Color Difference Equations for Solid Colors

1. CIELAB Color Space

Internationally standardized color systems, like the widely used CIELAB system, combine data of standardized illuminant, standardized observer, and spectral reflection data in three color components describing the lightness, hue and chroma of a color. These color components are used to communicate and document absolute color values as well as color differences.
The CIELAB system (Fig. 1), recommended by the CIE in 1976, consists of two axes a* and b* which are at right angles and represent the hue dimension. The third axis is the lightness L*. It is perpendicular to the a*b* plane. Within this system, any color can be specified with the coordinates L*, a*, b*. 
Alternatively, L*, C*, h° are commonly used.  C* (= Chroma) represents the intensity or saturation of the color, whereas the angle h° is another term to express the actual hue [1,2]. The correlates of hue and chroma are converting the rectangular a*b* axes into polar coordinates. Lightness, chroma and hue correspond to perceived color attributes, which are generally easier to understand [4].

C*_ab = (a*² + b*²)^1/2 
h_ab = arctan (b*/a*)

1.1 What do the "Color Differences" mean?

To keep a color on target a reference or standard needs to be established. Production runs can be compared to that standard and differences are recorded over time to determine a potential deviation at an early stage. Thus, color communication is done in terms of differences rather than absolute values. To determine the cause of a color difference, the individual colorimetric components ΔL*, Δa*, Δb* or ΔL*, ΔC*, ΔH* need to be recorded.
The differences are calculated by subtracting the standard values from the sample values:

Δ =  Sample – Standard

The resulting color differences may have descriptive words added to help in their understanding:

ΔH*_ab is introduced to provide consistency with the perceptual understanding that a color difference can be divided into a vector sum of a lightness, chroma and hue difference. It is calculated according to the following equations [2].  
 ΔH*_ab = 2(C*_ab,sample ·C*_ab,standard)^1/2 * sin( Δh_ab/2) or  ΔH*_ab = [( ΔE*_ab)² - ( ΔL*)² - (C*_ab)²]^1/2 
A positive sign indicates a counterclockwise change of hue, a negative sign a clockwise change. Consequently, for data interpretation it is necessary to know the color location of the standard.

1.2 Total color difference  ΔE*_ab

The total change of color, ΔE*_ab, is commonly used to represent a color difference. ΔE*_ab defines the Euclidean distance between sample and standard. There are two equivalent definitions [2].
ΔE*_ab = [(ΔL*)² - (Δa*)² - (Δb*)²]^1/2 
ΔE*_ab = [(ΔL*)² - (ΔC_ab*)² - (ΔH*_ab)²]^1/2
The same ΔE*ab value can be obtained for two sample sets, and yet they can look completely different. If the total color difference is divided equally between the individual color components, it is less obvious for the human eye than if the difference is dominant in one component. Such a variation is perceived more clearly.

Image 1 CIELAB color space

2. Color tolerances with CIELAB

The color differences that can be accepted must be agreed upon between customer and supplier. These tolerances are dependent both on the quality requirements of the application and technical capabilities of the process.
A common practice is to set tolerances on the individual color components. In practice, very often rectangular tolerances for ΔL*a*b* are used. They are also referred to as “box tolerances”. 
ΔL*a*b tolerances have a good agreement with visual perception for achromatic colors. Figure 2 shows a gray color with accepted and rejected samples. All accepted samples are within a circle as visual acceptance for achromatic colors is approximately the same for Δa* and Δb*. In this case, a box can well replicate a circle.
For chromatic colors the situation is different. Visual perception allows larger variations in the direction of chroma than in the direction of hue. The visual tolerance space is elliptical. Matching an ellipse with a box is not working very well. Either the tolerance box is too large which results in visually rejected samples being instrumentally accepted, or the tolerance box is too small which means that many samples that are visually rated as pass will be rejected (Fig. 3). Using “pie-shape” tolerances based on the polar coordinates ΔL* C* H* allow a better fit to visual perception (Fig. 4). However, there are still areas where a visually rejected sample would pass the specification.

