One field keeps what two rooms divide out

The reference lamp must not move designed an experiment. At mesopic light levels the rods add a signal to the cone pathways, and how strongly that signal enters the short-wavelength, blue-yellow pathway is a weight nobody has measured well; the rods’ route is priced by the lamp had found that under a white LED it changes a colour’s appearance by nearly a colour difference. An asymmetric match — a surface seen under one lamp matched by a setting under another — can measure it, because the two lamps move the match by different amounts as the weight changes. The design found that the reference lamp must be one the weight hardly moves: daylight against a phosphor LED needs 75 settings to fix the weight to a tenth, a pair of two sensitive lamps thousands.

Each half of that match was read with the observer adapted to its own lamp, as though the observer sat in one room long enough to adapt, then walked into the other. That is what makes the session slow; the essay said so, and named the cheaper arrangement. Put the two lamps on the two halves of one field, and the observer has one adaptation state and sets the match across the join in seconds. It predicted a cost: a common adaptation would remove part of what distinguishes the two lamps, the signal would fall, and the field would be worth using only if it fell by less than a factor of four in settings.

The signal rises #

Under one shared adaptation, the match moves with the S weight more than it does with two separate ones, for every pair of lamps: three times as much for daylight against the white LED, and up to 7.7 times for other pairs. The best pair now needs six settings, where the two-room design’s best needed seventy-five. The field stays the better experiment even if three quarters of each half’s adaptation is local.

• For daylight against the phosphor LED, the signal is 1.79 colour differences in one field against 0.58 in two rooms, and the settings fall from 75 to 8.
• Every one of the ten pairs of lamps gains, by between ×1.5 and ×7.7.
• The mechanism is in the adaptation. Adapted to its own lamp, the daylight half barely moves with the weight — 0.42 — because the adaptation divides each lamp’s rod contribution out with its white. Under the shared white both halves move further, 1.98 and 3.79.
• Letting the LED fill more of the field raises the signal further: with daylight on a tenth of it, 4.45.
• The choice of reference lamp matters less. In two rooms the best daylight pair beat the best pair of sensitive lamps by more than a factor of two; in one field, by 1.6.

Two rooms against one field #

The figure is every pair of the five lamps the earlier essay used — tungsten, daylight, a phosphor LED, a three-emitter LED and a fluorescent tube — by how far a match between them moves as the rod signal’s weight into the S channel goes from nothing to equal. The pale bars are the two-room design, the dark bars one field. The numbers are medians over forty-two surfaces, for a thirty-two-year-old observer with a rod signal a tenth of each cone’s peak sensitivity.

Every dark bar is longer than its pale one. The pair the earlier essay picked, daylight against the phosphor LED, goes from 0.58 to 1.79. Daylight against the three-emitter LED goes from 0.45 to 2.19, now the largest signal of all. The pairs of two sensitive lamps, which the two-room design ruled out, gain most: the three-emitter LED against the tube by 7.7 times, the phosphor LED against the tube by 6.0. The least gain is tungsten against the tube, 1.5 times, which was and remains the pair with the least to say.

In settings, the gain is squared. The two-room design’s best pair needed 75; the field’s best needs 6, daylight against the three-emitter LED, and daylight against the phosphor LED needs 8. Pairs that needed hundreds need tens: tungsten against daylight from 211 to 13, the phosphor LED against the tube from 514 to 15. The worst pair still needs over a thousand, because tungsten and the tube are lamps the S weight hardly separates under any arrangement.

A session of eight settings, each made across a join without waiting to adapt, is minutes. The two-room session was hours. The prediction put the break-even at a signal falling by half; it rises by three times.

Why the prediction had it backwards #

The prediction reasoned that adapting to a mixture of the two lamps would remove part of what distinguishes them. The measurement says that adapting to each lamp separately was removing it.

