Theoretical Considerations

Let's imagine an ideal observer—with respect to visual colorimetry—with cone fundamentals as shown in Figure 4.4. For any stimulus, the cone responses can be calculated via integration, shown in Eqs. (4.1)–(4.3):

4.1

4.2

4.3

where S_λ is an illuminant's spectral power distribution, R_λ is an object's spectral‐reflectance factor, and l_λ, m_λ, and s_λ are the cone fundamentals of the ideal observer. We need to build a visual colorimeter that can display a color that matches any stimulus the ideal observer encounters. The easiest way to build the colorimeter is by using three lights. Suppose that the lights are monochromatic (a single wavelength) with wavelengths of 450, 550, and 650 nm. As shown in Figure 4.5, varying the radiance of the 450 nm light results in changing the response of only the ideal observer's short‐wavelength receptor. The long‐ and middle‐wavelength receptors cannot “see” light at 450 nm, having no response at this wavelength. In a similar fashion, varying the 550 nm light only causes responses in the ideal observer's middle‐wavelength receptor; varying the 650 nm light only causes responses in the ideal observer's long‐wavelength receptor. Thus, any stimulus can be matched by varying the amounts of the 450, 550, and 650 nm lights. It is possible to “dial in” any response for this ideal observer. A visual colorimeter with primaries of 450, 550, and 650 nm could be used to standardize the color of any stimulus of interest for this ideal observer.

What happens when we apply this reasoning using a real observer? We would like to find three primaries that enable each cone response to be independently controlled. Evaluating a real observer's cone spectral sensitivities reveals that it is not possible to achieve this aim. About the best that can be achieved uses wavelengths of 400, 520, and 700 nm, shown in Figure 4.6. Although it is straightforward to choose wavelengths that stimulate only the short‐wavelength and long‐wavelength receptors, it is impossible to select a wavelength that stimulates only the middle‐wavelength receptor because of the large amount of overlap between it and the long‐wavelength receptor. As a consequence, any visual colorimeter that contains only three primaries cannot produce matches to every possible stimulus. (By extension, any three‐primary‐color system has the same limitation.)

The Color‐Matching Experiment

Let's suppose that a student has an interest in color science. She secures funding from the university and is hired as a summer intern to learn firsthand about colorimetry. Having knowledge of the theoretical considerations just described, she designs and builds a visual colorimeter, such as shown in Figure 4.7. The colorimeter produces two fields of light, forming a bipartite field. In the adjustment field, light results from the admixture of three primaries: 440, 560, and 620 nm. (Though not shown in the drawing, the radiance of each light is adjustable.) Unit amounts of each primary when combined produce a white equal to the color of the equal‐energy spectrum. The spectra are plotted in Figure 4.8. In the fixed field, a number of different lights can be used. A masking screen is used to control the size of the bipartite field, expressed as a field of view in degrees.

In the first experiment, the three lights are adjusted until a match is made to an incandescent light. Our intern has verified the principle of metamerism: that colors can be matched despite their differences in spectral properties. In her second experiment, she doubles the amount of light in both fields. She finds that although the two fields look brighter, they still match. She doubles them again with the same result. Next, she reduces the amount of light in both fields. Even though the two fields get dimmer, they continue to match. This matching property is known as proportionality.

In a third experiment, shown in Figure 4.9, the fixed field contains a combination of greenish‐blue light and the red primary at a moderate intensity. The resulting color has lower chromatic intensity than the greenish‐blue light alone. Again, she adjusts the primary radiances in the adjustment field until it and the fixed field are indistinguishable. If she increases the red primary by the same amount in both fields, the two fields continue to match, although their color changes. If she decreases the red primary by the same amount in both fields, the two fields continue to match. This matching property is known as additivity. She also notices that as she decreases the amount of red light in the adjustment field, chromatic intensity increases. The red primary modulates color between greenish‐blue and white, shown in Figure 4.10.

Our budding scientist has verified what are known as Grassmann's laws of additive color matching (Grassmann 1853; Wyszecki and Stiles 1982). Essentially, color matching follows the principles of algebra. She has also discovered a “trick” that can help her overcome the inability to select primaries that control each of the cone responses separately. Can you guess what it is?

