Transformation between Different Color Systems

As we have seen, any three linearly independent colors can be considered as the primary colors and used as the bases of the 3D color space. Therefore the same color may be represented by different tristimulus values under different color systems of different primaries. For example, consider color $L(\lambda)$ matched in the color space by two sets of primaries $P_j(\lambda)\;\;(j=1,2,3)$ and $Q_j(\lambda)\;\;(j=1,2,3)$:

$$L(\lambda) \equiv \sum_{j=1}^3 A_j P_j(\lambda) \equiv \sum_{j=1}^3 B_j Q_j(\lambda)$$

As all terms in this equation are matching colors, they must all cause the same receptor responses:

$$r_i=\int L(\lambda) S_i(\lambda) d\lambda =\sum_{j=1}^3 A_j ... ...3 B_j \int Q_j(\lambda) S_i(\lambda) d\lambda \;\;\;(i=1,2,3)$$

To find the relationship between the two sets of tristimulus values $[A_1, A_2, A_3]$ and $[B_1, B_2, B_3]$, we first consider matching the second set of primaries by the first set:

$$Q_k(\lambda) \equiv Q'_k(\lambda) =\sum_{j=1}^3 C_{kj} P_j(\lambda)\;\;\;(k=1,2,3)$$

Note that in general $Q'_k(\lambda)$ and $Q_k(\lambda)$ do not have identical spectral energy distributions, but they are matching colors represented by the same point in the 3D color space. Now we can use $Q'_k(\lambda)$'s as the primaries to match the color $L(\lambda)$ so that

$$r_i = \sum_{k=1}^3 B_k \int Q'_k(\lambda) S_i(\lambda) d\lambda = \sum_{k=1}^3 B_k \int \sum_{j=1}^3 C_{kj} P_j(\lambda) S_i(\lambda) d\lambda$$

$$= \sum_{k=1}^3 B_k \sum_{j=1}^3 C_{kj} \int P_j(\lambda) S_i(\lambd... ...mbda = \sum_{j=1}^3 A_j \int P_j(\lambda) S_i(\lambda) d\lambda \;\;\;(i=1,2,3)$$

The last equation is due to the fact that this color is also matched by the primary $P_j(\lambda)$'s as assumed originally, and can be written as

$$\sum_{j=1}^3 [ A_j-\sum_{k=1}^3 C_{kj} B_k ] \int P_j(\lambda) S_i(\lambda) d\lambda =0 \;\;\;(i=1,2,3)$$

As in general

$$\int P_j(\lambda) S_i(\lambda) d\lambda \neq 0$$

we must have

$$A_j=\sum_{k=1}^3 C_{kj} B_k \;\;\;(j=1,2,3)$$

Now the relationship between the two sets of primaries can be summarized as

• For the primaries:
  
  $$Q'_k=\sum_{j=1}^3 C_{kj} P_j\;\;\;(k=1,2,3)$$
  
  

• For the tristimulus values:
  
  $$A_j=\sum_{k=1}^3 C_{kj} B_k \;\;\;(j=1,2,3)$$
  
  

To express the above relationship in matrix form, define $\mbox{\bf P}(\lambda)$, $\mbox{\bf Q}(\lambda)$, $\mbox{\bf A}$ and $\mbox{\bf B}$: as 3D vectors

$$\mbox{{\bf P}(\lambda)}\stackrel{\triangle}{=} [P_1(\lambda), P_2(\lambda), P_3(\lambda)]^T$$

$$\mbox{{\bf Q}}(\lambda)\stackrel{\triangle}{=} [Q_1(\lambda), Q_2(\lambda), Q_3(\lambda)]^T$$

$$\mbox{{\bf A}}\stackrel{\triangle}{=}[A_1, A_2, A_3]^T$$

$$\mbox{{\bf B}}\stackrel{\triangle}{=}[B_1, B_2, B_3]^T$$

and $C$ as a 3 by 3 matrix

$$\mbox{{\bf C}}\stackrel{\triangle}{=}\left[ \begin{array}{lll... ...22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{array} \right]$$

and we have

$$\mbox{{\bf Q'}}(\lambda)=\mbox{{\bf C}} \mbox{{\bf P}}(\lambda)$$

and

$$\mbox{{\bf A}}=\mbox{{\bf C}}^T \mbox{{\bf B}}$$

where $\mbox{{\bf C}}^T$ represents the transpose of matrix $\mbox{{\bf C}}$. These linear relations are called projective transforms.