CIE's Spectral RGB Primaries

As an example, we want to find the spectral matching curves using a particular set of primaries composed of monochromatic (spectral) red, green blue, as recommended by the Commission Internationale de l'Eclairage (CIE):

• Red: $P_1(\lambda)=\delta(\lambda-\lambda_1)$, where 5 $\lambda_1=700$ nm;
• Green: $P_2(\lambda)=\delta(\lambda-\lambda_2)$, where $\lambda_2=546.1$ nm;
• Blue: $P_3(\lambda)=\delta(\lambda-\lambda_3)$, where $\lambda_3=435.8$ nm.

and the CIE reference white defined by a flat spectral distribution:

$$W(\lambda)=1$$

To find the spectral matching curves, we first find $a_{ij}$, the response of the ith type of cones to the jth primary:

$$a_{ij}=r_i(P_j)= \int S_i(\lambda) \delta(\lambda-\lambda_j) d\lambda=S_i(\lambda_j)$$

We next find the weights $A_j(W)$ to match the CIE reference white by solving the color matching equations for the reference white

$$\sum_{j=1}^3 A_j(W) T_j(W) a_{ij} = \int S_i(\lambda) W(\lambda) d\lambda = \int S_i(\lambda) d\lambda$$

where $T_j(W)=1$, as noted before.

Now we substitute these $A_j(W)$ $(j=1,2,3)$ and $a_{ij}=S_i(\lambda_j)$ into the matching equations for a spectral color $\delta(\lambda-\lambda')$ to get

$$\sum_{j=1}^3 A_j(W) T_j(\lambda) a_{ij} =\sum_{j=1}^3 A_j(W) T_j(\lambda) S_i(\lambda_j) =S_i(\lambda) \;\;\;(i=1,2,3)$$

Solving these equations we get $T_j(\lambda)\;\;\;(j=1,2,3)$ for any wavelength $\lambda$, i.e., the three spectral matching curves representing the amount of each of the spectral red, green and blue needed to match each spectral color, as shown in the plot. Note that the interpretation of the negative values for spectral red at certain wavelengths ( $438.1 nm < \lambda < 546.1 nm$) is the same as discussed before. Namely, spectral colors of wavelength in this range should be mixed with the spectral red to be matched with the mixture of the spectral green and spectral blue.