A rod signal adds to every colour a lamp lights, and it adds to the lamp’s white too. When a half is adapted to its own lamp’s white — the observer’s white, rods included — the adaptation divides each colour by that white, and the part of the rod signal the colour shares with the white divides out. What is left is only how much more or less the surface’s own rod signal is than its lamp’s. Under daylight, whose light the rods and the S cones catch in nearly the same proportion, that residue is small: the daylight half moves 0.42 as the weight opens. Under the phosphor LED, which puts its power where the rods are sensitive and the S cones are not, it is larger, 1.73.

Under the shared white, each half is divided by the average of the two lamps’ whites, which carries only the average of their rod contributions. Each half keeps its own lamp’s rod excess over that average, and both halves move further — 1.98 for the daylight half and 3.79 for the LED’s. The two-room match subtracted two small residues; the field subtracts two large excesses that differ more.

There is a way of putting this that does not need adaptation at all. An asymmetric match in two rooms is a statement about corresponding colours — which colour under one lamp looks like which under another, once the eye has adjusted to each — and a corresponding-colour match is built to discount whatever the two lamps do in common to every surface, the rods included. A match in one field is closer to a colorimetric match: two stimuli judged side by side, whose difference the eye reads almost directly. The rods are a fourth curve introduced the rods as an extra sensitivity added to the three cones, and a colorimetric match sees an extra sensitivity in full. A corresponding-colour match, by design, sees only what is left after the lamps’ whites have been discounted, which is exactly the part of the rods’ contribution that is not shared with the white.

That is the same observation the reference lamp must not move made about the reference lamp, arriving from the adaptation instead: an asymmetric match measures a difference, and anything that makes both sides move alike takes signal away. Adapting each side to its own lamp is such a thing. It is also the thing that makes an asymmetric match slow.

How local the observer’s adaptation is #

A bipartite field shares one adaptation only if the observer’s visual system pools over the whole field. It does not, entirely: adaptation is partly local, and an observer fixating near the join has retinal regions that see mostly one half. The honest model sits between the two cases.

The signal falls steadily as adaptation becomes more local, and reaches the two-room value only when each half is adapted wholly to its own lamp. For daylight against the phosphor LED it goes 1.79 with a shared adaptation, 1.64 with a quarter local, 1.52 with half, 1.17 with three quarters and 0.58 wholly local. In settings: 8, 10, 11, 19 and 75. Even at three quarters local, the field needs a quarter of the two-room settings.

The right end of the curve reproduces the two-room design exactly — the construction is the same with each half’s white set to its own lamp’s, and the two numbers agree to the last digit for every pair — which is the check that the two designs differ in nothing but where the observer is adapted. The curve’s shape is the useful part: the signal holds most of its gain until adaptation is mostly local, so a field that a real observer adapts to only partly still wins by a wide margin.

The fraction of an observer’s adaptation that is local in a bipartite field of this kind is itself measurable, and it depends on the field’s size and on how the observer looks at it. Large fields viewed with free gaze pool more; a small field with steady fixation on the join pools less. A session that wants the most signal should use a large field and free viewing.

Which lamp should fill the field #

A bipartite field need not be split evenly.

The less of the field daylight covers, the larger the signal: 4.45 with daylight on a tenth, 2.80 on a quarter, 1.79 on half, 1.41 on three quarters and 1.28 on nine tenths. When the LED fills most of the field, the observer’s shared white is nearly the LED’s, so the LED half is adapted almost as in its own room and the daylight half is read against a white it contributes little to — and daylight’s rod contribution, which its own white would have divided out, is left in.

That gives the design a second free choice after the pair of lamps. Make the test half small and daylight, and let the LED fill the surround. A daylight patch in an LED field is also the arrangement closest to how the question arises in practice — an object lit by daylight from a window, seen in a room lit by LEDs — which is a better reason than the signal for choosing it.

The reference matters less in one field #

The earlier essay’s principle was that the reference lamp must be one the S weight does not move, because a reference that moves cancels the test’s signal. In two rooms the best pair with daylight in it beat the best pair of two sensitive lamps by more than a factor of two.

