Is color perception linear?

Question

I'm learning about the trichromatic theory of color perception. Say a receptor detects a wavelength $\lambda\in\mathbb{R}$ and responds with $f(\lambda) = (r,g,b) \in \mathbb{R}^3$. This system is supposedly linear, so $f(n\lambda_1+\lambda_2) = nf(\lambda_1) + f(\lambda_2)$. However it is emphasized that the addition and scalar multiplication of wavelengths (left hand side) are not the "typical" ones on real numbers, and instead bear some physical meaning. The "addition" is really a simultaneous detection of two light sources with different wavelengths, and the "scalar multiplication" refers to a multiplied quantity of light at that wavelength.

My question is if this is contortion of the definition of linearity. If not, what vector space would $\lambda$ inhabit and would we need to supply a new addition definition for it?

Edit: I was asking specifically about linearity, if it's valid in a loose sense, and how to make it valid in a strict sense. I was already mostly knew the contents of the proposed duplicate post.

Answer 1

(Note: I am not an expert on human eye color response or color spaces, so take what I'm saying with a grain of salt. For example, I am not sure if the response to different wavelengths combines in a way that is equivalent to a simple addition of responses. And it's very possible my wikipedia-level understanding of rod-and-cone cells is overly simplified.)

Light is detected in the eye through photoreceptor cells. So-called cone cells are responsible for detecting color.

Most people have red, green, and blue cone cells, which respond to light in approximately the following fashion (image from wikipedia)

If you were to observe a superposition of two wavelengths, one at $530\ {\rm nm}$ and one at $570\ {\rm nm}$, then you would observe a kind of "superposition of the responses" of these two wavelengths. In particular, you would observe a combination of green light and red light, which would appear as a kind of yellow light.

However, there is a subtlety in using the word "superposition" here, which is what I believe is behind your question. In particular, you cannot add $530+570=1100\ {\rm nm}$, and say that the combined response was "as if" you had measured $1100\ {\rm nm}$. As you can see from the figure, at $1100\ {\rm nm}$ the eye would have practically no response at all.

Therefore, I think it makes sense to model the eye's response to a superposition of wavelengths $\lambda_1$ and $\lambda_2$ as something like $$ f(\lambda_1) + f(\lambda_2) $$ where $f$ maps the wavelength to a color-vector using the three cone response functions in the figure. But, you cannot conclude $f$ is linear in the sense that $$ f(\lambda_1) + f(\lambda_2) \neq f(\lambda_1+\lambda_2). $$