TETRACHROMATIC COLOUR SPACES
Spherical and Toroidal Hue Spaces
Alfredo Restrepo Palacios
Dept. Ing. Electrica y Electronica, Laboratorio de Se˜nales, Universidad de los Andes,
Cra. 1 No. 18A- 12, ML-427, Bogot´a, Colombia
Keywords:
Tetrachromacy, Colour Space, Hypercube, Spherical Hue, Toroidal Hue.
Abstract:
From a 4-hypercube colour space, spaces of the type hue-saturation-luminance are derived. The hue compo-
nent may have the topology of a 2-sphere, a 2-torus or a 3-sphere, in several possibilities we consider.
1 INTRODUCTION
Analogously to RGB colour space and the spaces
of the type hue-saturation luminance you can derive
from it, we start from a tetrachromatic colour hy-
percube and derive several spaces of the type hue-
colourfulness-luminance.
2 TETRACHROMATIC COLOUR
SPACES
Four types of a photoreceptor in an eye of an animal,
four spectral bands in an imaging system or four light
sources with different but perhaps overlapping spec-
tra that shine over the same spot, give rise to a four-
dimensional Cartesian cube of possible responses, or
stimuli.
2.1 The WXYZ Colour Hypercube
The hypercube I
4
, where I = {0}∪(0,1) ∪{1}, has
16 vertices, 32 edges, 24 square faces and, in addi-
tion to its inner 4-cell, 8 solid (i.e. three-dimensional)
cubes. The union of the solid cubes is the polytope
{4 3 3}, in Scl¨afli notation, which is a topological
3-sphere that is the boundary of the hypercube. The
14 vertices that remain after dropping out the vertices
[0000] and [1111], are the vertices of an icositetra-
hedron of 24 triangles that result from dividing 12
of the square faces along certain diagonals (Restrepo,
2012). This chromatic icositetrahedron is used anal-
ogously to the chromatic hexagon of the RGB cube
(Restrepo, 2011) to define a tetrachromatic hue; each
triangle corresponds to one of the 24 possible permu-
tations of the tetrad [wxyz]. In addition, 16 of the 24
square faces can be chosen that form a piecewise lin-
ear (PL) torus; the torus in turn can be used to define
tetrachromatic hue in a different way.
2.2 A Polytopal, 4D HSL Space
The 24 possible orderings of the coordinate tetrad
[wxyz] determine 24 families of hue. To each point
in the hypercube there corresponds the range ρ and
the midrange µ of its coordinates [w, x,y,z]. The
range ρ = max{w, x,y,z}−min{w, x,y,z} is a mea-
sure of the distance from the point [w, x,y,z] to the
achromatic segment {[t,t,t,t] ∈ I
4
: t ∈ [0, 1]} and is
taken to be the chromatic saturation. The midrange
µ =
1
2
(max{w,x,y, z}+min{w,x,y,z}) is a measure of
the distance to the black point [0,0,0,0] and is taken to
be the luminance of the point.
The pair (µ,ρ) lives on an isosceles triangle of
base 1 and height 1. Each point not in the achro-
matic segment is assigned a unique hue point d on
the chromatic icositetrahedron; d results from an ini-
tial move, roughly towards [0000], along a direction
parallel to the achromatic segment, followed by an
expansion from the point [0000] until the chromatic
icositetrahedron is reached. Let p = [p
1
, p
2
, p
3
, p
4
] =
[w,x,y,z] be a point in the hypercube with ρ 6= 0, and
let m = min{w,x, y, z} and M = max{w,x,y,z}; then,
d =
1
ρ
[w,x,y,z]−
m
ρ
[1,1,1,1]. Now, p lies on the chro-
matic triangle having vertices [0 0 0 0], [1 1 1 1] and
d, and each of the points on this triangle, not on the
achromatic segment, are assigned the same hue point
d. To name the hue point d, give the vertices of a tri-
angle in the chromatic icositetrahedron that contains
423
Restrepo Palacios A..
TETRACHROMATIC COLOUR SPACES - Spherical and Toroidal Hue Spaces.
DOI: 10.5220/0003846604230426
In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 423-426
ISBN: 978-989-8565-03-7
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Table 1: Vertices of triangles in the chromatic dodecahedron
and the corresponding ordering of coordinates. In the fourth
column, it is indicated the positions of the coordinates that
are fixed in the three vertices. Note that each pair of con-
secutive triangles (including the last and the first) share 2
vertexes.
