CIELUV

XYZ → CIELUV and CIELUV → XYZ conversions

By definition, 0 ≤ L* ≤ 100 .

The forward transformation

CIELUV is based on CIEUVW and is another attempt to define an encoding with uniformity in the perceptibility of color differences. The non-linear relations for L*, u*, and v* are given below:

${\displaystyle \begin{array}{cc}{L}^{\ast }& =\{\begin{array}{ll}(\frac{29}{3}{)}^{3}Y/Y_n,& Y/Y_n\le (\frac{6}{29}{)}^3,\\ 116\sqrt[3]{Y/Y_n}-16,& Y/Y_n>(\frac{6}{29}{)}^3,\end{array}\\ u^{\ast }& =13L^{\ast }\cdot (u^{\mathrm{\prime }}-u_n^{\mathrm{\prime }}),\\ v^{\ast }& =13L^{\ast }\cdot (v^{\mathrm{\prime }}-v_n^{\mathrm{\prime }}).\end{array}}$

The quantities u′_n and v′_n are the (u′, v′) chromaticity coordinates of a "specified white object" – which may be termed the white point – and Y_n is its luminance. In reflection mode, this is often (but not always) taken as the (u′, v′) of the perfect reflecting diffuser under that illuminant. (For example, for the 2° observer and standard illuminant C, u′_n = 0.2009, v′_n = 0.4610.) Equations for u′ and v′ are given below:

${\displaystyle \begin{array}{rlrl}{u}^{\mathrm{\prime }}& =\frac{4X}{X+15Y+3Z}& & =\frac{4x}{-2x+12y+3},\\ v^{\mathrm{\prime }}& =\frac{9Y}{X+15Y+3Z}& & =\frac{9y}{-2x+12y+3}.\end{array}}$

The reverse transformation

The transformation from (u′, v′) to (x, y) is:

${\displaystyle \begin{array}{cc}x& =\frac{9u^{\mathrm{\prime }}}{6u^{\mathrm{\prime }}-16v^{\mathrm{\prime }}+12}\\ y& =\frac{4v^{\mathrm{\prime }}}{6u^{\mathrm{\prime }}-16v^{\mathrm{\prime }}+12}\end{array}}$

The transformation from CIELUV to XYZ is performed as follows:

${\displaystyle \begin{array}{cc}{u}^{\mathrm{\prime }}& =\frac{1}{13}(u^{\ast }/L^{\ast })+u_n^{\mathrm{\prime }},\\ v^{\mathrm{\prime }}& =\frac{1}{13}(v^{\ast }/L^{\ast })+v_n^{\mathrm{\prime }},\\ Y& =\{\begin{array}{ll}(\frac{3}{29}{)}^3{L}^{\ast }{Y}_n,& L^{\ast }\le 8,\\ (\frac{1}{116}(L^{\ast }+16){)}^3{Y}_n,& L^{\ast }>8,\end{array}\\ X& =\frac{9u^{\mathrm{\prime }}}{4v^{\mathrm{\prime }}}Y,\\ Z& =\frac{12-3u^{\mathrm{\prime }}-20v^{\mathrm{\prime }}}{4v^{\mathrm{\prime }}}Y.\end{array}}$

Cylindrical representation (CIELCh)

CIELCh_uv, or HCL color space (hue–chroma–luminance) is increasingly seen in the information visualization community as a way to help with presenting data without the bias implicit in using varying saturation.

The cylindrical version of CIELUV is known as CIELCh_uv, or CIELChuv, CIELCh(uv) or CIEHLC_uv, where C*_uv is the chroma and h_uv is the hue:

${\displaystyle C_{uv}^{\ast }=\mathrm{hypot}(u^{\ast },v^{\ast })=\sqrt{(u^{\ast }{)}^2+(v^{\ast }{)}^2},}$

${\displaystyle h_{uv}=\mathrm{atan2}(v^{\ast },u^{\ast }),}$

where atan2 function, a "two-argument arctangent", computes the polar angle from a Cartesian coordinate pair.

Furthermore, the saturation correlate can be defined as

${\displaystyle s_{uv}=\frac{C^{\ast }}{L^{\ast }}=13\sqrt{(u^{\prime }-u_n^{\prime }{)}^2+(v^{\prime }-v_n^{\prime }{)}^2}.}$

Similar correlates of chroma and hue, but not saturation, exist for CIELAB. See Colorfulness for more discussion on saturation.

Color and hue difference

The color difference can be calculated using the Euclidean distance of the (L*, u*, v*) coordinates. It follows that a chromaticity distance of ${\displaystyle \sqrt{(\mathrm{\Delta }{u}^{\prime }{)}^2+(\mathrm{\Delta }{v}^{\prime }{)}^2}=1/13}$ corresponds to the same ΔE*_uv as a lightness difference of ΔL* = 1, in direct analogy to CIEUVW.

The Euclidean metric can also be used in CIELCh, with that component of ΔE*_uv attributable to difference in hue as ΔH* = √C*₁C*₂ 2 sin (Δh/2), where Δh = h₂ − h₁.