In one field that factor falls to 1.6. The phosphor LED against the tube, the best pair of sensitive lamps, gives 1.34 in one field against 2.19 for daylight against the three-emitter LED. The principle still holds — daylight is still the best reference — but it matters less, because in one field neither half’s rod contribution is divided out by its own white, so even two lamps that both move with the weight move by visibly different amounts.

What an experimenter can take from it #

Run the match in one field. It is faster by the whole adaptation time, and it carries three times the signal for the pair the earlier design chose, so it needs a ninth of the settings.

Use daylight against the three-emitter LED, or against the phosphor LED. Both give signals near two colour differences in an evenly split field, six to eight settings at half a colour difference of scatter.

Let the LED fill the surround, and keep the daylight test half small, which raises the signal further and is the everyday case the weight matters for.

And measure how local the adaptation is, because the gain depends on it. Two sessions — one in a large field with free gaze, one in a small field with fixation on the join — would bracket it, and a rod signal has no natural size is the reminder that the weight the session estimates is only defined once the rod signal’s normalisation is stated.

How the match was modelled #

The observers, lamps and surfaces are those of the two earlier essays: the site’s cone template for a thirty-two-year-old observer, a rod curve behind the same lens, and a rod signal of a tenth of each cone class’s peak sensitivity entering the L and M channels in full and the S channel with the weight being measured. The five lamps are the tabulation’s: a 2856 K tungsten radiator, a 6500 K radiator, a phosphor white LED, a three-emitter LED and a fluorescent tube.

Each half’s colour is the surface’s XYZ under its lamp, as the observer with rods computes it, divided by that lamp’s own luminance — the two halves are lit to equal luminance. In two rooms it is adapted by CAT16 from its own lamp’s white to D65; in one field, from the area-weighted mixture of the two lamps’ whites; with partly local adaptation, from a weighted mixture of the two. The signal is the median over forty-two surfaces of the colour difference between the match’s two ends as the S weight goes from nothing to equal, with the observer without rods as the baseline, exactly as the two-room design computed it. Settings are the square of half a colour difference of scatter over a tenth of the signal.

What this leaves out #

Adaptation is modelled as a von Kries scaling to a single white per half. A bipartite field has an edge, and the visual system’s response near an edge is not a scaling; induction across the join shifts each half’s appearance towards the complement of the other’s, which a CAT does not capture. Colour stops at the edge of sight is the reminder that the field’s position on the retina matters as well.

The mesopic weighting is fixed. Rod intrusion grows as light falls, and a session run at one luminance measures the weight at that luminance. The two designs differ in adaptation, not in light level, and the comparison holds at any level; the absolute signals do not.

And the model’s observer is one observer. The reference lamp must not move estimated settings against a stated scatter per setting, and a between-observer spread in the rod weight itself is what the experiment exists to find.

Still open: what the edge between the halves does #

The one thing a bipartite field has that two rooms do not is a boundary between the two lamps, and across a boundary colours induce each other: each half is pushed towards the complement of the other. If the two halves’ rod contributions differ, the induction across the edge carries that difference too.

The prediction is that induction adds to the signal rather than subtracting from it, because it pushes each half away from the other and the rod signal already separates them; if so, the edge makes the field better still, and a thin gap between the halves — the usual remedy for induction — would cost signal rather than clean it. The computation needs a spatial model: the two halves as a field with an edge, filtered by the visual system’s opponent channels, the way how fine a colour edge can be filtered an edge between two colours. The quantity worth reporting is the signal with and without a gap of a stated width between the halves.

A slow experiment was slow for a reason #

The habit is about what an experiment’s inconvenience is doing.

The two-room match was slow because the observer had to adapt to each lamp, and the obvious move was to find a design without the adaptation and accept whatever it cost. What it cost turned out to be nothing: the adaptation the slow design waited for was the very thing dividing the signal out of each half, so removing it made the experiment both faster and more sensitive.

The failure mode is to treat an experiment’s slow step as overhead — as a price paid to get the measurement, rather than as part of what the measurement is. The move is to ask what that step does to the quantity being measured. Here it normalised each half by its own lamp, and a measurement of how two lamps differ is not helped by first making them alike.