausts the maximum volume from
the color solid
There is a unique solution, the parts are red, green and blue
58

59
For the maximum
volume RGB crate:
tangents to edge color
curve at B and Y are
mutually parallel &
parallel to RC plane
&
tangents to edge color
curve at R and C are
mutually parallel &
parallel to YB plane

a view from the direction of
the achromatic axis
60
(notice the implied
regular hexagon)

1
0.75
0.5
0.25
0
1
0.75
0.5
0.25
0
400 nm 500 nm 600 nm 700 nm
400 nm 500 nm 600 nm 700 nm
400 nm 500 nm 600 nm 700 nm
The Schopenhauer cut loci
divide the spectrum into
“blue”, “green” and “red” parts.
Any color can be obtained by
way of a weighted sum of
these parts, in most cases a
convex combination will do, in
exceptional cases (slightly)
negative or weights (slightly)
over one are required.
61

W
K
The best fitting
“RGB crate”
inside the Color
solid
Since colors near
the boundary are
rare, almost any
natural color lies
inside the RGB
crate
62

63
The RGB crate
mapped on a unit
cube:
The color solid
appears like a
“rounded’’ cube.

The RGB crate for daylight in the spectrum cone
The full color locus is approximated with a hexagon
64

Ostwald’s semichromes

Wilmhelm Ostwald
Wilhelm Ostwald started color research after his
retirement. Although critical of Goethe/Schopenhauer
he worked much in their tradition – Ostwald managed
to formalize many of their concepts though
66

Ostwald asked a physicist:
“What is the spectrum of the best yellow paint?”
and received the answer:
“Reflect 580nm and nothing else!”
But “monochromatic paint” is black!
Then Ostwald put the best paints he could find before a
spectroscope and discovered the “semichromes”
67

The semichromes are
naturally ordered in a
periodic sequence
They contain all hues,
both Newtonean and
non-Newtonean
Ostwald arrived at the concept of “semichromes” by an
intuitive reasoning that can be considered a synthesis of
Goethe’s and Schopenhauer’s primitive notions
It was the first principled construction of
the “color circle” (topological circle) from
the “spectrum” (topological linear segment)
68

k
c
w
complementaries
In Ostwald’s description any color is associated with a
unique spectral reflectance
Thus any color is described via its “full color” (or
semichrome) and its color, white and black content
Full colors are “halves of daylight” (semichromes)
1
0
69

R
G B
{
{
R G B
d
R 80 G 60 B 30
80 60 30
exploded
K
R
Y
W
{
K
F
W
c=50, w=30, k=20
The Ostwald description also fits naturally in the RGB system
70

The “full colors” have maximum distance from the
achromatic axis of the Schrödinger color solid, thus they
are indeed “full”, that is the best possible for a given hue
71

72
The full color locus
is very similar to
the non-planar
hexagon formed by
the YRGCBMR edge
progression about
the RGB color cube

these full colors are the “best
(object) colors” for daylight
these “ultimate” colors are
“best colors” for some weird
illuminants (mixtures of two
complementary
monochromatic beams).
For typical illuminants (like
daylight) they are blacks.
The Ostwald
full color locus
(for the daylight
spectrum) in a
chromaticity
diagram.
73

Notice that:
The color solid depends upon the spectrum of the
illuminant (not just the color).
For instance, if you prepare a “white” illuminant from
two complementary monochromatic beams, the color
solid collapses to a planar parallelogram.
Conclusion:
The envelope of all color solids for a given white color is
the intersection of the spectrum cone and its inverted
copy from the white point.
74

W
K
W
K
The envelope of all color
W
solids for a given white
K
The “ultimate colors” form
the crease between the
spectrum cone & the
inverted spectrum cone
75

76
The spectrum double
cone volume is 1.79...
times that of the color
solid
The Ostwald double
cone volume is 0.84...
times that of the color
solid
The Schopenhauer
RGB-crate volume is
0.70... times that of the
color solid

“ultimate
color”
optimal
colors
full
color
inverted
spectrum cone
spectrum cone
generic
colors
white
a section
through the
achromatic axis
black
77

Spectrum locus (ultimate colors), Ostwald full colors and Schopenhauer
RGB “parts of daylight” as seen from the achromatic direction in a “true
picture” of color space
78

U
x
K
W
U c
Any color can be
written as the
interpolation between
black (K), white (W)
and some ultimate
color U (say):
x = c U + w W + k K
c : “Color content”
w: “White content”
k : “Black content”
79

As you vary “slit width” at
constant chromaticity you move
between black and white. Color,
white and black contents vary
There exists a unique “best” color
of maximum color content. It is
Ostwald’s “full color” or
“semichrome”
The edges of a semichrome are at
mutually complementary
wavelengths (hence the name)
80

100
75
50
25
0
G Y R M B C
The full colors
analyzed in
terms of the
ultimate colors
The cardinal colors analyzed in terms
of the ultimate colors. Yellow, blue and
red are the “best” colors, green
comes next, cyan and magenta are
embarrassing.

