Nature May Have Its Reasons

We used to ask the question why there are only three types of retina cone cells, and whether three types of cones are enough for color perception. Now we can show that three types of cones should be enough to uniquely determine the various daylight illuminations, the single most important natural light source for animals and humans in the long history of evolution.

Let the three types of cones respond to the daylight be

$$r_i=\int S_i(\lambda) I(\lambda) d\lambda =\sum_{k=1}^3 e_k \int S_i(\lambda) E_k(\lambda) d\lambda \;\;\;\;\;\;(i=1,2,3)$$

where $S_i(\lambda)$ is the sensitivity of the cones of the ith type.

If the response of the cones of the ith type to kth basis function can be assumed to be known and stored somewhere in the visual system

$$c_{ik}\stackrel{\triangle}{=}\int S_i(\lambda) E_k(\lambda) d\lambda \;\;\;\;(i,k=1,2,3)$$

then the three coefficients $\{e_1, e_2, e_3\}$ can be obtained by solving the following linear equation system (of three variables with three equations);

$$\left\{ \begin{array}{l} r_1=\sum_{k=1}^3 c_{1k} e_k \\ r_... ..._{2k} e_k \\ r_3=\sum_{k=1}^3 c_{3k} e_k \end{array} \right.$$

thus given retina cone responses and the coefficients c_ik, the spectral distribution of the daylight illumination can be uniquely determined.