# order vertex 1 vertex 2 vertex 3 fcs
1 wxyz 0001 0011 0111 1, 4
2 xwyz 0001 0011 1011 2, 4
3 xywz 0001 1001 1011 2, 4
4 xyzw 1000 1001 1011 1, 2
5 xzyw 1000 1010 1011 1, 2
6 xzwy 0010 1010 1011 2, 3
7 xwzy 0010 0011 1011 2, 3
8 wxzy 0010 0011 0111 1, 3
9 wzxy 0010 0110 0111 1, 3
10 zwxy 0010 0110 1110 3, 4
11 zxwy 0010 1010 1110 3, 4
12 zxyw 1000 1010 1110 1, 4
13 zyxw 1000 1100 1110 1, 4
14 zywx 0100 1100 1110 2, 4
15 zwyx 0100 0110 1110 2, 4
16 wzyx 0100 0110 0111 1, 2
17 wyzx 0100 0101 0111 1, 2
18 ywzx 0100 0101 1101 2, 3
19 yzwx 0100 1100 1101 2, 3
20 yzxw 1000 1100 1101 1, 3
21 yxzw 1000 1001 1101 1, 3
22 yxwz 0001 1001 1101 3, 4
23 ywxz 0001 0101 1101 3, 4
24 wyxz 0001 0101 0111 1, 4
it, together with barycentric coordinates with respect
to these vertices; see columns 3 to 5 of Table 1. To
find this icositetrahedron triangle face, consider the
ordering of the coordinates of the point and use the
second column of Table 1. By construction, at least
one of the coordinates i
1
of the hue point d has value
0 (p
i
1
= m) and at least one of the coordinates i
4
has
value 1 (p
i
4
= M); the remaining two coordinates can
be similarly labeled so that p
i
2
≤ p
i
3
. The triangle
that contains the hue point has vertexes q, r and s
with q
i
1
= r
i
1
= s
i
1
= 0, q
i
4
= r
i
4
= s
i
4
= 1, and q
i
2
=
0,q
i
3
= 0, r
i
2
= 0,r
i
3
= 1 and s
i
2
= 1,s
i
3
= 1. The hue
point can now be expressed as d = αq+ βr+ γs, with
0 ≤ α, β,γ ≤ 1 and α + β + γ = 1; then, γ =
p
i
2
−m
ρ
,
β =
p
i
3
−p
i
2
ρ
, and α = 1−γ −β =
M− p
i
3
ρ
. The colour
attributes of the point p in this space are thus the hue,
the chromatic saturation and the luminance, given by
d, ρ and µ, respectively.
2.3 A Double-cone Type Space
The polytopal space of Section 2.2 is now trans-
formed to a double-cone type, 4D space, in the sense
that the chromatic icositetrahedron is rounded up to a
2-sphere. The hue point is now named with spherical
coordinates, with a latitude angle φ and an azymuth
angle θ. The chromatic icositetrahedron is made to
correspond to this hue sphere, where the vertex [0 1
1 1] corresponds to the north pole and the vertex [1 0
0 0] corresponds to the south pole, by assigning the
14 vertexes of the chromatic icositetrahedron to 14
points uniformly distributed over the sphere, as indi-
cated in Table 2. With φ ∈[0,π] and θ ∈ [0,2π) ∪{∗}
(φ = 0 and φ = π respectively correspond to the
north and south poles where θ is ”left undefined
as ∗” ) the 14 points uniformly distributed on the
sphere are those with (θ,φ) ∈ {(∗,0),(∗, π)} ∪
{(
π
3
,0),(
π
3
,
π
3
),(
π
3
,
2π
3
),(
π
3
,
3π
3
),(
π
3
,
4π
3
),(
π
3
,
5π
3
)} ∪
{(
2π
3
,0),(
2π
3
,
π
3
),(
2π
3
,
2π
3
),(
2π
3
,
3π
3
),(
2π
3
,
4π
3
),(
2π
3
,
5π
3
)}.