When one is performing a color‐matching experiment, the question arises as to what colors to match. Rather than thinking about a test light as a spectral power distribution with power that is continuous over the entire wavelength range, it is helpful to think of it as a combination of monochromatic (or nearly so) lights, each suitably weighted, shown in Figure 4.11. If 301 sources, each of equal radiance and of 1 nm width and centered at 400, 401 nm, and so on through 700 nm were simultaneously projected into the fixed field, they would produce a color that matches a continuous spectrum with equal radiance at each wavelength. Using this reasoning, every color stimulus can be thought of as a combination of light at individual wavelengths. Thus, it is sufficient to perform a color‐matching experiment in which an observer matches the two fields at each wavelength. In practice, the experiment is performed at every 10 nm throughout the visible spectrum. Interpolation is used to estimate the 1 nm data.

Our industrious intern next builds a device that produces variable monochromatic light at constant spectral radiance, shown in Figure 4.12. As she varies its wavelength, she notices a large change in the brightness of each individual wavelength: green wavelengths appear much brighter than the other visible wavelengths. She decides to perform a brightness experiment in which the monochromatic light is set to 555 nm. Next, the blue primary in the adjustable field is varied (while the other primaries are turned off) until a brightness match is made. This is repeated for the green and red primaries. This type of experiment is called heterochromatic brightness matching and despite how difficult this visual task is to perform consistently, our intern found that about 50 times more radiance for the blue primary and about three times more radiance for the red primary were required to match the brightness at 555 nm. The green primary had similar radiance to 555 nm. That is, if all four lamps were adjusted to the same radiance and displayed next to one another, the blue primary would appear the dimmest and 555 nm would appear the brightest. Recalling that our cone receptors reduce in sensitivity toward each end of the visible spectrum, these results are reasonable.

Having completed a number of experiments, including verifying Grassmann's laws and brightness matching, the student is ready to perform a color‐matching experiment. The experimental task is determining the amounts of each primary necessary to match monochromatic light presented in succession throughout the visible spectrum. From her experiments verifying Grassmann's laws and a realization that monochromatic light has very high chromatic intensity, it is clear that she will have to use one or more of the primaries in the fixed field in order to reduce the chromatic intensity of the monochromatic light such that the two fields can be matched, shown in Figure 4.13. This “trick” enables the experiment to be completed.

Having to keep track of the amounts of each of the six primaries would be cumbersome. Instead, Grassmann's laws are used to define color matches with positive and negative quantities, essentially rearranging an algebraic equation. The results of the color‐matching experiment performed by our intern are shown in Figure 4.14 as a plot in which the amounts of each primary are functions of wavelength. These curves are called color‐matching functions. Notice that they have both positive and negative lobes. The negative lobes indicate the amounts of a given primary added to the fixed field. The net amounts of each primary required to match a color are called tristimulus values. Tristimulus values can be both positive and negative. At the end of the summer, the student is encouraged to apply for a PhD in color science.

The 1924 CIE Standard Photopic Observer

During the beginning of the twentieth century, it was noticed that physical measurements of the power of a light source (i.e. radiance or irradiance) correlated poorly with the source's visibility. Two lights that produced equal radiance often had very different brightnesses. Our intern found this out as well when comparing the brightness of each of her primaries (440, 560, and 620 nm) with 555 nm. Rather than perform heterochromatic brightness‐matching experiments, known to be difficult and imprecise, a technique known as flicker photometry was used. In flicker photometry, the two lights are displayed alternately rather than simultaneously. The rate of flicker is set such that only the visual system's white opposed by black opponent channel is stimulated. (The opponent channels' temporal responses are similar to their spatial responses, described in Chapter 2.) The radiance of the test lamp is varied until flicker is minimized. Based on performing flicker photometry or heterochromatic brightness matching for each wavelength of the spectrum in comparison to results at a reference wavelength (such as 555 nm), a brightness‐sensitivity function is derived. In 1923, Gibson and Tyndall compiled the results from a number of studies, about 200 observers in total. They calculated a weighted average based on several criteria (see Kaiser 1981). In 1924, the CIE adopted Gibson and Tyndall's “visibility curve,” known today as the CIE standard photometric observer or luminous efficiency function, plotted in Figure 4.15 and denoted by V_λ (CIE 1926, 2018).

