• cones, projection from spectrum to 3 responses
• relate to measurement equation in image formation
• **the response is a matrix–vector product**: discretize the spectrum into N wavelength bins → a vector E ∈ ℝᴺ; stack the three cone sensitivities as the **rows of a 3×N matrix C** (one row per L/M/S); the cone responses are then **r = C·E**, i.e. each r_k = Σ_i c_k(λ_i)·E(λ_i). In the real world wavelength is **continuous**, so it's really **infinite-dimensional** — the sum becomes the integral r_k = ∫ c_k(λ)·E(λ) dλ. [cone-response matrix figure]
• insist a given cone does not make the difference between wavelength. Example of monochromatic stimuli
• point out they are non orthogonal and lots of stuff can’t be negative.
• This make linear algebra more messy
• 💡 **Big lesson:** **color is non-orthogonal *and* non-negative** — overlapping cone axes plus the no-negative-light constraint are what make color algebra tricky (analysis needs a dual basis with negative coordinates).
• side bar: **non-orthogonal bases & the dual basis** — the cone "axes" are **not orthogonal** (L and M overlap heavily), so the comfortable orthonormal intuition fails. Picture two basis vectors **c₁, c₂ both in the positive quadrant** of the (canonical) spectrum space, each standing for one cone. **Synthesis** is easy — any point is a combination a₁c₁ + a₂c₂ (a parallelogram). But **analysis** — recovering the coordinates a₁, a₂ — is **not** a plain projection onto c₁, c₂; you must project onto the **dual (reciprocal) basis** c₁\*, c₂\* (defined by cᵢ\*·cⱼ = δᵢⱼ). For a non-orthogonal positive basis the **dual vectors point partly into the negative quadrants** — they have **negative coordinates** in the original space. That is exactly why "stuff can't be negative" makes color algebra messy: the natural *analysis* vectors are negative. (Only for an **orthonormal** basis do a basis and its dual coincide — then analysis = synthesis = projection.) [non-orthogonal dual-basis figure]
• color blindness. Linear algebra perspective (projection loses info)
• we are all color blind
• side bar — **animal vision and the divergence of opsins**: "color" is just whatever set of **opsins** (photopigments) an animal carries, and these have **diverged enormously** across major animal groups — many insects are UV–blue–green trichromats, birds and reptiles are often **tetrachromats** (a fourth, UV cone), most mammals reverted to **dichromacy** (primate trichromacy is *re-evolved*, via a gene duplication), and the **mantis shrimp** carries ~12–16 photoreceptor classes. So human red–green color blindness is one point in a vast space — every species is "color blind" relative to some other. The opsin family tree and its divergence are traced in the "Evolution of Eyes" chapter of *Vision* (Cambridge); the schematic opsin gene tree (ancestral opsin → rod RH1 + cone classes, human S/M/L with the recent L/M duplication) is [fig-opsin-tree]. [→ Animal eyes]
• the **multistage color model** — the brain re-codes color in stages (simplified; see Reinhard et al., *Color Imaging* for the full version):
• **stage 1 — cones**: three heavily-overlapping LMS responses (above)
• **stage 2 — opponent recoding** in the retina / LGN: LMS → **light–dark, red–green, blue–yellow** (decorrelates the channels, matches the four unique hues; → next section)
• **stage 3 — appearance level**: a higher organization closer to **hue / saturation / brightness** (an HSV-like arrangement) where context, adaptation and memory act (→ color appearance models, Color technology)
• **focal colors & naming**: color *names* cluster around shared **focal colors** across languages (Berlin & Kay; the World Color Survey) — evidence the recoding is perceptual, not arbitrary
• experimental support: **psychophysics** (hue-cancellation — Hurvich & Jameson) and **physiology** (opponent cells — De Valois)
equationscone response r_k = ∫ E(λ)·c_k(λ) dλ, k ∈ {L,M,S} (a projection)
metamerism = same r_k for different E(λ)
