Colour Processing in Tetrachromatic Spaces

Uses of Tetrachromatic Colour Spaces

Alfredo Restrepo Palacios

Laboratorio de Se

nales, Dpt. Ing. Electria y Electronica, Universidad de los Andes,

Cra. 1 No. 18A - 70, of. ML-427, Bogota, Colombia

Keywords:

Tetrachromacy, Multispectral Image, Visualization, Hue, 3-Sphere.

Abstract:

We exploit the geometry of the 4D hypercube in order to visualize tetrachromatic images.

1 INTRODUCTION

Tetrachromatic images i : N ×M → [0,1]

are images

where each pixel has four spectral components, each

component giving information regarding the energy

contents of the pixel in a given spectral band. We as-

sume that each component value of a pixel occurs in

the interval I = [0,1] and the total gammut of the pos-

sible colours a pixel can take can be modeled with

the hypercube I

, a 4D colour being a point [w,x, y, z]

of the hypercube. Two points of the hypercube are

the black (or ”schwartz”) vertex s := [0000], and the

white vertex w := [1111]; a subset of the hypercube is

A := {(t,t,t,t) : t ∈ [0,1]}, the achromatic segment

between s and w. See (Restrepo, 2012a) and (Re-

strepo, 2012b). Tetrachromatic images can be visu-

alized by feeding the RGB channels of a projector or

screen with 3 of the bands W, X, Y Z of the image, in

one or several of the of the 3!





= 24 possible ways

of doing this.

2 GEOMETRY AND 4D COLOUR

The (tridimensional) boundary ∂I

of the hypercube

⊂R

has a rich geometrical structure; it consists of





= 8 solid cubes with 16 vertices, 32 edges, and

24 square faces. A colour [w,x,y,z] is on T := ∂I

at least one of its coordinates is 0 or 1; each solid

cube consists of the points having a given coordi-

nate at value either 0 or 1; for example, the cube

{[w,x,y,z] ∈I

: w = 1}, which we denote as {w = 1}.

Indeed, we write ∂I

= {w = 0}∪ {w = 1}∪ {x = 0}∪

{x = 1}∪ {y = 0}∪ {y = 1}∪ {z = 0}∪ {z = 1}. ∂I

is a piecewise linear (PL) tridimensional sphere that

can be homeomorphed to a more standard, round S

In the 2-skeleton of the complex structure of T = ∂I

you ﬁnd 24 PL 1-spheres (one per face), 8 PL 2-

spheres and 3 PL Heegaard tori. Geometrically, these

manifolds can be used to deﬁne an orientation of the

points in the hypercube that, with corresponding coor-

dinate systems, is used to deﬁne several types of hue

for 4D colours.

2.1 Tint

To give spherical coordinates (d,Θ) to any point

p ∈ I

, denote the central point of the hypercube as

g = [

], let d be a measure of the distance be-

tween p and g (e.g. the max of the absolute values

of the components of p −g), and let Θ ∈ T be the

point where the ray from g through p leaves the hy-

percube. Call Θ the tint, or generalized hue, and call

d the colourfulness, or generalized saturation of p. In

this sense, T is the set of tints. Note that the vertices s

(”black”) and w (”white”) are fully colourful and are

tints.

2.2 Chromatic Hue

A pair of vertices of the hypercube is said to be a

pair of opposing vertices if the coordinates of one

are the ”negated” version of the coordiantes of the

other, for example, [0000] and [1111], or [0101] and

[1010]. Eight PL 2-spheres, that are dodecahedra

of square faces, result by considering the faces that

do not meet a given pair of opposing vertices. Each

of these 2-spheres serves as an equatorial 2-sphere

for ∂I

; for our purposes, the most relevant is the

one having as opposing vertices s and w. Call it the

chromatic dodecahedron D = {w = 0,x = 1} ∪ {w =

1,x = 0}∪{w = 0,y = 1}∪{w = 1,y = 0}∪{w = 0, z =

126

Restrepo Palacios A..

Colour Processing in Tetrachromatic Spaces - Uses of Tetrachromatic Colour Spaces.

