# Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

# CIE u′, v′ Uniform Chromaticity Scale Diagram and CIELUV Color Space

• János Schanda
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_12

## Definitions

### Uniform Chromaticity Scale Diagram (UCS Diagram)

The CIE 1976 uniform chromaticity scale diagram is a projective transformation of the CIE x,y chromaticity diagram yielding perceptually more uniform color spacing (see CIE colorimetry standard [1]). It is produced by plotting, as abscissa and ordinate, respectively, quantities defined by the equations:
$${\boldsymbol{u}}^{\boldsymbol{\prime}}=\frac{4\boldsymbol{X}}{\boldsymbol{X}+15\boldsymbol{Y}+3\boldsymbol{Z}}$$
(1)
$${\boldsymbol{v}}^{\boldsymbol{\prime}}=\frac{9\boldsymbol{Y}}{\boldsymbol{X}+15\boldsymbol{Y}+3\boldsymbol{Z}}$$
(2)
where X, Y, Z are the tristimulus values of the test color stimulus based on the CIE 1931 standard colorimetric system defined in ISO 11664-1/CIE S 014-1 [2].
Equivalent definition:
$${\boldsymbol{u}}^{\boldsymbol{\prime}}=\frac{4\boldsymbol{x}}{-2\boldsymbol{x}+12\boldsymbol{y}+3}$$
(3)
$${\boldsymbol{v}}^{\boldsymbol{\prime}}=\frac{9\boldsymbol{y}}{-2\boldsymbol{x}+12\boldsymbol{y}+3}$$
(4)
where
$$\boldsymbol{x}=\frac{\boldsymbol{X}}{\boldsymbol{X}+\boldsymbol{Y}+\boldsymbol{Z}} \mathbf{and}\ \boldsymbol{y}=\frac{\boldsymbol{Y}}{\boldsymbol{X}+\boldsymbol{Y}+\boldsymbol{Z}}\ .$$
(5)

### Uniform Color Space

The CIE 1976 L*u*v* color space is a three-dimensional, approximately uniform color space produced by plotting in rectangular coordinates, L*, u*, v*, quantities defined by the equations
$${\boldsymbol{L}}^{*}=116\ \boldsymbol{f}\left(\frac{\boldsymbol{Y}}{{\boldsymbol{Y}}_{\mathbf{n}}}\right)-116$$
(6)
$${\boldsymbol{u}}^{*}=13\ {\boldsymbol{L}}^{*}\left({\boldsymbol{u}}^{\boldsymbol{\prime}}-{\boldsymbol{u}}_{\mathbf{n}}^{\boldsymbol{\prime}}\right)$$
(7)
$${\boldsymbol{v}}^{\boldsymbol{\prime}}=13{\boldsymbol{L}}^{*}\left({\boldsymbol{v}}^{\boldsymbol{\prime}}-{\boldsymbol{v}}_{\mathbf{n}}^{\boldsymbol{\prime}}\right)$$
(8)
where
$$\boldsymbol{f}\left(\frac{\boldsymbol{Y}}{{\boldsymbol{Y}}_{\mathbf{n}}}\right)={\left(\frac{\boldsymbol{Y}}{{\boldsymbol{Y}}_{\mathbf{n}}}\right)}^{{1}\!\left/ {3}\right.} \mathbf{if} \frac{\boldsymbol{Y}}{{\boldsymbol{Y}}_{\mathbf{n}}}>{\left(\frac{6}{29}\right)}^3$$
(9)
$$\boldsymbol{f}\left(\frac{\boldsymbol{Y}}{{\boldsymbol{Y}}_{\mathbf{n}}}\right)=\frac{841}{108}\left(\frac{\boldsymbol{Y}}{{\boldsymbol{Y}}_{\mathbf{n}}}\right)+\frac{4}{29} \mathbf{if} \frac{\boldsymbol{Y}}{{\boldsymbol{Y}}_{\mathbf{n}}}\le {\left(\frac{6}{29}\right)}^3.$$
(10)
In these equations, Y, u′, and v′ describe the test color stimulus, and Yn, un, and vn describe a specified white stimulus.

If the angle subtended at the eye by the test stimulus is between about 1° and 4°, the tristimulus values X, Y, Z calculated using the color-matching functions of the CIE 1931 standard colorimetric system should be used. If this angular subtense is greater than 4°, the tristimulus values X10, Y10, Z10 calculated using the color-matching functions of the CIE 1964 standard colorimetric system should be used. The same color-matching functions and the same specified white stimulus should be used for all stimuli to be compared with each other.

### Correlates of Lightness, Saturation, Chroma, and Hue

Approximate correlates of the perceived attributes lightness, saturation, chroma, and hue are calculated as follows:
 CIE 1976 lightness L* as defined by Eq. 6 CIE 1976 u,v saturation (CIELUV saturation) $${s}_{\mathrm{uv}}=13{\left[{\left({u}^{\prime }-{u}_{\mathrm{n}}^{\prime}\right)}^2+{\left({v}^{\prime }-{v}_{\mathrm{n}}^{\prime}\right)}^2\right]}^{1/2}$$ CIE 1976 u,v chroma (CIELUV chroma) $${C}_{\mathrm{uv}}^{*}={\left[{\left({u}^{*}\right)}^2+{\left({v}^{*}\right)}^2\right]}^{1/2}$$ CIE 1976 u,v hue angle (CIELUV hue angle) $${h}_{\mathrm{uv}}= \arctan \left({{v}^{*}}\!\left/ \!{{u}^{*}}\right.\right)$$

where the CIELUV hue angle lies between 0° and 90° if u* and v* are both positive, between 90° and 180° if v* is positive and u* is negative, between 180° and 270° if v* and u* are both negative, and between 270° and 360° if v* is negative and u* is positive.

