Calibration with a Reference White

The absolute magnitudes of the weights $A_j(L)$ is directly affected by the intensity of the given color $L(\lambda)$. To calibrate various intensities, we normalize $A_i(L)$'s by comparing them against a reference white light source $W(\lambda)$.

As discussed before, we define the tristimulus values of a given color $L$ as

$$T_j(L)=\frac{A_j(L)}{A_j(W)}\;\;\;\;\;(j=1,2,3)$$

where $A_j(W)$'s are the weights for the three primaries to match the reference white $W(\lambda)$. $A_j(W)$'s can be obtained by solving the matching equations for the reference white:

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

In particular, the tristimulus values for the reference white itself is always 1:

$$T_j(L)\vert _{L=W}=\frac{A_j(L) \vert _{L=W}}{A_j(W)}=\frac{A_j(W)}{A_j(W)}=1$$

Now the color matching equations for an arbitrary color $L(\lambda)$ become

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

Solving these equations, we can get $T_j(L),\;\;(j=1,2,3)$ for matching any given color $L(\lambda)$.