A fourth dimension has a shape

Reconstruction studies report how much of a collection of spectra three basis functions capture, and the number is always a variance. A variance cannot say which fourth dimension a collection has, and which one it has decides almost everything.

The claim #

A fourth reflectance dimension is expensive when its structure is at the scale on which the cone fundamentals differ from one another, and cheap otherwise — and where that scale ends is decided by the light rather than by the observer.

• At the same amplitude the cost varies by a factor of nine across four plausible fourth basis functions.
• Two frequencies cost exactly nothing, and they are the two the family already has.
• The cost rises to a maximum and falls again. Structure finer than the fundamentals’ own scale integrates to nearly nothing against three broad sensitivities.
• Under a smooth source the cost has fallen by a factor of 2.2 by seven half-cycles. Under a three-line source it has not fallen at all, because the source has structure of its own for the surface’s structure to beat against.
• A narrow feature costs most at about 25 nanometres wide, and most when it sits where two fundamentals are crossing rather than where either peaks.

Why a variance figure cannot answer this #

The standard statement about surface reflectance is that three basis functions reconstruct a measured collection to within measurement error, or capture ninety-something per cent of its variance. That is a true and remarkable fact, and it is the reason linear models of surface colour work at all.

It is also a statement in the wrong space, in the way a variance is not a worst case. The quantity it summarises is the residual left in the spectrum, integrated over wavelength with equal weight everywhere. The quantity that matters for the matrix theorem is the residual left in the three cone signals after a change of light, which is that same spectral residual passed through two projections: first onto three broad, heavily overlapping sensitivities, then through the difference between two illuminants.

Those projections are not close to isometries. They annihilate some spectral shapes almost completely and pass others nearly intact, and the ratio between the two extremes is the subject of this essay. So a collection whose fourth component carries one per cent of the variance can be three-dimensional for practical purposes or not, depending entirely on what that one per cent looks like.

The sweep, and its own control #

Take a cosine of k half-cycles across 380–780 nm as the fourth basis function, hold its amplitude at ten per cent of each surface’s own brightness, and measure what is left after the exact 3×3 that the three-dimensional family makes exact.

The first thing the curve does is touch zero, twice.

At one and two half-cycles the “fourth” function is the family’s second and third basis function. Adding a coefficient along it enlarges the set — there are now three times as many surfaces — without enlarging its span, so the family is still three-dimensional and the theorem still holds exactly.

That is the sweep’s control and it is exact rather than approximate. A construction that had drifted — a fourth function added in the wrong place, a matrix quietly refitted to the enlarged set — would produce a curve that looked entirely plausible and had no zeros in it. The two zeros are the evidence that the rest of the curve is measuring what it says — the same shape of control as the five surfaces that answer nothing.

Where the cost is #

Two things in that table and the second is the one worth carrying.

The curve has a maximum. Very slow structure is nearly a smooth tilt, which a matrix carries perfectly; very fast structure oscillates several times under each broad cone sensitivity and integrates to nearly nothing. In between there is a band where a feature moves the three channels differently, and that band is where the cost lives. For tungsten it is centred around four to six half-cycles — a period of 130 to 200 nanometres, which is the scale on which the three fundamentals genuinely differ from one another.

The two lights disagree about where the band ends. Past seven half-cycles the tungsten curve has fallen by a factor of 2.2 from its peak. The triphosphor curve has not fallen at all — it is at its highest there, and is still at 2.2 at nine half-cycles.

Why the light sets the resolution #

The observer is the same in both columns. The same three cone fundamentals, the same integration, the same 5 nm grid. What differs is the spectrum of the light.

A smooth source multiplies the surface by a slowly varying function. Fine structure in the surface is then integrated against a broad sensitivity times a smooth weight, and a rapid oscillation cancels to nearly nothing — twice, once under each light — so the difference between the two lights sees almost none of it.

A three-line source does not. The triphosphor tube’s spectrum is three narrow emission peaks with very little between them, so integrating the surface against it is close to sampling the surface at three wavelengths. A surface feature that happens to fall on a line is seen at full strength; one that falls between lines is not seen at all. Fine structure therefore does not cancel — it aliases, and which way it aliases depends on how the surface’s period beats against the lamp’s line positions.

That is the origin of the jaggedness in the narrow-source column, which is real rather than numerical: 5.0 and 7.0 half-cycles are high, 8.0 is low, 9.0 is high again, because the beat between the surface’s period and the lamp’s three lines goes in and out of phase. A smooth source has no lines to beat against and its curve is smooth.

So the spectral resolution at which a surface’s structure begins to matter is a property of the illuminant. That is a genuinely useful sentence, and it is the reason three numbers cannot see a line is a statement about a particular pairing of light and surface rather than about the eye alone.

Where a narrow feature has to sit #

Swap the cosine for a Gaussian band 25 nm wide and sweep it across the spectrum.