At the 6+6 triangles with a vertex assigned to a
pole, the computation of the spherical coordinates is
slightly different than at the remaining 12 triangles
where the barycentric coordinates are plainly used
with the help of Table 2. For the afore mentioned tri-
angles with a vertex [0111] or [1000], the barycentric
coordinates α,β γ are transformed to η = 1−γ (dis-
tance to ”vertex γ”) and ξ = 1−
β
1−γ
(angular measure
from line through ”vertex γ” and hue point, and line
through ”vertexes γ and β”), β is chosen so that the
angle increases when the hue point moves from ver-
tex β to vertex α, and γ is the barycentric coordinate
corresponding to the pole. For example, to the hue
point in triangle with vertices q = [0010], r =[0110]
and s =[1110] and corresponding barycentric coordi-
nates α = 0.6, β = 0.3, and γ = 0.1, there corresponds
(θ,φ) = α(2π, π/3) + β(5π/3,π/3) + γ(5π/3,2π/3)
= (5.6π/3,1.1π/3) and to hue point in triangle with
vertices q = [0100] and r = [0110] s = [0111] (north
pole) , and corresponding barycentric coordinates
α = 0.6, β = 0.3, and γ = 0.1, there corresponds
η = 0.9, ξ = 1/3 and (θ,φ) = (ξπ/3+ 5π/3,ηπ/3) =
(5.25π/3,0.4π/3). In this way, for the double-cone
type space, the colour components are the luminance
µ, the chromatic saturation ρ, the azymuth hue θ and
the latitude hue φ.
To convert back angular hue coordinates to
icositetrahedron coordinates, consider 3 cases: the
spherical triangle has no vertex at a pole, or it has
a vertex at the north pole (φ = 0), or it has the south
pole (φ = π) as a vertex. In the first case, we compute
barycentric coordinates of the point with respect to
the vertices (θ
i
,φ
i
) and then get the point (w
i
,x
i
,y
i
,z
i
)
by computing the corresponding barycentric combi-
nation using the corresponding vertices of the chro-
matic icositetrahedron. For triangles with a vertex at
the north pole, we use the map (θ,φ) 7→(θ
φ
π/3
,φ), and
compute the barycentric coordinates of the mapped
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
424

point with respect to the triangle with vertices (0,0),
((n − 1)π/3,π/3), and (nπ/3, π/3). For the south
pole we use the map (θ,φ) 7→ (θ
π−φ
π/3
,φ) and the tri-
angle with vertices (0,π), ((n − 1)π/3, 2π/3), and
(nπ/3, 2π/3).
Table 2: The 14 vertices of the chromatic dodecahedron are
made to correspond to 14 points uniformly distributed over
the sphere S
2
. The points on the sphere are indicated with
spherical coordinates θ ∈ [0,2π) and φ ∈ [0.π].
# vertex θ φ
1 0111 ∗ 0
2 0010 0 π/3
3 0011 π/3 π/3
4 0001 2π/3 π/3
5 0101 3π/3 π/3
6 0100 4π/3 π/3
7 0110 5π/3 π/3
8 1010 0 2π/3
9 1011 π/3 2π/3
10 1001 2π/3 2π/3
11 1101 3π/3 2π/3
12 1100 4π/3 2π/3
13 1110 5π/3 2π/3
14 1000 ∗ π
2.4 Runge Space
A ”round” space is obtained by deforming the hyper-
cube into the 4-ball {(w
′
,x
′
,y
′
,z
′
) ∈ R
4
: w
2
+ x
2
+
y
2
+ z
2
≤ 1}. Let [w,x,y,z] be a point in the hyper-
cube, shift the hypercube so that intermediate gray
ends up at the origin of 4-space R
4
and rescale so
that the maximum values of the coordinates is 1 and
the minimum is -1. Let [w
′
,x
′
,y
′
,z
′
] = 2[w−0.5,x −
0.5,y−0.5,z−0.5] be the coordinates of the resulting
hypercube [−1, 1]
4
.
