A sharp edge is bought with depth

The limits assume a pigment that switches instantly charged the object-colour solid for a reflectance that jumps between nought and one at a wavelength, and replaced the jump with a limit on how fast a reflectance may change. It was careful to say what that limit was: “the weakest statement of a pigment’s bluntness”. The stronger one it named and did not write is a band-shape limit — the set of reflectances a chemistry can actually produce, which is not a set of slowly-changing curves but a set of sums of absorption bands.

The two are not the same kind of statement. A slope limit admits a staircase, a comb, a reflectance that climbs to one over forty nanometres and falls again ten nanometres later. Nothing made of molecules does that. A colorant absorbs in bands whose shape is set by the transition that produces them, absorbances add when colorants are mixed, and the reflectance is the exponential of minus the total. So the honest admissible set is the one where the reflectance is the exponential of minus a flat absorbance plus a sum of Gaussian absorption bands, each band having a centre, a full width at half its maximum of at least forty nanometres, and a peak absorbance — at most three of them, which is what a mixture of three colorants gives. Written down and maximised, it does not behave as the earlier essay expected.

The stronger constraint is the weaker one #

Three forty-nanometre absorption bands reach further than a forty-nanometre slope limit in 94 of the census’s 154 directions, and less far in 24. The median direction keeps 0.992 of its ideal reach against the slope limit’s 0.986, and the tenth percentile 0.958 against 0.938.

• A band’s width is not the width of the edge it draws. A forty-nanometre band at an absorbance of 3 draws a reflectance edge 32 nanometres wide; the same band at an absorbance of 12 draws one 21 nanometres wide.
• What converts the first into the second is the loading. The same chemistry, more of it in the film, gives a steeper edge — the exponential saturates, so the reflectance has already reached nought where the absorbance is still climbing.
• The census reads the loading directly. Held at forty-nanometre bands throughout, the median direction goes from 0.962 of its ideal reach at an absorbance of 2.5 to 0.992 at 12, and the tenth percentile from 0.670 to 0.958.
• Below an absorbance of 2.3 there is no edge at all, because the band never brings the reflectance down to a tenth.
• Where the band limit does worse is where the optimum has more structure. With two runs of one sign it reaches further in 72 of 84 directions; with three, in 22 of 34.

What a band is allowed to be #

The search is over the band parameters rather than over the reflectance, which makes it a different kind of computation from the dynamic program the slope census uses. A sum of Gaussians is not convex in its own centres, so the maximum is found by Nelder–Mead from a set of starts built from the direction’s own weighting — one band on each region where the weighting is negative, which is where a reflectance wants to be dark — and the best of the starts is taken.

Being a search rather than a program, it needs a check that it is finding what it claims to find, and the check is nested: allowing a third band must never reach less than the search found with two, and two never less than one. That is a necessary condition and not a sufficient one, and the first version of the search failed it on two directions of twenty-six, by a per cent and a half, because the simplex in three bands wandered below what it had already found in two. It passes now because each search starts, among other places, from the previous one with a null band added, which makes the nesting true by construction. A bound that falls when the constraint is loosened is not a bound, and the honest reading of that first failure is that the search was the thing being measured.

The figure above is the arithmetic the whole result rests on, and it is a page of algebra rather than a computation. A reflectance of $e^{-kg}$ passes nine tenths where the Gaussian $g$ reaches $0.105/k$ and a tenth where it reaches $2.303/k$, and a Gaussian of full width $w$ reaches a value $g$ at a distance $\tfrac{w}{2}\sqrt{\ln(1/g)/\ln 2}$ from its centre. Both of those distances shrink as k grows, and the difference between them — the edge — shrinks faster than either. A deep band is a steep band. The three curves fall through their own dashed lines at an absorbance of about 3 and keep falling; by 12 each band draws an edge little more than half its own width.

The two limits, over the whole census #

With the arithmetic in hand the census is not surprising, but it is worth seeing the size of it.

The band curve sits above the slope curve through the whole of the middle of the distribution and the two meet at the top, where 36 directions lose nothing to either. The band limit costs 73 directions more than a per cent against the slope limit’s 89. At the very bottom they meet again — the worst direction keeps 0.761 under both — which says the hardest direction in the census is hard for a reason neither limit relieves.

