CIRCULAR PROCESSING OF THE HUE VARIABLE

A Particular Trait of Colour Image Processing

Alfredo Restrepo Palacios, Carlos Rodríguez and Camilo Vejarano

Laboratorio de Señales, Dpt. Ing. Eléctrica y Electrónica

Universidad de los Andes, Bogotá, Colombia

Keywords: Colour image processing, angular data, hue, median filter, range filter, von Mises distribution, mathematical

morphology.

Abstract: Novel tools for colour image processing are presented. Unlike many magnitudes dealt with in engineering,

the hue variable of a colour image is circular and requires a special treatment. Special techniques have been

advanced in statistics for the analysis of data from angular variables; likewise in image processing for the

processing of the hue variable. We give a definition of the median and of the range of angular data and

apply their running versions on images to smooth them and to detect hue edges. We also give definitions of

hue morphology; one based on the topological concept of lifting and on grey level morphology; another

definition is wholly given in a circular context.

1 INTRODUCTION

We consider a specific aspect of colour image

processing, namely the processing of the H variable

of the HVS colour system. The H variable is an

angular variable, i.e. one that lives in the circle

which is the one dimensional sphere S

, that we

interpret as the one-point compactification of the

interval (–π, π]. S

is orientable; we assume that the

orientation is positive when it is counterclockwise;

in this sense, colours are positively sequenced as in

red, orange, yellow, citrine, green, cyan, blue violet,

red etc.; see Figure 1. Unless otherwise stated, in the

examples, we assume V = S = 0.8 (constant value

and saturation). The tools given here can be

combined with other tools that process the V and S

variables.

The elements of S

cannot be linearly ordered in

any way compatible with its topology; thus, a

circular version of most statistics already defined for

linearly ordered data is usually not obvious. A

typical problem for the definition of location

statistics is that for certain uniformly distributed

samples it is best to leave them undefined; e.g. the

average of a sample of equally spaced angles. Also,

the problem of multiplicity is more ubiquitous than

in the case of a linearly ordered range space (e.g., in

the linearly ordered case, the median of an even

sized sample). We consider four sample statistics:

the circular average, a circular median, a circular

range and the circular concentration, and their

corresponding running versions.

Red=0

Yellow=0.5š

Green=š

Blue=-0.5š

Figure 1: The space S

of hues.

A 2D image (i.e. a 2D signal) h:A→B is a

function whose domain set A is two dimensional;

the image is discrete if A is countable, it is digital if

its range set B is finite. In the case of discrete

images, we use as domain set a product of integer

intervals. An integer interval /n, m/ is the set of the

integers less than or equal to m and larger than or

equal to n. We concentrate here on images whose

range set is S

; we call them hue images and picture

them as computer images in the HSV colour system

with S and V constant. The elements of the domain

set are called pixels and the elements of the range set

are the possible values of the pixels.

Several proposals for doing hue mathematical

morphology have been advanced e.g. (Peters II,

Restrepo Palacios A., Rodríguez C. and Vejarano C. (2007).

CIRCULAR PROCESSING OF THE HUE VARIABLE - A Particular Trait of Colour Image Processing.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IFP/IA, pages 69-76

 SciTePress

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1997), (Comer and Delp, 1999), (Hanbury and Serra,

2001), (Hanbury and Serra, 2001), (Vejarano, 2002)

and others. We give one based on the uniqueness of

the lifting of a map (a map is a continuous function)

with domain a simply connected space to the

universal cover of the range space (Christenson and

Voxman, 1998); another, based on intrinsic aspects

of the circular relation of angular data.

2 STATISTICS OF LOCATION

AND DISPERSION FOR

DIRECTIONAL DATA

Most of the time we assume angles (i.e. hues) to be

either numbers in the interval (-π, π] or numbers in

the interval [0, 2π); the choice by default is (–π, π].

Except in the case of liftings, we do not use multiple

code representations (equivalent mod-2π) for the

same angle.

It is convenient to use the complex number e

jφ

a representation the angle

φ. Assuming the function

arctan to have range (-π/2, π/2); the angle arg(z) of a

nonzero complex number (arg(z) is not defined if z =

0+j0) is defined as:

arg(z)= arctan(Im(z)/Re(z)) if Re(z)>0

arg(z)= –π + arctan(Im(z)/Re(z)) if Re(z)<0

arg(z)= π/2 if Re(z)=0 and Im(z)>0

arg(z)= –π/2 if Re(z)=0 and Im(z)<0

Let [

, … φ

] be a sample of N angles; unlike a

set, in a sample there may be repeated data. Let [z

… z

], with φ

=arg(z

), be the N-tuple of the

corresponding complex numbers on the unit circle.