DOI: 10.5220/0004289001260129

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2013), pages 126-129

ISBN: 978-989-8565-47-1

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

1}∪{w = 1,z = 0}∪{x = 0, y = 1}∪{x = 1, y = 0}∪{x =

0,z = 1}∪{x = 1, z = 0}∪{y = 0, z = 1}∪{y = 1,z = 0}.

Each of the faces of D has a ”primary” w, x, y or

z at its fullest value 1, and another at its minimum

value 0. Each square face of D is subdivided into

two triangles so that the points in each triangle obey

the same ordering of their coordinates; e.g. the

triangle with vertices [0,1,0,1] [0,1,0,0] [1, 1, 0, 1]

of points [w, x,y, z] with y ≤ w ≤ z ≤ x, and the

triangle [1,1,0,0] [0,1,0,0] [1,1,0,1] of points

[w,x,y,z] with y ≤ z ≤ z ≤ w, are the subdivision

of the face {x = 1,y = 0} of points [w, x,y,z] with

min{w,x,y,z} = y and max{w,x,y, z} = x. There

are 24 such ordering triangles; together, they give

the subdivision of D called the chromatic icositetra-

hedron IT; on each ordering triangle, the relative

contribution of the primaries is ﬁxed; each ordering

triangle represents a family of hue. To get the hue

family corresponding to a colour not in A, ﬁnd out

the permutation that orders its coordinates. More

precisely, the hue h of [w, x, y,z] is the point h in IT

that is obtained as h =

[w,x,y,z] −

[1,1,1,1] where

ρ is the chromatic saturation given by the range of

the primaries, and ν is the min. Each chromatic point

[wxyz] is in a unique chromatic triangle w −s −h.

Indeed [wxyz] = (1 −ζ)s + ρh + ν[1111] is an ex-

presion in barycentric coordinates [1 −ζ,ν,ρ] in the

plane spanned by the points s, w and h.

2.3 Hue in a Rhombic Dodecahedron

When the points of R

are projected along the di-

rection [1111] onto the 3-subspace (through the ori-

gin)

the chromatic dodecahedron projects, with-

out selﬁntersections, to a (2D) rhombic dodecahe-

dron

. The achromatic segment projects to the

central point of the rhombic dodecahedron and the

cubes in T project to overlapping parallelepipeds in

the (solid) rhombic dodecahedron. the orthonor-

mal points a = [

,−

], b =

[0,

,−

] and c = [0, 0,

,−

] are

a basis that gives 3D coordinates to the projection

space. The coordinates of the projections of the ver-

tices of D are shown in Table 1.

The abc coordinates of the intersection of the ray

This is computed by subtracting the average of the co-

ordinates from each coordinate.

The rhombic dodecahedron is a Catalan solid, i.e. a

polyhedron that is dual to an Archimedean solid; in this

case, to the cuboctahedron, which has 12 vertices, 24 edges,

8 triangle faces and 6 square faces; two triangles and two

squares meet at each vertex.

Table 1: The 14 vertices of the chromatic dodecahedron are

projected onto the 3-subspace normal to [1,1,1,1]. Then,

the projections are given 3-space coordinates in the third

column.

vertex projection [a, b, c]

0111 [−

] [-0.8660, 0, 0]

0010 [−

,−

] [-0.2887, -0.4082, 0.7071]

0011 [−

,−

] [-0.5774, -0.8165, 0]

0001 [−

,−

] [-0.2887, -0.4082, -0.7071]

0101 [−

,−

] [-0.5774, 0.4082, -0.7071]

0100 [−

,−

] [-0.2887, 0.8165, 0]

0110 [−

,−

] [-0.5774, 0.4082, 0.7071]

1010 [

,−

] [0.5774, -0.4082, 0.7071]

1011 [

,−

] [-0.2887, -0.8165, 0]

1001 [

,−

] [0.5774, -0.4082, -0.7071]

1101 [

,−

] [0.2887, 0.4082, -0.7071]

1100 [

,−

] [0.5774, 0.8165, 0]

1110 [

,−

] [0.2887, 0.4082, 0.7071]

1000 [

,−

] [0.8660, 0, 0]

from the center of the rhombic dodecahedron trhough

the projection of a chromatic point, and the bound-

ary of the rhombic dodecahedron, gives an alternate

hue η. The distance from the center of the rombic

dodecahedron to the projection point is a measure of

chromatic saturation σ; also, the projection [λ, λ, λ,

λ] on A of [w, x, y, z], λ :=

w+x+y+z

, gives a measure

of luminance. Thus σ =

+ x

+ y

+ z

−4λ

. In

this way an alternate colour space

to that with the ρµ

triangle results.

