chapter 2 transformations

7/30/2019 Chapter 2 Transformations

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Chapter 2 Colour Spaces and Transformation

Algorithms

Colour Spaces and Transformation Algorithms

2.1 Introduction

In this chapter the Red-Green-Blue (RGB) colour space will be presented and the

transformation algorithms to alternative colour spaces will be described. Colour spaces can be

grouped into their applications and can be categorised as follows:

For display technology, a model suited to display a diversity of colour based onmixtures of light intensities to produce a range, orgamut, of hues.

For printed matter where pigments are combined together to generate a diversity of

colours.

For television technology where the priority is minimising the bandwidth and

maintaining backward compatibility with black and white (B&W) TV sets.

For artists who derive the way mind processes colour into perceptual colour models

where the elements of colour (hue, saturation and lightness) can be separated.

In industry that needs to standardise colour in order to be uniform when describing or

matching colour.

2.2 The RGB colour cube

The RGB colour model is primarily used in colour Cathode Ray Tube (CRT) monitors, raster

graphics devices and colour cameras. It is by far the most common used technology for


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computer monitors. The RGB primaries are additive, which means that individual contributions

of each primary are added for the creation of a new colour via the modulation of intensities of

light. The model is based on the tristimulus theory of vision and is a hardware-oriented model.This model employs a Cartesian coordinate system to give the location of every colour

combination. It can be represented using a normalised colour cube as depicted in Figure 2.1b.

Achromatic pixels lie on the line that extends from Black (0,0,0) to White (1,1,1), which is

known as the grey-scale line. Equal contributions ofR, G and B ( BGR == ) will always

produce an achromatic, or colourless, pixel. Any pixels falling out of the grey-scale line are

known as chromatic, or coloured, pixels. The six coloured corners of the cube in Figure 2.1b

show the primary colours Red (1,0,0), Green (0,1,0) and Blue (0,0,1), whilst the complementary

colours are Yellow (1,1,0), Cyan (0,1,1) and Magenta (1,0,1). Figure 2.1b also contains an

inner cube, 50% smaller than the external, which shows how colours get darker as they are

mixed with equal amounts of black and white and move towards point (0,0,0). Figures 2.1a and

2.1c illustrate what the outer cover or skin of each individual cube would look like. These

covers are also projected and rotated as viewed from point (1,1,1) towards (0,0,0). It is useful to

see that a hexagon has been generated which is the basis of the perceptually oriented colour

systems described in Section 2.3.3.

Figure 2.1: The RGB colour cube including its surrounding skins for the outer shell a) and

inner shell c) cubes shown in b)

Cyan (0,1,1)

Green (0,1,0)

Yellow (1,1,0)

Blue (0,0,1)

Red (1,0,0)

White

(1,1,1)

Black

(0,0,0)

Magenta (1,0,1)

a) b) c)


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2.3 Conversions from RGB to alternative colour spaces

Before introducing the conversion from RGB to alternative colour spaces or models, the

terminology that will be used throughout the rest of the chapter to describe the individual

components of colour will be described [Poy99].

Hue: The apparent colour of light, which is determined by the dominant wavelength as

perceived by the observer. It represents the redness, greenness and blueness,

depending on the relative mixtures and it is given as an angle 0 to 360.

Saturation: Is the purity of the colour or the proportion of white contained in the colour. Lightness: Is the amount of energy an observer perceives from a light source (self-

luminous objects).

Value/Brightness: Is the amount of black contained in a colour. The colour with the

highest intensity determines the amount of black. 100% would therefore be the

brightest colour containing no black, 0% would be black (reflecting objects).

Intensity: The total light across all frequencies, that is a measure over some interval of

the electromagnetic spectrum of the flow of power that is radiated from, or incident on

a surface.

Luminance: According to CIE [CIE86] it is a radiant power weighted by a spectral

sensitivity function that is characteristic of human vision.

2.3.1 RGB to television colour spaces

These colour spaces are widely used for the broadcast of TV signals throughout the world.When colour TV was first introduced, reduction of bandwidth and backward compatibility with

B&W TV sets was a very significant consideration. The human eye is far more insensitive to

colour changes than changes in luminance [Hea94], this enables the chrominance bandwidth to

be significantly smaller than that for the luminance. There are 3 colour spaces employed in TV:

YIQ, YUV and YCrCb. The Ycomponent is the luminance that is used in B&W television sets.

Also, two chrominance components are contained based on colour difference signals: R-G (i.e.

Red Green) and B-Y (i.e. Blue Yellow), which are two of three colour difference signals

existing, the third can be derived from the other two. TV colour spaces present advantages in


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the sense that the transformation needed to obtain them are simple and a separation of

luminance and colour can therefore be obtained and used without being correlated.

2.3.1.1 RGB to YIQ

This colour space is the basis of the National Television System Committee (NTSC) colour

broadcasting TV signal coding system used in countries such as Canada, US, Mexico and

Japan. It decouples the luminance channel Y from the chrominance channels Iand Q, which

convey the required information on hue and saturation of the colours. I is known as the In-

Phase signal and operates in the orange/cyan region whilst Q, the Quadrature operates in the

green/magenta region. The steps involved making a transformation from RGB to YIQ is as

described next [Wu96].