Image 2 ± dL*a*b* tolerances for achromatic colors

Image 3 ± dL*a*b* tolerances for chromatic colors

Image 4 Polar dL*C*H* tolerances for chromatic colors

3. Limitations of CIELAB System

The intention of the CIELAB system was to create a uniform color space, meaning the same ΔE*_ab value is perceived as the same magnitude of difference, no matter which color is evaluated. But this was not entirely successful. Human beings are very sensitive to slightest variations for achromatic (or neutral) colors such as white, gray or pastel colors, while the same small change for a high chromatic yellow for example is hardly recognized. 
Figure 5 shows the CIELAB color space divided into a multiple number of ellipsoidal micro-spaces. All colors within one ellipse are perceived as the same color. It can clearly be noticed that the size and shape of the ellipses are different dependent on the hue. In addition, chromatic colors have larger ellipses than achromatic colors and the tolerance in the hue direction is smaller than in the chroma direction. Last but not least, lighter shade colors can have larger tolerances than similar darker shades (Fig. 6). Therefore, tolerances need to be defined by color families and differently for the individual color components ΔL*a*b*C*H*. 
Very common is to monitor the individual color difference components separately in a line chart related to their tolerance. But in reality, a 3-dimensional tolerance box is created around the standard and the color component deviations must be considered as a whole. All samples within the box are passing. However, the maximum deviation of a sample in the box varies depending on whether only one, two or three color components use up the maximum tolerance. For example, a ΔL*Δa*Δb* tolerance of 1 results in a maximum tolerance of ΔE* = 1 for one component, ΔE* = 1.4 for two components or ΔE* = 1.7 for three components (Fig. 7). Numerically all three samples pass the box tolerances, but visually there is a big difference. 

Image 5 Example of tolerance ellipses in CIELAB dependent on chroma and hue

Image 6 Example of tolerance ellipsoids in CIELAB dependent on lightness

Image 7 dE distance within a 3D tolerance box

4. Weighted color difference equations for solid colors

The only tolerance space that ensures the same distance to the standard is a sphere (Fig. 8). Over the years, new color difference equations for solid colors were developed. The goal was to improve the agreement with visual perception and use the same Pass/Fail criteria (ΔE) for all colors. The prerequisite for an overall ΔE limit is that all components must have the same tolerance. This can only be achieved with a weighted color difference equation.
Weighted color difference (ΔE_w) means rescaling of the color difference components with a scaling factor. The rescaling adjusts the size and shape of the ellipses dependent on the location of the standard in color space and the direction of differences between the color pairs. 
The scaling factors are calculated by two types of coefficients:

• S-factors (also called weighting functions) to adjust the calculated color differences for variations in perceived color differences dependent on the location in color space [3]  
• k or g factors (parametric coefficients or application factors) referring to conditions influencing color difference judgment e.g. experimental viewing conditions or different applications like batch approval or production QC [3] 

A weighted color equation not only has the advantage that one ΔE limit can be used for all colors, but as the individual color components are weighted, they can always be compared to a limit of one and do not need to be compared to their color specific tolerances.

ΔL_W = ΔL* / Scaling Factor_L            
ΔC_W = ΔC* / Scaling Factor_C            ΔE_W = [(ΔL_W)² + (ΔC_W)² + (ΔH_W)²]^1/2 
ΔH_W = ΔH* / Scaling Factor_H    
Scaling Factor = k x S-Factor

For the development of weighted color equations different sets of experimental data have been used. These data sets play an essential role because the equations are essentially optimized to those data sets. When the equations are later applied to other data sets, they may not perform as well [4].

Image 8 A sphere within the box

5. Weighted color equation ΔECMC

CMC is an acronym for the Color Measurement Committee of the Society of Dyes and Colorists of Great Britain which was largely responsible for the development of the equation in 1984. It is based on the visual evaluation of colored polyester thread pairs [4] and is today specified in the following standards:

• British Standard BS6923
• American AATCC Test Method 173
• ISO International Standard 105-J03

Modifications were introduced to correct two anomalies: a discontinuity of chromatic differences close to the achromatic axis and the overpredictions of lightness differences close to black (L* < 16) [4]. 
The CMC acceptance volume takes the shape of an ellipsoid with the semi-axis lS_L, cS_C and S_H in the direction of lightness, chroma and hue differences (Fig. 9). The total volume of the ellipsoid is equal to a ΔE_CMC of 1.0. As the region of color space changes, the size of the ellipse changes, but the ΔE_CMC remains 1.0 (Fig. 10). Thus, it is possible to use a single value for ΔE_CMC as a pass/fail tolerance for all colors.
ΔE_CMC is typically written as ΔE_CMC (2:1), where (2:1) corresponds to the variables (l:c) in the equation. With these values the ratio of lightness, chroma, and hue are defined so that they correlate with visual assessment of textile samples [5]. 

Image 9 dECMC equation and weighting functions

Image 10 dECMC tolerance ellipses in CIELAB [8]

