A gain is not an observer

The previous essay emptied the stimulus’s factor of the pairing. This one empties the observer’s, and the way it empties explains a number in the ladder that had looked out of place.

The claim #

Two observers whose sensitivities differ by a constant factor in each cone are the same observer, exactly, because the white-point normalisation is that factor’s inverse.

• Measured at 1.1 × 10⁻¹³ ΔE₀₀ for gains of 1.6, 0.7 and 2.4 — an identity rather than a small number.
• It is von Kries’ hypothesis used as arithmetic, not as a model of adaptation. No claim about how an eye adapts is involved; the cancellation is in the definition of a relative excitation.
• It explains the ladder’s bottom row. Cone optical density has the largest reported spread of the individual-variation parameters and the smallest cost, because most of what it does is a gain.
• And it is exact only in the eye’s own coordinates. The same condition imposed in a published cone space leaves 13.0 ΔE₀₀ standing.

The arithmetic #

An observer’s relative excitation for a stimulus is the cone’s response to the stimulus divided by the same cone’s response to the white:

eᵢ = ∫ φ lᵢ / ∫ w lᵢ

Multiply lᵢ by a constant gᵢ. Both integrals are linear in the sensitivity, so both are multiplied by gᵢ, and the ratio is unchanged. Three cones, three independent constants, three unchanged ratios.

Everything downstream is a function of those three numbers, so nothing downstream moves. The colour is identical, and it is identical for any gains at all — a factor of a thousand in one cone and a factor of a thousandth in another give exactly the same answer.

No approximation is made anywhere in that. It is not a first-order result and it does not require the gains to be near one, which is why the measured residual is at the floating-point floor rather than at some tolerance.

What is not a gain is nearly everything. The identity is narrow and the narrowness is the point. Three constants is three numbers, and a set of sensitivities is three functions of wavelength — so the class of differences the identity absorbs is three-dimensional inside an infinite-dimensional space.

Everything that changes a curve’s shape survives. A lens that yellows absorbs more in the blue than in the red, which is not a gain — and the eye’s own filters are exactly that kind of change. A macular pigment absorbs in a band centred at 460 nanometres, which is not a gain. A peak wavelength that moves slides a whole curve along the axis, which is not a gain. A rod signal added to all three channels is not a gain either, because it adds rather than multiplies and the addition has its own spectral shape.

So five of the six departures in this round’s ladder are untouched by the identity. The sixth is not, and that is what makes it interesting.

Why cone optical density is the cheapest departure #

The table of reported individual variation in this collection has four entries, and one of them looks alarming: the axial optical density of the cone outer segment, reported near 0.4 with a standard deviation of about nine per cent of it, and individuals from 0.25 to 0.6.

A density change is a large change to a sensitivity. The absorptance of a pigment layer is 1 − 10^(−D·τ(λ)) where τ is the normalised template, so doubling D doubles the absorbance everywhere the absorbance is small — a gain of two — and does almost nothing where it is already saturating, which is at the peak.

That is self-screening, and this collection has an essay about it: a denser cone has a broader sensitivity, not a taller one. The consequence for this round is the complement of that sentence. The part of a density change that is a uniform scaling — which is most of it, since the exponential is nearly linear over most of the curve — is a gain, and gains are free. What is left is the broadening, and the broadening is what costs.

Measured, the density departure at two standard deviations is 1.198 ΔE₀₀ on a red pigment under daylight, the smallest of the six, and 0.566 at the median over forty-two surfaces, again the smallest. A parameter with the largest relative spread and the smallest effect, and the identity says precisely why.

The distribution is more convincing than the single example, because a single example could be a sample that happened to be insensitive to density. Over forty-two surfaces the density’s median is 0.566 ΔE₀₀ against the age’s 1.943 and the macular’s 1.610, and its ninety-fifth percentile is 1.707 against their 5.752 and 4.081. It is the smallest departure at every point of the distribution, not just at the middle.

That consistency is what makes the explanation credible. A parameter that was merely lucky on some samples would cross the others somewhere in the family; one whose dominant component is structurally invisible stays below them everywhere, and this one does.

Where the exactness stops #

The identity holds when the white is divided out in the same cones the gains were applied to. That is a sentence with a hidden variable in it, and the variable is worth a large part of this round.

