Two floors trade the screen for the halo

A dimming panel has no single black gave ΔEITP a declared floor — the light under every pixel before the picture adds any — and asked what a display’s own black does to it. For an LCD or an OLED the answer was simple: the display’s black and the room’s veil add, ΔEITP sees only their sum, and one declared floor reads every grey step exactly. A local-dimming LCD broke that. Its black is its backlight leaking through closed pixels, and its backlight is dimmed zone by zone to follow the picture, so under a dark scene with one highlight the floor in the dimmed zones is a hundredth of the floor beside the highlight. In a dark room, with a veil of 0.01 cd/m², a 1,000 cd/m² panel’s floors ranged twenty-six times across one scene, and the best single declared floor misread some five-per-cent grey step by a factor of 2.6.

That essay closed on the obvious repair. A panel knows which of its zones are dimmed. A colour engine told the backlight map could apply the dimmed floor to one part of the picture and the lit floor to the other. The prediction was that two floors would bring the error within a quarter, because the zones of a real picture cluster at the two ends of the backlight’s range — dimmed and full — with the halo between them a small share of the screen. If they did not cluster, the halo’s width, set by the panel’s diffuser rather than by the picture, would become the thing a tolerance had to state.

Two floors do not do it either way #

The zones do cluster: behind every diffuser measured, 61 to 99 per cent of the screen sits at one of the two ends. Two floors still fail. Placed to hold the worst zone down, they bring the error from 0.79–0.96 in log to 0.22–0.46 — under half, but within a quarter in only one case of nine — and misread more than 85 per cent of the screen by over ten per cent. Placed exactly on the two clusters, they read 61 to 99 per cent of the screen exactly and misread the halo, 0.7 to 28.5 per cent of it, by up to a factor of 2.7, which in one case is worse than a single floor. To keep every zone within ten per cent takes three floors behind the sharpest diffuser and six or seven behind the others, for all three pictures alike.

• The prediction half holds. The picture’s zones cluster, and the halo is a small share of the screen.
• A small share is not a small error. The halo’s zones take every floor between the clusters, and a declaration that reads the clusters exactly misreads the halo as badly as one floor misread the whole scene.
• The number of floors is the diffuser’s: the same for a moon, a window and a line of subtitles, and set by how many distinct backlight levels the diffuser leaves.
• What a panel should declare is its map. The floors fold exactly, zone by zone, as they did for a whole screen.

A picture, zone by zone #

The earlier census described a scene by its two extremes, the floor in a dimmed zone and the floor beside a highlight, and spread its floors evenly between them. A real picture puts its zones somewhere particular, and whether two floors can work depends on where. So the panel here is drawn as a grid of 32 by 18 dimming zones — 576, a count in the range local-dimming televisions use — each driven at the level its brightest content needs and never below one per cent. The diffuser that makes the backlight even also spreads it: each zone’s backlight is the largest of its neighbours’ drives, weighted by a Gaussian in the distance between zone centres, with a standard deviation of half a zone, one zone or two.

Three pictures are stated rather than sampled. A small highlight on black — a moon in a night sky — drives one zone to full and leaves the rest dimmed. A window filling a fifth of a dark room drives a block of 110 zones to 80 per cent against an interior at 3 per cent. A line of subtitles drives sixteen zones along the bottom to full against a frame at 2 per cent.

Each zone’s floor is the room’s veil, 0.01 cd/m² in a dark room — the light that a display in a room is a smaller display found eating a panel’s range as the room brightens — plus the panel’s black at that zone’s backlight — a third of a candela at full backlight on a 1,000 cd/m² panel with 3,000:1 native contrast, as the earlier census set it — nearly ten times the range a black that is not black found a four-colour press reaching against its own paper. A guessed veil halves the error is where the veil came from and how it is priced; here it sets the lowest floor any zone can have, 0.0133 cd/m², against 0.343 beside the moon.

Where the floors fall #

The clustering the proposal expected is there. Sorted, each picture’s floors run flat for most of the screen at its dimmed level, rise through a short slope and run flat again at full backlight: behind a diffuser of one zone, the moon’s dimmed zones are 96 per cent of the screen and its lit ones one per cent, the window’s 64 and 26, the subtitles’ 83 and 9. What is between — the halo — is 3.5, 9.4 and 8.0 per cent.

But the slope is continuous. The halo’s zones do not sit at a third level; they take every floor between the two clusters, because a Gaussian’s tail passes through every value on its way from full to one per cent. It has no edge, which is the same property a pool with an edge had to replace before a model of filling-in could say where a region stops. That is the property the rest of this essay turns on. A declared floor is judged by the worst grey step it misreads, and a zone halfway up the slope is misread by whichever cluster’s floor it is given — a floor five times too high or five times too low.

The pictures differ in how much halo they make — a window’s long edge makes more than a moon’s single zone — and in where their clusters sit. They do not differ in the shape of the slope, which is the diffuser’s.

More floors, one at a time #

One floor misreads some step by 0.79 to 0.96 in log, a factor of 2.2 to 2.6, the earlier census’s number. Split the zones by backlight into two groups, each given the floor that serves it best, and the worst error falls to 0.22–0.46: a factor of 1.25 to 1.58, under half of one floor’s in every case, but within a quarter only for the window behind the sharpest diffuser. A third floor takes it to 0.22–0.31 behind the wider diffusers, a fourth to 0.15–0.21.

Behind the sharpest diffuser the curve drops to nothing at three floors. That diffuser’s Gaussian is so narrow that a lit zone reaches only its four nearest neighbours appreciably and its diagonals barely; the screen carries three or four distinct backlight levels, and three floors can give each its own. Behind the wider diffusers the screen carries six to twenty levels, and the curve comes down slowly, passing ten per cent at six or seven floors.

