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CIELAB to sRGB Hex Conversion

Scope and assumptions

MeatColor::summarize_lab() converts mean CIELAB coordinates with:

colorspace::hex(colorspace::LAB(L, a, b))

The conversion path is CIELAB → CIEXYZ → linear-light sRGB → encoded sRGB → hexadecimal.

This calculation assumes:

  • the input is CIE 1976 L*a*b* (CIELAB), not Hunter Lab;
  • the CIELAB values use the same D65/2° reference white as the colorspace package default; and
  • the intended output is the sRGB display color space.

Colorimeter exports can depend on illuminant, observer angle, and color-space settings. Confirm that the instrument exported CIELAB values under D65 with the 2° standard observer before interpreting the generated hex values as sRGB. Values measured relative to another white point or observer require an appropriate conversion before the sRGB matrix is applied.

Step 1: Convert CIELAB to CIEXYZ

For CIELAB coordinates $L^{\ast}$, $a^{\ast}$, and $b^{\ast}$, define:

$$ f_y = \frac{L^{\ast} + 16}{116}, \qquad f_x = f_y + \frac{a^{\ast}}{500}, \qquad f_z = f_y - \frac{b^{\ast}}{200} $$

Use the exact CIELAB constants:

$$ \epsilon = \frac{216}{24389} \approx 0.008856451679 $$

$$ \kappa = \frac{24389}{27} \approx 903.2962963 $$

The inverse CIELAB transfer function is:

$$ f^{-1}(t) = \begin{cases} t^3 & \text{if } t^3 > \epsilon \\ \dfrac{116t - 16}{\kappa} & \text{otherwise} \end{cases} $$

The relative tristimulus values are:

$$ x_r = f^{-1}(f_x), \qquad y_r = \begin{cases} f_y^3 & \text{if } L^{\ast} > 8 \\ \dfrac{L^{\ast}}{\kappa} & \text{otherwise} \end{cases}, \qquad z_r = f^{-1}(f_z) $$

The boundary $L^{\ast}=8$ follows from $\kappa\epsilon=8$.

For the D65 reference white used by colorspace, scaled so $Y_n=100$:

$$ (X_n, Y_n, Z_n) = (95.047,\ 100.000,\ 108.883) $$

Then:

$$ X = X_n x_r, \qquad Y = Y_n y_r, \qquad Z = Z_n z_r $$

XYZ components are not each restricted to $[0,100]$. In particular, the D65 reference-white value $Z_n$ is greater than 100, and colors outside the sRGB gamut can produce values outside the usual display range.

Step 2: Convert CIEXYZ to linear-light sRGB

The colorspace implementation uses XYZ values scaled to $Y_n=100$ and the following D65 matrix:

$$ \begin{bmatrix} r \\ g \\ b \end{bmatrix} = \begin{bmatrix} 3.240479 & -1.537150 & -0.498535 \\ -0.969256 & 1.875992 & 0.041556 \\ 0.055648 & -0.204043 & 1.057311 \end{bmatrix} \begin{bmatrix} X/100 \\ Y/100 \\ Z/100 \end{bmatrix} $$

Here $r$, $g$, and $b$ are linear-light components, not yet encoded sRGB values.

Step 3: Apply the sRGB transfer function

For each linear-light component $v$, the IEC sRGB encoding function is:

$$ V = \begin{cases} 12.92v & \text{if } v \le 0.0031308 \\ 1.055v^{1/2.4} - 0.055 & \text{if } v > 0.0031308 \end{cases} $$

The current colorspace C implementation uses 0.00304 as its branch threshold. This small implementation difference does not affect the worked example below, but the formula above gives the standard sRGB threshold.

Step 4: Quantize and encode as hexadecimal

For an in-gamut encoded component $V$:

$$ C_8 = \mathrm{round}(255V) $$

Each resulting integer is written as a two-digit hexadecimal value and concatenated:

hex = #RRGGBB

MeatColor currently calls colorspace::hex() with its default fixup = FALSE. Therefore, if any rounded component lies outside 0–255, colorspace returns NA rather than clipping the color. Calling colorspace::hex(..., fixup = TRUE) would instead clamp out-of-range components to the nearest endpoint.

Verified example: CIELAB(50, 25, -25)

First:

$$ f_y = 0.5689655172,\qquad f_x = 0.6189655172,\qquad f_z = 0.6939655172 $$

All three values use the cubic branch:

$$ x_r = 0.2371370239,\qquad y_r = 0.1841865185,\qquad z_r = 0.3342055621 $$

Scaling by the D65 reference white gives:

$$ X = 22.53916271,\qquad Y = 18.41865185,\qquad Z = 36.38930421 $$

Using the matrix implemented by colorspace gives linear-light sRGB:

$$ r = 0.2658411096,\qquad g = 0.1421921876,\qquad b = 0.3597087397 $$

After sRGB encoding:

$$ R = 0.5524516066,\qquad G = 0.4130411268,\qquad B = 0.6340203340 $$

Quantizing to 8-bit components gives:

$$ (R_8,G_8,B_8) = (141,105,162) $$

Therefore:

Result: #8D69A2

This result matches:

colorspace::hex(colorspace::LAB(50, 25, -25))
#> "#8D69A2"

Interpretation limits

  • A screen swatch is an sRGB approximation of measured color, not a color-managed reproduction of the physical meat sample.
  • CIELAB coordinates are meaningful only with their reference illuminant, observer, and measurement geometry.
  • Out-of-gamut CIELAB colors cannot be represented exactly in sRGB.
  • Keep full precision through the intermediate calculations and round only when producing the final 8-bit channels.

References