The Luminance component Yis obtained giving individual weights toR, G andB based on how

they are perceived by a human observer and is given by:

BGRY 1144.05866.02989.0 ++= (2.1)

In order not to overload the transmitter with the modulated colour difference signals when they

are added to the luminance signal, an amplitude reduction is done and labelled Uand Vsignals

(i.e. weighted colour difference signals):

)(493.0 YBU = (2.2)

)(877.0 YRV = (2.3)

To minimise the bandwidths of the quadrature-modulated signals, U and V are not directly

modulated on to the blue and red sub-carrier phases. Instead they are advanced by 33 and two

further signals are obtained,Iand Q:

= 33sin33cos UVI (2.4)

+= 33cos33sin UVQ (2.5)


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By directly combining Equations (2.1) to (2.5) the following results will be obtained:

BGRYBYRI 3218.02741.05959.0)(27.0)(74.0 == (2.6)BGRYBYRQ 3113.05227.02113.0)(41.0)(48.0 +=+= (2.7)

Placing Y,Iand Q in a matrix form, yields [Ber87]:

=

B

G

R

Q

I

Y

*

311.0523.0211.0

313.0274.0596.0

114.0587.0299.0

(2.8)

YIQ can be transformed into an IHS (see Section 2.3.3.1) or analogous spaces for diverse

colour image processing applications [Yan96] obtaining Hue (H) and Saturation (S). This is

done by using the next cartesian to polar conversion equations:

)/arctan( IQHIQ = (2.9)

22 QISIQ += (2.10)

2.3.1.2 RGB to YUV

YUV is the basic colour space used for Phase Alternating Line (PAL) and Sequential Colours

with Memory (SECAM) TV broadcasting in countries such as United Kingdom, Germany,

Brazil, France, etc [Rob97]. Yis the luminance channel and is obtained as in equation (2.1). U

and Vare the chrominance components calculated from the colour difference signals (B-Y) and(R-Y) and are the same as for YIQ that were obtained using Equations (2.2) and (2.3):

BGRYBU 437.0289.0147.0)(493.0 +== (2.11)

BGRYRV 100.0515.0615.0)(877.0 == (2.12)

Rearranging and defining Y, Uand Vin a matrix form yields:


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=

B

G

R

V

U

Y

*

100.0515.0615.0

437.0289.0147.0

114.0587.0299.0

(2.13)

As with YIQ, YUV can be transformed into a pseudo-IHS colour space obtaining Hue Hand

Saturation Sby using the following cartesian to polar conversion equations [For98]:

)/arctan( UVHUV = (2.14)

22 VUSUV += (2.15)

2.3.1.3 RGB to YCrCb

This is an independent coding system, which is appropriate for digital coding TV images. It

contains three components, the luminance Y that is obtained using equation (2.1), and the

chromatic information divided in Colour-Red Cr and Colour-Blue Cb are obtained using the

following equations [Kas92]:

)(564.0 YBCb = (2.16)

)(713.0 YRCr = (2.17)

Using equation (2.1) with equations (2.16) and (2.17) gives:

GGRCr 0816.04182.05.0 = (2.18)

BGRCb 5.03308.01686.0 += (2.19)

Defining the results for YCrCb in a matrix form yields:

=

B

G

R

C

C

Y

b

r *

5.0331.0169.0

082.0418.05.0

114.0587.0299.0

(2.20)


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An alternative implementation used by Altera [Alt97] uses slightly different matrix values

when converting gamma-corrected RGB data to YCrCb. This new set of constants is used to

provide a more natural image suited for the new TV sets with new phosphors coatings on theCRT. This variation is also considered in our design and is the following (including the

conversion to polar coordinates):

+

=

128

128

16

*

439.0291.0148.0

071.0368.0439.0

098.0504.0257.0

B

G

R

C

C

Y

b

r (2.21)

)/arctan( rbCrCb CCH = (2.21a)

22

brCrCb CCS += (2.21b)

2.3.2 RGB to hardcopy oriented colour spaces

Here colour spaces used for the creation of colour hardcopy or printed matter is considered.

Applications for this colour model include colour printers (ink-jet and lasers.), newspapers,

magazines, books, etc.

2.3.2.1 RGB to CMY

Cyan C, Magenta Mand Yellow Yare the complements of Red, Green and Blue respectively.

They are considered to be subtractive colours as they remove colour from white light, instead of

adding to brightness. Mixing equal proportions of pigments will give the following colours:

yellow and cyan will produce green, cyan and magenta will produce blue, magenta and yellow

will produce red and all mixed together will give black. One main application is for hardcopy

devices that deposit coloured pigments on paper.

The simplest matrix representation for this colour space is [For98]:


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=

B

G

R

Y

M

C

1

1

1

(2.22)

A variant of this colour model includes blackKand is known as the Cyan-Magenta-Yellow-

Black CMYK colour space that is widely used by many colour printers. The reason for

including black is to provide a rich dark black that otherwise cannot be obtained by mixing

equal amounts ofC,Mand Yon their own.

2.3.3 RGB to perceptually oriented colour models

The perceptually oriented colour models are basically used in computer graphics [Fol90] and

therefore are also known as computer graphics colour spaces or computer generated colour

models. These separate colour information from the luminance obtaining three parameters,

namely hue, saturation and value. An advantage of these colour spaces is that they closely

relate on how humans perceive colours and this is demonstrated in Figure 2.2. When colours

have a value V=1 and saturation S=1 they are said to be pure. When a pure colour is mixed

with white alone its saturation is decreased and a tint of that particular colour is obtained (e.g.

pink is an unsaturated red). If the colour is mixed with black only, it makes it darker and a

shade is created. Equal proportions of black and white (grey) mixed with a pure colour produce

a tone shown in the middle of the triangle on Figure 2.2.

Figure 2.2: Artist visualisation of colours (red cross-section)

Black

White

VVALUE

SSATURATION

Shad

es

Tints

Pure Hue point,

with no contents

of black or white.

S=1, V=1

Tones


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For the use in computer graphics and colour image processing applications the individual

components of these colour models can be visualised in Figure 2.3. Saturation S gives an

indication of the purity of the colour. The further out from the centre, the more pure (saturated)the colour will be. To describe colour, Hue His described as an angle from 0 to 360, where

Red is represented by 0, Green by 120 and Blue by 240. Moving the circle upwards would

make the hues lighter and moving the circle downwards darker. This component is shown in

Figure 2.3 and is given different names depending on the colour space employed, e.g.

