  Volume 8 Issue 1 October 2004  #### Appendix B: Calculating color rendering metrics

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, CEE 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. Previous © 2004 Rensselaer Polytechnic Institute. All rights reserved. Next