We respectively call

Z :=

i=1

∑

and Φ = (

i=1

∑

) mod-2π

the complex sum and the angular sum of the angles.

Clearly, arg(Z) ≠ Φ. For example, consider 3π/4 and

5π/4.

2.1 Sample Mean or Average

We give the standard definition of the average of a

sample of angles; see e.g. (Nicolaidis and Pitas,

1998) and (www.higp.hawaii.edu, 2006).

Let [

, … φ

] be a sample of N angles; if their

complex sum Z is 0+j0, we leave the average of the

sample undefined, otherwise, the average of the

sample is set to be the angle arg(Z) of the complex

sum.

For the texture images at the top in Figure 2,

after the application of the running average (we use

windows of size 3x3) the images at the bottom

result. When the average of a window is undefined,

the central pixel was left unchanged. The textures

are based on the Von Mises distribution, given by:

f(φ) = (1/2πI

(κ)) exp(κ cos(φ–α))

where I

is a Bessel function that ensures that the pdf

integrates to 1, the parameter κ has to do with the

variance and α is the mean of the distribution.

Figure 2: A Von Mises’ texture images and a noisy image

(top) and the result after applying a moving average

(below).

2.2 Sample Median

As above, let [φ

, … φ

] be a sample of angular

data. Let d

be the distances between pairs (a pair is

a set of cardinality two) of consecutive angles φ

and

, given by d

= T(|φ

– φ

|), where:

T: [0, 2π) → [0, π]

has the graph indicated in Figure 3.

To find out which angles are consecutive, order the

angles in the sample, in their domain of

representation (-π, π], get the pairs of consecutive

angles in this ordering, and add an extra pair given

by the largest and the smallest angles in this interval.

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T(x)

0 π 2π

Figure 3: The function T, used to define a metric on S

If each component of the sample has the same

value, let the median be this common value. If the

sample is not constant, the set of distances {d

} has

a positive (i.e. larger than zero) maximum; if this

positive maximum is achieved for a unique pair of

angles

and φ

, subtract the 0-sphere {φ

, φ

} from

the 1-sphere S

; two connected components result

(Jordan theorem in dimension 1;) of these two

components, one contains the data; subtract the other

connected component, which we call the gap; the

resulting arc contains the data and and may now be

linearly ordered (see Figure 4); take the median of

the data in this arc, if the sample has even size, take

the (angular) average of the two central data.

Figure 4: The data determine a gap on S

, which is taken

off.

If the maximum of the distances between

consecutive angles is achieved more than once, the

definitions must be refined; see Figs. 5a. and 5b. For

one thing, the sample has several gaps of unique

length and it still may have a median. Proceed in two

steps to find it. Initially, for each of the multiple,

equally maximally sized gaps, obtain a median as

above. The resulting set of medians may further

have a median or it may not; it does not if this set of

preliminary medians determines again a unique

distance between consecutive points; in such a case

the sample is said to be uniformly spaced and to

have no median (nor a mean.) Otherwise, compute

the median of this set of preliminary medians, as

above, and call it the median of the sample.

A rule of thumb to check things is to slightly

separate repeated data. The definition given of the

sample median of angular data gives a unique

answer in cases when the median defined in

(Nicolaidis and Pitas, 1998) does not; see Figure 5c.

-a- -b- -c-

Figure 5: In a, there are three maximal gaps; in b, four. In

each case, they are of the same length.

In Figure 6, we show the result of applying a 3x3

hue median filter to the texture images at the top of

Figure 2.

Figure 6: The result of applying the median filter to the

images at the top in Figure 2.

2.3 Max and Min

Both the average and the median of a sample of hues

are hues and so are the max and the min as defined

below; nevertheless, the length of the gap, the

concentration and the range are angles but no hues:

they are differences of hues. Two different pairs of

hues may have the same difference and a sample

may have a range but no median.

Let [

, … φ

] be a sample of angular data. If

the sample is uniformly spaced or if there are a

multiple gaps, the max and the minimum of the

sample are left undefined; otherwise, for a unique

gap, the maximum (max) and minimum (min) of the

sample are defined as follows: take the gap off as

above and, in the remaining arc, let the point most

ccw (counterclockwise) be the maximum of the

sample and the point most cw (clockwise) be the

minimum. For example, for the sample [orange, red,

yellow], red is the min and yellow is the max.

