Round Colour Space for Pentachromacy

Circularity of Multidimensional Hue

Alfredo Restrepo Palacios

Laboratorio de Se˜nales, Dept. Ing. El´ectrica y Electr´onica, Universidad de los Andes,

Carrera 1 No. 18A-70; of. ML-427, Bogot´a 111711, Colombia

Keywords:

Pentachromacy, Hypercube, Chromatic Saturation, Luminance, Equatorial Sphere, S

, Permutations, S

Hamiltonian Circuit, Cayley Graph.

Abstract:

We generalize previous results to dimension 5 and further. The geometry of the 5-hypercube [0,1]

gives a

model for colour vision in the case of 5 photoreceptor types and a colour space corresponding to the combina-

tion of ﬁve primary lights. In particular, we focus on the (topologically spherical) boundary of the hypercube

and on an equatorial sphere within the boundary, roughly orthogonal to the achromatic segment. In the poly-

topal and double-cone type spaces, we consider a tridimensional hue component; in the round Runge space

we consider a 4-dimensional colourfulness component.

1 INTRODUCTION

The three properties of colour, luminance, hue and

chromatic saturation can be seen as geometric prop-

erties of points in the RGB cube; properties that de-

pend on the position and orientattion of the points in

the cube with respect to both the

black point

and the

achromatic segment

(Restrepo, 2011). This approach

generalises to dimension 4 (Restrepo, 2012a), (Re-

strepo, 2012b), (Restrepo, 2013b), (Restrepo, 2013a)

and to dimensions 5 and further. A model for colour

vision in the case of 5 photoreceptor types, or a colour

space corresponding to the additive combination of

ﬁve primary lights is presented here. The approach

allows to do pentachromatic colour image processing

and the study of pentachromatic metamerism.

Pentachromacy is the case of the visual system

of many animals, e.g. pigeons (David M. Hunt

and Davies, 2009) and dragonﬂies and ﬂies (Kelber,

2006).

For the visualization of multispectral images with

ﬁve bands, pentachromacyis likewise relevant as pen-

tachromatic colour processing followed by the RGB

visualisation of three out of the ﬁve channels makes

explicit important aspects of the image (Restrepo and

Maldonado, 2015). Likewise, in the screen ilumi-

nation industry, it is also useful to have models for

more than three primary lights (Shmuel Roth, 2010),

(Roger P. A. Delnoij, 2012) in the visible spectrum.

2 GEOMETRY AND COLOUR

We extend the geometric characterisation of trichro-

matic colour to a model for pentachromatic colour

that is based on the geometry of the hypercube [0,1]

This interpretation provides a basis for the processing

and visualisation of pentachromatic images as well as

a plausible model for the study of the colour vision

systems of pentachromatic animals.

Let the interval [0,1] model the set of possible in-

tensities of each of ﬁve primary lights in an additive

colour combination or, of the possible response levels

of each of ﬁve photoreceptors. In this way the

cubic

colour space

[0,1]

⊂ R

models the set of possible

primary combinations or of photoreceptor responses;

thus,

colours

are modelled as points in the hypercube.

The points of R

are denoted either as (v,w,x, y, z) or

as p = (p

, p

). The position of the colour

points, relative to an

equatorial

3-sphere σ in the 4-

spherical boundaryΣ := ∂[0,1]

of the 5-cube; σ is the

basis for the deﬁnition of tridimensional

pentachro-

matic hue

while Σ is the basis for the deﬁnition of

4-dimensional

pentachromatic kolor

The boundary Σ of the 5-cube is the set of

the colour points having at least one 0-valued co-

ordinate or one 1-valued coordinate. The points

, p

) ∈ Σ are classiﬁed into 5 × 2 = 10

4-cubes depending on which coordinate p

is equal to

1 or equal to 0; e.g. {0vwxyz} and {1vwxyz}. As a

matter of fact, Σ is a PL (piecewise linear) 4-sphere,

Restrepo Palacios A..

Round Colour Space for Pentachromacy - Circularity of Multidimensional Hue.

DOI: 10.5220/0005301900980105

In Proceedings of the 10th International Conference on Computer Vision Theory and Applications (VISAPP-2015), pages 98-105

ISBN: 978-989-758-089-5

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

a topological S

, that in addition to these 10 4-cubes,

can be

celled

into 40 3-cubes, 80 squares, 80 edges

and 32 points; these numbers result as follows: by

ﬁxing 2 of the coordinates p

and p

of points in the

5-cube with values in {0,1}, you get the 2





= 40

3-cubes of Σ; ﬁxing 3 coordinates in {0, 1}, you get

the 2





= 80 squares of Σ; ﬁxing 4 coordinates,





= 80 edges and, ﬁxing all 5 coordinates,

= 32 points or vertices.

Each of the 32 vertices of Σ, that is points

, p

) with p

∈ {0,1} is a vertex of





= 10 3-cubes in the cell complex; e.g. (00000)∈

{00xyz}. Each of the 80 edges is an edge of





6 3-cubes; for example, {v0000} ⊂ {v00yz}. Each of

the 80 squares is a face of 3 3-cubes; for example,

{vw000} ⊂ {vwx00},{vw0y0},{vw00yz}.