Mixtures of a full color with white and black
(Gustav Fechner’s “veiling triangle”)
82

Eisblau
Ublau-Eisblau
Ublau-Veil
Seegrün-Eisblau
Veil
Ublau 12 850, 50 25, 25 25< 25
Seegrün-Laubgrün
Rot-Veil
Laubgrün
Rot-Kress
Gelb-Laubgrün
Gelb-Kress
Kress
Ublau
12
The Ostwald atlas is a conceptual entity
You can program it from first principles
using the standard observer color
matching matrix
83

The Munsell atlas is a pure
eye measure construct
whereas Ostwald’s atlas is a
conceptual entity.
These objects have entirely
distinct ontologies.
(Notice in the drawing: Dark/black and bright/white
are confused, the branches have no natural length.)
84

Ostwald’s color atlas has nothing to do with “eye
measure”: From the standard observer tables you can
implement it on your computer.
85

The Ostwald construction is a conceptual one. Thus
there is no need to guard a master copy.
You can simply program it on your computer, all you
need is the matrix of the projection.
Unfortunately the color atlas in common use today (the
Munsell atlas) is an ad hoc eye measure construction.
It can only be “cloned” from a unique master copy.
The computer implementation (“CIE Lab*-space”) is an
awful mix of magical numbers and arbitrary functions
that somehow “fit” the eye measure.
86

Ostwald’s “Principle of
Internal Symmetry”

Ostwald’s “Principle of Internal Symmetry” defines a
canonical “bisection” of arcs along the color circle based
on color mixture
88

it can be shown that an “honest pie slices’’ method
yields equivalent results to Ostwald’s P.I.S.
89

The “Principle of Internal Symmetry” is defined in the
absence of a metric.
It can be defined in the context of Maxwell’s
colorimetry based on mere indiscriminability of beams.
Moreover it is fully independent of any “eye
measure” (thus not psychology!).
Physiologically it depends only on the action spectra of
the retinal photopigments.
91

The “Principle of
Internal Symmetry”
is easy enough to
apply.
Empirically, we
find that eye
measure results
differ as much
from each other
as either one
does from the
predictions of
inner symmetry.
470 nm
450 nm
380 nm
480 nm
490 nm
500 nm
510 nm
620 nm
660 nm
520 nm
530 nm
600 nm
540 nm
590 nm
550 nm
560 nm
570 nm
580 nm
92

Euclid explains how to obtain a regular scale by
recursive, mechanical bisection (internal symmetry):
It tends to “look regular” though the seen
intervals are of course qualia rather than
geometrical extents.
A filled interval “looks longer” then an empty
one of the same geometrical extent:
There is no necessary correspondence between
qualities and quantities.
93

The “ Wyszecki Hypothesis”
and how to make it come true

According to “Wyszecki’s Hypothesis” any spectrum s can be
uniquely decomposed in a “fundamental” and a “black” component:
s = f + k, where f is “causally effective’’ and k not (a “metameric
black”).
F
F ′
k
k ′
K
s s = f + k = f ′ + k ′
f
f ′
95
This is easily seen to
be false though:
Such a decomposition
can be made in
infinitely many ways!

There are two ways to force “Wyszecki’s Hypothesis” to be true:
• define a metric on the space of spectra: Then the unique
orthogonal complement of the black space is “fundamental space”
• promote some complement of the black space to “fundamental
space” by fiat
In either case you have infinitely many choices and you cannot
decide on colorimetric grounds.
In the former case you need considerations of physics
In the latter case you need to take properties of “perceived
color” (experience) into account
96

In the 1970’s Cohen developed the first method.
Unfortunately, he did not notice that he (implicitly!)
introduced a metric on the spectra.
His “Matrix–R” (projector on fundamental space is
defined as R = A ( A T A) -1 A T ( A the color matching
matrix), the transpose requires a scalar product.
The Cohen method is very powerful and popular in
applied settings.
It is hard to defend the choice of metric on the
grounds of physical optics though.
97

An obvious (though apparently unrecognized) alternative is to
promote an arbitrary three dimensional subspace (though
transverse to he lack space) to “fundamental space”.
This also makes “Wyszecki’s Hypothesis” come true: Any
spectrum can be uniquely decomposed into a “fundamental” and
a “black component”. Spectra that share the fundamental
component have the same color, the black components are fully
irrelevant.
This method does not require any metric, but is no less
arbitrary than Cohen’s choice. The choice can be made on the
basis of color science (not colorimetry!) rather than physics
though.

Since the Schopenhauer/Ostwald representation
rather closely approximates “eye measure”, it is as
good a candidate for “fundamental space” as any.
Then the Cartesian metric in 3D color space can be
extended to a metric for the space of spectra by
declaring all dimensions of the black space to be
isotropic (thus the metric is degenerate).
Mirabile dictu: Confined to color space the 1 st and 2 nd
methods yield essentially identical results.
99

1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0.2
400nm 500nm 600nm 700nm
400nm 500nm 600nm 700nm
400nm 500nm 600nm 700nm
a spectrum
its fundamental component
its black component

0.10
0.05
0.00
400
500
600
700
400
500
600
700
Different from Cohen’s Matrix–R the projector on
fundamental space is not symmetric, the projection is
oblique.
4
3
2
1
0
400nm
500nm
600nm
700nm
400nm
500nm
600nm
700nm

A few remarks on
“perceived colors”

colorimetry is not the same as
“Color Science” (which is not a science) 103

The squares need not be adjacent for the effect to occur.
These are problems of psychology, colorimetry yields no
handle on them.

“Color Science” starts
where colorimetry stops.
Runge traced trichromacy
to the Trinity.
105

thank you for your
attention
j.j.koenderink@phys.uu.nl
106