2.4 Concentration and Range

Let [φ

, … φ

] be a sample of angular data; let the

concentration C of the sample be given by the

magnitude of the complex sum Z divided by N,

C: = (1/N)|Z|

CIRCULAR PROCESSING OF THE HUE VARIABLE - A Particular Trait of Colour Image Processing

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C ranges between 0 and 1, and clearly is a measure

of the concentration of the data. If the complex sum

of the sample is 0, the sample has no average but it

has (zero) concentration. The name given here to

this statistic is not standard (mean resultant length,

in (www.higp.hawaii.edu, 2006)).

For a constant sample, the range is set to be 2π;

otherwise, proceed as in Section 2.2 and define the

range ρ of the sample as:

ρ = 2π – max{d

} = 2π – length(gap)

ρ clearly is a measure of the dispersion of the data.

The moving range and moving concentration are

used to get maps of hue edges, from the image at the

top, left, in Figure 7.

Figure 7: Hue edge maps.

3 HUE IMAGES

A continuous 2D image (as opposed to a discrete

image as defined in Section I) is a function with

domain set a product I

×J of real intervals The

continuity referred to here is of the domain; thus, a

continuous image may be a discontinuous function.

If the range set is S

, the values of the function are

hues.

The existence of certain mathematical tools for

handling continuous functions makes the

corresponding images important from a theoretical

viewpoint. For example, the lifting of a continuous

function to a simply connected cover of the range

set, is unique. Thus, even though the discontinuities

of a function carry important information, it may be

convenient to restrict attention to continuous, and

even differentiable, functions. After all, the rates of

change can be arbitrarily large. In fact, the largest

possible jump of a hue image has value π and it

corresponds to a change from a hue to its opposing

hue, which may correspond as well to its

complementary colour. (Two colours are said to be

complementary if their additive mixture produces an

achromatic colour).

But, perhaps more to the point, in digital image

processing, one considers discrete images.

Moreover, in practice, discrete images are digital (as

defined in Section I); nevertheless, we disregard

here the discrete nature of the range set. The range

set of the hue component of an image, in all cases

will be assumed to be S

Since the basic definitions of mathematical

morphology are given in terms of the operations of

taking the maxima and minima of sets, it is assumed

that the range set has at least the structure of a

linearly ordered set (and the domain set, that of a

lattice). This is the case of grey level images, for

example, but not the case of hue images. We explore

below two approaches for performing morphology

on angle-valued functions; one is based on the

topological concept of the lift of a function while the

other works in the natural ambient space of the

graph of the function; both are based on grey level

morphology. Initially, we consider briefly the

geometry of the ambient space of the graph of the

hue function.

3.1 The Graph of a Function from a 2D

Interval to the Circle

Consider 1D continuous hue images. The graph of a

function f:I

→S

lives on I×S

which is a cylinder

and can be also thought of an annulus (the annulus is

homeomorphic but not isometric); if the function is

continuous, the graph is connected. See Figure 8.

RedGreen

Blue

Yellow

RedGreen

Yellow

Blue

Figure 8: Two pictures of the graph of the function

f:I→S

, corresponding to the coloured line, shown

discontinuous above.

The graph of a function f : I×J →S

lives on

×J×S

which can be thought of a solid cylinder; if

the function is continuous, the graph is connected.

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

Figure 9: The product I

×S

For each s ∈ S

the intersection K

of the graph of

f with the plane {(x, s) : xI

×J}  I×J×S

has as

connected components points, arcs, Jordan curves

and unions of these. We may give a partial order to

the set of the simple closed curves in

:= K

∗ ∂(I×J)

= {(x, s) : xI

×J, f(x) = s} ∗ ∂(I×J)

A simple closed curve C

bounds a region on its

interior (i.e. the region farthest from ∂(I

×J)); if

another Jordan curve C

in T

lies in this region, we

write C

< C

. This is a partial order.

3.2 Lifting Angular Maps

is the universal covering space of S

(Christenson

and Voxman, 1998), as projection map we have

p(x):= [x]

2π

where [x]

2π

is the only number in [0, 2π)

equivalent to x, mod-2π. See Figure 10.