Out of the 40 3-cubes of Σ, 20 do not have neither

of the points s := (0 0 0 0 0) or w := (1 1 1 1 1) as

vertices (w and s stand for white and black (schwartz),

respectively.) The union of these 20 cubes is an

equa-

torial

for Σ, called the

hue sphere

, and denoted as

σ. The points of σ are precisely those points of Σ

having at least one coordinate at value 0 and at least

one coordinate at value 1. The equatorial σ is linked

in S

= R

∪ {∞} with the line (circle through ∞)

through the points s and w. The

achromatic segment

is the line segment that joins s and w and is given

by φ := {λs+ (1− λ)w : λ ∈ [0, 1]}; it consists of the

colour points having equal coordinates. In a sense that

is made precise below, loosely speaking we say that σ

and φ are orthogonal.

The equatorial σ, of codimension 2 in R

, is used

to deﬁne pentachromatic hue by giving coordinates to

the points of [0, 1]

on the basis of σ, i.e. by locating

the colour point with respect to σ (and φ). There are

other ways to deﬁne a pentachromatic hue as there

are other closed 3-manifolds in the cell complex Σ,

that are unions of 3-cubes of Σ but, in a sense, σ is the

canonical choice.

In fact, since each of the 20 3-cubes of σ can be

triangulated into 3! = 6 tetrahedra, according to the

6 possible orderings of the

free

3 coordinates (there

are two

ﬁxed

coordinates p

, p

, with p

, p

∈ {0,1},

in each such cube), each of the resulting 120 tetra-

hedra contains the points of σ having one of the 120

possible orderings of their (ﬁve) coordinates, where

the minimal coordinate has value 0 and the maximal

one has value 1. To each point in one such tetra-

hedron of σ, there corresponds a triangle of points

in the hypercube having such an ordering of their

coordinates, this time without the restriction of the

minimal coordinate being 0 and the maximal being

1, and the union of such triangles is the 5-simplex

of the points in the hypercube sharing such order-

ing of their coordinates. For example, the ordering

≤ p

corresponds to the tetrahe-

dron {p

= 0 ≤ p

≤ p

= 1} in the cube

= 0, p

= 1} ⊂ σ. Thus, in this triangulation of σ

into 20×6 = 120 tetrahedra, the interior of each tetra-

hedron precisely corresponds to each of the elements

of the symmetric group S

, which in turn corresponds

as well to the interior of the 5-simplex of points in the

5-hypercube having that ordering of their coordinate

values.

We say that a colour point h = [h

]

is a

hue

if at least one of its coordinates h

is 0-valued

and at least one of its coordinates h

is 1-valued,

that is, the points of σ are the possible

hues

of the

colour points not on the achromatic segment φ, called

chromatic colour points

. The set of colour points

having a given hue h is the

hue triangle

that is the

cone of the achromatic segment φ and the hue point h

given by {[γ + βh

,γ + βh

]

: β+ γ ≤ 1,0 ≤ β,γ ≤ 1}. There are 120

hue families

into which the hue points h ∈ σ can be classiﬁed.

All colours of a given hue family have the same

ordering regarding the relative contributions from

each of the photoreceptors, or primaries. An analogy

with the trichromatic RGB case is for example the

family of the oranges

, where R ≥ G ≥ B. In the

trichromatic case, a hue is either unary or binary, in

the pentachromatic case, a hue may be unary, binary,

trinary or quaternary, correspondingly depending

on whether it is a vertex, or is in the interior of

a segment, a triangle or a tetrahedron of the PL,

equatorial, hue sphere σ. Hues at the boundary of a

3-cube of σ belong to exactly two families.

The hypercube [0, 1]

is triangulated into 120 5-

simplices by taking the topological join of the achro-

matic segment φ and each of the 120 tetrahedra in the

triangulation of σ. (For our purposes, the join of two

subsets A and B of R

is the union of the line seg-

ments connecting points of A with points of B. ) Thus,

in each of these 5-simplices, the hues of the colour

points belong to the same hue family.

To each chromatic colour point p =

, p

) there corresponds a unique

hue

h ∈ σ, in fact, p belongs to the unique

hue

triangle

having as one side φ and as opposing vertex

the point h = (h

) ∈ σ with coordinates

− m

(1)

where m := min{p

: i ∈ 5} and ρ := max{p

: i ∈

5}−min{p

: i ∈ 5}; for example, the hue of p = [0.1,

0.3, 0.5, 0.7, 0.9] is h = [0, 0.25, 0.5, 0.75, 1].

RoundColourSpaceforPentachromacy-CircularityofMultidimensionalHue

The connectivity of the cubes and tetrahedra of

σ is topologically captured by a connectivity graph.

Consider the graph having as nodes the tetrahedra

(equivalently the elements of S

) of the chromatic

sphere σ, where two nodes are joined by a branch

precisely when the two corresponding permutations

differ by a transposition of two consecutive elements.

This corresponds to two tetrahedra being connected

by a triangular face. When a colour changes from a

hue family to another, the two corresponding order-

ings differing by a transposition of two consecutive

elements, we say that a

mild change

of hue family

has occurred. We show below a table corresponding

to a Hamiltonian circuit in the graph of hue families

connected by mild changes, that cycles through the

120 families. In this way, we give a cyclic order to

the hue families, analogous to the trichromatic case,

where you can give a cyclic order to the 6 hue fami-

lies and in fact (but this only in the trichromatic case)

a cyclic order to the hues themselves.

