Appendix B: Calculating color rendering metrics

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Volume 8 Issue 1	October 2004

Color rendering index (CRI) - A measure of the degree of color shift that objects undergo when illuminated by a lamp, compared with those same objects when illuminated by a reference source of comparable correlated color temperature (CCT). A CRI of 100 represents the maximum value. A lower CRI value indicates that some colors may appear unnatural when illuminated by the lamp. Incandescent lamps have a CRI above 95. The cool white fluorescent lamp has a CRI of 62; fluorescent lamps containing rare-earth phosphors are available with CRI values of 80 and above. Correlated color temperature (CCT) - A specification of the apparent color of a light source relative to the color appearance of an ideal incandescent source held at a particular temperature and measured on the Kelvin (K) scale. The CCT rating for a lamp is a general indication of the warmth or coolness of its appearance. As CCT increases, the appearance of the source shifts from reddish white toward bluish white; therefore, the higher the color temperature, the cooler the color appearance. Lamps with a CCT rating below 3200 K are usually considered warm sources, whereas those with a CCT above 4000 K usually considered cool in appearance. Efficacy - The ratio of the light output of a lamp (lumens) to its active power (watts), expressed as lumens per watt. Spectral power distribution (SPD) - A representation of the radiant power emitted by a light source as a function of wavelength. Blackbody radiator - A temperature radiator of uniform temperature whose radiant output in all parts of the spectrum is the maximum obtainable from any temperature radiator at the same temperature. Such a radiator is called a blackbody because it absorbs all the radiant energy that falls upon it. All other temperature radiators can be classed as non-blackbodies. Non-blackbodies radiate less in some or all wavelength intervals than a blackbody of the same size and the same temperature. Chromaticity - The dominant or complementary wavelength and purity aspects of the color taken together, or of the aspects specified by the chromaticity coordinates of the color taken together. It describes the properties of light related to hue and saturation, but not luminance (brightness). Color appearance - The resultant color perception that includes the effects of spectrum, background contrast, chromatic adaptation, color constancy, brightness, size and saturation. Color consistency - The measure of how close in color appearance random samples of a lamp or source tend to be. Color matching - The action of making a color appear the same as a given color. Often used as a method of evaluating the ability of a light source to render colors faithfully. Color stability - The ability of a lamp or light source to maintain its color rendering and color appearance properties over its life. The color properties of some discharge light sources may tend to shift over the life of the lamp. Full-spectrum index (FSI) - A mathematical measure of how much a light source's spectrum deviates from an equal energy spectrum, based on the slope of its cumulative spectrum. Full-spectrum color index (FSCI) - A mathematical transformation of full-spectrum index into a zero to 100 scale, where the resulting values are directly comparable to color rendering index. An equal energy spectrum is defined as having an FSCI value of 100, a “standard warm white” fluorescent lamp has an FSCI value of 50, and a monochromatic light source (e.g., low pressure sodium) has an FSCI value of 0. Gamut area - A measure of color rendering based upon volume in color space. It is the range of colors achievable on a given color reproduction medium (or present in an image on that medium) under a given set of viewing conditions. Hue - The attribute of a light source or illuminated object that determines whether it is red, yellow, green, blue, or the like. Isotemperature - A set of coordinates within which all points have the same temperature. In a color space diagram, isotemperature lines represent lights with identical correlated color temperatures. Metamers - Lights of the same color but of different spectral power distribution. Photopic - Vision mediated essentially or exclusively by the cones. It is generally associated with adaptation to a luminance of at least 3.4 cd/m². Primary - Any one of three lights in terms of which a color is specified by giving the amount of each required to match it by additive combination.

This appendix discusses four major color rendering metrics: color rendering index (CRI); full-spectrum index (FSI); full-spectrum color index (FSCI); and color gamut area (GA). Each color rendering metric emphasizes a slightly different aspect of color rendering.

Color rendering index

Color rendering index is a measure of a light source's ability to show object colors "realistically" or "naturally" compared to a familiar reference source, either incandescent light or daylight. CRI is calculated using eight reference samples in CIE 1995 color space (Technical Report No. 13.3-1995) when illuminated by a given light source (Boyce 2003). The CIE 1995 color space is used because equivalent distances in this color space are assumed to be "perceptually equal."

Full-spectrum index

Full-spectrum index (FSI) is a mathematical measure of how much a light source's spectrum deviates from an equal-energy spectrum. The following is a step-by-step procedure for calculating FSI:

1. Begin with a relative spectral power distribution (SPD). Normalize the values so the total power from 380 to 730 nm is equal to 1.

2. Calculate the cumulative power as a function of wavelength from short to long wavelengths.

3. Calculate the squared difference of the cumulative power distribution of the SPD and the cumulative power distribution of an equal energy spectrum, also with a total power of 1, over the wavelength range 380 to 730 nm.

Where the cumulative equal energy spectrum, C_EE is calculated as

4. Integrate the squared difference from 380 to 730 nm.

5. Circularly shift the SPD values a delta wavelength unit and repeat steps 2 through 4 for one complete circular cycle. A circular shift amounts to moving the first SPD value in the series to the end of that series. For example, using a delta wavelength of 1 nm and an SPD defined from 380 to 730 nm, the value at 380 nm is moved to the 730 nm position and the former 381 nm value now becomes the new 380 nm value. This is repeated until the series starts with what was originally the 730 nm value. In this example, this is repeated 350 times for the delta and wavelength limits.
With an SPD defined over a finite wavelength interval (in this case 380 to 730), a circular shift can be implemented by making the SPD periodic and extending over twice the wavelength interval.

6. FSI equals the average of the resulting integrated squared differences from step 4 when circularly shifted over one complete cycle. As the delta wavelength shift values approach infinitesimals the average would be calculated as an integral. Putting this together into integral form yields the following:

For practical computation the delta wavelength shift values are small wavelength increments and the average is computed with a summation. The LRC used a delta wavelength shift value of 1 nm.

Full-spectrum color index

Full-spectrum color index (FSCI) is a mathematical transformation of full-spectrum index into a zero to 100 scale. The resulting values compare directly with color rendering index. FSCI is a variant of FSI that has an inverse scale starting at 100 and scaled so that a warm white fluorescent lamp has a value of approximately 50, and any values less than zero (e.g. monochromatic light) are set to zero. FSCI is calculated as follows:

Gamut Area

Gamut area (GA) is more commonly used in Japan than in North America. In principle, GA is defined as the area enclosed within three or more chromaticity coordinates in a given color space. GA is usually calculated from the area of the polygon defined by the chromaticities of eight CIE standard color samples in CIE 1995 color space (Technical Report No. 13.3-1995) when illuminated by a given light source (Boyce 2003). The CIE 1995 color space is used because equivalent distances in this color space are assumed to be "perceptually equal." In general, the larger the GA, the more saturated the object colors will appear.