F p

×J f S

Figure 10: R

is the universal cover of S

Now to lift a continuous function f:I×J→S

is to

find a continuous function F:I

×J→R

such that

p(F(x)) = f(x). If we specify F(0, 0) = f(0, 0), F is

unique.

The max minus the min of the values taken by a

lift F of a hue function f is said to be the degree of f.

It is rare that in an image of a common scene, the

degree of the hue be larger than 2π; it implies more

than one set of colours as in a rainbow, with the

same ordering of colours along some path.

To lift f is, in a sense, to unfold its graph. The

smallest curves in T

, according to the partial

ordering defined in Section 3.1, are the starting point

for an algorithm that finds the lift F. Find these

smallest simple closed curves and then define F on

the regions bounded by these components. Initially,

on these smallest regions set F:=f. Then, depending

on whether F is positive or negative on a region, for

the next region, define F as f+2π or f-2π. On a

region of order n, add or subtract 2nπ.

4 MATHEMATICAL

MORPHOLOGY FOR A

CONTINUOUS ANGULAR

VALUED FUNCTION

The (unique) lifting F:I×J→R

of a hue map

f:I

×J→S

, has as range set R

, which is linearly

ordered. We define MOP(f) as p(mop(F)); where

mop is a standard grey level morphological operator

(Heijmans, 1994), (Serra, 1998) and MOP is the

corresponding hue, morphological operator being

defined.

For example, the hue image in Figure 11

corresponds to the function with graph (plotted in a

flat 3D ambient space) as shown in Figure 12; the

lifting of this function is also shown in Figure 12;

applying (“grey level”) dilation/erosion and

opening/closing, with the structuring element in

Figure 13, and projecting back on S

with the

projection p = (mod 2π), we get the image shown in

Figure 11.

Figure 11: Hue morphology based on lifting.

As we see, for the given structuring element, in

the erosion, reds become violet, in general there is a

cw shift of hues which, for the given image, gives an

impression of migration of hues in the (opposite)

ccw direction (violets migrating toward red regions.)

CIRCULAR PROCESSING OF THE HUE VARIABLE - A Particular Trait of Colour Image Processing

Figure 12: Hue function and lifted version (false colours).

Figure 13: Structuring element.

4.1 Algorithms for Lifting Discrete

Images

For a function f:N×M→S

, where N and M are

discrete intervals, several algorithms can be

proposed to obtain functions F:N

×M→R

such that

p(F(x)) = f(x); however, the difference between two

such F´s (we chose to call them “lifts”) is not

necessarily constant. One kind of algorithm is based

on the idea of sweeping the domain set N

×M

another kind is based on the idea of interpolating the

discrete image and obtain a continuous image.

4.1.1 Lifting Along Paths

Consider initially a coloured line, that is, a function

with domain set an integer interval /0, N-1/. We start

at either one of the extrema of the interval, say at 0.

Initially, we set F(0) = f(0), to compute the

remaining values of F, we proceed as follows.

Assume F has been defined up to pixel i; next, let

i+1

= F

+ Δ(δ

) where δ

:= f

i+1

- f

and Δ is

defined as Δ(δ) = δ, if |δ| < π; Δ(δ) = 2π + δ, if δ < –

π; and Δ(δ) = δ – 2π if δ > π. Clearly, f = F (mod-

2π). The algorithm does not give a unique lift. The

lift obtained starting from the right may be different.

Nevertheless, the two lifts are equivalent, i. e. their

difference is a constant congruent with 0 (mod-2π).

Then, the operators of grey-level morphology can be

applied, before projecting back on S1, See Figure

14. Now consider 2D, discrete, hue images f: /1, N/

x /1, M/

→ S

; a lift F of f is such that F(i, j) = f(i, j)

mod-2π. Sweeping the domain set along a simple

path gives Δs between neighbours that depend on the

path chosen to do the lift.

Figure 14: A 1D hue function and its lift, a structuring

element and the resulting function, the original function

and the projected processed function. The original 1D

image and the resulting one on the right.

For example, consider the image and the two

sweeping paths below

0 1.5π π

0.5π 0 0.5π

π 1.5π 0

the corresponding resulting lifts are

2π 1.5π 1π 0 -0.5π π

2.5π 0 0.5π 0.5π 0 0.5π

3π 3.5π 4π π -0.5π 0

4.1.2 Interpolating Hue Images

The (e.g. linear) interpolation of a function defined

on a rectangular grid ZxZ to a function defined on

ZxZ is not uniquely defined; consider 4 neighbour

pixels as in the corners of the square, notice that

there are several, possibly conflicting, ways of

interpolating along the diagonals. From an image

with a very good resolution, that won’t loose much

from a process of decimation as it is redefined on a

(non regular) hexagonal grid, discard the pixels (i, j)

for which i+j is odd, as shown in Figure 15.