3 ROUNDER COLOUR SPACES

For colour processing, it is better to transform the hy-

percube into a rounder space. Initially, we obtain a

space of the ”hexcone” type, that although not round

is a starting point to obtain the rounder

double-cone

type

space and a round ball type space. Such spaces

are more intuitive and less prone, under colour trans-

formations, to end up at ”forbidden colours” due to

a careless change of coordinates. A useful geometric

technique to obtain rounder spaces is spinning.

The

spin

Sp(N,M,R

) in R

of a subset N of the

upper half plane of R

around

a set M ⊂ R

of dimen-

sion 1 with parametrisation M = {[m

(t), m

(t)] : t ∈

P ⊂ R

} is given by

Sp(N,M,R

) := {[x,y.m

(t), y.m

(t)] : (x,y) ∈ N}

Usually, the coordinates m

of M are taken to be in

the interval [-1, 1]; for example, the spin of the trian-

gle {[x,y] ∈ R

: x ∈ [−1,1],y = 1− |x|} around a cir-

cle in R

, is the double cone {[x,(1− |x|)cos(t),(1−

|x|)sin(t)] : x ∈ [−1,1],t ∈ [0,2π)} and the spin of the

same triangle around a hexagon is a hexcone. In a

sense, you are spinning the ﬂag of the upper half of

, that contains the triangle, with pole the x axis.

Similarly, the

spin

Sp(N,M,R

) in R

of a subset

N of the upper half plane of R

around

a set M ⊂ R

usually of dimension 3, with parametrisation M =

{[m

),m

)] :

) ∈ P ⊂ R

} is given by

Sp(N,M,R

) := {[v, w.m

),w.m

w.m

),w.m

)] : (v,w) ∈ N,(t

) ∈

P}.

The equatorial σ sits in R

in codimension 2. By

projecting σ onto the 4-hyperplane Π that is orthog-

onal to the achromatic segment φ you get again a PL

, denoted σ

∗

, this time embedded (in Π) in codi-

mension 1. The equatorial sphere σ has been made

”ﬂat”, and each ray in Π from the origin intersects a

circumscribing, round S

, which is key to getting a

round hue space.

In fact, for this projection π : σ → Π, π(σ) is an

embedding of σ. To see that it is injective, notice that

π(s) = π([s

]) = [s

−k,s

−

k,s

−k], where k is the average of the s

’s. Therefore,

if π(

s) = π(

s) then

s−

s =

k−

k, that is,

s and

s differ by a constant vector, but since the points of

σ have exactly one 0-valued coordinate and one 1-

valued coordinate, the remaining coordinates being

strictly between 0 and 1, the only possibility is that

such constant vector be the origin.

The function f : R

→ R

with

, p

) 7→ (µ,ρ), where ρ is the range of

the coordinate values and µ :=

max{p

:i∈5}+min{p

:i∈5}

is the midrange, bijects each of the hue triangles

(whose union is [0, 1]

) to a canonical, isosceles

”µρ-triangle”, also called the

luminance-saturation

triangle

T := f([0,1]

). T is an isosceles triangle

with base {(µ,ρ) ∈ R

: µ ∈ [0,1],ρ = 0} and height

{(µ,ρ) ∈ R

: µ = 0.5,ρ ∈ [0,1]}. We say that µ

measures the luminance of a colour on each hue

triangle and that ρ measures its chromatic saturation.

The 5-hypercube is the (union of the) collection

of the hue triangles, each of which has as common

side the achromatic segment φ and oposing vertex a

point in σ; the fact that the hue triangles have diffrent

shapes differentiates this construction of the hyper-

cube [0,1]

from a spin.

The spin of the µρ triangle around σ

∗

is (analog

to the the hexcone space) the 5D 120-cone H :=

Sp(T,σ

∗

The spin of the triangle T around the standard

(Euclidean, round) sphere S

⊂ R

gives the double-

cone type space. This space makes explicit the fact

that, corresponding to maximal 1 and minimal 0

values of luminance (at the vertices of the cone),

there is no hue nor chromatic saturation to be added.

By deforming the triangle T into a semicircle, D

and spinning it around the round S

of R

, you get

a solid ball B

, called Runge space. The process of

spinning adds the hue to the luminance and saturation

of T. In Runge space, the hue and the luminance are

made explicit but not so is the chromatic saturation;

instead, we call the distance from the central middle

gray the

kolorfulness

or vividness of a colour point

VISAPP2015-InternationalConferenceonComputerVisionTheoryandApplications

100

and the corresponding point in the S

boundary of the

ball the

kolor

or chromaticity ( deﬁned for achromatic

colours such as s and w as well) of the colour point.

The kolor is the point of the 3-sphere at which the spin

was made and the kolorfulness is the distance to the

central point

middle gray

of the 5-ball.

4 PENTACHROMATIC HUE

In the trichromatic case, the hue is a

cyclic variable

(Restrepo and Estupinan, 2014); in the tetrachromatic

case, the hue is bidimensional and spherical; in the

pentachromatic case the hue is tridimensional, spheri-

cal. In both the tetrachromatic and the pentachromatic

cases, it is not the hues but

families of hue

that can be

cyclically ordered. There are 24 families of hue in the

tetrachromatic case, and 120 families of hue in the

pentachromatic case.

The ordering of the relative contributions of each

of the ﬁve primaries giving rise to a coloured light

beam can be seen as a broad property of its colour;

colours are thus characterised as belonging to one of

120 possible

families of hue

. Likewise for the rela-

tive contributions of ﬁve photoreceptorswith different

photopigments at a small retinal spot.