Divide the white out in tristimulus values instead — form X, Y and Z first, then divide by the white’s X, Y and Z — and the identity fails. A gain applied in the cones is a diagonal matrix in cone space and is not diagonal by the time it has been through the three-by-three that reaches tristimulus values, so the ratio it was supposed to preserve is not the ratio being taken.

Divide it out in CAT16’s cone space, which is what this collection does everywhere and what most appearance work does, and the identity fails again by 13.0 ΔE₀₀ on a red pigment under daylight. That is not a rounding error and it is not a fault in CAT16; it is the cost of doing a division in a basis that is not the one the gains live in.

A gain needs a basis established the same thing from the direction of adaptation models, where the question is which cone space a chromatic adaptation transform should work in and the answer is decided by fitting corresponding-colour data. This is the other half of it: there is a basis in which one particular condition is exactly zero, and it is the eye’s own, and it is not the basis anybody computes in.

Which is the right arithmetic #

Two computations disagree by thirteen colour differences and only one of them can be what an eye does, so the question is which.

The honest answer is that neither is, and the reason is instructive. Von Kries’ hypothesis is a hypothesis. The claim that adaptation is a gain on each cone independently is a model, it is known to be approximate, and the corresponding-colour data that would settle how approximate are the data this collection has been recording the absence of for six rounds. Using it as arithmetic here is a definition rather than a claim: the relative excitation is defined as the ratio, and the identity follows from the definition.

What follows for the results of this round is a division of labour. The identity is a statement about a definition and is exact. The departures are computed in this collection’s usual arithmetic, which adapts in CAT16’s space, because that is the unit everything else here is quoted in and comparability across nineteen rounds is worth more than a cleaner identity. That is the same decision the unit audit reached about colour-difference formulae: a published quantity’s unit is a commitment rather than a preference.

The two are reported side by side rather than reconciled, and the gap between them is published as a measurement of its own: dividing the white out in CAT16’s cones or in the observer’s own moves an ordinary sample by 15.8 ΔE₀₀ at the median over the surface family.

Why the two published routes agree and the third does not #

That figure has a shape worth reading carefully, because it is not the shape a reader would guess.

Dividing the white out in tristimulus values and dividing it out in CAT16’s cone space are two quite different operations, and they agree to 0.59 ΔE₀₀ at the median. Dividing it out in the observer’s own cone space — the physiologically motivated choice, the one in which the identity holds — differs from both by more than fifteen.

The reason is that CAT16’s cone space is a sharpened basis, chosen to fit corresponding-colour data, and sharpening moves it away from the physiological cones rather than towards them. The published transforms are not attempts to find the receptors; they are attempts to fit adaptation behaviour, and the fit has taken them somewhere else.

So the field’s two standard arithmetics agree with each other and disagree with the physiology, which is a coherent state of affairs rather than a scandal. They are fitted to what people report under changes of light, and if the eye’s adaptation is not a pure cone gain then the best-fitting gain will not be in the cone basis. The identity is exact in the coordinates where nothing was fitted.

Recomputing the conditions under a second light is the check that separates the two halves of that figure. An identity that held only under one illuminant would be a coincidence of that illuminant’s spectrum; all eight hold under both, at the same floating-point floor. The two limits change — the gain condition in CAT16’s basis is 13.0 under daylight and a different number under tungsten — because they are measurements rather than theorems, and a measurement of a mismatch between two bases depends on what is being looked at through them.

That is the sharpest available demonstration that the classification into identities and limits is a real distinction and not a matter of how small the numbers happen to be. An identity does not move when the experiment changes.

What it means for the ladder #

The identity reorganises the six departures into two kinds, which is more useful than their ordering.

This is also where the essay on what a still eye stops seeing meets the arithmetic: two parts of one retina differ in macular density, which is a shape change, and not in a gain, which is why they are two observers rather than one seen at two brightnesses.

Three are shape changes with a location. The macular pigment absorbs in a band at 460 nanometres, the lens absorbs below 500, the peaks slide the curves along the axis. Each has a place on the wavelength axis where it lives, and which stimuli it can reach is decided by that place.

One is mostly a gain, with a shape residue. The density’s cost is the broadening left over after the scaling has been divided out.

One is an addition rather than a multiplication. The rod term adds a fourth curve, and no normalisation can remove an addition — which is why the rods cost 1.60 ΔE₀₀ despite the added signal being a tenth of the cone response. The rods have their own sensitivity curve and its peak is nowhere near any cone’s.