LuminanceL, Value V, BrightnessB or IntensityI.

Figure 2.3: Perceptually oriented colour models

2.3.3.1 RGB to IHS

Known as the Intensity I, Hue H, Saturation Scolour model. It was seen in Section 2.2 thatRGB is defined with respect to a colour cube. The IHS model is defined with respect to a

colour triangle, as seen in Figure 2.4. The vectorOPis related to the intensity, equivalent to the

grey-scale line of the RGB colour cube. The triangle with vertices atR, G andB axes are known

as the Maxwell triangle [Wys00]. The intersection point p with the OPvector determines the

Hue Hand Saturation S. Hue is derived in them same way for all perceptually colour spaces

and is the most computational demanding operation of all.

Saturation Luminance (L), Value (V),

Brightness (B), Intensity (I)

Hue

Dark

Light


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Several solutions on obtaining H will be presented next based on the results published by

diverse authors. Each of these were carefully analysed and evaluated in order to pick the one

best suited for hardware realisation.

Figure 2.4: The IHS colour model and the Maxwell triangle

It can be shown [Str87] [Led90] that one of the first derivations ofHfor perceptually oriented

colour models can be represented by:


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existent when one of the values ofR, G orB are unequal. Therefore, a pixel having BGR ==

would contain no hue value and would be considered achromatic.

Figure 2.5: Procedure to obtain Hue for perceptually oriented colour spaces using

trigonometric functions

Bajon [Baj86] proposed a method to obtain Hwithout the use of trigonometric functions. If a

pixel is chromatic, the minimum of the RGB components is obtained. This minimum will give

an indication as to where Hlies and three outcomes are possible (refer to the colour circle in

Figure 2.3):

1. If B=min the resulting colour is between magenta and yellow or between 300 and

60 in the colour circle.

2. If R=min the resulting colour is between yellow and cyan or between 60 and 180 in

the colour circle.

3. If G=min the resulting colour is between cyan and magenta or between 180 and 300

in the colour circle.

Start R=G=B ?

G>=B andR>B ?

H5

3tan

13

R B( )

R G( ) B G( )+

+:=

G>R ?

Is B minimum?

Is R minimum?

G is minimum

His undefined(achromatic pixel)

NextPixel

N

Y

Y

Y

N

N

H

3tan

13

G R( )G B( ) R B( )+

+:=

H tan1

3B G( )

B R( ) G R( )+

+:=


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The procedure to obtain Hue without using trigonometric functions is illustrated in Figure 2.6.

Figure 2.6: Procedure to obtain Hue for perceptually oriented colour spaces without the

use of trigonometric functions

The value forIis obtained averaging the individual contributions ofR, G andB as shown:

BGRI3

1

3

1

3

1 ++= (2.24)

And finally, the saturation component is obtained as follows:

BGR

BGRminS

++= ),,(31 (2.25)

2.3.3.2 RGB to HSV

This is the Hue (H), Saturation (S) and Value (V) colour model. It is also known as the single

cone colour model or as Smiths [Smi78] HSV model. Sometimes this model is also known as

Start R=G=B ?

min(R,G,B)=B ?

min(R,G,B)

=R ?

Is B minimum?

Is R minimum?

G is minimum

His undefined(achromatic pixel)

NextPixel

Y

Y

Y

HG B( )

3 R G+ 2B( ):=

HB R( )

3 G B+ 2R( )

1

3+:=

HR G( )

3 R B+ 2 G( )

2

3+:=

N

N

N


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the HSB, with B for brightness instead of using V. The set of coordinates for this colour model

is cylindrical, and a hexcone is defined within this space as observed in Figure 2.7.

Figure 2.7: The HSV hexcone colour model

Value runs from black (bottom) to white (top) along the centre of the hexcone, this is similar to

the diametrical grey-scale line for the RGB colour cube. Hue is measured as an angle ranging

from 0 to 360 with red at 0, yellow at 60, green at 120, cyan at 180, blue at 240 and

magenta at 300. Saturation is a ratio ranging from 0 (Vaxis) to 1. The top of the HSV hexcone

can be seen as a projection along the principal diagonal of the RGB cube looking from white to

black (see Figure 2.1a). Achromatic colour is defined as having S= 0 and undefinedH.

Hue can be calculated as described in Section 2.3.3.1. A more generalised and simpler method

of obtaining H for perceptually oriented colour models was proposed by Levkowitz et al.

[Lev92] [Lev93] and will be described next. It divides the colour into 6 sectors (see Equation2.26) each labelled k, ranging for 0 to 5 according to the dominant wavelength of the Hue. It

then obtains the Hue fraction sector offset )(cf (see Equation 2.27) depending on whether the

colour is primary (when )(ck is even) or complementary (when )(ck is odd). It should be

observed that is necessary at this point to sort the RGB values to obtain min, mid and max

respectively. Finally, Hue is scaled (Equation 2.28) so it can give a value between 0 and 360.

Value

S

H

black (0)

white (1)

0o

60o120o

180o

240o 300o


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>>>>

>>

=

GreenlowestGBRif

BluehighestGRBif

RedlowestRGBifGreenhighestRBGif

BluelowestBRGif

RedhighestBGRif

ck

5

4

32

1

0

)( (2.26)

=

oddisckifcc

cc

evenisckifcc

cc

cf

)()min()max(

)mid(-)max(

)()min()max(

)min()(mid

)( (2.27)

))()((*60)( cfckcHue += (2.28)

Value Vfor HSV is obtained as follows:

},,min{min BGR= , },,max{max BGR= (2.29)

max=V (2.30)

Saturation Sfor HSV will be obtained for chromatic pixels [Bur 89] as shown in Figure 2.8.

