# Lecture 5 - Tristimulus Values

## The Principle on which the Computer Graphics Use of Colour is Based.

If the photoreceptors provide the same signal to the brain, the perceptual systems of the brain will provide the same visual experience.'

• This principle is reasonable ceteris paribus. Examining how reasonable it is in reality is one interest of this course.

This principle is made operational by the principle of univariance: the signal provided by any photoreceptor depends only on the number of photons absorbed.'

• There is a large amount of empirical evidence, explicit and implicit, for the truth of the principle of univariance.

## Colour Matching & Tristimulus Values

#### Expressing a colour as a spectral power distribution

• C ~ \sum_l a_l C_l

#### Matching the monochromatic colours

• C_l ~ \sum_i b_li P_i
• C ~ \sum_l a_l \sum_i b_li P_i ~ \sum_i ( \sum_l a_l b_li ) P_i ~ \sum_i p_i P_i
• where p_i = \sum l_a b_li ( i = 1,2,3 or i = X,Y,Z ) are called tristimulus values, which uniquely specify a colour match with respect to a set of primaries, P_i
• Remember that a_l is the spectral power distribution of C, and that b_li are the tristimulus values of a unit amount of monochromatic light, called the colour matching functions

#### Measuring tristimulus values

• Not that if I have a light there are two ways to determine its tristimulus values
1. Obtain three standard primaries, and determine by eye how much of each is needed to match C
2. Obtain the spectral power distribution of the colour and integrate (sum) it weighted by the colour matching functions
• The second method is the one most commonly used in practice
• To make this work in practice everybody must agree on a standard set of primaries
• If you are using the first method it must be easy to create the standard primaries
• That is, they must be physically realisable
• If you are using the second method it is not necessary to create the primaries
• But it should be easy to do the sum.
• In 1930 it was easy to add numbers mechanically, but not to subtract them
• The spectral power distribution is never negative: a_l >= 0
• Primaries were chosen so that b_li >= 0

## 3D Colour Geometry

We said earlier that the linear algebra Grassmann created includes many colours that cannot be realised physically. We will now invetigate the set of physically possible colours.

The physically realisable colours are a subset of a three dimensional colour space. Construct it as follows:

• Put black ( a_l = 0 ) at the origin.
• Choose the standard CIE primaries as the axes.

Then, all physically realisable colours lie in the +++ octant. Why?

#### Spectral colours

Choose a wavelength in the visible region of the spectrum.

• When we look at light of this wavelength we see a colour.
• If we reduce the amount of light the colour we see goes in a straight line toward black
• Why is the line straight?
• If we increase the amount of light the ray extends away from black, to infinity if you wish.
• Note that you can read the direction of this ray off the colour matching functions

Choose a second wavelength, and draw the line from black through it.

• It goes in a different direction.
• By taking affine combinations of any points on the two rays,
• affine combination means: a C_1 + ( 1 - a ) C_2 for all 0 <= a <= 1
• we get a triangular' part of a plane
• The restriction on a occurs because we can't have a negative number of photons

If we continue taking more wavelengths we fill out a cone' the cross-section of which is determined by the colour matching functions.

• There are good physical reasons for believing that the colour matching functions are continuous and differentiable.
• Experiments seem to bear this out.

The 2D manifold that is the surface of the cone has blue at one end, red at the other.

• They do not extend in the same direction.
• The surface is closed by the mixtures of the bluest blue and the reddest red
• This is a continuum of purple colours

#### The gamut of physically realisable colours

This is a volume that is neither easy to draw nor easy to understand.

• Where we cut it off depends on the metaphysical assumptions you want to make about the nature of colour.

One things is clear

• Cross-sections of this figure look pretty much the same, varying only in brightness.
• What do they resemble?

## 2D Colour Geometry

#### Chromaticity coordinates

An increase in brightness increases all tristimulus values by the same amount

• (X, Y, Z) --> (aX, aY, aZ)

We can take out the brightness by doing any transformation that produces values independent of a. The standard transformation, used for a century, is

• x = X / ( X + Y + Z )
• y = Y / ( X + Y + Z )
• z = Z / ( X + Y + Z )
• x + y + z = 1
• Because x > 0, y > 0 and z > 0, x, y & z are all < 1 and x + y < 1

(x, y) are called the chromaticity coordinates of a colour. The 2D map of chromaticities is called the chromaticity diagram.

• The chromaticity disgram is limited to a triangle that is 1/2 of the unit square in the xy plane.

The transformation to chromaticity coordinates is a projective transformation, which gives it an attractive property.

• Colours produced by additive mixing of a pair of colours lie on a straight line in the 3D colour geometry.
• Projective geometry maps straight lines to straight lines.
• Therefore, if we plot the the chromaticity of two colours the chromaticity of any additive mixture of the two colours lie on the line joining the two chromaticities.

#### Spectrum locus

Where do the spectral colours lie? We can practically read this off the colour matching functions.

• Wavelengths > 560 nm (red-orange-yellow) : x + y = 1, end-point at lim( y/x )
• Wavelengths between 520 nm & 560 nm (yellow-green) : curves a little away from x + y = 1
• About 500 nm (bluish green) : x ~ 0
• Wavelengths < 500 nm (greenish blue-blue-violet) : y -> 0, x increases slowly.

This curve is called the spectrum locus; the straight line that closes the curve is called the purple line.

The long straight line from 780nm to 550nm indicates that humans are essentially dichromats for these wavelengths.

#### Black body curve

The colours emitted by black bodies lie on a curve:

• along the spectrum locus in the reds and oranges,
• curving away from the spectrum locud in the oranges and yellow,
• passing through white to bluish white.

The temperatures of the points on the diagram are

• A: 2750K, the colour of the light emitted by a 100 watt incandescent light bulb
• C: 5500K: the colour of the light emitted by the sun

The standard lights Dx are black-body radiators at xK. x is called the colour temperature.

#### Concepts related to the white point

There is a region of colours called white.

• Which one depends on the ambient illumination.
• One is chosen and called the white point.

The dominant hue of a chromaticity is the wavelength at which a line from the white point through the chromaticity intersects the spectrum locus.

• It is an operational definition sharpening the fuzzy concept of hue

The excitation purity of a chromaticity is a measure of how far a chromaticity is from the white point.

• It is an operational definition sharpening saturation.