In the case of human colour vision, the percep-

tual proper of uniqueness is made explicit (in Marr’s

jargon (Marr, 1982)) at V4 and not at the receptoral

level nor in the retina, the thalamus or V1 (Zeki,

1993). In fact, since the cone responses overlap (as

do the spectral responses of camera ﬁlters), it is not

possible to have a stimulus eliciting response from

only one cone type; instead of using the term unique-

ness of hue, to such hypothetical case, we refer to

as (receptoral) the

unariness

of a hue. Further reti-

nal, and cortical processing differentiates further the

hue in these hue families of a receptoral level; in our

trichromatic case, the receptoral family L ≥ M ≥ S

is subdivided into 2 cortical-hue families: oranges

and citrines (greenish yellows) separated by cortical,

unique yellow ( L = M). The cortical uniques and hue

families are then unique red, reddish oranges, central

orange, yellowish oranges, unique yellow, yellowish

citrines, central citrine, greenish citrines and unique

green.

Also in the trichromatic case, the vertices of the

hue hexagon

, an equatorial S

in the PL S

given

by ∂[0, 1]

, called the

chromatic hexagon

in (Re-

strepo, 2011), belong either to {red green, blue} or to

{yellow, violet, cyan} (at the receptoral level, yellow

is a binary colour). With the modiﬁcation of yellow

to a unique and the corresponding addition of the bi-

nary families orange and cetrine, the hue hexagon be-

comes an octagon with vertices {red, yellow, green,

blue} (the uniques) and segments {orange, cetrine,

cyan, violet} (the binaries). In the trichromatic case,

the

uniqueness

of a hue is the opposite of its

binari-

ness

: the closer you get to a unique hue (red, green,

yellos or blue) the farther you are from either of the

central binary

hues (orange, cetrine, violet or cyan).

Geometrically in σ, deﬁne

uniqueness

as the distance

to a closest unique and

binariness

as the distance to a

closest central binary hue.

In the pentachromatic case, a vertex of σ is said to

be a

unary

hue; at a point in the interior of an edge

you have contributions from two primaries and the

hue is said to be

binary

, on a triangle in the complex

σ, you have contributions from 3 primaries and the

hue is said to be

trinary

and inside (i.e. in the interior

of) a tetrahedron of σ, the hue is said to be

quater-

nary

. There are 5 unary hues, e.g. [10000], which are

the vertices of σ having 4 zero-valued coordinates, 10

binary hue families, e.g. [1w000] which are the seg-

ments of σ made of points having 3 0-valued ccor-

diantes; 10 trinary hue families, e.g. [1wx00], which

are the triangles of σ of points having 2 0-valued co-

ordinates and 5 quaternary hue families, e.g. [1wxy0],

which are the tetrahedra of σ having 1 0-valued coor-

dinate. Since a colour point with no zero coordinate

is not a hue (not an element of σ), there are no pentary

hues, such a colour has an achromatic component.

Every colour point can be writtten uniquely either

as an achromatic colour e.g. [0.1,0.1, 0.1,0.1,0.1], or

an achromatic colour plus a colour of a quaternary hue

e.g. [0.1, 0.2,0.2,0.2, 0.2], or an achromatic colour

plus a colour of a quaternary hue plus a colour of a

trinary hue, e.g. [0.1,0.2, 0.4,0.4,0.4], or an achro-

matic colour plus a colour of a quaternary hue plus a

colour of a trinary hue plus a colour of a binary hue

e.g. [0.1,0.2,0.4, 0.7,0.9], or as an achromatic colour

plus colour of a quaternary hue plus a colour of a tri-

nary hue plus a colour of a binary hue plus a colour

of a unique hue as follows: [0.1, 0.2, 0.4, 0.7, 0.9] =

[0.1, 0.1, 0.1, 0.1, 0.1] + [0, 0.1, 0.1, 0.1, 0.1] + [0, 0,

0.2, 0.2, 0.2] + [0, 0, 0, 0.3, 0.3] + [0, 0, 0, 0, 0.2], of

hues, correspondingly,

undeﬁned

, [01111], [00111],

[00011] and [00001].

For a given a 5-tuple q = [q

] ∈

[0,1]

there is at least one (of the 120 possible) per-

mutation p : {0,1, 2, 3, 4} → {0,1,2,3,4} such that

a nondecreasing ordering q

p(0)

≤ q

p(1)

≤ q

p(2)

≤

p(3)

≤ q

p(4)

results. As already said, in the equatorial

3-sphere σ, the set of 5-tuples so ordered by a given

permutation is a tetrahedron; thus the 120 permuta-

tions p determine a triangulation of the hue sphere

σ into 120 tetrahedra. Correspondingly, the 120 per-

mutations determine a triangulation of [0,1]

into 120

RoundColourSpaceforPentachromacy-CircularityofMultidimensionalHue

101

5-simplices where each 4-simplex is the topological

join of the achromatic segment and the corresponding

tetrahedron in the triangulation of σ.

The permutations have the structure of a group;

algebraically, they are the elements of the symmetric

group S

and, perhaps not an obvious fact, its ele-

ments can be

cyclically sequenced

so that two con-

secutive permutations differ by a transposition of con-

secutive coordinates (Johnson, 1963). From the view-

point of the wavelength domain, two consecutive co-

ordinates of a colour are better related than another

pair of ccordinates. The swapping of two consecu-

tive colour coordinates is called a

mild, hue-family

change

. Also, it so happens that the two tetrahedra of

σ that correspond to such a permutation have exactly

a triangular face in common; also, you can visit each

tetrahedra of σ exactly once going from one tetrahe-

dron to the next through such triangles. Equivalently

stated, there is a Hamiltonian circuit in the Cayley

graph of S

(Witte, 1982).