The “brightness” of a source is calculated by multiplying its spectral properties by the V_λ function, wavelength by wavelength, followed by integration, and finally multiplying by a normalization constant, K_m, shown in Figure 4.16 and Eqs. (4.6) and (4.7)

4.6

4.7

The normalizing constant, K_m, equals 683 lm/W, known as the maximum luminous efficacy. A lumen is a quantity that weights light by the luminous efficiency function, V_λ. When a light source's spectral irradiance, E_λ, is measured, illuminance, E, has units of lux (lx, lumens per square meter). When spectral radiance, L_λ, is measured, luminance, L, has units of candelas per square meter (cd/m², or lumens per square meter per steradian). A candela is a unit of luminous intensity defined as 1/683 W/Sr. Illuminance is used to define the amount of light falling on a surface expressed as lux or foot‐candles (fc, lumens per square foot). Foot‐candles multiplied by 10.76 equals lux. Luminance is used to define the amount of light generated by a source. Essentially, luminance measurements are not dependent on distance; illuminance measurements are. See Wyszecki and Stiles (1982) and McCluney (2014) for greater details on photometry and radiometry.

We have referred to brightness in quotations because of the large body of experimental evidence that indicates that a source's photometric measurement does not correlate with its perceived brightness (Kaiser 1981; Wyszecki and Stiles 1982; CIE 1988). A computer display appears much brighter in a darkened room than in one that is fully lit. If two lights are adjusted to have the same photometric quantity but one is highly chromatic while the other appears white, the chromatic source appears brighter. Despite the known limitations of photometry, it is still used extensively to define the level of illumination, and in other industrial applications as diverse as gloss and photographic film speed.

The 1931 CIE Standard Colorimetric Observer

Two experiments were performed during the 1920s in England that measured the color‐matching functions of a small number of color‐normal observers. Guild (1931) measured seven observers and Wright (1928, 1929) measured 10 observers. Both experiments employed the same viewing conditions, a bipartite field subtending a 2° field of view that was surrounded by darkness. In 1931 at a meeting of the Colorimetry Committee of the CIE, delegates representing a number of countries agreed to adopt a color‐matching system based on the Guild and Wright experimental results. The committee concluded that the agreement was sufficiently close to provide both independent validation and reasonable population estimation. The average data were calculated based on a set of primaries that would provide excellent repeatability when building laboratory visual colorimeters. The primaries selected were 435.8, 546.1, and 700 nm. (In imaging, these are known as the CIERGB primaries.) The first two wavelengths correspond to two of the mercury emission lines. It was reasoned that 700 nm would stimulate only the long‐wavelength receptor and therefore, small errors in setting this wavelength accurately would have a negligible effect on color matching. Another requirement when calibrating a colorimeter is defining the white that results from the admixture of the three primaries, each set to a unit amount. The white produced by an equal‐energy light source was selected. This set of color‐matching functions, shown in Figure 4.17 is known as , , and . They define the tristimulus values of the spectrum colors for this particular set of primaries. The bar over each variable implies average, as in .

At this time period, the General Electric‐Hardy Recording Spectrophotometer (Hardy 1929; Oil 1929) was near completion and soon to be marketed. This instrument produced an analog signal that traced an object's spectral reflectance or transmittance. It was envisioned, correctly, that this signal could be interfaced with mechanical or electrical devices that would be used to perform tristimulus integrations. This would allow colorimetry to evolve from a visual system to a computational system. However, since the , , and color‐matching functions all have both positive and negative tristimulus values, such devices would have to have six channels, greatly increasing their complexity and cost. If a second set of primaries was defined that resulted in all‐positive color‐matching functions, these devices could be built practically since only three channels would be required.

A second concern about , , and was also raised. Photometry using the visibility curve was already in use by the lighting industry. Since this system was based on different experimental techniques and a different set of observers, it was possible that two colors having the same tristimulus values could have different photometric values. Furthermore, for some applications, it might be necessary to calculate both colorimetric and photometric values — four integrations.

These two major concerns were alleviated by deriving an approximately linear transformation such that the color‐matching functions were all positive and that one of the color‐matching functions would be the 1924 CIE standard photometric observer function, V_λ. Having all‐positive color‐matching functions meant that the corresponding primaries would be physically nonrealizable. Variables X, Y, and Z were used for the new system to clarify that the primaries were not actual lights, resulting in color‐matching functions labeled as , , and , plotted in Figure 4.18. These color‐matching functions are known as the CIE 1931 standard colorimetric observer, or simply, the standard observer.

The details of how the transformation from , , to , , was derived can be found in Fairman, Brill, and Hemmendinger (1997, 1998). We will focus on several of them. First, , , and , and V_λ were normalized to unit area as shown in Eq. (4.10)

4.10

The color‐matching function was arbitrarily defined to equal V_λ

4.11

Ideally, is a linear combination of , , and . Least squares (i.e. pseudo‐inverse) was used to calculate the scalars of each color‐matching function, shown in Eq. (4.12):

4.12

where superscript + denotes pseudo‐inverse. These scalars define the position of primary Y.