2.4 Tori

The tint of a colour p different from g is given by Θ =

g+χ(p−g) where χ =

2max{|w

|,|x

|,|y

|,|z

where w

w−0.5, x

= x−0.5, y

= y−0.5 and z

= z−0.5. The

indexes i of the coordinates Θ

of Θ = [Θ

,Θ

]

of value 0 or 1 indicate the cube Θ is at; for example,

if Θ

= 0, then Θ ∈{x = 0}.

A coordinate system for the points in an S

re-

sults by considering the Heegaard splitting of genus

1. It uses two angles and a ”signed radius” r ∈[−1,1],

rather than the better-known, spherical coordinates of

three angles. A Heegaard torus splits the 3-sphere

into two open solid tori and their common boundary.

Out of the 24 square faces, 16 faces can be chosen

that together are a Heegaard torus for T = ∂I

; this

can be done in three ways since the 8 cubes in T

can be grouped in





= 3 ways, into two groups

of four cubes each, so that each group is a solid torus.

To get a B&W image from a color image, in the trichro-

matic case, it gives better visual results to use the max (as

in the HSV colour system) than to use the average.

ColourProcessinginTetrachromaticSpaces-UsesofTetrachromaticColourSpaces

127

Here, we consider the solid tori V

:= {z = 0}∪{y =

1}∪{z = 1}∪{y = 0} and V

:= {w = 0}∪{x =

1}∪{w = 1}∪{x = 0}.

The boundaries of V

and V

are the torus H; H

can be seen as the union of four square pipe segments

in two ways; each pipe segment (topological cylinder

or annulus) is a stack of 1-squares that are meridians

for the solid torus in question and longitudes for the

other solid torus. For the solid torus V

we have the

pipes of square meridians with vertices

:= {(0, 0, s,0), (1, 0,s, 0), (1,1, s, 0),(0, 1, s,0) : s ∈ [0,1)} (z=0),

:= {(0, 0, 1,s), (1, 0,1, s), (1,1, 1, s),(0, 1, 1,s) : s ∈ [0,1)} (y=1),

:= {(0, 0, 1 −s, 1),(1, 0, 1 −s, 1),(1, 1, 1 −s, 1),(0, 1,1 −s,1) : s ∈[0, 1)} (z=1) and

:= {(0, 0, 0,1 −s),(1,0, 0,1 −s),(1, 1, 0,1 −s),(0,1, 0,1 −s) : s ∈[0, 1)} (y=0),

similarly, the boundary of the V

is given by the pipes

of square meridians with vertices

:= {(0,t,0, 0),(0,t,1, 0), (0,t,1,1), (0,t,0,1) : t ∈ [0,1)} (w=0),

:= {(t,1,0, 0),(t, 1,1,0), (t,1, 1,1), (t,1, 0,1) : t ∈ [0,1)} (x=1),

:= {(1, 1 −t,0,0), (1,1 −t, 1, 0),(1, 1 −t,1,1), (1,1 −t, 0, 1) : t ∈ [0,1)} (w=1) and

:= {(1 −t, 0, 0,0), (1 −t,0, 1, 0),(1 −t, 0,1,1), (1 −t,0, 0, 1) : t ∈ [0,1)} (x=0).

As we remarked above, H = ∪P

= ∪Q

. Each

point of T is either in the open solid torus T

, in the

open solid torus T

, or in their common boundary H.

The subindex n of the pipe segment together with the

value of t or s, as in n.t, or n.s, give an angular mea-

sure that ranges from 0 to 4, mod-4.