The crossing at the bottom is the first hint that the two are not one constraint at two strengths. If they were, one curve would lie under the other everywhere, and the difference would be a single number.

Which directions those 73 are is the same answer the slope census gave and for the same reason. A limit written in energy charges the reds and the essay before it both found the saturated greens paying most, because an optimal green is a pass-band with an edge on each side and every limit on sharpness is charged twice for it. That does not change here: what changes is the price, not who pays it. A limit that acts on the edges will always fall hardest on the reflectances that have two of them, whatever the limit is written in.

Ninety-four points above the diagonal, twenty-four below, thirty-six on it, and the scatter either side is up to 0.07 of the ideal reach. No factor converts one limit into the other, and no monotone function does either: there are pairs of directions at nearly the same slope-limited reach whose band-limited reaches differ by more than the whole spread of the slope-limited ones. The two constraints are measuring different things and happen to agree about the ranking most of the time.

One direction, drawn twice #

The mechanism is easiest to see in the direction where the gap is largest.

The slope-limited reflectance is a trapezium. It rises over exactly forty nanometres, holds at one across the whole positive region, and falls over exactly forty — that is the best a slope limit can do and it is flat where it can be flat.

The trapezium is worth one more sentence, because it is the shape the slope census’s whole distribution is made of. A slope limit’s optimum is always a trapezium or a sum of them — rise at the maximum rate, hold, fall at the maximum rate — since any reflectance that is below one where it could be one is leaving weight unclaimed. That is why the dynamic program is exact and fast, and it is also why the slope limit admits shapes no chemistry produces: nothing stops the program from putting two trapezia five nanometres apart.

The band-limited one is not flat anywhere. Its edges are far steeper than forty nanometres, because the absorbance is at the depth cap and the exponential has saturated; but its top is rounded, because the absorption bands either side of the pass region have tails that reach into it, and a Gaussian tail does not stop. So the band-limited reflectance buys its edge and pays for it with the middle, and in this direction the edge is worth more. That is the trade in one picture: the slope limit has flat tops and blunt edges, and the band limit has sharp edges and rounded tops.

What the loading buys #

Since the edge is set by the depth, the census has a parameter in it nobody thought of as a colour parameter.

The tenth-percentile direction gains twenty-nine points of its reach across that sweep, from 0.670 to 0.958, and the median six. The chemistry has not changed: every band in every one of those censuses is forty nanometres wide. What has changed is how much colorant the film holds, and it moves the solid’s reachable part further than any plausible change of chemistry would.

This is the same exponential the colour is in the thickness followed into hue — that essay found that doubling the depth of an absorbing medium squares its transmittance rather than halving its colour, and that a translucent object therefore has no one colour. Here the same squaring does something to a bound: a thicker film is not a darker version of a thinner one, it is a sharper one, and sharpness is what the object-colour solid pays for.

It also revises the essay before it. A limit written in energy charges the reds computed a census with the transition width scaled as the square of the wavelength, on the grounds that an absorption band’s width is set in energy rather than in nanometres. That is right about the band and it is now visibly not enough, because the quantity the census is sensitive to is the edge, and the edge is the band width divided by something that depends on the loading. A census that is exact about how band width varies with wavelength and silent about loading is precise in the smaller of its two terms.

Where three bands are not enough #

The directions the band limit treats worse are not random.

A direction whose optimum is a single edge loses nothing to either limit — 36 of 154, the long-pass and short-pass directions where a reflectance has all the room it needs on one side. At two runs the band limit reaches further in 72 of 84 directions and at three in 22 of 34. The advantage falls as the structure rises, which is what a constraint on shape rather than on sharpness would do.

The census under daylight contains no direction with more than three runs, so the case the band limit should lose outright — a comb, four or more alternations — is not in it. That is a limit of this census rather than a finding about materials, and it is the one place a reader should not extrapolate: a slope limit permits a comb and three absorption bands cannot draw one, so wherever combs are optimal the ordering should reverse. Under a lamp with more structure they might be; three lines spare a slow pigment is the census where the weighting changes sign more often, and the band limit has not been run under it.

How the search was set up #

Three bands, each with a centre, a width at half maximum of at least forty nanometres and a peak absorbance between nought and twelve, plus a flat absorbance in the same range. Nelder–Mead from five structured starts and the previous band count’s answer, 700 iterations each, with the widths and depths clamped inside unpack so the simplex may walk outside the feasible set without the objective going undefined.

