Editing Color rendering index (section)

== Example ==
{{missing information|section|Can we briefly do a R96<sub>a</sub> too?|date=November 2021}}
The CRI can also be theoretically derived from the spectral power distribution (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&nbsp;nm increments {{harvtxt|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 value]]s using the 1931 standard observer. Calculation of the [[inner product]] of the SPD with the standard observer's color matching functions (CMFs) yields (''X'',&nbsp;''Y'',&nbsp;''Z'') = (109.2,&nbsp;100.0,&nbsp;38.9) (after normalizing for ''Y''&nbsp;= 100). From this follow the ''xy'' chromaticity values:

[[Image:CIE 1960 UCS, FL4.svg|thumb|right|upright=0.8|The tight isotherms are from 2935&nbsp;K to 2945&nbsp;K. FL4 marked with a cross.]]

<math display="block">x = \frac{109.2}{109.2 + 100.0 + 38.9} = 0.4402,</math>

<math display="block">y = \frac{100}{109.2 + 100.0 + 38.9} = 0.4031.</math>

The next step is to convert these chromaticities to the [[CIE 1960 color space|CIE 1960 UCS]] in order to be able to determine the CCT:

<math display="block">u = \frac{4 \times 0.4402}{-2 \times 0.4402 + 12 \times 0.4031 + 3} = 0.2531,</math>

<math display="block">v = \frac{6 \times 0.4031}{-2 \times 0.4402 + 12 \times 0.4031 + 3} = 0.3477.</math>

[[Image:CIE illuminant F4 and a blackbody of 2938K.svg|thumb|Relative SPD of FL4 and a [[black body]] of equal CCT. Not normalized.]]

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

<math display="block">\begin{align}
 \text{DC} &= \sqrt{(0.2531 - 0.2528)^2 + (0.3477 - 0.3484)^2} \\
           &= 8.12 \times 10^{-4} < 5.4 \times 10^{-3}.
\end{align}</math>

We can verify the CCT by using [[Color temperature#Approximation|McCamy's approximation algorithm]] to estimate the CCT from the ''xy'' chromaticities:

<math display="block">\text{CCT}_\text{est.} = -449 n^3 + 3525 n^2 - 6823.3 n + 5520.33,</math>

where <math>n = \frac{x - 0.3320}{y - 0.1858}</math>.

Substituting <math>(x, y) = (0.4402, 0.4031)</math> yields ''n''&nbsp;= 0.4979 and CCT<sub>est.</sub>&nbsp;= 2941&nbsp;K, which is close enough. ([[Color temperature#Robertson's method|Robertson's method]] can be used for greater precision, but we will be content with 2940&nbsp;K in order to replicate published results.) Since 2940&nbsp;< 5000, we select a Planckian radiator of 2940&nbsp;K as the reference illuminant.

The next step is to determine the values of the test color samples under each illuminant in the [[CIE 1964 color space|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'',&nbsp;''v'' coordinates of the reference illuminant as white point):

{| class="wikitable"
|-
! colspan = 2 | Illuminant || TCS1 || TCS2 || TCS3 || TCS4 || TCS5 || TCS6 || TCS7 || TCS8
|-
! rowspan = 3 | 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
|-
! rowspan = 3 | 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
|-
! rowspan = 3 | CIE FL4<br />(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 <math>R_i = 100 - 4.6 \Delta E_{UVW}</math>.

{| class="wikitable"
|-
! || TCS1 || TCS2 || TCS3 || TCS4 || TCS5 || TCS6 || TCS7 || TCS8
|-
! <math>\Delta E_{UVW}</math> 
| 12.99 || 7.07 || 2.63 || 13.20 || 12.47 || 9.56 || 7.66 || 19.48
|-
! R<sub>i</sub>
| 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.

[[File:CIE CRI TCS under FL4.svg|thumb|upright=2.8|center|The cyan circles indicate the TCS under the ''reference'' illuminant. The short, black, vectors indicate the TCS under the ''test'' illuminant, before and after chromatic adaptation transformation (CAT). (The vectors are short because the white points are close.) The post-CAT end of the vector lies NW, mirroring the chromaticity vector between the reference and test illuminants.

The special CRIs are reflected in the length of the dotted lines linking the chromaticities of the samples under the reference and chromatically adapted test illuminants, respectively. Short distances, as in the case of TCS3, result in a high special CRI (87.9), whereas long distances, as in the case of TCS8, result in a low special CRI (10.4). In simpler terms, TCS3 reproduces better under FL4 than does TCS8 (relative to a [[black body]]).]]