Figure 2.8: Obtaining Saturation for HSV

k=0

k=1k=2

k=3

k=4 k=5

0

60120

180

240 300

Start max = min ?

(achromatic pixel)

His undefined

S= 0

Finish

Y

Smax min

max:=

N


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2.3.3.3 RGB to HLS

Gerald Murch at Tektronix [Tek90] developed the Hue (H), Luminance (L) and Saturation (S)

colour space. An advantage of this colour model is that it decouples the colour information

from the Luminance permitting individual processing of the channels. Hue, Luminance and

Saturation are defined by a double-hexcone obtained by rotating and projecting the RGB colour

cube as shown in Figure 2.9.

Figure 2.9: The HLS double-hexcone colour model

Hue is an angle going from 0 to 360 as for the HSV model. Luminance extends from 0

(black) to 1 (white), where the centre line in the double-hexcone is the grey-scale line.

Saturation is also a ratio of how much white is contained in a colour. Grey-levels in this model

have S= 0, and fully saturated Hues can be found at S= 1 andL = 0.5. For this colour model,

the Hue is also calculated as described in the previous two sections. In order to obtain L the

average of the minimum and maximum values is obtained [Gla90] [Gol97]:

},,min{min BGR= , },,max{max BGR=

S

H

L

black (0)

white (1)


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2

minmax+=L (2.31)

To obtain S, first it has to be determined whether the pixel is achromatic or not. If the pixel is

coloured, it is then established if it belongs to the bottom or top half of the double-hexcone.

This is illustrated in Figure 2.10.

Figure 2.10: Obtaining Saturation for HLS

A major drawback of this model is that two colours of equal perceived brightness will have

different values ofL [San98].

2.3.4 RGB to uniform distributed colour models

These colour models are the basis of industrial colorimetry where a way of defining

colours based on coordinates is desired. In order to define a perceptually uniform colour

space based on the results of human colour matching, the Commission Internationale de

LEclairage (CIE) was created in 1931 [CIE86]. Since then, various revisions have

occurred in 1964, 1976, etc. Basically two spaces were defined whose spaces are

defined in terms of CIE tristimulus values with respect to a reference white point. The

Start max = min ?

(achromatic pixel)

His undefined

S= 0

Finish

Y

L


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CIELa*b* coordinate system recommended for the use under reflected light conditions,

and the CIELu*v* coordinate system recommended in additive light source conditions.

They are based on the idea of having a colour space in which perceived colour

differences would be recognised as equal by the human eye and will correspond to equal

Euclidean distances [Taj83].

To further illustrate this, a representation of the Hue, Saturation and Lightness

(explained in Section 2.3.3) within a 3-D perceptual colour space is shown in Figure

2.11. Were (a*, b*) correspond to coordinates of the CIEL*a*b* colour space and (u*, v*)

to CIEL*u*v*. Luminance L* is equal for both [Rob 88]. The bottom section of Figure

2.11 defines a colour using Cartesian coordinates and the top defines colours using polar

coordinates.

Figure 2.11: Hue, Saturation and Lightness representation within a 3D Euclidean

perceptual colour space

The CIE also defined 3 special supersaturated primaries, X, YandZ. They do not correspond

to real colours, but they do have the property that all real colours can be represented as positive

combinations of them. The CIEXYZ colour system was created based on these three primaries

to describe any colour. The Y component is the description of colour as a luminance

component. TheXandZgive the spectral weighting curves that have been standardised by CIE,

based on statistics from experiments involving human observers [Jac94]. The magnitudes of

these components are proportional to physical energy, but their spectral composition

a*/u*

b*/v*

L*

HS

Lightness


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corresponds to the colour matching characteristics of human vision. These values are obtained

by the matrix multiplication shown in Equation 2.32 [Wys00]:

=

B

G

R

Z

Y

X

*

950227.0119193.0019334.0

072169.0715160.0212671.0

180423.0357580.0412453.0

(2.32)

The American Society for Testing and Materials (ASTM) [ASTM96] standardises values ofX,

Y and Z for computing the colour of objects using the CIE system for several illuminant

conditions (e.g. Cis used for average daylight). Some of these values are shown in Table 2.1.

The value ofYhas been normalised to 100 for every case.

Illuminant Tristimulus Values

X Y Z

A 109.85 100.00 35.58

WF35 103.81 100.00 49.94

D50 96.45 100.00 82.49

D65 95.04 100.00 108.89

D93 89.74 100.00 130.77

C 98.07 100.00 118.23

Table 2.1: Tristimulus values ofX, Yand Zdepending on the illuminant for a white

reference point

2.3.4.1 RGB to CIExyY

This colour model is used to normalise the values ofXand Ydescribed in Section 2.3.4 giving

( )ZYXXx ++= and ( )ZYXYy ++= with Y as luminance. Sometimes to make the

diagram of the model easier to comprehend, it is useful to describe pure colour in the absence

of brightness. This gives a plane with pure colours with the values ofx and y acting as

coordinates generating point (x,y) in the chromaticity diagram shown in Figure 2.12.

The CIE chromaticity diagram of Figure 2.12 has several interesting properties that will be

mentioned next:

Any colour along P1 to P2 can be obtained by mixing appropriate amounts of P1 and

P2.


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Appropriate combinations of P3, P4 and P5 can generate any colour within the triangle

defined by P3, P4 and P5. Therefore P3, P4 and P5 could be R, G andB respectively to

have the RGB set of colours, also known as thegamut.

Proper combinations of P6 and P7 can generate white as long as the line crosses the

point defined as white in the chart.

By extending P6, which is the complement of P7 to the edge of the diagram, gives the

dominant wavelength of the colour (i.e. 480nm).

The line running from red to blue is known as the purple line. The colours along this

line are non-existent and therefore have no associated wavelengths.