Because Y defines luminance, the other primaries are without luminance. This constrained the locations of primaries X and Z along a specific line. Two additional constraints were imposed to locate X and Z on this line. The first was to define the volume of the new XYZ space to just encompass the gamut of real colors. The second was to maximize the number of wavelengths where equaled zeroes. This resulted in the transformation matrix shown in Eq. (4.13)

4.13

Because there was a small amount of residual error in estimating , color‐matching functions , , and were first normalized by multiplying by n_λ such that was identical with V_λ. The final transformation is shown in Eqs. (4.14) and (4.15):

4.14

where

4.15

The CIEXYZ system with color‐matching functions of , , and is often referred to as the 1931 standard observer or the 2° observer and is assumed to represent the color‐matching results of the average of the human population having normal color vision and viewing stimuli with a 2° field of view. When we compare color‐matching functions with cone fundamentals (e.g. Figure 4.6), it is clear that color‐matching functions should not be referred to as the eye's spectral sensitivities. As we will show below, they can be referred to as linear transformations of the eye's spectral sensitivities.

The 1964 CIE Standard Colorimetric Observer

We describe in Chapter 2 how the structure of the eye is different in the central region of the retina, the fovea, than in the surrounding regions. The experiments leading to the 1931 CIE standard observer were performed using only the fovea, which covers about a 2° angle of view. Because of experimental limitations during the 1920s, it was much easier to produce uniform bipartite fields if they were kept small. Also, the light that could be generated by visual colorimetry was somewhat dim. Since the fovea does not contain rods, the resulting color‐matching functions should be equally applicable for colors viewed at typical levels of illumination. That is, Grassmann's law of proportionality should be upheld.

However, there are a number of applications in which stimuli subtend a much larger field of view. The CIE encouraged experiments that would both determine whether the 2° observer would accurately predict matches for larger fields of view and validate the continued use of the 1931 standard observer.

During the 1950s Stiles, at the National Physical Laboratory in England, performed a pilot experiment in which 10 observers' color‐matching functions were determined for both 2° and 10° fields of view. The two fields of view are compared in Figure 4.20. The CIE Colorimetry Committee concluded that for practical colorimetry, the 1931 standard observer was valid for small‐field color matching; however, for large‐field color matching, research should continue (Wyszecki and Stiles 1982).

Stiles and Burch (1959) measured the color‐matching functions of 49 observers with a 10° field of view. Very high levels of illumination were used to minimize rod intrusion and further computations eliminated these nearly negligible rod effects. Speranskaya (1959) measured the color‐matching functions of 27 observers, also with a 10° field of view, but at considerably lower levels of illumination. The CIE removed the effects of rod intrusion in Speranskaya's data and weight averaged the two data sets, resulting in the 1964 CIE standard colorimetric observer (Wyszecki and Stiles 1982; Trezona and Parkins 1998; CIE 2018). It is usually referred to as the 1964 standard observer or the 10° observer. Its color‐matching functions are notated as , , and compared with the 1931 standard observer in Figure 4.21. A word of warning: is not the same as , or V_λ, and the corresponding tristimulus value Y₁₀ does not directly represent a color's luminance. Its use is recommended whenever color‐matching conditions exceed a 4° field of view.

Clearly, the 10° observer has a firmer statistical foundation since it is based on many more observers. Furthermore, large‐field color matching has higher precision (Wyszecki and Stiles 1982). Anecdotally, it is often believed that the 10° observer correlates more closely with visual evaluations when judging the color difference within metameric pairs. However, this is likely observer dependent. Most industries that manufacture colored products use the 1964 standard observer. For imaging applications, the 1931 standard observer is used.

Cone‐Fundamental‐Based Colorimetric Observers

In Chapter , we describe that there is a range of color vision among color‐normal observers. In 2006, the CIE published a model in which cone fundamentals can be calculated for average observers ranging in age from 21 to 80 years old and from a 1° to a 10° field of view (CIE 2006a). However, cone fundamentals are not all‐positive color‐matching functions. Work is underway by the CIE to provide a standard practice to convert cone fundamentals to the XYZ system, enabling the practical use of the 2006 CIE model. Thus far, matrices have been published that convert cone fundamentals for 32 years of age and either a 2° or 10° field of view, notated as , , and , , , respectively. Both standard observers have color vision equivalent to the average population of 32‐year‐old color‐normal observers. The transformations are shown in Eqs. (4.16) and (4.17), respectively (CIE 2018)

4.16

4.17

The 2° standard observer is based on the 1924 CIE standard photometric observer, as we described above. With time, it became clear that the weighting used by Gibson and Tyndall when deriving V_λ was in error, particularly in the short‐wavelength region where visibility was underestimated. For full‐spectrum lights, ignoring this error is inconsequential. Today, where stimuli can be narrow band such as LED lighting and laser primary digital projection, this error cannot be ignored. The 1931 standard observer and the 32 years of age and 2° field of view cone‐fundamental observer are compared in Figure 4.22, showing the magnitude of error. When accurate photometric quantities are required, either the cone‐fundamental or 1964 standard observer should be used.