For Θ in an open torus, there is a distance r 6= 0

from the boundary of the solid torus the tint is at;

the distance from the boundary is measured with the

product metric; that is, for example, for the piece of

solid torus bounded by pipe P

, a tint point [w,x,t, 0]

is at distance 0.5 −max{|w −0.5|,|x −0.5|} from its

boundary. Also, there are two 1-squares in pipes

say P

and Q

with corresponding parameters s and

t such that one of them (a meridian) bounds a two-

square the tint is in, and the other intersects the ﬁrst

1-square at a point u on H that is closest to Θ

. Let

u = (φ, ψ) := (n.s,m.t) in H be the toroidal hue of

p. If Θ is on H, let r = 0. Denote Θ as (φ,ψ,r);

with the understanding that if r = ±0.5 (i.e. if Θ is

precisely on the axis or core of a solid torus), exactly

one of the angles φ or ψ is left undeﬁned and only the

longitude of the corresponding solid torus that con-

tains Θ is needed and a coordinate corresponding to

the meridian is left undeﬁned. For example, the tint

of [0.9, 0.2, 0.3, 0.4] is [1, 1/8, 1/4, 3/8] = (3.625,

2.875, 0.25), corresponding to pipes P

and Q

, with

s = 5/8 and t = 7/8.

On a disk, with the Euclidean metric, each point differ-

ent from the center has a unique point on the circle boundary

that is closest; in a square though, with the product metric,

for each point on the diagonals, there are two points on the

square perimeter that are closest to the point on the diago-

nals, 4 if it is the center of the square.

2.5 Spinning

We generalize Artin’s concept of spinning is spinning

with an S

to spinning with a sphere S

. Given a sub-

set E of R

(such as the ρµ triangle) with a closed sub-

set F (such as the µ edge), form the topological space

(E ×S

)/ ≈, where each set of the form {f }×S

f ∈F, is identiﬁed to a point. Artin’s method provides

a geometric embedding of subsets F of R

, in R

, as

{(x,y,z cos θ,z sin θ) : f = (x,y,z) ∈ F,θ ∈[0,2π)}.

2.6 Runge Ball

A 4D round space is obtained by deforming the hy-

percube into the standard 4-ball {(w

) ∈ R

+ x

+ y

+ z

≤ 1}. This can be done in several

ways; one is to spin the ρµ triangle, deformed to a

semicircle, around S

, with hinge the µ basis of the tri-

angle, where S

is derived from the chromatic dodec-

ahedron; another is to spin the midray (that that orig-

inates at intermediate gray) with S

, with hinge the

point of intermediate gray. In the ﬁrst case we have a

space with coordinates the luminance, the chromatic

saturation and a 2D (the equatorial sphere derived

from the chromatic dodecahedron) spherical hue; in

the second case, we have a space with coordinates

given by the generalized saturation r and a general-

ized 3D hue given vy the S

that is derived from the

boundary of the hypercube.

Let [w,x,y,z] be a point in the hypercube, shift the

hypercube so that intermediate gray ends up at the ori-

gin of 4-space R

and rescale so that the maximum

values of the coordinates is 1 and the minimum is -1.

Let [w

] = 2[w −0.5, x −0.5, y −0.5, z −0.5]

be the coordinates of the resulting hypercube [−1,1]

The lightness in this space is given by the angle

with the achromatic axis: λ = arcos

√

= arcos

w+x+y+z−2

√

+1−(w+x+y+z)

. Rather than us-

ing a chromatic saturation measure i.e. a dis-

tance measure to the achromatic line segment, we

use a distance g obtaining a measure of colour-

fulness in the sense of ”ungrayness”. Let Λ =

max{|w

|,|x

|,|y

|,|z

|}; if Λ 6= 0, the point on the

boundary of the hypercube that is in the same di-

rection is

] (at least one of its coordi-

nates has value of 1); let d =

+ x

+ y

+ z

and normalize by this length (with the result that

the hypercube is deformed into a 4-ball), get-

ting the point s = [s

] :=

]

whose distance from the center of the ball is κ =

√

−1

√

= Λ. Thus

VISAPP2013-InternationalConferenceonComputerVisionTheoryandApplications

128

κ = max{2w −1,2x −1,2y −1,2z −1} is the colour-

fulness of the point [w,x,y,z]. χ =

2Λ

3 PROCESSING

By colour processing a digital tetrachromatic image,

we mean the application of a law to each pixel in the

image, producing a new tetrachromatic image. The

image is then to be visualized or fed to a computer vi-

sion algorithm. By appropriately modifying the hue,

it is possible to visualize tetrachromatic images in

such a way that certain aspects are made conspicuous.