Two maxima, near 500 and 575 nm, with a deep minimum between them at 550 and a long fall to nothing in the far red.

The maxima are not where any sensitivity peaks. They are where two of them are changing relative to one another. A bump under a single dominant sensitivity multiplies one channel and leaves the ratios between channels alone, which is exactly what a matrix can absorb; a bump where two sensitivities cross changes a ratio, and a ratio that changes differently under two lights is what no fixed matrix can carry. The minimum at 550 is where the long- and medium-wave fundamentals are most nearly parallel, so a feature there scales both together.

The fall beyond 700 nm is the mundane half: there is very little sensitivity left, so a feature there does nothing at all under either light. Everything a fourth dimension does, it does between about 420 and 620 nanometres.

And how wide #

An interior maximum, for two reasons pulling opposite ways. Narrower than about fifteen nanometres and the feature contains too little light to move a tristimulus integral at all. Wider than about forty and it starts to look like the smooth reweighting a matrix carries exactly.

The peak at 25 nm is worth noticing because it is not a number this exercise put in. It is the same order as the separation between the two long-wave pigment peaks, which is the scale on which the whole long/medium distinction operates, and a feature at that scale is precisely the one that can move L and M differently.

The smooth source’s curve is not smooth either #

The jaggedness in the narrowband column is attributed to a beat between the surface’s period and the lamp’s three lines, with the contrast drawn explicitly: a smooth source has no lines to beat against and its curve is smooth. The tungsten column beside it is not.

Its values from three to nine half-cycles run 0.639, 1.139, 0.350, 1.539, 0.694, 0.389, 0.131. From four to five it falls by a factor of 3.3 and from five to six it rises by 4.4 — neighbouring points on a curve described as smooth, differing by more than a factor of four.

Measured as the mean absolute log-ratio between consecutive points, the two columns come out at 0.95 for tungsten and 1.10 for the triphosphor tube — fifteen per cent apart, on a quantity that the essay’s account says should differ by a great deal. The triphosphor column’s single largest step is bigger, at ×12.3 against ×4.4, and that is the whole of the difference.

So the mechanism is very likely right and the evidence for it is much weaker than the framing suggests. Both curves are jagged at this sampling, and one is slightly more so. The dip at five half-cycles in the tungsten column is unexplained by anything in the essay and is nearly as large as anything the narrowband source produces.

Two readings are available and the essay does not choose between them. The sweep is eighteen evaluations across a range where the cost varies by a factor of twelve, so the structure may be real and undersampled — in which case both curves have fine structure and a finer sweep would show it. Or the tungsten dip is an artefact of where the cosine’s nodes fall relative to the fundamentals’ crossovers, which would be a real effect and a different one from the lamp’s lines. Either way the smooth-against-jagged contrast is not what these eighteen numbers support, and the claim that does survive is the one about the fall-off: past seven half-cycles tungsten has dropped by 2.2 from its peak and the triphosphor has not dropped at all. That comparison uses the envelope rather than the fine structure and is unaffected by all of this.

The width sweep has a floor, and the peak is modest #

The width sweep is described as an interior maximum with two mechanisms pulling opposite ways — too little light below fifteen nanometres, too much like a smooth reweighting above forty. The second half does not continue.

The cost at 60 nanometres is 0.049 and at 140 it is 0.048: a difference of two per cent across a more than twofold widening. The curve does not fall towards zero at the wide end; it flattens onto a floor at about 0.048, which is 38 per cent of the peak.

That floor is the interesting number, because it says a feature broad enough to be very nearly a smooth reweighting still costs more than a third of what the worst-case feature costs. A matrix carries a smooth reweighting exactly is true of a reweighting that is genuinely smooth across the whole band; a 140-nanometre Gaussian at 550 is not, and the residual it leaves is not small.

The peak is correspondingly less dramatic than the framing suggests. Against the two wings — 42 per cent of the peak at 8 nanometres, 38 at 140 — the interior maximum is a factor of 2.5, not a resonance. The essay’s practical conclusion survives it and is worth restating with the right magnitude: the width of a real pigment band moves the cost by a factor of two and a half across the whole plausible range, which is much less than the factor of nine the four named shapes span and means width is the least of the three variables.

Two maxima carried by one sample each #

One caution about the position sweep, which is the source of the essay’s most quotable structural claim — maxima at the crossovers, a minimum where two fundamentals are parallel.

The sampling is uneven: 400, 425, 475, 500, 525, 550, 575, 625, 700, 750, with gaps of 25, 50, 25, 25, 25, 25, 50, 75 and 50 nanometres. The minimum at 550 is one sample deep, between 0.450 and 0.506, and the maximum at 575 is one sample wide with nothing sampled between it and 625, where the cost has fallen by a factor of 11.8.