### Color Differences

The CIE 1976 L*u*v* color difference, ΔE uv * , between two color stimuli is calculated as the Euclidean distance between the points representing them in the space:
$$\begin{array}{l}\Delta {E}_{\mathrm{uv}}^{\ast }={\left[{\left(\Delta {L}^{\ast}\right)}^2+{\left(\Delta {u}^{\ast}\right)}^2+{\left(\Delta {v}^{\ast}\right)}^2\right]}^{1/2}\\ {}\mathrm{or}\ \Delta {E}_{\mathrm{uv}}^{\ast }={\left[{\left(\Delta {L}^{\ast}\right)}^2+{\left(\Delta {C}_{\mathrm{uv}}^{\ast}\right)}^2+{\left(\Delta {H}_{\mathrm{uv}}^{\ast}\right)}^2\right]}^{1/2}\end{array}$$
(11)
where $$\Delta {H}_{\mathrm{uv}}^{*}=2{\left({C}_{\mathrm{uv},1}^{*}{C}_{\mathrm{uv},2}^{*}\right)}^{{1}\!\left/ \!{2}\right.} \sin \left(\Delta {h}_{\mathrm{uv}}/2\right).$$

For further details and calculating differences of color coordinate components, see Eq. 1.

## Overview

The CIE 1976 uniform chromaticity scale diagram and the CIE 1976 L*u*v* color space have been agreed by the CIE in 1976 as a modification of the CIE1960 UCS diagram (u,v diagram) and CIE 1964 uniform color space (U*, V*, W* space). The u,v diagram was devised by David MacAdam [3] and recommended by the CIE in 1959 [4]. Observations have shown that the u,v diagram could be made more uniform if v coordinate would be modified. The original equations for u, v are given in Eq. 12.
$$u=4X/\left(X + 15Y + 3Z\right) \mathrm{and} v=6Y/\left(X + 15Y + 3Z\right);$$
thus, u′ = u, and v′ = 3v/2 [5].
Figure 1 shows MacAdam ellipses in the u′, v′ diagram. As can be seen, the u′, v′ diagram is not absolutely equidistant (the ellipses are not circles), but its uniformity is by an order of magnitude better as that of the x, y diagram.
The u, v diagram described only chromaticities of colors of equal luminance. In 1963, CIE extended this to a three-dimensional spacing perceptually more nearly uniform than that provided by the (XYZ) system. “The recommended coordinate system is formed by plotting the variables U, V, and W along orthogonal axes where U, V, and W are defined in terms of the tristimulus values X, Y, Z, as
$$W=25{Y}^{1/3}\hbox{--} 17, 1\le \mathrm{Y}\le 100;U=13W\left(\mathrm{u}\hbox{--} {\mathrm{u}}_{\mathrm{n}}\right);\;V=13W\left(v\hbox{--} {v}_{\mathrm{n}}\right)$$
where u, v, un, and vn are described in the previous paragraphs [6].

The establishment of the CIE uniform chromaticity scale diagram and the CIELUV space was the result of a long-lasting development. The experts, at the CIE 1976 meeting, proposed two systems, the CIELUV and the CIELAB systems, but on the then available visual data, they were unable to decide between the two. During the past 30–40 years, the CIELAB system and its derivatives (see CIELAB in this volume) proved to be a better approximation of color perception; nowadays, the CIELUV space has only very limited use, but the u′, v′ diagram is still in general use, as it represents a linear diagram, in which chromaticities can easily be calculated. Thus, e.g., if the chromaticity of a blue LED and of a phosphor emission that is excited by the LED is given, it is easy to visualize the chromaticities that can be produced by the additive mixture of the two colors; they are located on the straight line drawn between the two chromaticity points.

## References

1. 1.
IE: Joint International Standard: Colorimetry – Part 5: CIE 1976 L*u*v* colour space and u′, v′ uniform chromaticity scale diagram. CIE S 014-5/E:2009/ISO 11664–5 (2009)Google Scholar
2. 2.
CIE: Joint International Standard: Colorimetry – Part 1: CIE Standard Colorimetric Observers. CIE S 014-2/E:2006/ISO 11664–2:2007(E)Google Scholar
3. 3.
MacAdam: Projective transformations of I. C. I. color specifications. JOSA. 27, 294–299 (1937)Google Scholar
4. 4.
CIE: Official Recommendation of the Committee W-1.3.1 – Colorimetry on “a diagram yielding colour spacing perceptually more nearly uniform than the (x y) diagram.” In: Proceedings of the Brussels Session 1959. CIE Publication 4–7, vol. A, pp. 3–37 & 91–94. CIE (1960)Google Scholar
5. 5.
CIE: Progress report of CIE TC – 1.3 Colorimetry. Compte Rendu 18e Session, Londres 1975. CIE Publication 36, pp. 161–172. (1975)Google Scholar
6. 6.
CIE: Official Recommendations of the Committee E-1.3.1 – Colorimetry. In: Proceedings of the Vienna Session 1963. CIE Publication 11. vol. A, pp. 35 & 112–113. CIE (1963)Google Scholar