Color Rendering Index - Example

Example

The CRI can also be theoretically derived from the SPD of the illuminant and samples since physical copies of the original color samples are difficult to find. In this method, care should be taken to use a sampling resolution fine enough to capture spikes in the SPD. The SPDs of the standard test colors are tabulated in 5 nm increments CIE (2004), so it is suggested to use interpolation up to the resolution of the illuminant's spectrophotometry.

Starting with the SPD, let us verify that the CRI of reference illuminant F4 is 51. The first step is to determine the tristimulus values using the 1931 standard observer. Calculation of the inner product of the SPD with the standard observer's color matching functions (CMFs) yields (X,Y,Z)=(109.2,100.0,38.9) (after normalizing for Y=100). From this follow the xy chromaticity values:

The next step is to convert these chromaticities to the CIE 1960 UCS in order to be able to determine the CCT:

Examining the CIE 1960 UCS reveals this point to be closest to 2938 K on the Planckian locus, which has a co-ordinate of (0.2528, 0.3484). The distance of the test point to the locus is under the limit (5.4×10−3), so we can continue the procedure, assured of a meaningful result:

\begin{align} DC&=\sqrt{ (0.2531-0.2528)^2+(0.3477-0.3484)^2 } \\ & =8.12 \times 10^{-4} < 5.4 \times 10^{-3} \end{align}

We can verify the CCT by using McCamy's approximation algorithm to estimate the CCT from the xy chromaticities:

, where .

Substituting yields n=0.4979 and CCTest. = 2941 K, which is close enough. (Robertson's method can be used for greater precision, but we will be content with 2940 K in order to replicate published results.) Since 2940 < 5000, we select a Planckian radiator of 2940 K as the reference illuminant.

The next step is to determine the values of the test color samples under each illuminant in the CIEUVW color space. This is done by integrating the product of the CMF with the SPDs of the illuminant and the sample, then converting from CIEXYZ to CIEUVW (with the u,v coordinates of the reference illuminant as white point):

Illuminant TCS1 TCS2 TCS3 TCS4 TCS5 TCS6 TCS7 TCS8
Reference U 39.22 17.06 −13.94 −40.83 −35.55 −23.37 16.43 44.64
V 2.65 9.00 14.97 7.88 −2.86 −13.94 −12.17 −8.01
W 62.84 61.08 61.10 58.11 59.16 58.29 60.47 63.77
CIE FL4 U 26.56 10.71 −14.06 −27.45 −22.74 −13.99 9.61 25.52
V 3.91 11.14 17.06 9.42 −3.40 −17.40 −15.71 -10.23
W 63.10 61.78 62.30 57.54 58.46 56.45 59.11 61.69
CIE FL4
(CAT)		 U 26.34 10.45 −14.36 −27.78 −23.10 −14.33 9.37 25.33
V 4.34 11.42 17.26 9.81 −2.70 −16.44 −14.82 −9.47
W 63.10 61.78 62.30 57.54 58.46 56.45 59.11 61.69

From this we can calculate the color difference between the chromatically adapted samples (labeled "CAT") and those illuminated by the reference. (The Euclidean metric is used to calculate the color difference in CIEUVW.) The special CRI is simply .

TCS1 TCS2 TCS3 TCS4 TCS5 TCS6 TCS7 TCS8
12.99 7.07 2.63 13.20 12.47 9.56 7.66 19.48
Ri 40.2 67.5 87.9 39.3 42.6 56.0 64.8 10.4

Finally, the general color rendering index is the mean of the special CRIs: 51.

Read more about this topic:  Color Rendering Index

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