Figure 2.12: The CIExyY chromaticity diagram properties

There is a drawback, however, usingx andy coordinates, which is of being non-uniform. Equal

geometric steps result in different perceptual steps in different regions of the space. To alleviate

this problem the new coordinates u and v were introduced. This produced a five fold


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improvement in uniformity [San98] and resulted in the CIEuvY chromaticity diagram.

Equations 2.34 and 2.35 are used to obtain uand v, respectively by normalising and weighting

the tristimulus valuesX, YandZ.

ZYX

Xu

315

4'

++= (2.34)

ZYX

Yv

315

9'

++= (2.35)

2.3.4.2 RGB to CIEL*a

*b

*

This colour model was specified in 1976 by CIE [CIE86]. For the CIEL*a

*b

*uniform colour

space,L* represents Lightness, whilst a

* and b* are the coordinates in the chromaticity diagram.

The value ofL*

is orthogonal to the (**

,ba ) plane described in Figure 2.11 and is based on the

cube root of the Luminance. This model is defined directly in terms of the XYZ tristimulus

(Equation 2.32). To obtain the values L*, a

* and b* the following equations are used [Pra91]

[Con97]:

16*1160

*

=

Y

YfL (2.36)

=

00

* 500Y

Yf

X

Xfa (2.37)

=

00

* 200Z

Zf

Y

Yfb (2.38)

Where,

=

008856.0116

16292.903008856.0

)(

3/1

xifx

xifxxf (2.40)

Here,X0, Y0 andZ0 are the coordinates of the white reference point, which varies depending on

the illuminant shown in Table 2.1; a*

denotes relative redness-greenness and b*

yellowness-


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blueness. If the colour difference of two sets ofL*a

*b

* coordinates (L*

0, a*

0, b*0) and (L

*1, a

*1,

b*1) is desired it can be obtained using Equation 2.40 and is known as the deltaEunits:

( ) ( ) ( )2*0*12*

0*1

2*0

*1

* bbaaLLELab ++= (2.40)

The polar forms of Hue H and Chroma C* are useful for some applications, they are compiled

as follows [Gon95]:

=

*

*

arctana

bH (2.41)

( ) ( )2*2** baC += (2.42)

2.3.4.3 RGB to CIEL*u

*v

*

CIEL*u

*v

*is used primarily for industries considering additive mixing for applications such as

colour displays, TV and lighting. The Lightness value L*

is again obtained as in Equation 2.36,

where the u* and v* coordinates are defined as follows [Rob86] [Str90]:

( )''13 0* uuLu = (2.43)

( )''13 0* vvLv = (2.44)

Where uand v are obtained as expressed in Equations 2.34 and 2.35, respectively and '0u and

'0v are the coordinates of the reference white point.

The equation to determine the perceptual colour difference between two luminous sources in

CIEL*u

*v

*, with points at (L

*0, u

*0, v

*0) and (L

*1, u

*1, v

*1) is:

( ) ( ) ( )2*0*12*

0*1

2*0

*1

* vvuuLLELab ++= (2.45)

The following equations are used to obtain polar coordinates (i.e. HueH and Chroma C*):


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45

=

*

*

arctanu

vH (2.46)

( ) ( )2*2** vuC += (2.47)

2.4 Generalisation of a conversion algorithm from RGB to an

alternative colour space

This generalisation contemplates the implementation in hardware where a structured and

uniform approach is desired. After carefully analysing all colour models transformations, a

series ofstages have been identified that cover all transformations described so far and are not

restricted to future implementations of other colour spaces. Each of these three stages contains

several scenarios, which are called cases and depending on the case, the operation is modified

to achieve the desired result. A brief description of the stages and cases for each transformation

are given in Table 2.2.

Stage 1 Stage 2 Stage 3Matrix Multiplication Functional Mapping Final Representation

Cases:

1. Weighted coefficient

matrix multiplication with

RGB.



sorted RGB, obtaining the

colour sextant k.



RGB and gamma

correction.

Cases:

1. Final expression is

obtained directly.

2. Conditional arithmetic

and intermediate variable

generation (e.g. ,, ).

3. Equation partitioning

(start solving an

expression, leaving the

remaining for Stage 3,Case 1).

Cases:

1. Equation partitioning

(finish solving the

partially generated

expression in Stage 2,

Case 3).

2. Obtaining further results

from final expressions in

Stage 2 (i.e. obtaining

polar coordinates).

Table 2.2: Generalisation stages for the universal colour transformation functions

The subsequent sections will detail 12 different representations, namely YIQ, YUV, YC rCb,

CMY, IHS, HSV, HLS, CIEXYZ, CIExyY, CIEuvY, CIEL*a*b* and CIEL*u*v*. The initial

aim of the work was to determine a generalised processing structure for all.


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46

Stage 1

This stage is known as the Matrix Multiplication (MM) stage. For the majority of thetransformations, a coefficient matrix multiplication is needed to give weighted values forR, G

andB. For those conversions that do not implicitly require a matrix multiplication (e.g. CMY

and HSV), one was provided in order to have a uniform structure for Stage 1. In this way by

only changing the values of the coefficients in the matrix all transformations can be included.

Second, a variation for transformations to perceptually oriented colour models where a matrix

multiplication with sorted values ofR, G andB to obtain a min, midand max is required and is

also considered in this first stage. Finally, for perceptually oriented colour models, a colour

sextant khas to be identified and included as described in Equation 2.24. In summary, Stage 1

consists of scalar multiplier with weighted values, an RGB sorter and a means of obtaining the

value for the colour sextant k. The following three cases have been identified for Stage 1:

1. A floating-point matrix multiplication to produce weighted values forR, G andB. The

size of the matrix is always 33 and the range of its coefficients has been found to be

0.10.1 abc , where cab is the weighted coefficient at row a, column b.