The depth cap of twelve is an absorbance of 12, which is an optical density of about 5.2 and a deep but attainable loading for a pigmented film. It is stated rather than fitted, and the sweep above is there because it is load-bearing: it is the parameter that sets the edge, so quoting one census at one cap and calling it the band limit would hide the thing that matters.

The census is every twelfth direction of the 2,400-direction sweep with ideal support above five units, which is 154 directions against the slope census’s 305 at every sixth — the search costs about a thousand times what the dynamic program costs per direction, and the coarser sweep is what that buys. Every ratio is against the same ideal solid under D65 that the slope census uses.

What this leaves unmeasured #

A Gaussian is not a measured band shape. Real electronic absorption bands are asymmetric, usually with a tail towards short wavelengths, and a vibrational progression puts shoulders on them. The asymmetry would change which directions three bands can draw and would not change that a deep band draws a steep edge, which is a property of the exponential rather than of the shape.

The depth cap is one number for all three bands. A real mixture has each colorant at its own loading and they are bounded together by the film’s total solids, not separately, so the true constraint is a budget across the bands rather than a cap on each.

And scattering is absent. Everything here is absorption in a clear film over a perfect reflector. A pigmented paint scatters as well as absorbs, and the Kubelka–Munk relation between the two is not the exponential used here — it flattens the reflectance at both ends and would take back part of what the loading buys. Paint is not a filter is where that difference is set out, and its consequence for this census is specific: scattering puts a floor under the reflectance in the absorbing region, which costs a stop-band direction exactly where the deep loading was buying it most.

The lamp is daylight throughout. Every ratio here is against the D65 solid, and the earlier censuses showed that a bound over surfaces is a bound under a light: three lines spare a slow pigment found a slope limit costing a third as much under a three-emitter lamp, and the gap has to be dark, not the line narrow found how fragile that is. The band limit has not been run under any of them.

Still open: whether a measured colorant library sits where this says #

The whole of this is a model of a colorant, and a library of measured reflectances would replace it. The measurement is one a dyehouse or a pigment supplier already has in a drawer: a few hundred reflectance curves of single colorants at known loadings.

The fit is three numbers per band — centre, width, depth — and the quantity worth tabulating is not any of them but the edge each colorant draws at the loading it is sold at, because that is what the census is sensitive to. The prediction is that the measured edges will spread far more widely than the measured band widths do, since the edges carry the loading as well as the chemistry, and that the spread will be ordered by tinting strength rather than by hue.

The stronger test comes free with it. If the fitted band widths cluster in energy rather than in nanometres, as an absorption band should, then the energy-uniform census is right about its own term; if the fitted edges do not cluster in either, then loading dominates the whole effect and the wavelength dependence the previous essay computed is a correction to a smaller quantity than the one it was correcting.

A constraint named after its parameter is not a constraint on its parameter #

The habit is about what a stated limit actually limits.

“A band no narrower than forty nanometres” sounds like a statement about sharpness, and it is a statement about a Gaussian’s width. What the census is sensitive to is the reflectance edge, and the map from the first to the second runs through a parameter — the depth — that the constraint does not mention. So a limit stated in the natural units of the material was a limit on something else, and it was the second quantity that had to be compared against the slope limit’s own forty nanometres.

The move is to derive the quantity the objective actually reads from the quantity the constraint is written in, before comparing two constraints at the same number. Here that derivation is four lines and it inverts the expected answer. Without it, two limits with the same number in them were assumed to be two strengths of the same thing, and they are not even ordered.

It is worth saying what survives. The band limit is still the more realistic constraint, and its numbers are still the ones a colorant maker should read, because a sum of absorption bands is what a colorant is and a slope limit is a convenience. What does not survive is the ordering — the idea that writing the realistic constraint would take more away. It took less, and the reason it took less is a fact about materials rather than an artefact: real colorants draw edges steeper than their band widths, and the slope censuses here have been charging them for a bluntness they do not have. No surface can be that colourful is the essay about how much a bound over surfaces can move; this moves it the other way.

The failure mode is comparing two models at equal values of a parameter they name differently. The number agrees, the units agree, and the thing the number measures is not the same thing.