The Mathematical Description of Colour · Colour models Trichromatic models (RGB) Cylindrical (or...

The Mathematical Descriptionof Colour

Angel Jose Riesgo

Universidad de Oviedo

ariesgo@yahoo.com

Tuesday, 21 November 2017

Angel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 1 / 49

1 Introduction

2 Colour as a 3-dimensional quantity

3 Mixing colours

4 The 1931 CIE Colour Space

5 Measuring colour differences: the CIE LAB /

CIE LUV colour spaces

Introduction

Pixels: The building blocks of digital images

In this talk I will be talking about:

The geometric nature of colour

The various colour models: RGB, HSL,

CIELAB / CIELUV

The standard CIE XYZ colour space

Measuring differences between colours

Colour as a 3-dimensionalquantity

The oldest colour model: names!

Name-based colour modelsNatural language (“red”, “green”,

“cerveny”, “zeleny”)

Standardised codes based on a printed

reference model. Example: Pantone.

Such colour models are obviously not

appropriate for digital image processing.

Colours as three-dimensional vectorsColours are typically represented as points

in a 3-dimensional compact space C ⊂ R3

(or C ⊂ N3).

Example: the RGB colour model

Why is colour three-dimensional?

Two explanationsThe physiology of the human eye: three

types of cones.

Intuitively, colour can be decomposed in

three distinct elements.

And what are the three distinct elements that

make up colour?

Let’s begin with the spectral colours:

Spectral colours are defined by the wavelength of

the light:

From red (380 nm) to violet (760 nm)13.

This is a one-dimensional space [380, 760] (or

[0, 1]).

But it doesn’t account for all the colours!

Where is brown or maroon?

Compare the following two spectra:

The difference is in the intensity of the light. We

call this property luminance (also luminosity,

brightness or value).

Fixed wavelength and variable luminance:

We now have a second dimension, but there are

still missing colours.

Where is pink? ...Or white?

White light occurs when all the wavelengths are

mixed:

= + + + + + +

Through “whitening” we can get additional

colours:

= 0.5 + 0.5

Fixed wavelength and variable addition of white

light:

= λ + (1− λ)

This property is called saturation or chromaand is the third dimension of colour.

But whitened colours are not the only ones that

mixing wavelengths produce.

By mixing red and violet, we get an additional

range of quasi-spectral colours:

= λ + (1− λ)

These are the purples.

So, the spectral colours can be complemented

with the purples, giving rise to a colour wheel:

This way, we have completed the first dimension

of colour, the hue.

When we express colour with the three values

representing hue, luminance and saturation,

we have a general form of the HSL colour

model.

Such models are also often called cylindrical or

conical models.

Colour modelsTrichromatic models (RGB)

Cylindrical (or conical) models (HSL, HSV,

HSI, etc.)

Luminance-bichromatic models (CIE LAB,

CIE LUV)

There are various standard colour spaces for

each one of these general models.

Mixing colours

We saw that white light can result as a

combination of many wavelength values:

= + + + + + +

But also:

Any given colour can be replicated through

many alternative distributions of wavelengths.

For example, orange:

= 0.5 + 0.5

= 0.66 + 0.33

These are called metamers.

The existence of metamers indicates that colour

is a psychological phenomenon, not easy to

explain in purely physical terms.

Because of that, we cannot represent colour

simply as a distribution of wavelength values

plus luminance.

The 1931 CIE Colour Space

The 1931 CIE Colour SpaceBased on the experiments by D. Wright and

J. Guild in the 1920’s24

Standardised in 1931 by the CIE

(Commission internationale de l’eclairage)3

Wright and Guild’s experimental setupThree lamps with primary monochromatic

colours red, green and blue illuminating the

same area.

Additional lamp with variable

monochromatic colour illuminating an

adjacent area.

A volunteer modifies red, green and blue

intensities to match a fixed monochromatic

colour.

Video with a good explanation of Wright and

Guild’s experiments1:

https:

//www.youtube.com/watch?v=KDiTxWcD3ZE

Wright and Guild’s experiment results in the

colour-matching functions r(λ), g(λ), b(λ)7.

400 450 500 550 600 650 700 750

frequency

Negative values of r(λ) and g(λ) =⇒ Not all

spectral colous reachable from the R , G , B

primaries.

r(λ), g(λ), b(λ) + normalisation → CIE RGBspace (trichromatic model with “imaginary

colours”)

Further normalisation conditions (white point)

→ CIE XYZ space (luminance-bichromatic

model)

The X , Y (luminance), Z coordinates can

additionally be transformed into:

X + Y + Z

These x , y , z coordinates have no units. It is

common to use x and y together with Y for the

luminance.Angel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 32 / 49

In the xyY space, if we fix the luminance Y we

can represent the chromaticity on the xy plane.

This is the chromaticity diagram.

Chromaticity diagram

Image by user BenRG of Wikipedia, originally shared under a free licence(https://commons.wikimedia.org/wiki/File:CIE1931xy_blank.svg).

The chromaticity diagram gives us some very

powerful insights into the geometric nature of

colour.

The hue as boundary

Colour mixes as linear combinations

The reach of trichromatic models

During the 20th century, the many

device-dependent RGB colour spaces were

defined in terms of the CIE XYZ space.