2. A 33 sorted matrix multiplication with coefficients also in the range 0.10.1 abc is used. For this case, a multiplication will be realised on the sorted values ofR, G, and

B, i.e. },,min{ BGRmin = , },,{mid BGRmid= and },,max{ BGRmax = . A value for

the colour sextant k, 50 k , will be obtained from the sorted values of RGB. These

sorted values will also be used in Stage 2.

3. After the 33 matrix multiplication has been performed with R, G and B, adding an

integer offset to the result is desired for gamma correction in colour spaces like YCrCb.

Stage 2

This stage is known as the Functional Mapping (FM) section. Its primary function is to simplify

whenever possible the transformations by means of reducing the number of bits needed to

generate a partial or full result. This is achieved by analysing the equations or algorithms

involved in the process of making the colour transformations. During this stage, non-

computational demanding operations like addition, subtraction, basic multiplication and

division will be performed. In addition, decision trees will be included for those colour


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47

transformations that require them, e.g. HSV and HLS. This stage sometimes works in

conjunction with Stage 3 to partition expressions in order to simplify the operations required to

obtain a result. The following 3 cases were identified for Stage 2:

1. In some cases, the colour transformation from RGB to an alternative colour model can

be determined by the matrix multiplication alone, thus final expressions will be

generated directly without further simplification.

2. Conditional arithmetic is a special case needed for some transformations, namely the

perceptually oriented colour models that use an if-then-else structure to generate

results. This decision is made based on the outcome of certain results obtained from

Stage 1. For instance, to obtain the Saturation S for the HLS colour model it is

important to know in which half of the double-hexcone the Luminance L lies (see

Figure 2.10).L is obtained from the matrix multiplication and by considering its most

significant bit is will give an indication on which half of the hexcone it lies on. In this

way the if-then-else (conditional arithmetic statement) can be implemented as follows:

if 07 =L then // if bit 7 ofL is zero, then 128


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48

Stage 2 generating a temporary result. Stage 3 will be waiting for the partial result

generated from Stage 2 to produce a final expression. As an example, suppose we wish

to obtain the Hue for a perceptually oriented colour model using equations 2.26 to 2.28when the dominant wavelength is green, i.e. 2=k . The equation that would therefore

be required is:

+=minmax

minmid60 kHue

Where min, mid and max are the sorted values for RGB respectively. This equation

could be split into two separate operations by introducing an intermediate variable

Htemp:

minmax

minmid

=tempH

Now, Stage 2 can solve Htemp and then pass this partial result to Stage 3, which will

obtain the final result )(60 tempHkHue += .

Stage 3

Stage 3 is called the Final Representation (FR) stage. This stage can act as an auxiliary section

to finish solving elaborate algorithms that have been partitioned in the previous stage. Also, it

can be used to implement functions that generate values based on the results generated in Stage

2. An example would be obtaining Hue oH and Chroma *C from a*

and b*

for the CIEL*a

*b

*

colour space. Three cases were identified for Stage 3 to cope will all transformations, these are:

1. Splitting or breaking up complex equations or algorithms which were partially solved

in Stage 2 generating a partial solution that will then be used by this stage to produce a

final result. To illustrate this, Hue (H) will be obtained for the HSV single-cone model

and the following pseudo-code will indicate the operations performed by each stage.

/* Stage 2: Partitioning the algorithm to obtainHtemp */

if even (k) then // check if primary wavelength is red, green or blue


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49

Htemp = (midmin) / (max min);

else // colour is yellow, cyan or magenta

Htemp = (max mid) / (max min)

/* Stage 3 UsingHtemp and kto finalise calculations, obtainingH*/H= 60 * (k+Htemp);

2. Obtaining further results from the ones obtained in Stage 2. This case is best suited to

obtain polar coordinates from rectangular coordinates for several colour models, e.g.

Hue (H) and Saturation (S) for YIQ can be obtained as shown in Equations 2.9 and

2.10.

Based on this analysis and considering Table 2.2, we will define the transformations from RGB

to 12 alternative colour spaces in Sections 2.4.1 to 2.4.12. The generalisation used is presentedas a table showing each of the three stages and the cases considered for each.X1, Y1 andZ1 are

used as identifiers in Stage 1 to avoid confusion withX, YandZused for the CIE-based colour

spaces.

2.4.1 RGB to YIQ generalised transformation

Stage 1Case 1

=

B

G

R

ccc

ccc

ccc

Z1

Y1

X1

333231

232221

131211

(Eq. 2.8)

Stage 2Case 1

Z1Q

Y1I

X1Y

===

(Eq. 2.8)

c11 = 0.299 c12 = 0.587 c13 = 0.114

c21 = 0.596 c22 = 0.274 c23 = 0.322

c31 = 0.211 c32 = 0.523 c33 = 0.312

Stage 3

Case 2

22

1)/(tan

QISaturation

IQHue

+=

= (Eqs. 2.9 & 2.10)

Observations: YIQ is an example of a straightforward implementation of the generalisation

procedure. First, in Stage 1, the valuesX1, Y1 and Z1 are obtained by multiplying each RGB

pixel by the weighted values c11 to c33. Second, in Stage 2, the final values are obtained directly

from the matrix multiplication alone. Last, in Stage 3, the polar coordinates are obtained from

the results obtained in Stage 2.