We say that each tetrahedron of σ determines a

family of hues in the sense that in each tetrahedron

the relative contributions of the primaries is ﬁxed.

Also, you have that the 120 hue families are cycli-

cally ordered. It is the families of hue rather than the

hues themsalves that are cyclically ordered, unlike the

trichromatic case where the hue sphere is triangulated

into 6 intervals (Restrepo, 2011), and each interval is

linearly ordered, the points within each tetrahedron of

σ do not have a linear order.

As we said in Section 2, the S

boundary Σ of

[0,1]

consists of 10 4-cubes; each such 4-cube con-

nects with 8 other 4-cubes via each 3-cube of its

boundary. For example, the 4-cubes {v = 0} and

{z = 1} connect via the 3-cube {v = 0,z = 1}. In the

corresponding connectivity graph, having as nodes

the 4-cubes and as branches the connecting 3-cubes,

there are several Hamiltonian circuits.

Also, the 120 tetrahedra of the S

equatorial hue

sphere σ can be grouped into 20 3-cubes; for exam-

ple, the 3-cube {v = 0,w = 0} groups the tetrahedra

{v = 0,w = 0,x ≤ y ≤ z}, {v = 0,w = 0,y ≤ x ≤ z},

{v = 0,w = 0,y ≤ z ≤ x}, {v = 0,w = 0,z ≤ y ≤ x},

{v = 0,w = 0,z ≤ x ≤ y} and {v = 0,w = 0,x ≤ z ≤ y}.

That is, σ can also be

celled

into 20 3-cubes, on

each of which, as in the trichromatic case, the colour

points can be given the trichromatic attributes of hue

saturation and value. Each 3-cube in σ connects with

each of 6 other 3-cubes via a 2-square. For example,

the 3-cubes {v = 1, x = 0} and {x = 0,z = 1} connect

via the square {v = 1, x = 0,z = 1}; there are several

Hamiltonian circuits in the graph corresponding to

this cellular decomposition of σ into 3-cubes. in such

a way that each 3-cube (the union of six tetrahedra)

is visited exactly once and every pair of consecutive

3-cubes have a square (the union of two triangles)

face in common.

We indicate an iterative procedure of obtaining

Hamiltonian circuits in the graph of S

. Let e

denote

the identity element of S

. In Figure 1 the permuta-

tions of two elements are used to get the permutations

of a set of three elements in such a way that you get

a factorisation, into transpositions of consecutive el-

ements, of e

from that of e

. The symmetric group

with three elements is generated by the transposi-

tions t

:= (0,1) and t

:= (1,2). Denote the action

of a transposition t acting on an ordered set, or triple

p = [p

, p

] as t(p), or simply tp; thus, for exam-

ple, (1, 2)[0, 1, 2] = [0, 2, 1]. From Figure 1, the

permutations of [0,1,2] are cyclically ordered as

Figure 1: The lines mustard and red represent the symmet-

ric group S

. A new element is positioned at each possible

position for after each transposition in the previous permu-

tation to get S

. The iteration of this gives a Hamilton circuit

on each S

[0,1,2]

[1,0,2] = (0,1).[0,1,2]

[1,2,0] = (1,2)(0,1).[1,2,0]

[2,1,0] = (0,1)(1,2)(0,1).[0,1,2]

[2,0,1] = (1, 2)(0, 1)(1, 2)(0, 1).[0,1,2]

[0,2,1] = (0,1)(1,2)(0,1)(1,2)(0,1).[0,1,2]

[0,1,2] = (1,2)(0,1)(1,2)(0,1)(1,2)(0,1).[0,1,2]

The last ordering being the initial one. Thus, at

the last line you have a factorization of the identity e

of the symmetric group S

= Π

3!−1

i=0

= (1,2)(0,1)(1,2)(0,1)(1,2)(0,1)

into 6 transpositions of consecutive elements, either

(0,1) or (1,2). In fact, the 6 permutations

:= Π

i=0

, j ∈ /0,5/

of S

are thus cyclically ordered. We simplify further

the notation by writing only the ﬁrst element of each

transposition in the factorisations, for example e

1.0.1.0.1.0.

The 24 permutations in S

are likewise cyclically

ordered. By interleaving the sequences of transposi-

tions 0:=(0, 1), 1:=(1,2), 2:=(2,3), 2:=(2, 3), and its re-

verse, alternatively in between each of the 6 transpo-

sitions in the factorisation of e

∈ S

and taking care

of adding a one in the notation of the transpositions in

, after the end of an inserted sequence 2, 1, 0, but

not after an inserted sequence 0, 1, 2, (extending the

idea in Figure 1) you get

VISAPP2015-InternationalConferenceonComputerVisionTheoryandApplications

102

= Π

4!−1

i=0

= [0.1.2.(0).2.1.0.(1+1).0.1.2.(0).2.1.0.(1+1).0.1.2.(0)