Today, the most widely used RGB colour space

is the sRGB standard12.

sRGB is specified in terms of CIE XYZ through a

transformation matrix and γ correction

(non-linear stretching of the luminance11).

Many cylindrical and conical colour spaces such

as HSL, HSI, HSV, etc. have been defined based

on simple geometric transformations of a

reference RGB space.

They don’t fully separate luminance, chroma and

hue, and are problematic for digital image

processing.

A better alternative is the Improved HLS(IHLS) colour space by Allan Hanbury and Jean

Serra6.Angel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 40 / 49

Measuring colour differences:the CIE LAB / CIE LUV colour

spaces

Measuring the difference between two colours is

very important in digital image processing (for

example, when comparing the output of an

algorithm with the desired result).

Can we simply use the Euclidean distance on an

RGB-space?

As a broad approximation, yes. But there are

better colour spaces.

The perceived difference in colours is a

psychological phenomenon, so we need a

transformation of space based on experimental

In 1942, David MacAdam carried out some

colour perception tests with volunteers9.

The result of these tests were the MacAdamellipses on the chromaticity diagram.

Chromaticity diagram with MacAdamellipses

Image by user PAR commonswiki of Wikipedia, originally shared under a free licence.(https://commons.wikimedia.org/wiki/File:CIExy1931_MacAdam.png).

Mathematical problemDefine a coordinate transformation

XYZ → X ′Y ′Z ′ such that the MacAdam ellipses

become circles in the transformed space.

The problem is difficult. It cannot be solved

while preserving some of the fundamental

properties of the CIE XYZ space (like mixed

colours as linear combinations, Euclidean

geometry).

Two good approximations are the CIE LABand CIE LUV colour spaces (1976)14.

There are many software libraries that handle

these colour space conversions. In MATLAB

there are the rgb2lab and lab2rgb functions.Angel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 47 / 49

The problem of defining a good perceptual

colour distance remains open10.

Recent proposalsCIEDE2000 Colour formula8

J. Gravensen 20165

Thank you!

[1] Craig Blackwell. Video on YouTube.

https://www.youtube.com/watch?v=

KDiTxWcD3ZE

[2] A. D. Broadbent. Calculation from the

original experimental data of the CIE 1931

RGB standard observer spectral

chromaticity co-ordinates and color

matching functions. http://www.cis.

rit.edu/research/mcsl2/research/

broadbent/CIE1931_RGB.pdf

[3] CIE website - General Information

http://www.cie.co.at/index.php/Angel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 49 / 49

LEFTMENUE/About+us/General+

Information

[4] H. S. Fairman, M. H. Brill and H.

Hemmendinger. How the CIE 1931

colormatching functions were derived from

the Wright-Guild data. Color Res Appl

1997. Retrieved from

http://infocom.uniroma1.it/

~gaetano/texware/Full-How%20the%

20CIE%201931%20Color-Matching%

20Functions%20Were%20Derived%

20from%20Wright-Guild%20Data.pdf.Angel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 49 / 49

[5] Jens Gravensen. The metric of colour

space Graphical Models. Volume 82,

November 2015, Pages 77–86

doi:10.1016/j.gmod.2015.06.005

[6] Allen Hanbury, Jean Serra. A 3D-polar

coordinate colour representation suitable

for image analysis. Technical Report

PRIP-TR-077, TU Wien 2002.

http://cmm.ensmp.fr/~serra/

notes_internes_pdf/NI-230.pdf

[7] I. Kasajima, Plotting Colors on Color

Circle: Interconversion between XYZAngel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 49 / 49

Values and RGB Color System, The

Scientific Pages of Analytical and

Bioanalytical Chemistry, Volume 1, Issue

1, (2017). (http://

thescientificpages.org/Articles/

analytical-and-bioanalytical-chemistry/

the_scientific_journal_of_

analytical_and_bioanalytical_

chemistry_tsjabc-1-001.pdf)

[8] M.R. Luo, G. Cui, B. Rigg. The

Development of the CIE 2000 Colour

Difference Formula: CIEDE2000 ColorAngel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 49 / 49

Research & Application. Volume 26, Issue

5, October 2001, Pages 340–350

doi:10.1002/col.1049

[9] D. L. MacAdam. Visual sensitivities to

color differences in daylight, J. Opt. Soc.

Am. 32, 247 (1942).

[10] D. Malacara Color Vision and Colorimetry:

Theory and Applications, Second Edition

SPIE Press. 2011

[11] John Novak. What every coder should

know about gamma. Blog post. 2016.

http:Angel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 49 / 49

//blog.johnnovak.net/2016/09/21/

what-every-coder-should-know-about-gamma/

[12] M. Stokes, M. Anderson, S. Chandrasekar

and R. Motta. A Standard Default Color

Space for the Internet - sRGB https:

//www.w3.org/Graphics/Color/sRGB

[13] G. Waldman. Introduction to Light: The

Physics of Light, Vision, and Color

(Expanded and revised edition). Dover

Books on Physics, Mineola, New York

(2012).

[14] G. Wyszecki and W. S. Styles. ColorAngel Jose Riesgo (U. Oviedo) Colour spaces 21 November 2017 49 / 49

Science: Concepts and Methods,

Quantitative Data and Formulae. John

Wiley, New York (1982).

The Mathematical Description of Colour · Colour models Trichromatic models (RGB) Cylindrical (or...

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