Colour Space Transformations

Linear Transformations

Converting between linear RGB spaces is a matrix operation:

$\begin{bmatrix} X \\ Y \\ Z \end{bmatrix} = \mathbf{M}_{\text{RGB}\to\text{XYZ}} \begin{bmatrix} R \\ G \\ B \end{bmatrix}$

RGB to XYZ Matrix

For sRGB (ITU-R 709 primaries):

$\mathbf{M}_{\text{709}\to\text{XYZ}} = \begin{bmatrix} 0.4124 & 0.3576 & 0.1805 \\ 0.2126 & 0.7152 & 0.0722 \\ 0.0193 & 0.1192 & 0.9505 \end{bmatrix}$

Between RGB Spaces

To convert from sRGB to Rec. 2020:

$\begin{bmatrix} R_{2020} \\ G_{2020} \\ B_{2020} \end{bmatrix} = \mathbf{M}_{\text{XYZ}\to\text{2020}} \cdot \mathbf{M}_{\text{709}\to\text{XYZ}} \begin{bmatrix} R_{709} \\ G_{709} \\ B_{709} \end{bmatrix}$

Steps

1. Linearise: Remove gamma encoding
2. Convert to XYZ: Apply $\mathbf{M}_{\text{RGB}\to\text{XYZ}}$
3. Convert to target: Apply $\mathbf{M}_{\text{XYZ}\to\text{RGB}}^{-1}$
4. Re-encode: Apply target gamma

Computing RGB to XYZ Matrix

Given primaries $(x_R, y_R)$, $(x_G, y_G)$, $(x_B, y_B)$ and white point $(x_W, y_W)$:

Step 1: Form Primary Matrix

$\mathbf{P} = \begin{bmatrix} x_R/y_R & x_G/y_G & x_B/y_B \\ 1 & 1 & 1 \\ (1-x_R-y_R)/y_R & (1-x_G-y_G)/y_G & (1-x_B-y_B)/y_B \end{bmatrix}$

Step 2: Solve for Scaling Factors

$\begin{bmatrix} S_R \\ S_G \\ S_B \end{bmatrix} = \mathbf{P}^{-1} \begin{bmatrix} x_W/y_W \\ 1 \\ (1-x_W-y_W)/y_W \end{bmatrix}$

Step 3: Construct Transform

$\mathbf{M}_{\text{RGB}\to\text{XYZ}} = \begin{bmatrix} S_R x_R/y_R & S_G x_G/y_G & S_B x_B/y_B \\ S_R & S_G & S_B \\ S_R(1-x_R-y_R)/y_R & S_G(1-x_G-y_G)/y_G & S_B(1-x_B-y_B)/y_B \end{bmatrix}$

Mapping to a Display with Custom Primaries

Given:

• Target spectrum $\mathbf{L}$
• Display primary spectra $\mathbf{P}_{\text{RGB}}$ (N×3 matrix)
• CIE XYZ matching functions $\mathbf{S}_{\text{XYZ}}$ (N×3 matrix)

Algorithm

1. Find XYZ of target:

$\mathbf{C}_{\text{XYZ}} = \mathbf{S}_{\text{XYZ}}^\top \mathbf{L} = \begin{bmatrix} X \\ Y \\ Z \end{bmatrix}$

2. Construct primary matrix:

$\mathbf{M}_{\text{RGB}\to\text{XYZ}} = \mathbf{S}_{\text{XYZ}}^\top \mathbf{P}_{\text{RGB}}$

3. Solve for RGB:

$\begin{bmatrix} R \\ G \\ B \end{bmatrix} = \mathbf{M}_{\text{RGB}\to\text{XYZ}}^{-1} \mathbf{C}_{\text{XYZ}}$

4. Apply display encoding:

$\begin{bmatrix} R' \\ G' \\ B' \end{bmatrix} = \begin{bmatrix} R^{1/\gamma} \\ G^{1/\gamma} \\ B^{1/\gamma} \end{bmatrix}$

Luma Formula

Luma (display-encoded luminance approximation):

$Y' = 0.2126 R' + 0.7152 G' + 0.0722 B'$

Important Note

Luma ≠ Luminance:

• Luma uses gamma-corrected (display-encoded) values
• Luminance uses linear values
• Luma is an approximation for video processing

Gamut Mapping

When converting between spaces with different gamuts:

Situation	Action
Colour inside both gamuts	Direct mapping
Colour outside destination	Gamut mapping required

Gamut Mapping Strategies

Strategy	Description
Clipping	Clip to nearest in-gamut colour
Compression	Reduce saturation proportionally
Perceptual	Preserve relationships, compress into gamut

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Summary

• Linear RGB to XYZ conversion uses 3×3 matrices
• Transform between RGB spaces via XYZ intermediate
• Matrix derivation requires primaries and white point
• Luma approximates luminance from gamma-encoded values

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Past Paper Questions

2024 Paper 3 Question 4: How would you convert a colour from sRGB to Rec. 2020? What steps are involved?

2023 Paper 3 Question 4: Derive the RGB to XYZ transformation matrix given the primaries and white point.

2022 Paper 3 Question 3: What is the difference between luma and luminance?