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50

2.4.2 RGB to YUV generalised transformation

Stage 1Case 1

=

B

G

R

ccc

ccc

ccc

Z1

Y1

X1

333231

232221

131211

(Eq. 2.13)

Stage 2Case 1

Z1V

Y1U

X1Y

===

(Eq. 2.13)

c11 = 0.299 c12 = 0.587 c13 = 0.114

c21 = 0.147 c22 = 0.289 c23 = 0.437

c31 = 0.615 c32 = 0.515 c33 = 0.100

Stage 3

Case 2

22

1)/(tan

VUSaturation

UVHue

+==

(Eqs. 2.14 & 2.15)

Observations: As for YIQ, this is a straightforward implementation of the generalisation

procedure. First, in Stage 1, the valuesX1, Y1 and Z1 are obtained by multiplying each RGB

pixel by the weighted values c11 to c33. Second, in Stage 2, the final values are obtained directly

from the matrix multiplication alone. Last, in Stage 3, the polar coordinates are obtained from

the results obtained in Stage 2.

2.4.3 RGB to YCrCb generalised transformation

Stage 1

Case 3

+

=

128

128

16

333231

232221

131211

B

G

R

ccc

ccc

ccc

Z1

Y1

X1

(Eq. 2.21)

Stage 2

Case 1

Z1C

Y1C

X1Y

b

r

==

=(Eq. 2.21)

c11 = 0.257 c12 = 0.504 c13 = 0.098

c21 = 0.439 c22 = 0.368 c23 = 0.071

c31 = 0.148 c32 = 0.291 c33 = 0.439

Stage 3

Case 2

22

1 )/(tan

br

rb

CCSaturation

CCHue

+=

=

(Eqs. 2.21a & 2.21b)

Observations: Of all the transformations considered, this is the only case were a gamma

correction is used in Stage 1. After the weighted values R, G and B are obtained, the gamma

correction values are added. The rest of the methodology is identical to the one described for

YIQ.


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51

2.4.4 RGB to CMY generalised transformation

Stage 1Case 1

=

B

G

R

ccc

ccc

ccc

Z1

Y1

X1

333231

232221

131211

(Eq. 2.22)

Stage 2Case 1

Z1Y

Y1M

X1C

==

=

255

255

255

(Eq. 2.22)

c11 = 1.0 c12 = 0.0 c13 = 0.0

c21 = 0.0 c22 = 1.0 c23 = 0.0

c31 = 0.0 c32 = 0.0 c33 = 1.0

Stage 3

Not Needed

Observations: Although a matrix multiplication is not needed for Stage 1, an identity matrix is

used to provide consistency with the generalisation procedure. Also, not all colour

transformations require Stage 3, as can be observed here.

2.4.5 RGB to IHS generalised transformation

Stage 1

Case 2

=

B

G

R

sort

max

mid

min

, Obtaining k(Eq. 2.26)

=

max

mid

min

ccc

ccc

ccc

Z1

Y1

X1

333231

232221

131211

(Eq. 2.24)

Stage 2

Case 2

maxmidmin

minmax

kmidmaxkminmid

++==

=

oddisif

evenisif

(Eqs. 2.27 & 2.25)

Case 1

25531

=

=

minS

X1I

(Eqs. 2.24 & 2.25)

Case 3

=tempH

c11 = 1 / 3 c12 = 1 / 3 c13 = 1 / 3

c21 = 0.0 c22 = 0.0 c23 = 0.0

c31 = 0.0 c32 = 0.0 c33 = 0.0

Stage 3

Case 1

60+= tempHkH (Eq. 2.28)

Observations: In Stage 1, a sorter is used on the individual RGB values to obtain the minimum

(min), medium (med) and maximum (max) values that will then be used to obtain X1, Y1 and

Z1. The values given for c11 to c33 are to obtain intensity Ifor Equation 2.24. This stage also

obtains the colour sextant kdescribed in Equation 2.26. Variables with Greek letters (e.g. ,


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52

and ) are introduced to indicate temporary variables used for simplifications in Stage 2. Hue is

obtained using Equations 2.26 to 2.28. To obtain , a conditional statement is implemented and

is based on the outcome of the colour sextant k(Equation 2.27). Since HueHis rather elaborate

to obtain, it has been partitioned into Stages 2 and 3. Case 3 of Stage 2 will be generating a

partial calculation ofH, namedHtemp, which will then be used by Stage 3 to get the final value

forH.

2.4.6 RGB to HSV(B) generalised transformation

Stage 1Case 2

=

B

G

R

sort

max

mid

min


=

max

mid

min

ccc

ccc

ccc

Z1

Y1

X1

333231

232221

131211

(Eq. 2.30)

Stage 2Cases 2

minmax

kmidmax

kminmid

=

=

oddisif

evenisif

(Eq. 2.27)

Case 1

255=

=

maxS

Z1V

(Eq. 2.30 & Fig. 2.8)

Case 3

=tempH

c11 = 0.0 c12 = 0.0 c13 = 0.0

c21 = 0.0 c22 = 0.0 c23 = 0.0

c31 = 0.0 c32 = 0.0 c33 = 1.0

Stage 3

Case 1

60*tempHkH += (Eq. 2.28)

Observations: The same observations made for the IHS generalisation in Section 2.4.5 also

apply here. The values given for c11 to c33 are for obtaining value Vfor Equation 2.30.


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53

2.4.7 RGB to HLS generalised transformation

Stage 1Case 2

=

B

G

R

sort

max

mid

min


=

max

mid

min

ccc

ccc

ccc

Z1

Y1

X1

333231

232221

131211

(Eq. 2.31)

Stage 2Case 2


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54

2.4.9 RGB to CIExyY generalised transformation

Stage 1Case 1

=

B

G

R

ccc

ccc

ccc

Z1

Y1

X1

333231

232221

131211

(Eq. 2.32)

Stage 2Case 2

Z1Y1X1 ++= (Sec. 2.3.4.1)Case 1

255

255

=

=

=

Y1y

X1x

Y1Y

(Sec. 2.3.4.1)

c11 = .412453 c12 = .35758 c13 = .180423

c21 = .212671 c22 = .71516 c23 = .072169c31 = .019334 c32 = .119193 c33 = .950227

Stage 3

Not Needed

Observations: The values ofx andy in Stage 2 are multiplied by 255 to generate values ranging

from 0 to 255. The multiplication could be ignored if normalised values from 0 to 1 were

desired.

