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