And one is not a change to the observer at all. The basis is three curves for one space.

That taxonomy predicts the ladder’s order better than the parameters’ reported spreads do, and it is the sort of thing that only becomes visible once the identity has separated what cancels from what does not.

The walk is worth running on the density in particular, because the identity removes most of what the parameter does and leaves a residue. A residue that behaved differently from the other five would suggest the removal had changed the mechanism rather than only its size.

It does not. The residual broadening is a pairing like every other departure, straight in the excitations to the same two per cent, with a smaller slope.

Across lights the density’s row moves less than the two blue-absorbing rows do, because a broadening acts on the whole curve rather than on a band and so is less sensitive to where a lamp puts its power.

What was computed, and how #

The gains are 1.6, 0.7 and 2.4, chosen to be large and unequal — a small perturbation would have made a limit indistinguishable from an identity, which is the failure this collection has had to guard against repeatedly.

The identity is asserted at 10⁻¹⁰ ΔE₀₀ and measures 1.1 × 10⁻¹³. The same gains through the CAT16 route are asserted only to be finite and measure 13.0, and that number is reported as a limit rather than as a residual because it is a real disagreement between two arithmetics rather than an approximation to zero.

The three routes are computed over the same forty-two surfaces under the same light with the same observer, so nothing but the division differs. The assertion the family carries is that the two published routes agree with each other more closely than either agrees with the observer’s own, which is a claim about where the field’s conventions sit and which the fit could have contradicted.

There is a practical reading of all this for anybody choosing which observer parameter to measure on a real person, and it is the reverse of the obvious one. Cone optical density is the hardest of the four to measure — it needs a bleaching experiment or a careful fit to matching data — and it is the one whose value matters least. Macular density is measurable in a few minutes with a flicker-photometric method and matters three times as much. Lens transmittance can be estimated from age alone and matters most of all.

Measurability and importance run in opposite directions here, which is a common and awkward arrangement. It is worth stating because the individual-variation literature reports all four parameters with equal prominence, and a reader assembling a personalised observer would reasonably spend effort in proportion to that prominence rather than in proportion to what the effort buys. The elasticity of a conclusion to each of its inputs is the quantity that should govern, and it is almost never published beside the inputs.

Where the model stops #

The identity is about a definition and says nothing about whether an eye adapts that way. Everything known about chromatic adaptation says it does not, exactly: there are second-site effects, there is a contribution that does not scale, and the best-fitting transforms are not diagonal in any physiological basis.

The density model here is a single axial density applied to a normalised template, which is the standard construction and ignores the fact that a real outer segment has a wavelength-dependent path length and that the cones’ inner segments act as waveguides. Both of those would add shape to what is modelled as a scaling.

And the gains chosen are arbitrary. A gain arising from a physical cause — a real density change, a real change in cone diameter — is never a pure gain, and the identity applies to the pure part of it only.

The generalisation #

The habit is about finding the group of transformations a measurement is blind to.

Every measurement system has one, it is usually smaller than people assume, and knowing it converts vague statements about robustness into exact ones. Here it is three positive numbers out of an infinite-dimensional space of curve changes: tiny, and enough to absorb most of one of the six parameters anybody worries about.

The move that finds it is to ask what the output is a function of rather than what the input is. Here the output depends on the sensitivities only through three ratios, so any change preserving the ratios is invisible, and the change preserving the ratios is a per-channel gain.

The failure mode is to assume that a parameter with a large reported spread matters in proportion to it. A spread is a fact about the population and a cost is a fact about the arithmetic, and the two are related by whatever the measurement happens to be blind to.

Who found it, and when #

Johannes von Kries proposed in 1902 that adaptation acts as an independent gain on each receptor class, and the hypothesis has been the backbone of every chromatic adaptation transform since. Its use here is not as his hypothesis but as the algebra that hypothesis suggested, which is a distinction the literature does not always keep.

Self-screening’s effect on cone sensitivity shape has been understood since the microspectrophotometry of the 1960s and 1970s, and the standard treatment — an axial density applied to a normalised absorbance — is what the CIE’s 2006 fundamental observer uses, with different densities for the two- and ten-degree fields. That difference in density is one of the two things this collection models a field size as.

Where the ladder goes next #

Two of the identities are done and the third is a theorem rather than a measurement. Three curves spanning one space are one observer, which sounds obvious and settles a question the literature keeps reopening.