The lightness in this space is given by the angle
with the achromatic axis: λ = arcos
w
′
+x
′
+y
′
+z
′
2
√
w
′2
+x
′2
+y
′2
+z
′2
= arcos
w+x+y+z−2
2
√
w
2
+x
2
+y
2
+z
2
+1−(w+x+y+z)
. Rather than us-
ing a chromatic saturation measure i.e. a distance
measure to the achromatic line segment, we use a dis-
tance measure from the central point of intermediate
gray and we obtain a measure of colourfulness in the
sense that it is a measure of ”ungrayness”. The main
difference with chromatic saturation is that any pri-
mary, say [1000], is as colourful as the black and the
white points [0000] and [1111]; in fact, any point on
the boundary of the hypercube is fully colourful. Let
Λ = max{|w
′
|,|x
′
|,|y
′
|,|z
′
|}; if Λ 6= 0, the point on the
boundary of the hypercube that is in the same direc-
tion is
1
Λ
[w
′
,x
′
,y
′
,z
′
] (at least one of its coordinates
has value of 1); let d =
1
Λ
p
w
′2
+ x
′2
+ y
′2
+ z
′2
and
normalize by this length (with the result that the hy-
percube is deformed into a 4-ball), getting the point
1
d
[w
′
,x
′
,y
′
,z
′
] whose distance from the center of the
ball is
κ =
√
w
′2
+x
′2
+y
′2
+z
′2
Λ
−1
√
w
′2
+x
′2
+y
′2
+z
′2
= Λ. Thus
κ = max{2w−1,2x−1,2y−1,2z−1}is the colour-
fulness of the point [w,x, y, z].
The hue is as in double cone space.
2.5 A Toroidal Hue Space
The eight cubes of the boundary of the hypercube can
be grouped into two solid tori in three ways; con-
sider for example the solid tori YZ = {z = 0}∪{y =
1}∪{z = 1}∪ {y = 0} and WX = {w = 1}∪{x =
0}∪{w = 0}∪{x = 1}. Each cube of solid torus YZ
intersects each cube of solid torus WX at a square
face; for example, the square {y = 0} ∩ {w = 1};
there are 16 such square faces where the solid tori
touch and together they form the PL chromatic 2-
torus. For each point p = [w,x,y,z] in the hypercube,
different from intermediate gray, find a correspond-
ing hue point h on the chromatic 2-torus; consider the
intersection e of the ray emanating from intermedi-
ate gray and going through p and the boundary of the
hypercube; call e the exit point from the hypercube.
Let [w
′
,x
′
,y
′
,z
′
] := [w −0.5, x −0.5, y −0.5,z −0.5]
and let χ =
1
2
1
max{|w
′
|,|x
′
|,|y
′
|,|z
′
|}
; the coordinate(s) of
f := χ[w
′
,x
′
,y
′
,z
′
] + [0.5, 0.5,0.5, 0.5] in the set {0,1}
indicate the solid cube(s) in the boundary of the hy-
percube e is; for example, the cube {w = 0} when
f
1
= 0. Then find the point h in the boundary of
the corresponding solid torus (WX in the example),
which is the chromatic 2-torus, that is closest to e.
The hue point is now best specified by the remain-
ing coordinate that is largest in magnitude; in the
example, take max{|y
′
|,|z
′
|} (otherwise, if the exit
point is in the torus YZ, take max{|w
′
|,|x
′
|}); this in-
dicates which face of the chromatic torus e is clos-
est to. In the example, if max{|y
′
|,|z
′
|} = |z
′
| and
z
′
< 0, then the closest face is {w = 0} ∩{z = 0},
if max{|y
′
|,|z
′
|} = |z
′
| and z
′
> 0, the closest face is
{w = 0} ∩{z = 1}. Once a face in the chromatic
torus that contain the hue point is identified, the point
p is assigned hue coordinates ω and η (angles mod-
1), as indicated in Table 3. The quantity ρ := χ
−1
measures the distance of the point p from intermedi-
ate gray, call it the chromaticity. To this measure of
distance of the max, there corresponds balls centered
at intermediate gary and with a shape that is a scaled
version of the hypercube. (When the max is taken
without shifting the hypercube, a measure of distance
TETRACHROMATIC COLOUR SPACES - Spherical and Toroidal Hue Spaces
425

to the point black results, and the balls are corners of
a hypercube.) The quantity τ = −2max{ |w
′
|,|x
′
|} or
τ = 2max{|w
′
|,|x
′
|}, depending on whether the exit
point is in solid torus YZ or WX, respectively, is a
signed measure of the distance of the exit point to the
chromatic torus; τ = 0 if e = h; |τ| measures the dis-
tance from the center of the corresponding solid cube
the ray leaves the hypercube. For example, the cen-
ter of the solid cube {w = 0} has equal contributions
from the primaries x, y and z and none of the primary
w; the point is ”a dark xyz”, on the other hand, the
center of the solid cube {w = 1} is a ”desaturated w”.
Table 3: Values of the coordinates ω and η depending on
the solid torus the hue point is.