.2.1.0.(1+1)]

where the transpositions in parenthesis are those cor-

responding to e

; thus,

= [0.1.2.0.2.1.0.2.0.1.2.0.2.1.0.2.0.1.2.0.2.1.0.2]

where the T in Π

indicates a reversing in the order-

ing of the factorization Π. A cyclic ordering of the

elements of S

is then {π

| j ∈ {0,1,2,...23}}, with

= Π

i=0

This procedure can be generalized. A factoriza-

tion of e

∈ S

into 5 × 24 = 120 transpositions of

consecutive elements is

= Π

5!−1

i=0

= [α 0 α

1’ α 2 α

0’ α 2 α

1’ α 0 α

2’ α 0 α

1’

α 2 α

0’ α 2 α

1’ α 0 α

2’ α 0 α

1’ α 2 α

0’ α 2 α

1’ α 0 α

2’]

where α = 0.1.2.3 and a prime, as in n

′

, denotes

n+ 1, that is,

= [α 0 α

2 α 2 α

1 α 2 α

2 α 0 α

3 α 0 α

2 α 2

1 α 2 α

2 α 0 α

3 α 0 α

2 α 2 α

1 α 2 α

2 α 0

= [0 1 2 3 0 α

2 0 1 2 3 2 α

1 0 1 2 3 2 α

2 0 1 2 3

0 α

3 0 1 2 3 0 α

2 0 1 2 3 2 α

1 0 1 2 3 2 α

2 0 1 2 3

0 α

3 0 1 2 3 0 α

2 0 1 2 3 2 α

1 0 1 2 3 2 α

2 0 1 2 3

0 α

= [0 1 2 3 0 3 2 1 0 2 0 1 2 3 2 3 2 1 0 1 0 1 2 3 2 3 2

1 0 2 0 1 2 3 0 3 2 1 0 3 0 1 2 3 0 3 2 1 0 2 0 1 2 3 2 3 2

1 0 1 0 1 2 3 2 3 2 1 0 2 0 1 2 3 0 3 2 1 0 3 0 1 2 3 0 3 2

1 0 2 0 1 2 3 2 3 2 1 0 1 0 1 2 3 2 3 2 1 0 2 0 1 2 3 0 3 2 1 0 3]

and a ciclic ordering of the 120 elements of S

{π

| j ∈ {0, 1, 2,...119}}, with

= Π

i=0

, j ∈ /0,119/ (2)

5 CODING AND DECODING THE

HUE

In the 120-cone space, the luminance and the chro-

matic saturation are coded in the µρ triangle while the

hue is coded as a point in the PL hue sphere σ. This

code of σ is the basis for the deﬁnition of hue in the

of the double-cone type space.

The cyclic ordering of S

given by Equation 2, to-

gether with the initial ordering v ≤ w ≤ x ≤ y ≤ z,

denoted shortly as vwxyz, determine a cyclic ordering

of the orderings of the 5-tuples [v,w,x,y,z] of coordi-

nates. Thus, an ordering of [v,w,x,y,z] determines a

number j given by the position in the list (see Table 1)

of the cyclic ordering of S

, and viceversa. Equation

Table 1: A cyclic ordering of the 120 orderings of the 5-

tuple [vwxyz].

# ordering # ordering # ordering

0 vwxyz 40 vwzxy 80 vwyzx

1 wvxyz 41 wvzxy 81 wvyzx

2 wxvyz 42 wzvxy 82 wyvzx

3 wxyvz 43 wzxvy 83 wyzvx

4 wxyzv 44 wzxyv 84 wyzxv

5 xwyzv 45 zwxyv 85 ywzxv

6 xwyvz 46 zwxvy 86 ywzvx

7 xwvyz 47 zwvxy 87 ywvzx

8 xvwyz 48 zvwxy 88 yvwzx

9 vxwyz 49 vzwxy 89 vywzx

10 vxywz 50 vzxwy 90 vyzwx

11 xvywz 51 zvxwy 91 yvzwx

12 xyvwz 52 zxvwy 92 yzvwx

13 xywvz 53 zxwvy 93 yzwvx

14 xywzv 54 zxwyv 94 yzwxv

15 xyzwv 55 zxywv 95 yzxwv

16 xyzvw 56 zxyvw 96 yzxvw

17 xyvzw 57 zxvyw 97 yzvxw

18 xvyzw 58 zvxyw 98 yvzxw

19 vxyzw 59 vzxyw 99 vyzxw

20 vxzyw 60 vzyxw 100 vyxzw

21 xvzyw 61 zvyxw 101 yvxzw

22 xzvyw 62 zyvxw 102 yxvzw

23 xzyvw 63 zyxvw 103 yxzvw

24 xzywv 64 zyxwv 104 yxzwv

25 xzwyv 65 zywxv 105 yxwzv

26 xzwvy 66 zywvx 106 yxwvz

27 xzvwy 67 zyvwx 107 yxvwz

28 xvzwy 68 zvywx 108 yvxwz

29 vxzwy 69 vzywx 109 vyxwz

30 vxwzy 70 vzwyx 110 vywxz

31 xvwzy 71 zvwyx 111 yvwxz

32 xwvzy 72 zwvyx 112 ywvxz

33 xwzvy 73 zwyvx 113 ywxvz

34 xwzyv 74 zwyxv 114 ywxzv

35 wxzyv 75 wzyxv 115 wyxzv

36 wxzvy 76 wzyvx 116 wyxvz

37 wxvzy 77 wzvyx 117 wyvxz

38 wvxzy 78 wvzyx 118 wvyxz

39 vwxzy 79 vwzyx 119 vwyxz

1 gives the j-th ordering π

[vwxyz] of the 5-tuple; on

the other hand, to recover the number j from a per-

mutation of [vwxyz], we proceed as follows.