Displays

The spectral radiance measurements of each primary of an LED backlight liquid‐crystal display, each at six different levels, are shown in Figure 4.29. Each primary has similar spectral characteristics. The relationship between these spectra is defined in Eqs. (4.30)–(4.32):

4.30

4.31

4.32

where R, G, and B are scalars modulating the intensity of each primary between zero and maximum. This property is known as scalability.

The relationship between spectral radiance and tristimulus values is linear, shown in Eqs. (4.33)–(4.35) for the red primary. Similar equations can be written for the green and blue primaries

4.33

4.34

4.35

Linearity between R and XYZ, G and XYZ, and B and XYZ, where RGB is defined using Eqs. (4.30)–(4.32), means that the tristimulus values of each primary define a line in the 풳풴풵 three‐space, shown in Figure 4.30.

A second aspect to explore is whether the sum of individual measurements of the three primaries is identical to a measurement of the three primaries displayed simultaneously, shown in Eq. (4.36)

4.36

This is verified experimentally, sampling the display's color gamut. This property is known as additivity.

A simple way to analyze scalability and additivity is to plot the chromaticities based on measurements of each primary at several intensities and their sum, producing a gray scale. There should be four points, one for each primary and one for the neutral. This is shown for two displays in Figure 4.31, one that is scalable and additive and one that is not.

For the display that is scalable and additive, also known as having stable primaries, we can predict tristimulus values for any color combination from measuring the spectral radiance of each primary, that is, three measurements. This is shown in Eq. (4.37)

4.37

The white point of the display, X_n, Y_n, and Z_n, is calculated by setting the display to R = G = B = 1, shown in Eq. (4.38)

4.38

Each primary's chromaticities remain stable for any setting. Accordingly, the primary tristimulus matrix can be expanded into a product of a chromaticity matrix and a luminance matrix, shown in Eq. (4.39) for a display's white point, expressed by chromaticities and peak luminance

4.39

By rearranging Eq. (4.39), the luminance of each channel can be calculated for any desired white point, shown in Eq. (4.40)

4.40

Usually, the luminance ratio is specified. By defining L_n = 1 and performing the multiplication on the left‐hand side of Eq. (4.40), the luminances of each channel are calculated, shown in Eq. (4.41)

4.41

Finally, the tristimulus matrix is calculated as shown in Eq. (4.42)

4.42

Cone Fundamentals

Thus far, we have used the well‐established (Smith and Pokorny 1975) cone fundamentals when plotting the eye's spectral sensitivities. However, these fundamentals are not linear transformations of CIE color‐matching functions as shown above in Figure 4.22 and there are a number of applications where LMS functions that are linear transformations of CIE color‐matching functions are useful. The set derived by Hunt and Pointer (1985) based on research by Estevez on color‐defective vision is in most common use, known as the HPE cone fundamentals. The transformation matrix is given in Eq. (4.49) and the cone fundamentals compared with the Smith and Pokorny set are plotted in Figure 4.33

4.49

L^* Lightness

The achromatic property of a color is one of the most important color attributes. Accordingly, scales that relate physical measurements with our perceptions of lightness, blackness, whiteness, etc. have received great attention over the years. As we described in Chapter , the Munsell value scale was the first achromatic scale to have both careful visual and instrumental definitions. Accordingly, it became the basis for a CIE lightness scale. When the Munsell system was published as colorimetric data (Newhall, Nickerson, and Judd 1943), luminance factor was a function of value, the function a fifth‐order polynomial. Although this was useful for producing color chips with specific value, the inverse transformation, used to relate luminance factor and lightness, required the use of a precalculated published table (Nickerson 1950).

Glasser et al. (1958) determined that the relationship between luminance factor and value was well modeled using a cube‐root psychometric function plus an offset. This led to the CIE L^* formula, given in Eqs. (4.50) and (4.51) (CIE 2018):

4.50

where

4.51

The relationship between luminance factor and L^* is plotted in Figure 4.34.