Figure 15: The decimation of a rectangular grid that give a

hexagonal grid.

On each triangle linear interpolation is used.

Once again, there are multiple possible linear

interpolations, some of them of the same cost. By

cost we mean the spent arc length on the circle.

Consider first interpolation along a line. Here there

are only two possible options: for x and y are angles

in [0, 2π), assuming x≤y, there are two choices:

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T(|x-y|) then the second choice is less expensive.

The parameter α varies between 0 and 1.

Let x, y and z be three pixels which are the vertices

of a triangle and let u=f(x), v=f(y) and w=f(z) be the

corresponding hues and assume that, as numbers in

[0, 2π), u≤v≤w. There are three possible linear

interpolations for the colours of the points in the

triangle; for each point in the triangle let λ

, λ

the (unique) barycentric coordinates of the point; the

interpolations are given by [λ

(u+2π)+λ

v+λ

2π

[λ

u+λ

v+λ

2π

and [λ

u+λ

v+λ

(w–2π)]

2π

. The

choice depends on the cost of each interpolation and

on the choices made for neighbour triangles.

Once a continuous function is obtained, the

image has a unique lift.

Figure 16: Interpolations of triangles given the vertices red

red green and violet-orange-green; and of lines given

extremes red-citrine and red-cyan.

4.2 Circular Mathematical

Morphology

An alternate way of applying morphological

operators to the hue component of colour images is

to interpret the max and min operators and the

addition operation in the context of a circular

variable, as in Section 2.3. We translate addition as

counter clockwise rotation, i.e. as addition of real

numbers followed by the operation of taking modulo

2π. Then, apply the standard definitions of grey level

image morphology. Occasionally, the data in, say a

3x3 window, have multiple gaps and there is not a

max nor a min, and a special treatment must be

given to the pixel at hand. For this sort of sample,

we choose to leave the corresponding pixels

unchanged.

As can be observed in Figure 17, in the erosion,

the blue of the wall became cyan (a negative shift)

while in the dilation, violet (a positive shift). The red

arrow in a yellowish background grew larger in the

erosion and thinner in the dilation. In Figure 18, two

Von Mises textures are shown; due to the shape of

the structuring element, the inner square grows and

becomes bluer with an erosion; with a dilation, it

shrinks and becomes more yellow; both textures lost

contrast.

Figure 17: Circular morphological operations.

Figure 18: Circular morphological operations.

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CIRCULAR PROCESSING OF THE HUE VARIABLE - A Particular Trait of Colour Image Processing

5 CONCLUSIONS

We have extended the definition and applied circular

versions of statistics commonly used for linearly

ordered data; in particular, the sample average,

median, gap, min, max, concentration and range. We

applied their running versions as smoothers and edge

detectors of colour images. As expected, for the

noises tried, the median filter works better visually

than the average. We consider that the edge detector

with best performance is the one given by 1-T(gap).

As in the case of the phase of complex numbers,

undeniable useful, in some cases these statistics

must be left undefined. Two methods to apply

morphological operators to angle valued signals are

presented. These novel tools for colour image

processing are likely to be useful.

Unlike previous versions of colour morphology, we

consider only the hue component, leaving the

components of saturation and value unaltered. Also,

we respect the circular nature of the hue variable

while taking advantage of grey level morphology.

The processing of the hue component alone

illustrates the effect of the tools which are particular

due to the circular nature of the hue variable.

The cases of undefined statistics and

morphological operators are more common,

although on a, say 5x5 window, it is probably hard

to find 25 hues uniformly distributed. We have

chosen to leave the corresponding pixels unaltered

but other choices are possible.

We have given algorithms for the interpolation

of hue valued functions on triangular meshes as well

as on 1D discrete domains. We found an unexpected

lack of algorithms for the lifting of angle functions,

this seems to be a fertile field of research.

The field of color image processing is important

in computer vision tasks such as the detection of

malaria in blood films (Ortiz et al., 2005) and also in

tasks where the aesthetic quality of the processed

image is important such as in commercial colour

photography and in digital document restoration.

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VISAPP 2007 - International Conference on Computer Vision Theory and Applications