6. Weighted color equation ΔE*94

In 1995 the CIE published a new color difference formula, called ΔE*94, for use in industrial pass/fail color difference evaluations. It is based on the CIELAB color system with added corrections for variation in perceived color difference resulting from variation in the chroma of the standard. For the equation development existing sample sets were used (BFD-P, RIT-DuPont and Witt–BAM) including different materials like textiles and glossy painted panels [4]. It is published in the CIE Technical Report 116: Industrial Color Difference Evaluation [6]. 
The CIE94 equation also produces an ellipsoid for color acceptance. A conservative approach was taken by only including corrections that could be reliably estimated. The most reliable determined effects of color standard location are a decrease in perceived chroma and hue difference with increasing chroma. It was assumed that the CIELAB L* scale was correct [6]. Please refer to figure 11 and 12 for the complete equation and an illustration of the tolerance ellipses. Reference viewing conditions were defined such as luminance level, background color and presentation technique. For these conditions the parametric factors k_L, k_C, k_H are equal to 1. They may be adjusted for varying conditions.
CIE94 is a further development of 1976 CIELAB. As CIELAB works well for achromatic colors, the CIE94 weighting functions for achromatic colors are equal to 1 and ΔE*94 gives the same result as ΔE*_ab. For brilliant colors the situation is different. Figure 13 shows a sample with a chroma difference of 3 and another sample with a hue difference of 3. ΔE*_ab is the same for both, namely 3. But visually the difference in hue is much more obvious to the human eye than the difference in chroma. When using the dE*94 equation this effect is taken care of:

• Scaling factors are larger than 1 because the human eye is more tolerant to changes for chromatic colors.
• Scaling factor for chroma is larger than scaling factor for hue because chroma variations are not perceived very well and thus, are less weighted.

Consequently, the sample with the chroma difference is accepted whereas the sample with the hue difference is rejected.

Image 11 dE*94 equation and weighting functions

Image 12 dE*94 tolerance ellipses in CIELAB [8]

Image 13 dE*94 result for samples with chroma and hue difference

7. Weighted color equation ΔE00

As further research also proved a hue dependence of color difference a new CIE technical committee was assigned the task to carry out a new working program for further improvements of CIELAB color space. Large data sets of Luo 1986, Berns 1991, Kim 1997 and Witt 199 were used for the studies. The result was the publication of CIEDE2000 (ΔE₀₀) in the technical report CIE 142 in 2001. It is recommended as a replacement for the CIELAB and CIE94 equation [6]. 
The ΔE₀₀ formula includes the following major improvements in comparison to CIELAB:

• Weighting functions for lightness, chroma and hue were the weighting function for hue (S_H) includes a chroma and hue correction (Fig. 14 + 15)
• Rescaling of the CIELAB a*-axis to improve the performance for neutral colors. 
• A rotation function to improve the performance for blue colors. 

The last two optimizations clearly distinguish the ΔE₀₀ equation from the ΔE*94 equation. Therefore, these two differences are discussed in detail in the following.

Improvements for neutral colors
For neutral colors the visual perception volume in the a*b* plane is not a circle but an ellipse with its major axis oriented in the b* direction [7]. To account for this effect ΔE₀₀ introduced the G function (Fig. 16). For low chroma the modification increases the magnitude of a’ values compared to CIELAB a* values. The maximum modification is 50%. At higher chroma the modified a’ value approaches CIELAB a* value [3]. The highest impact of the G function is for C*_ab < 30 units. 
A practical example is shown in figure 17. Two gray samples with a difference in a* are compared to the standard. Visually they are rated as being out of specification. When applying the ΔE*94 equation, one sample is accepted (within the green tolerance circle), the second sample is in the warning range. Only when using ΔE₀₀, the instrumental results agree with visual judgement. Both samples are failing because the tolerance space is optimized to an ellipse.

Improvements for blue colors
The visual studies also showed an interaction between chroma and hue differences in the blue region of color space. The result is a significant tilt of the tolerance ellipse. The major axis does not point towards the zero point but is rotated anti-clockwise (Fig. 18). To account for this effect ΔE₀₀ introduced a rotation function. It has a significant impact only for high chromatic blue colors with a hue value of approx. 275° ± 25° [3]. 
A practical example is shown in figure 19. Two blue samples with a hue value of approx. 270° are compared to the standard. Visually they are rejected but when using the ΔE*94 equation they are well within specification. Only when applying ΔE₀₀, the rotation of the tolerance ellipse causes them to fail. 

Image 14 dE00 equation and weighting functions

Image 15 dE00 tolerance ellipses in CIELAB [8]

Image 16 Rescaling of a* axis for dE00

Image 17 dE*94 and dE00 results for gray samples with difference in a*

Image 18 Tilting of tolerance ellipses in the blue region for dE00 [8]

Image 19 dE*94 and dE00 results for blue samples

8. Summary

CIELAB color space as of 1976 is still widely used in many industries. However, since many companies are placing more importance on instrumental color measurement, weighted color difference equations are increasingly finding their way into company specifications. The perceived advantages of being in better agreement with visual perception and the possibility to use the same ΔE limit for all colors gain the upper hand over the routines of the past. 
BYK-Gardner’s spectro2guide spectrophotometer (Fig. 20) together with the data analysis software smart-chart offers flexible data analysis with the latest color difference equations. You can easily toggle between ΔE*_ab, ΔE_CMC, ΔE*94 and ΔE₀₀ to define your color tolerances in a comprehensive and professional way.