Getting Color Right in Products

Color is a feature of many products’ branding and benefits. Here I discuss how the “easy way” is not preferred for tolerancing color –that is quantifying  how close one measured color is to another either for quality control or for quantifying cleaning product efficacy. To do this appropriately in clients’ applications, we use a validated calculation template, and its formulas adapt readily to scripts in data pipelines. The template generates both ΔE* and CMC ΔE* color difference statistics –with the latter recommended for its superior correlation to how humans perceive color differences. Note that officially, the difference measures have a “*” in their names, but we will omit that for simplicity here.

Color Basics and the Easy (But Not Preferred) Way to Quantify Differences

Color is typically quantified by three numbers that can be measured by lab instruments or online sensors. The three sub-measures are L* (Lightness/darkness), a* (so-called red-to-green), and b* (yellow-to-blue). They locate a set of measurements in 3-D space as diagrammed below. High L-color of 100 is white and low L-color of zero is black. All points in between on the L-axis (e.g. a* and b* = 0) are a shade of gray ranging from very light to very dark.

The a and b rectangular coordinates are also commonly converted to polar coordinates –L,C,H space. Here the radius length is referred to as Chroma, C. It reflects the intensity of the color starting as a shade of gray at the a-b origin and increasing in visual intensity as C increases. The hue angle from 0 to 360 degrees, h, quantifies the color’s shade from red to green to blue to purple in a continuous scale while rotating around the a-b origin. The meaning of hue can be understood from the diagram. A hue angle of 45 degrees is a shade of red (e.g. the bold arrow on the diagram is at 45 degrees). Similarly, a hue angle of 135 degrees is green; 225 degrees is blue and so on.

Wikimedia Commons CIELAB chroma.svg

For quantifying color differences, it is neither convenient nor consumer relevant to individually control L, a and b (or L, C and h). Who wants to simultaneously track three numbers on a specification when one will do? The easy and historically-used method is to compute the ΔE color difference whose equation is shown below. It is the Euclidean distance between a color and its target/standard. The circle in the plot shows constant ΔE around a target in the a-b plane if L* is fixed. In 3-D space, these boundaries are spheres. ΔE is easy to calculate, but it doesn’t conform to how humans perceive color differences. We discuss CMC ΔE next. It was developed to overcome the shortcomings of ΔE, and the graph shows a CMC ΔE tolerance boundary for comparison. Note that the absolute limits are not comparable between the two systems. To display similar-sized regions, we chose 4.0 for one and 7.0 for the other.

CMC ΔE: Quantifying Color Differences in a Consumer-Relevant Way

Consumer relevance comes from switching to using the CMC ΔE statistic. Its tolerances are stretched and resized to correspond to human perception. This makes CMC DE boundaries elliptical in the a-b plane and in the out-of-plane L* direction –making them ellipsoids in three dimensions. They are oriented along lines of constant hue angle and are stretched in the Chroma direction as shown in the graph above.

The elliptical shape accounts for the fact that humans don’t perceive Chroma differences as readily as they do L* or especially hue differences. Adding to the complexity, human perception also depends on proximity to the a-b origin and to black (low L*) or white (high L*) in the third dimension. At least this latter point makes sense qualitatively. It’s difficult to tell what color an object is in a dark room.

The graph below shows color tiles generated to illustrate differences. These were generated by the excellent, website and have the same CMC ΔE of 4.0 versus the target. The differences are subtle at this level (and possibly especially viewed on a computer monitor). At this hue angle of 21 degrees (e.g. red), the +h* sample is slightly yellower and -h* is a bit more purplish. The -C* tile tends towards grayish versus +C*, and +L* and -L* tend towards lighter and darker respectively. The latter difference is made especially noticeable by placing those two tiles side-by-side. Acceptability would depend on the application.

The formula for CMC ΔE (See the Wikipedia Color Science article or ASTM standard D2244-16) is understandably complex based on the above discussion, but it is straightforward in Excel or in data pipeline scripts. Equations are shown at the bottom of this post, and a previous Tech Notes blog discusses some intricacies of h calculation. To safeguard calculations, our template stores the CMC ΔE and other formulas in a “refresh from” recipe sheet that keeps things updated and correct as data are entered and analyzed.

A rule of thumb from literature and personal experience is that, for non-textured/smooth samples in the stringent setting of well-lit, side-by-side testing, a CMC ΔE of 0.75 is the threshold where differences become perceptible with statistical significance in a consumer panel. Under these conditions, a CMC ΔE of 2.0 is clearly noticeable leading to comments like, “That one looks a bit yellower” or “That one is the same color but somewhat duller.”  If the product is more textured or if the lighting is not perfect, it takes a bigger difference to be noticeable.

For a given product context, CMC ΔE limits can be validated by consumer panel protocols such as triangle testing where products (for example makeup foundation shades, paints or molded plastics) are generated that vary CMC ΔE. In such a panel, panelists are asked to choose which of three products are different from the other two.

Some situations are harder to test because of the difficulty of generating samples having a specified difference. Tolerances for liquid products can often be set by conducting design-of-experiment (DOE) panel testing. This gets into what would be a separate blog topic about color formulation, but liquid products of a specified color difference can be made in test quantities using a mixing model that calculates the amounts to blend of mono-pigment versions of the product.

A watchout in panel testing is ensuring that the test design is appropriate for the in-market context. If the product is textured/matte finish or consumers are unlikely to experience true “side-by-side” or “well-lit” conditions, a higher tolerance is appropriate. In the case of cleaning efficacy, the goal is to bring the substrate back to its original color –imperceptibly different from the pristine material. This might lead to a side-by-side context in market if the consumer will see both cleaned and uncleaned zones.

If you are not yet using CMC ΔE as your color difference statistic, you should be! Please reach out if we can help!



CMC ΔE Excel Formula

CMC ΔE Mathematical Formula Per ASTM D2244-16