Approximately uniform color‐difference spaces recommended by the CIE use an asterisk to differentiate them from other formulas that use the same letters. Robertson (1990) describes the derivation of L^* and Berns (2000) provides a historical review of lightness formulas. Because of the −16 offset, black colors (i.e. luminance factor less than about 0.009) could lead to negative lightness values. As a consequence, a straight line from zero to the tangent of the cube‐root function was used to avoid negative values (Pauli 1976) and as a result, a logical operator is necessary when calculating L^*.

The need for an offset resulted from defining the psychometric function as a cube root. By fitting an exponential psychometric function, the Munsell value scale can be predicted equally well without an offset, shown in Eq. (4.52) (Fairchild 1995)

4.52

By replacing 10 with 100, a lightness function results that could be used instead of Eqs. (4.50) and (4.51).

u′v′ Uniform‐Chromaticity Scale Diagram

The lighting and display industries have used the xy chromaticity diagram since its inception in 1931. This enabled the specification of color that was independent of luminance. Regions in chromaticity space were defined for signal lights. Broadcasting defined the colors of the red, green, and blue primaries and white point of televisions using chromaticities. The need to specify acceptable ranges in chromaticities quickly led to improved projections that had better visual uniformity. Rather than projecting tristimulus values onto a unit plane, shown in Figure 4.24, tristimulus values were projected obliquely. The geometry was determined by using color‐discrimination data where observers scaled differences in chromaticities (Judd 1935; MacAdam 1937).

As an example, two lines are drawn in 풳풴풵 space, one about four times longer than the other, shown in Figure 4.35. The endpoints represent a color‐difference pair. Each pair has the same perceived color difference. Depending on our viewing direction, the differences in line length vary considerably. The optimal viewing direction, equivalent to a specific oblique projection, occurs such that the two lines are equal in length. At this projection, the calculated distance predicts the perceived distance.

Projections with improved visual spacing are known as uniform chromaticity scale (UCS) diagrams. The current CIE recommendation is u′ (“u‐prime”), v′, the formulas shown in Eqs. (4.53) and (4.54). See Berns (2000) for its historical development

4.53

4.54

It is also possible to calculate u′ and v′ from x and y and its inverse, shown in Eqs. (4.55)–(4.59). The third coordinate is w′, calculated in similar fashion to z

4.55

4.56

4.57

4.58

4.59

Plots similar to those shown in Figure 4.26 are shown for u′v′Y in Figure 4.36. There is clear improvement compared with xyY.

CIELUV

The volume of nonfluorescent reflecting colors in xyY, shown in Figure 4.28, or u′v′Y, shown in Figure 4.36, has a somewhat conical shape with its apex at Y = 100. As a consequence, two color‐difference pairs, each pair having the same chromaticity difference but dissimiliar average luminance factor, will appear unequal in color‐difference magnitude. This can be corrected by scaling the chromaticity diagram as a function of luminance factor such that the cone becomes a cylinder. Hunter (1942) used this approach in the design of a tristimulus colorimeter and corresponding color space.

The u′v′ UCS and L^* were combined, resulting in CIELUV (Robertson 1990), having rectangular coordinates of L^*, u^*, and v^*. The formulas for L^*, u^*, and v^* are shown in Eqs. (4.60)–(4.63) (CIE 2018):

4.60

4.61

where se 4.76,77

4.62

4.63

Because a white point can have a wide range of coordinates within u′v′, u^* and v^* result from a translation about the reference white, notated as and . This results in coordinates that are positive and negative and neutrals with u^* = v^* = 0. The Munsell colors at value 3, 5, and 8 are plotted in Figure 4.37.

Historically, CIELUV was used when both a UCS diagram and corresponding color‐difference space were desired. Because of improvements in color‐difference formulas that are not based on CIELUV, as we explain below, maintaining correspondence is not seen as advantageous and the use of CIELUV had diminished appreciably.

CIELAB

A second approach was taken to develop an approximately uniform color‐difference space, also during the early 1940s. Adams's (1942) approach was to transform tristimulus values directly without the intermediate step of a UCS. He used XYZ data for the Munsell system to develop and evaluate his new color space. Adams first rescaled the tristimulus values so that neutral colors would have equal tristimulus values, shown in Eqs. (4.64)–(4.66) (We introduce the prime superscript as a teaching aid.)

