Tristimulus Colour Representation

The Key Insight

Any colour can be matched by combining three linearly independent primary colours.

This is the foundation of colour science and all colour display technology.

Colour Matching Experiments

Historical Experiments (1930s)

• 2° visual field (fovea only)
• ~12 observers with normal colour vision
• Derived the CIE XYZ colour matching functions

Experimental Setup

An observer views a split screen:

• One side: A test colour (monochromatic light)
• Other side: Adjustable mixture of three primaries

The observer adjusts the primary intensities until both sides match.

Colour Matching Functions

Three functions $\bar{x}(\lambda)$, $\bar{y}(\lambda)$, $\bar{z}(\lambda)$ describe how much of each primary is needed to match monochromatic light at each wavelength.

Tristimulus Values

Given a light spectrum $L(\lambda)$, the tristimulus values are:

$X = \int L(\lambda) \bar{x}(\lambda) d\lambda$

$Y = \int L(\lambda) \bar{y}(\lambda) d\lambda$

$Z = \int L(\lambda) \bar{z}(\lambda) d\lambda$

Why Three Primaries?

Human Trichromacy

Human vision is trichromatic because:

• Three types of cones (L, M, S)
• Each cone type produces one value
• Three numbers fully describe colour perception

Linear Independence

Three primaries must be linearly independent:

• No primary can be matched by a mixture of the other two
• Guarantees a unique solution for any colour

Primaries are Arbitrary

Any three linearly independent colours work:

• Different display technologies use different primaries
• RGB is common but not unique
• XYZ primaries are imaginary (not physically realisable)

Grassmann’s Laws

Additivity

If colour A matches B and C matches D, then A+C matches B+D.

Scalar Multiplication

If A matches B, then kA matches kB for any k > 0.

Transitivity

If A matches B and B matches C, then A matches C.

Practical Application

Display Technology

Displays exploit trichromacy:

• Three phosphors/LEDs/subpixels (R, G, B)
• Each pixel is a colour match for the intended spectrum
• Viewer perceives the target colour

Limitation

The match is metameric:

• Display spectrum differs from real object spectrum
• But L, M, S cone responses are identical
• Viewer cannot distinguish

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Summary

• Any colour can be matched by three primaries (trichromacy)
• Colour matching functions $\bar{x}$, $\bar{y}$, $\bar{z}$ map spectra to XYZ
• Three primaries work because humans have three cone types
• Grassmann’s laws govern colour addition

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Past Paper Questions

2024 Paper 3 Question 3: Explain the principle of tristimulus colour representation. Why do three primaries suffice?

2023 Paper 3 Question 4: What are colour matching functions? How are they measured?

2022 Paper 3 Question 3: State Grassmann’s laws of colour matching.