The linear (i.e. noncircular, nonspherical) coordi-

nates such as colourfulness, chromatic saturation and

luminance, are transformed via exponential-law maps

. The hue may be independently processed by au-

tomorphisms either of the 3-sphere, a hue sphere or

of a hue torus. As the hue surfaces are rotated or

otherwise automorphed, the colours of a tetrachro-

matic image may change in interesting ways when

trichromatically visualized. The automorphisms re-

spect the continuity; the rotations are isometries and

respect the antipodicity or complementary colours as

well. The simplest modiﬁcation type of the hue of 4D

colour is given by rotations of the 2-sphere, of the

3-sphere, or of the Heegaard torus. The rigid mo-

tions of s ∈ S

or equivalently, the rotations of R

are implemented by pre- and post-multiplying by unit

quaternions p, q, as in psq, s ∈ S

. The rigid mo-

tions of S

are implemented by pre and post multy-

plying a pure quaternion s times a unit quaternion q

and its conjugate, as in qsq

∗

. The space of rigid mo-

tions of S

has the group structure SO(4); it is the

topoloical space S

×RP

for which S

×S

is a dou-

ble cover.

The rigid motions of S

can be coded as a

pair (θ

,θ

) ∈ S

×S

in the sense that a unit quater-

nion is being pre and post multiplied by unit quater-

nions. The space H of the quaternions can be seen as

or as C

. For C

, the analogous case of an orthog-

onal transformation is that of a unitary transformation

that, rather than preserving the structrure of the in-

ner product in R

, it preserves the standard hermitian

form (z

).(w

) = z

¯w

+ z

¯w

. The set of uni-

tary transformations has the group structure SU(2). A

point of S

can be denoted as a pair (z

) ⊂ C

with

¯z

+ z

¯z

= 1.

For toroidal hue, for PL rotations, the 1D squares

with sides parallel to the axes w and x are meridians of

the yz solid torus and longitudes of the wx solid torus;

The set of rotations of the plane is the group SO(2)

which has the topology of S

while the set of rigid motions

of S

(of rotations of R

) is the group SO(3) which has the

topology of RP

Figure 1: p=[1/2, 1/2, -1/2, -1/2], q=[-1/2, -1/2, 1/2, 1/2],

γ = 0.6; bands 1 (in R), 3 (in G), 4 (in B). p=[1/2, 1/2, 1/2,

-1/2], q=[1/2, -1/2, 1/2, 1/2], γ = 1.0; bands 1, 2 and 3.

p=[1/2, 1/2, 1/2, -1/2], q=[-1/2, 1/2, 1/2, 1/2], γ = 1.0; bands

2, 3 and 4. p=[1/2, 1/2, 1/2, -1/2], q=[1/2, -1/2, 1/2, 1/2],

γ = 1.0; bands 1, 2 and 4.

the 1D squares with sides parallel to the axes y and z

are meridians of the wx solid torus and longitudes of

the yz solid torus. Similarly for the other cases. Shifts

around such squares implement modiﬁcations of hue.

4 CONCLUSIONS

Tetrachromatic colour spaces ﬁnd applications in the

visualization of 4-spectral images. Its use in satellite

imaginery (Landsat, 2012) is very likely providing al-

ternate ways to the mere feeding of the visualizing

RGB channels with permutations of the image wxyz

channels. Also, as a technique for computational pho-

tography, the explotation of IR and UV bands is likely

to be of use in different ways. Further work remains

to be done in the exploration of automorphisms of

spheres and tori different from isometries. Depend-

ing on the application diferent types of tetrachromatic

colour processing will be needed.

REFERENCES

Landsat (2012). http://zulu.ssc.nasa.gov/mrsid/tutorial/

landsat%20tutorial-v1.html.

Restrepo, A. (2012a). Tetrachromatic colour space. SPIE

Electronic Imaging, San Francisco.

Restrepo, A. (2012b). Tetrachromatic colour spaces. Visapp

Rome.

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