2.4.10 RGB to CIEuv

Y generalised transformation

Stage 1

Case 1

=

B

G

R

ccc

ccc

ccc

Z1

Y1

X1

*

333231

232221

131211

(Eq. 2.32)

Stage 2

Case 2

Z1Y1X1

Y1

X1

315

9

4

++===

(Eqs. 2.34 & 2.35)

Case 1

255*'

255*'

=

=

=

v

u

Y1Y

(Eqs. 2.34 & 2.35)

c11 = .412453 c12 = .35758 c13 = .180423

c21 = .212671 c22 = .71516 c23 = .072169

c31 = .019334 c32 = .119193 c33 = .950227

Stage 3

Not Needed

Observations: Values 'u and 'v in Stage 2 are multiplied by 255 to span their range from 0 to

255. This multiplication could be ignored if normalised values from 0 to 1 were desired.


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55

2.4.11 RGB to CIEL*a

*b

*generalised transformation

Stage 1

Case 1

=

B

G

R

ccc

ccc

ccc

Z1

Y1

X1

*

333231

232221

131211

(Eq. 2.32)

Stage 2

Case 1

=

=

=

3/13/1

*

3/13/1

*

3/1

*

200

500

16116

nn

nn

n

Z

Z1

Y

Y1b

Y

Y1

X

X1a

Y

Y1L

(Eqs. 2.36 to 2.38)

c11 = .412453 c12 = .35758 c13 = .180423

c21 = .212671 c22 = .71516 c23 = .072169c31 = .019334 c32 = .119193 c33 = .950227

Stage 3

( )( ) ( )2*2*

**1

/tan

baChroma

abHue

+==

(Eqs. 2.41 & 2.42)

Observations: It should be noted that in Stage 2 only X1, Y1 and Z1 are variables originated

from Stage 1, while Xn, Yn andZn are constants and are the values for the white reference point

given in Table 2.1.

2.4.12 RGB to CIEL*

u*

v*

generalised transformation

Stage 1

Case 1

=

B

G

R

ccc

ccc

ccc

Z1

Y1

X1

*

333231

232221

131211

(Eq. 2.32)

Stage 2

Case 2

Z1Y1X1

Y1

X1

315

9

4

++===

(Eqs. 2.34 & 2.35)

Case 1

16116

3/1

*

=

nY

Y1L (Eq. 2.36)

Case 3

=

=

'0

*

'0

*

vv

uu

temp

temp

(Eqs. 2.43 & 2.44)

c11 = .412453 c12 = .35758 c13 = .180423

c21 = .212671 c22 = .71516 c23 = .072169

c31 = .019334 c32 = .119193 c33 = .950227

Stage 3

Case 1

***

***

13

13

temp

temp

vLv

uLu

=

=(Eqs. 2.43 & 2.44)


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56

Observations: The values of '0u and'0v in Stage 2 are the coordinates of the white reference

point for CIEuv

Y.

2.5 Maximum and minimum values for alternative colour spaces

It is important to know the range of values that each and every variable can take during the

transformation to alternative colour spaces. R, G andB are integers with values from 0 to 255

(8-bit). Testing every possible combination for RGB gives a total of 224

or 16.7 million colour

representations. With the speed of computers these days, it is now possible to check everycombination ofR, G andB to obtain the ranges that should be considered when designing the

hardware.

A program included in the CD-ROM was used to convert RGB values to all 12 alternative

colour spaces. It tests all possible combinations for RGB, whilst it monitors maximum and

minimum values for all variables. At the end, it obtains all the results presented in Table 2.3.

A series of considerations were made for obtaining these minimum and maximum values,

which are described here. An achromatic pixel contains no colour information and occurs only

when BGR == , or for perceptually oriented colour spaces when maxmin = . These pixels do

not contain hue or saturation information, i.e. 0,0 == SH . Several ranges of values are fixed

for all colour models, which are:

2550and2550,2550 BGR // RGB are always 8-bit values

2551and2550,2540 maxmidmin // If 255=min or 0=max anachromatic pixel is obtained 50 k // Colour sextant

Table 2.3 is colour coded and includes all variables needed for the generalisation described in

Sections 2.4.1 through 2.4.12 for the colour transformation procedure with their minimum and

maximum values. The coloured identifier in the first column reflects the final parameter to be

obtained. Every transformation obtains three parameters. These are generated either in Stage 2

or Stage 3, hence the colour in these two stages correspond to ranges of the desired parameter


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(i.e. The minimum and maximum values for parameterUin the YUV colour model will be

111.2 and 111.2 respectively). Whenever an X appears in the table, it indicates that particular

variable is not` utilised.

2.6 Conclusions

This chapter first looked at the conventional method of performing transformations from RGB

to alternative 12 colour models that form part of a colour system. These will be implemented in

hardware in Chapter 3 and include:

Hardware oriented systems based on TV: YIQ, YUV, and YCrCb.

Hardware oriented systems based on printing: CMY.

Perceptual user oriented systems: IHS, HSV, and HLS.

Perceptually oriented systems: CIEXYZ, CIExyY, CIEuvY. CIEL*a*b*, CIEL*u*v*.

Next a generalisation method was proposed based on a uniform structure that could handle the

diversity of transformation procedures and methodologies. It was finally determined that this

could be achieved by dividing the problem into 3 parts called stages. Each stage handles the

requirements to transform RGB into an alternative colour space using cases. Later, the

generalisation is presented as a series of tables indicating the operations that will be carried out

by every stage and case.

Finally this chapter gives the ranges of values (from minimum to maximum) that every

parameter and intermediate variable used in the generalisation will take.

chapter 2 transformations

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