4.64

4.65

4.66

Since neutrals colors have X′ = Y′ = Z′, then (X′ − Y′) and (Z′ − Y′) are zero. Plotting colors this way would result in coordinates that are both positive and negative, in a similar fashion to Hunter (1942). Adams reasoned that the nonlinearity between Munsell value and luminance factor should be applied to all the tristimulus values, finally resulting in his chromatic value space.

Nickerson and Stultz (1944) modified Adam's space by optimizing constants using their color‐difference data. With the suggestion by Glasser and Troy (1952) to rearrange one of the chromatic axes to result in opponent properties similar to those of Hunter, ANLAB resulted, shown in Eqs. (4.67)–(4.69):

4.67

4.68

4.69

where f_V(x) is the nonlinear relationship requiring table lookup.

The CIE Colorimetry Committee met in 1973 and by combining ANLAB and the Glasser et al. (1958) cube‐root psychometric function, CIELAB was “born,” shown in Figure 4.38. Following the meeting, the final coefficients were determined (Robertson 1990). The new color space was named CIELAB with rectangular coordinates of L^*, a^*, and b^*. The formula is shown in Eqs. (4.70)–(4.73):

4.70

4.71

4.72

where

4.73

Inverting CIELAB to tristimulus values is shown in Eqs. (4.74)–(4.77):

4.74

4.75

4.76

where

4.77

The Munsell renotation colors at value 3, 5, and 8 are plotted in Figure 4.39. The circularity is excellent and the hue lines are well spaced. However, the blue and purple lines are quite curved. Because the Munsell system has five principal hues, they do not line up with the a^* and b^* axes. Therefore, the color names red, green, yellow, and blue should not be associated with +a^*, −a^*, +b^*, and −b^*, respectively. In fact, the CIE documentation defining both CIELAB and CIELUV never refers to the chromatic axes with color names (CIE 2018).

Rotation of CIELAB Coordinates

Although CIELAB (and CIELUV) are rectangular systems, it is straightforward to define positions using polar‐cylindrical coordinates. In this manner, there are approximate correlates of hue, h_ab, lightness, L^*, and chroma, . The forward and reverse formulas for CIELAB hue and chroma are shown in Eqs. (4.78)–(4.82)

4.78

4.79

4.80

4.81

4.82

CIELAB hue angle and chroma are shown in Figure 4.40. By convention, CIELAB hue angle is defined in degrees with the following values for each semiaxis: +a^* = 0° (360°), +b^* = 90°, −a^* = 180°, and −b^* = 270°. The same caution about color names applies to h_ab. The NCS elementary hues red, yellow, green, and blue have h_ab values of approximately 25°, 90°, 165°, and 260°, respectively. Chroma is the distance from the neutral point (a^* = b^* = 0) to the sample. We prefer the use of a^*b^* for near neutrals and for chromatic colors.

Thus far, we have considered lightness and chromaticness independently. However, colors in mixture vary in both dimensions simultaneously. Billmeyer and Saltzman (1966) described terms used by textile dyers and paint formulators, shown in Figure 4.41. There is a rotation within a plane of constant hue.

We pointed out in Chapter that NCS is arranged as a double cone. For some hues, this geometry is evident when the colors of constant nuance are plotted in CIELAB chroma versus lightness, for example, in Figure 4.42 for NCS R10B.

Berns (2014a) introduced two CIELAB variables, vividness, , and depth, , in order to increase the utility of CIELAB in relating coordinates with perception. Their formulas, both forward and inverse, are shown in Eqs. (4.83)–(4.88)

4.83

4.84

4.85

4.86

4.87

4.88

The inverse formulas are based on triangular coordinates where the third coordinate is calculated by knowing the coordinates of two other coordinates from among L^*, , , and . Their geometric meanings are diagrammed in Figure 4.43. Colors that become “cleaner” or “brighter” increase in CIELAB vividness. Colors that become “deeper” or “stronger” increase in CIELAB depth. The words “vivid” and “deep” have been used as adjectives to identify regions of the Munsell system (Kelly and Judd 1976). In the textiles industry, depth or depth of shade is a metric to aid in determining whether a batch of dyestuff has similar money value to previous batches (Smith 1997). Dye concentration is well correlated with (Berns 2014a).

When Berns was preparing visualizations of chroma, he noticed that if a color had the same lightness as the background, it became less distinct, seeming to fade away. This effect resulted irrespective of the starting color at maximum chroma or the color of the background, whether achromatic or colored. This led to a third dimension, clarity, , defined in Eq. (4.89) and its reverse defined in Eqs. (4.90) and (4.91)

4.89

4.90

4.91

An orange color is reduced in depth, clarity, chroma, and vividness in Figure 4.44 as an example. The background for the clarity calculation is the gray background used in the graph.