None of that makes the claim wrong — the crossover explanation predicts the shape found, and finding it is evidence. It does mean the two maxima’s positions are known to about twenty-five nanometres and their heights are lower bounds, since a peak between samples would be higher than either neighbour. A structural claim identified by a single sample deserves the sweep the essay’s own closing section recommends, and the same argument it makes for sweeps over named cases applies one level down: forty evaluations at 5-nanometre spacing would cost about as little as the eighteen already run and would settle where the maxima are rather than which of ten samples is largest.

What to do with it #

When a linear model is adopted, characterise the fourth component rather than counting it. A collection reported as “three components, 99.4 per cent of variance” is not enough information to decide whether a matrix will do. The useful report is the fourth component’s shape: where it has energy, on what scale, and whether the intended illuminants have structure of their own.

Under narrowband sources, do not assume the model’s dimension is enough. Three-primary displays, laser projectors, RGB LED lighting and triphosphor tubes all sample the spectrum rather than weight it, and every one of them extends the range of surface structure that matters. The census row that responds most to a fourth dimension is the three-primary display, by ninety per cent.

And the cheap direction is real. Real pigments absorb in bands, and a band at 550 nm 25 nm wide is the cheapest place a real feature can be. The measured cost of a fourth dimension in a collection of ordinary painted surfaces is likely to be at the low end of this range, which is an argument for the three-dimensional model rather than against it — but it is an argument that has to be made from the shape and has never been made from the variance.

Where the model stops #

The four shapes and the three sweeps are constructions. None of them is a measured fourth principal component, and this essay is not evidence about what real collections carry — only about what any given fourth component would cost.

The sweeps are also taken on one lattice — itself a quadrature rule rather than a sample — at one amplitude, on two changes of light out of fourteen. The frequency curve’s fine structure under a narrowband source is specific to that lamp’s three line positions; a different triphosphor blend would put the peaks elsewhere. What generalises is the shape of the argument — a maximum at the fundamentals’ own scale, a fall-off that a smooth source imposes and a line source does not — and not the position of any particular peak.

What the sweeps cost to run #

A note on why these are sweeps rather than four points, because the difference decides what can be said.

Four named shapes give four numbers and support the statement the shape matters. They cannot say which shapes, cannot show the maximum, cannot show the fall-off, and cannot contain the two exact zeros that make the whole family credible. A curve does all four.

The cost is nothing worth mentioning. The frequency sweep is eighteen evaluations of the matrix residual over a 375-member set — about seventy milliseconds, memoised — and the position and width sweeps are twenty-three more. That is well under a second for three curves that replace four points and carry their own control.

The general point is that a parametrised family is almost always cheaper than it looks and almost always says more than the same effort spent on named cases. This collection has arrived at it before, on a wall’s band width — where profiling one parameter with the others optimised turned a boundary problem into an interior one and revealed a turnover nobody expected. The same move here revealed two exact zeros and a maximum, neither of which was in the four named shapes.

Who found it, and when #

The band-limiting argument is the same one that makes metamerism possible at all: three broad sensitivities are three low-pass filters, and everything in a spectrum above their cut-off is invisible to them. Cohen’s and Wyszecki’s work on the fundamental metamer, and the whole apparatus of metameric black, is the formal version — a spectrum splits into a part the observer sees and a part it cannot, and the second is arbitrarily large.

What is added here is that the cut-off is not a property of the observer alone. A metameric black under one light is not a metameric black under another, which is the definition of metamerism failing, and this essay is that fact measured on a dial: sweep the surface’s spatial frequency and the light’s spectral structure decides where the cut-off falls. That connection was available in the metamer machinery this site has had since its second phase, and was not drawn until a fourth dimension gave something to sweep.

What a fourth component would have to be measured as #

The essay’s practical claim is that a variance figure is the wrong summary, and it is worth saying what the right one would be, since somebody with a measured collection could produce it.

Three numbers rather than one. Where the fourth component has its energy, on what spectral scale, and how much of it there is. The third is the variance figure everybody reports; the first two are what decide whether it matters.

Reported against the observer rather than against the spectrum. A fourth component’s energy at 750 nm is worth nothing whatever its variance, and at 500 nm it is worth twenty times as much per unit. So the useful summary weights the component by the crossover regions of the cone fundamentals — which is a fixed weighting anybody can apply, computed once and applied to any collection.

And with the intended illuminants named. The band that costs is set by the source, so a collection destined for daylight viewing and one destined for triphosphor lighting need different readings of the same fourth component.

None of that needs new measurement. It needs the fourth principal component of an existing collection to be published as a curve rather than as a percentage, which is the ordinary situation of a summary statistic being reported where the object itself would fit in the same space.

Where the ladder goes next #

One row of the census gets better as the surfaces gain a dimension, and it is the row that looks most fragile. That inversion is worth its own rung.