As a ﬁrst step,

reduce

a given ordering of

[v, w,x,y,z] to an ordering of [y, z] by deleting the let-

ters v, w and x from the given ordering; for exam-

ple, zyxwv 7→ zy; then, notice if this corresponds to

an even or an odd permutation (odd in the example)

of yz; next, reduce the initial permutation by deleting

RoundColourSpaceforPentachromacy-CircularityofMultidimensionalHue

103

only the letters v and w, obtaining a permutation of

[x,y,z]; in the previous example, you get zyxwv 7→ zyx

and notice if the reduced permutation (xyz 7→ zyx in

the example) is even or odd (again odd, in the exam-

ple) next notice the parity of the reduced-by-v permu-

tation (wxyz 7→ zyxw, even, in the example); ﬁnally,

check the parity of the original (complete) permuta-

tion (vwxyz 7→ zyxwv, even, in the example.) Notice

also, as if going from a reduced ordering to a less re-

duced ordering, the

position

at which the added ele-

ment is inserted; this position is sometimes measured

from left to right and sometimes from right to left. If

you are at an even ordering (e.g. 5432) the new ele-

ment (e.g. 1) is assumed to be inserted from left to

right (thus, in 54321, the 1 has been inserted at the 4-

th position) while, if you are based at an odd ordering,

the inserted element is assumed to have been inserted

from right to left (thus, 54 being an odd permutation

tells us that in 543, the 3 has been inserted at the 0-th

position). These positions are the weight factors t

the computation of the number j of the permutation,

that is,

j =

∑

k=2

(3)

where n = 5, and t

∈ /0, k − 1/ is calculated as

described above.

Given a colour point, for example [0.5, 0.4, 0.3,

0.2, 0.1], with Equation 3, we ﬁnd that we are in the

tetrahedron number 64 of σ; also, its luminance and

its chromatic saturation are given by µ =

0.1+0.5

= 0.3

and ρ = 0.5− 0.1 = 0.4. Next, we ﬁnd where in tetra-

hedron 64, the point [1, 0.75, 0.5, 0.25, 0] is (see

Equation 1); that is, we compute the barycentric co-

ordinates of point [1, 0.75, 0.5, 0.25, 0] in the tetra-

hedron with vertices [10000], [11000], [11100] and

[11110] which, in a sense, are the barycentric coordi-

nates of [0.75, 0.5, 0.25] in the tetrahedron with ver-

tices [1000], [1100], [1110] and [1111] of R

. Since

[1,0.75, 0.5,0.25,0] = α[1000] + β[1100] + γ[1110] +

δ[1111] and α+ β+ γ + δ = 1, you can get α,β,γ and

δ.

6 HUE PROCESSING

A simple yet useful way of processing pentachromatic

images is to process separately the hue. The remain-

ing pair of colour attributes of the luminance and the

saturation can likewise be processed separately using

e.g. exponential laws (A. Restrepo, 2009). The lu-

minance and saturation can be jointly processed with

ﬂows

of the points in the luminance-saturation trian-

gle. Likewise, the kolor can be processed as a rotation

or a homeomorphism of S

and the kolorfulness with

an exponential law, separately. In the remaining of

this section we concentrate on the separate process-

ing of the hue, by ”PL rotations” of σ, a tool for the

transformation of pentachromatic hue.

By a

PL rotation

of σ we mean the following. Ini-

tially, σ is projected on the 4-subspace Π of R

that is

orthogonal to φ. Call σ

∗

the embedding π(σ). Then,

Π is rotated; then, the rotated version of σ

∗

(a PL S

)

is ”ﬁtted back” (see below) on σ

∗

. Then, σ

∗

is back

projected to σ.

The projection π is linear and each tetrahedron of

σ projects to a tetrahedron of σ

∗

. A point of σ such

as h = [0,w, x, y,1], with average η =

v+x+y

, has an

image π(h) in σ

∗

given by h

∗

= [−η,w− η, x− η,y−

η,1− η] which has zero average. Notice that the or-

dering of the coordinates of h is the same as that of h

∗

deﬁnes the ray on R

given by {th

∗

:t ∈ [0,∞)}. To

each ray in Π ⊂ R

there corresponds a unique point

of σ

∗

(and viceversa).

The the rigid motions of S

⊂ R

(the rotations of

) can coded with the help of unit quaternions, as

in rot(s) = asb, with |a| = |b| = 1. The correspond-

ing group is known as SO(4). After ⊂ Π is rotated,

to each point h

∗

∈ σ

∗

there corresponds a new point

∗∗

∈ Π, not necessarily in σ

∗

. We ﬁnd an appro-

priately corresponding in σ

∗

as follows. The order-

ing of the coordinates of h

∗∗

determines a tetrahedron

of σ

∗

that contains the point intersection of the ray

through h

∗∗

and σ

∗

; this intersection point is the a

”rotated” version of h

∗

. We compute the barycen-

tric coordinates of this intersection point with respect

to the vertices of the tetrahedron. Then we com-

pute in the correspoding tetrahedron of σ, the cor-

responding barycentric combination, with the com-

puted coordinates gives the ”PL rotated” version of

h. The embedding of σ

∗

= π(σ) ⊂ Π of σ has an

inverse given by the addition of the constant vector

such that the resulting largest coordinate has value 1

and the smallest one has value 0. For example, con-

sider the hue h = [0,0.4,0.5, 0.6,1] of average 0.5 and

projection h

∗

= [−0.5,−0.1, 0, 0.1,0.5] that is rotated

to g

∗

= [0.5,0.1, 0, −0.1,−0.5] via multiplication by

the orthogonal matrix [00001; 00010; 00100; 01000;