The NCS eight principal hues at maximum chromaticness are reduced in CIELAB chroma, depth, vividness, and clarity, shown in Figure 4.45. In a recent experiment, over 100 naïve observers assessed about 120 samples using the words “saturation,” “vividness,” “blackness,” and “whiteness.” There was high correlation between CIELAB depth and “saturation,” CIELAB clarity and “vividness,” CIELAB vividness and “blackness,” and CIELAB lightness and “whiteness” (Cho, Ou, and Luo 2017). CIELAB chroma had poorer correlation with “saturation” and “vividness,” indicating the importance of having CIELAB variables that vary in both lightness and chroma.

Whiteness

Whiteness is associated with a region in color space within which objects are recognized as white; degree of whiteness is, in principle, measured by the degree of departure of the object from a “perfect” white. Unfortunately, there are strong prejudices associated with industrial and national preferences which have prevented agreement on exactly what the “perfect” white is, and which directions of departure from it (toward blue, yellow, red, or green) are preferred or avoided. One consistent trend is that bluish whites appear whiter than neutral whites for samples having identical lightness. This is observed in Figure 4.54. In some circumstances, increasing blueness at the expense of reducing lightness increases whiteness.

With the introduction of fluorescent whitening agents (FWAs), it became possible to increase blueness without a reduction in lightness. FWAs absorb UV radiation and emit bluish light. These are used extensively, for example, in paper products, detergents, and plastics to increase whiteness. As fluorescent materials, the spectral properties of the illumination strongly affect the material's whiteness. This occurs for both visual evaluations and instrumental measurements.

Methods exist to evaluate whiteness using visual and instrumental measurements. See Smith (1997) for a historical review. Instrumentally, either spectral or colorimetric data can be analyzed. The CIE method uses colorimetric data (CIE 2018). It is also an ISO standard (2004). Research by Ganz (1976, 1979) and Griesser (1994) are the basis for a CIE whiteness index W—or commonly WI—where both luminance factor and hue are used in the calculation, shown in Eqs. (4.108) and (4.109) for the 1931 and 1964 standard observers, respectively

4.108

4.109

The illuminant in both cases is illuminant D65. These formulas assume that luminance factor for the perfect reflecting diffuser is 100.

The index is based on luminance factor, dominant wavelength, and purity, shown in Figure 4.25. With an increase in purity, a correlate of chromatic intensity, whiteness decreases. Depending on the dominant wavelength, a correlate of hue, whiteness will either increase or decrease. The hue weighting is along a blue‐yellow axis where blue is favored over yellow. Ganz and Pauli (1995) derived an approximate whiteness index in CIELAB based on the CIE WI; whiteness was a function of only L^* and b^*. The CIE whiteness index for the five colors is listed in Figure 4.53. The central color is neutral. The bluish sample has a higher whiteness than the neutral while the other colors have lower values. The yellow sample has the lowest whiteness.

The CIE method also includes the calculation of tint, T_W, shown in Eqs. (4.110) and (4.111)

4.110

4.111

When T_W is positive, the sample is greenish; when T_W is negative, the sample is reddish.

For materials suspected of being fluorescent, the spectrophotometer's light source must closely approximate D65 in both the UV and visible regions of the spectrum (Bristow 1994; Lin, Shamey, and Hinks 2012). We describe the measurement of fluorescent materials in detail in Chapter 6.

There are two obvious limitations of the CIE whiteness index. The first is the limitation to a single illuminant — D65. The use of a chromatic adaptation transform overcomes this limitation (Wei et al. 2017). The second limitation is the use of a color space with poor visual uniformity — xyY. However, since the region of color space is quite limited, a uniform color space is unlikely to lead to marked improvement.

Yellowness

The presence of small amounts of yellowness in nearly white (or nearly colorless) samples is both common and, usually, objectionable; therefore, considerable attention has been paid to devising uniform yellowness scales. The ASTM (2015b) has adopted a yellowness index, YI, for both standard observers and illuminants C and D65 that is based on CIE tristimulus values, the formula shown in Eq. (4.112)

4.112

There are two constants, C_X and C_Z. Their values depend on the illuminant and observer combination, listed in Table 4.2. These scales correlate with visual perception only for colors seen as yellow or blue; in the latter case, the yellowness index is a negative number. They should never be used to describe colors that are visibly reddish or greenish.