# cube η ω
1 {x = 0}∩{ y = 1} z w
2 {x = 0}∩{z = 1} 0.25(y+ 1) w
3 {x = 0}∩{ y = 0} 0.25(z+ 2) w
4 {x = 0}∩{z = 0} 0.25(y+ 3) w
5 {w = 0}∩{y = 1} z 0.25(x+ 1)
6 {w = 0}∩{ z = 1} 0.25(y+ 1) 0.25(x+ 1)
7 {w = 0}∩{y = 0} 0.25(z+ 2) 0.25(x + 1)
8 {w = 0}∩{ z = 0} 0.25(y+ 3) 0.25(x+ 1)
9 {x = 1}∩{ y = 1} z 0.25(w+ 2)
10 {x = 1}∩{z = 1} 0.25(y+ 1) 0.25(w+ 2)
11 {x = 1}∩{y = 0} 0.25(z+ 2) 0.25(w+ 2)
12 {x = 1}∩{z = 0} 0.25(y+ 3) 0.25(w+ 2)
13 {w = 1}∩{y = 1} z 0.25(x+ 3)
14 {w = 1}∩{z = 1} 0.25(y + 1) 0.25(x+ 3)
15 {w = 1}∩{y = 0} 0.25(z+ 2) 0.25(x+ 3)
16 {w = 1}∩{z = 0} 0.25(y + 3) 0.25(x+ 3)
3 CONCLUDING REMARKS
Hering’s opponent process model for colour vision
explicitly uses the chromatic uniques red and green,
and yellow and blue and the achromatic uniques black
and white, rather than the primaies RGB. It has the ad-
vantage of giving yellow the important role it has in
our colour vision and the perception of natural scenes.
The transformation [R,G,B] 7→ [R−G, 0.5(R + G) −
Y,(R + G + B)/3] can be seen perhaps as a princi-
pal component analysis that enhances the information
contents of the code. It is not clear how to carry on
the primary-unique dichotomy to the tetrachromatic
case; it surely should depend on the application of a
tetrachromatic vision system. In principle, there are
several possibilities; for example, you could say that
y + z is a unique (analogously to the case of unique
yellow being an additive mixture of a spectral red and
a spectral green; i.e. that the channel M+L is the re-
ceptoral basis for perceptual unique yellow). That is,
you could propose a collection of channels that are
derived from receptoral signals w, x, y and z, perhaps
linearly.
R
4
can be partitioned into a collection of open 2D
flags with pole the line w = x = y = z, and the line
itself. For the hypercube, this translates into a parti-
tion into a collection of triangles with the achromatic
axis on one side and opposite vertex in the chromatic
icositetrahedron. The points on each of these trian-
gles, not in the achromatic segment, are said to have
the same hue; hue being then the orientation with re-
spect to the achromatic axis which in turn is deter-
mined in a discrete fashion by the ordering or relative
importance of the primaries w, x, y and z.
Alternately, the hypercube can be decomposed
into its central point plus a collection of intervals with
one extreme at the central point and the other extreme
in the boundary of the hypercube, a PL S
3
; the bound-
ary of the hypercube is then partitioned into two solid
tori joined by a 16-face 2-torus; each point of the
S
3
determines a possible hue and this tridimensional
hue is characterized in terms of the two angles that
describe the 2-torus, plus a linear variable that says
how deep within one of the solid tori the point of
S
3
is. The trichromatic corresponding analogous case
is not commonly used: it corresponds to considering
that each point on the 6 faces of the cube is a value
of hue; this results in a two-dimensional trichromatic
hue, characterized e.g. by latitude and azymuth (the
azymuth being a traditional trichromatic hue); the two
colour attributes would then be: hue and colourful-
ness (distance from intermediate gray); pretty much
along Runges’ model.
Tetrachromatic colour spaces such as these should
find applications in the visualization of 4-spectral im-
ages (e.g. satellite images) and in 4-spectral computer
vision systems. Also, the study of tetracrhomatic vi-
sion could benefit, e.g. in the production of visual
stimuli, and in the modeling of tetrachromatic percep-
tion.
REFERENCES
Restrepo, A. (2011). Colour processing in runge space.
In Electronic Imaging, San Francisco, January 2011.
SPIE.
Restrepo, A. (2012). Tetrachromatic colour space. In Elec-
tronic Imaging, San Francisco, January 2012. SPIE.
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
426