10000]; the ray through h

∗∗

is in the terahedron of σ

with vertices 11110, 11100, 11000, 10000 which it

intersects at the same point h

∗∗

= g

∗

and that is back

projected to point [.5 .1 0 -.1 -.5] + [.5 .5 .5 .5 .5] = [1

.6 .5 .4 0]]

VISAPP2015-InternationalConferenceonComputerVisionTheoryandApplications

104

7 CONCLUDING REMARKS

A model for colour vision in the case of 5 photore-

ceptor types as well as a colour space corresponding

to the additive combination of ﬁve primary lights are

presented here. The approach allows to do pentachro-

matic colour image processing and the study of pen-

tachromatic metamerism.

Pentachromacy is relevant in the study of the vi-

sion systems of many animals, e.g. pigeons (David

M. Hunt and Davies, 2009) and dragonﬂies and ﬂies

(Kelber, 2006). Most mammals are dichromatic (dol-

phins are monochromatic)and old-world monkeysare

trichromatic; the reduction in cone type of mammals

is probably related to the fact that they evolved as

nocturnal animals. Among vertebrates, only some

rodents and marsupials take advantage of ultraviolet

light. Light of short wavelengths is less absorbed by

water and penetrates deeper in water.

By feeding three of the ﬁve channels of a pen-

tachromatic image to the RGB inputs of a visualis-

ing device, useful information can be made explicit.

This, combined with pentachroamtic colour process-

ing should provide with a useful tool for the search

of objects with given spectral surface reﬂectance in

pentachromatic images.

The visualisation of multispectral images can be

done in time; for this, continuous changes of hue

(therefore of SO(4)) are useful (Lenz and Homma,

1996).

The colour vision of animals such as pigeons and

dragonﬂies can be tested according to these colour

components of luminance, saturation, 3D hue, kolor

and kolorfulness with the models provided and a ﬁve-

primary illuminating system that include two types of

UV, in addition to RGB. Pentachromatic metamerism

can also be studied along the lines of (Restrepo,

2014b), (Restrepo, 2014a). This has applications in

illumination, photography and animal vision as well.

In the screen ilumination industry, it is also use-

ful to have models for more than three primary lights

(Shmuel Roth, 2010), (Roger P. A. Delnoij, 2012) in

the visible spectrum. It is a fact that three lights can-

not reproduce all colours, mainly when seen isolated;

nevertheless when colours are seen in context, chro-

matic contrast gives the illusion of many more colours

that those that can be projected in isolated form.

REFERENCES

A. Restrepo, S. Marsi, G. R. (2009). Hsv-domain enhance-

ment of high contrast images. Visapp Lisbon.

David M. Hunt, Livia S. Carvalho, J. A. C. and Davies,

W. L. (2009). ??evolution and spectral tuning of visual

pigments in birds and mammals. In TEMPLATE’06,

1st International Conference on Template Production.

Phil. Trans. R. Soc. B.

Johnson, S. (1963). Generation of permutations by adjacent

transpositions. Math. Comput. 17, 83, 282-285.

Kelber, A. (2006). Invertebrate colour vision. In Inverte-

brate Vision, E. Warrant and D.E. Nilsson eds. Cam-

bridge University Press.

Lenz, R. and Homma, K. (1996). Rotational symmetry: The

lie group so(3) and its representations. In ICIP 96.

Marr, D. (1982). Vision. W.H. Freeman and Company, San

Francisco.

Restrepo, A. (2011). Colour processing in runge space.

SPIE Electronic Imaging, San Francisco.

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

Electronic Imaging, San Francisco.

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

Rome.

Restrepo, A. (2013a). Colour processing in tetrachromatic

colour spaces. Visapp Rome.

Restrepo, A. (2013b). Hue processing in tetrachromatic

spaces. SPIE Electronic Imaging, San Francisco.

Restrepo, A. (2014a). Tetrachromatic metamerism. Visapp

Lisbon.

Restrepo, A. (2014b). Tri and tetrachromatic metamerism.

SPIE Electronic Imaging, San Francisco.

Restrepo, A. and Estupinan, V. (2014). Color visualisation

of cyclic magnitudes. SPIE Electronic Imaging, San

Francisco.

Restrepo, A. and Maldonado, E. (2015). Visualisation of

tetrachromatic images. SPIE Electronic Imaging, San

Francisco.

Roger P. A. Delnoij, K. P. E. N. (2012). Illumination system

with four primaries. In US patent US 8174210 B2.

Shmuel Roth, G. C. T. L. (2010). Multi-primary dis-

play with spectrally adapted back-illumination. In US

patent US7714824 B2.

Witte, D. (1982). On hamiltonian circuits in cayley dia-

grams. Discrete Math. 38, 99-108.

Zeki, S. (1993). A Vision of the Brain. Blackwell Scientiﬁc

Publications, London.

RoundColourSpaceforPentachromacy-CircularityofMultidimensionalHue

105