RGB ↔ CIELAB Conversion Guide

This explains the full mathematical conversion pipeline between sRGB and CIELAB (Lab) color spaces.

1. Conversion Pipeline Overview

RGB CIELAB

1. sRGB Linear RGB
2. Linear RGB XYZ
3. XYZ CIELAB

CIELAB RGB

1. CIELAB XYZ
2. XYZ Linear RGB
3. Linear RGB sRGB

2. sRGB to Linear RGB

sRGB values are gamma‑compressed. Convert them to linear light:

$$C_\text{lin} = \begin{cases} \frac{C_\text{srgb}}{12.92}, & C_\text{srgb} \le 0.04045 \\ \left(\frac{C_\text{srgb} + 0.055}{1.055}\right)^{2.4}, & C_\text{srgb} > 0.04045 \end{cases}$$

This is applied independently to (R), (G), and (B).

3. Linear RGB to XYZ

Using the sRGB color space matrix with a D65 white point:

$$\begin{bmatrix} X \\ Y \\ Z \end{bmatrix} = \begin{bmatrix} 0.4124564 & 0.3575761 & 0.1804375 \\ 0.2126729 & 0.7151522 & 0.0721750 \\ 0.0193339 & 0.1191920 & 0.9503041 \end{bmatrix} \begin{bmatrix} R_\text{lin} \\ G_\text{lin} \\ B_\text{lin} \end{bmatrix}$$

4. XYZ to CIELAB

Normalize XYZ by the D65 reference white:

$$X_n = 0.95047,\quad Y_n = 1.00000,\quad Z_n = 1.08883$$ $$x = \frac{X}{X_n},\quad y = \frac{Y}{Y_n},\quad z = \frac{Z}{Z_n}$$

Define the nonlinear function:

$$f(t) = \begin{cases} t^{1/3}, & t > \left(\frac{6}{29}\right)^3 \\ \frac{t}{3\left(\frac{6}{29}\right)^2} + \frac{4}{29}, & t \le \left(\frac{6}{29}\right)^3 \end{cases}$$

Then compute Lab:

$$L^* = 116 f(y) - 16$$ $$a^* = 500 \left[f(x) - f(y)\right]$$ $$b^* = 200 \left[f(y) - f(z)\right]$$

5. CIELAB to XYZ

The inverse of (f(t)):

$$f^{-1}(t) = \begin{cases} t^3, & t > \frac{6}{29} \\ 3\left(\frac{6}{29}\right)^2 \left(t - \frac{4}{29}\right), & t \le \frac{6}{29} \end{cases}$$

Compute:

$$f_y = \frac{L + 16}{116}, \quad f_x = f_y + \frac{a}{500}, \quad f_z = f_y - \frac{b}{200}$$ $$X = X_n f^{-1}(f_x),\quad Y = Y_n f^{-1}(f_y),\quad Z = Z_n f^{-1}(f_z)$$

6. XYZ to Linear RGB

$$\begin{bmatrix} R_\text{lin} \\ G_\text{lin} \\ B_\text{lin} \end{bmatrix} = \begin{bmatrix} 3.240970 & -1.537383 & -0.498611 \\ -0.969244 & 1.875968 & 0.041555 \\ 0.055630 & -0.203977 & 1.056972 \end{bmatrix} \begin{bmatrix} X \\ Y \\ Z \end{bmatrix}$$

7. Linear RGB to sRGB

$$C_\text{srgb} = \begin{cases} 12.92\, C_\text{lin}, & C_\text{lin} \le 0.0031308 \\ 1.055\, C_\text{lin}^{1/2.4} - 0.055, & C_\text{lin} > 0.0031308 \end{cases}$$

Clamp results to ([0,1]) and scaled by 255 before converting to 8‑bit.

8. Summary

RGB Lab

• Remove gamma (sRGB linear)
• Convert to XYZ
• Normalize by D65
• Apply nonlinear transform
• Produce L*, a*, b*

Lab RGB

• Convert Lab XYZ via inverse nonlinear transform
• XYZ linear RGB
• Linear RGB sRGB (gamma)
• Clamp to valid output

9. References

• CIE 1976 L*a*b* Specification
• IEC 61966‑2‑1 sRGB Standard