This is the arithmetic a dimming panel has no single black found for one floor, applied to each group: a floor covers a span of about a factor of two within ten per cent, and a twenty-six-fold range needs as many such spans as it contains — unless the screen skips most of the range.

Two ways to place two floors #

Two floors chosen to hold the worst zone down are both compromises. The lower one serves the dimmed zones and the bottom half of the halo, so it sits above the dimmed zones’ own floor; the upper one serves the lit zones and the top of the halo, so it sits below theirs. Every zone is misread a little, and 88 to 100 per cent of the screen is misread by more than ten per cent. A declaration that minimises the worst case has made almost everything slightly wrong in order that nothing is badly wrong.

Two floors placed on the clusters make the opposite trade. The dimmed zones get their own floor, 0.0133 cd/m² under the moon, and the lit zones theirs, 0.343; each halo zone gets whichever is nearer in ratio. The clusters are then read exactly — 61 to 99 per cent of the screen — and the halo is misread, by more than ten per cent on 0.7 to 28.5 per cent of the screen. The halo’s worst zone is misread by 0.44 to 1.00 in log, as badly as a single floor misreads the whole scene; behind the widest diffuser, round the moon, it is misread by 1.00, worse than one floor’s 0.96. A zone halfway up the slope sits five times from either cluster, and neither cluster’s floor serves it.

So two floors buy either a screen that is nearly right everywhere or a screen that is exactly right except where it is badly wrong. Which is better depends on what a tolerance is for, but neither is what the proposal predicted: within a quarter everywhere.

The diffuser’s number #

All three pictures need three floors behind the sharpest diffuser, and six or seven behind the wider two. The moon needs six behind a diffuser of one zone and seven behind two; the window and the subtitles need six behind both. The pictures could hardly be more different — a single bright zone, a fifth of the screen at 80 per cent, a line along the bottom — and the count barely notices.

The count follows the number of distinct backlight levels the diffuser leaves on the screen: three or four behind the sharpest, six to eight behind a spread of one zone, sixteen to twenty behind two. It is not the same number — some levels lie close enough to share a floor — but it rises with it. A diffuser that spreads light further makes a gentler halo with more distinct levels, each of which needs its own floor.

This is what the earlier essay’s closing section anticipated if the zones did not cluster, arriving even though they do. The halo’s shape is the diffuser’s, it is a fixed property of the panel, and it is what a declaration of floors has to be sized for — not the picture.

How much of the screen is halo #

The halo’s share grows with the diffuser’s spread in every picture: round the moon from 1.4 per cent behind the sharpest diffuser to 3.5 and 13.9 per cent behind the wider two; round the window from 7.3 to 9.4 and 19.1; round the subtitles from 5.9 to 8.0 and 16.3. A diffuser spread wider hides the zone structure better — which is its purpose, since a visible zone boundary is the local-dimming panel’s characteristic artefact — and pays for it with more of the screen at an intermediate floor.

That is where the cluster declaration’s failures lie. Behind the widest diffuser, a declaration exact on the clusters misreads a sixth to more than a quarter of the screen by more than ten per cent. A tolerance has no light level found ΔEITP’s own strength is that it prices a difference absolutely; the price of that strength is that it needs the absolute floor, and a panel that hides its zones well spreads its floors over more of the screen.

What a local-dimming panel should declare #

Its backlight map. The earlier essay found that a display’s black and a room’s veil fold into one floor exactly, because ΔEITP sees only their sum; nothing in that argument depended on the floor being the same everywhere. A floor declared per zone — 576 numbers, which the panel’s controller computes every frame to drive its LEDs — reads every zone exactly, halo included. Any declaration coarser than the map is choosing between the two failures above.

If only a summary can be passed, the choice between the two failures is a choice about what the tolerance protects. A delivery check that must pass or fail a whole frame is served by the worst-case pair, and should expect most of the screen to be read more than ten per cent off and its worst zone twenty-five to sixty per cent off. A check on the picture’s important content — a face in the lit zones, a shadow detail in the dimmed ones — is served by the cluster pair, and should know which part of the screen it has given up: the halo, which sits exactly round the brightest content, where a viewer’s eye is most likely to be. The units disagree in the dim found the floor mattering most in the shadows; the halo is a shadow that sits next to a highlight.

How the floors were placed #

Each zone’s drive is its picture’s stated level, never below one per cent. Its backlight is the largest over zones within four standard deviations of its drive times $\exp(-d^2/2\sigma^2)$, where $d$ is the distance between zone centres in zones and σ the diffuser’s spread. Its floor is 0.01 cd/m² of veil plus 1,000/3,000 cd/m² times its backlight. A declared floor F’s error on a zone of floor A is the largest absolute natural logarithm of ΔEITP(F) over ΔEITP(A) for a five-per-cent D65 grey step at every quarter-decade level from 0.005 cd/m² to the white, as in the earlier census; a group’s best floor minimises the worst error over nine floors spanning the group’s range, by ternary search on its logarithm. The fewest groups within a tolerance are found greedily from the lowest floor, which is exact for intervals on a line; the least worst error for k floors bisects the tolerance until the greedy count reaches k. Halo zones are those whose backlight is more than one and a half times the picture’s lowest and less than half its highest.

What this leaves out #

The diffuser is a Gaussian and the backlight a maximum over neighbours. Real panels spread light with their own point-spread function, and their controllers set each zone’s LEDs by an algorithm that trades halo against clipping; a controller that adds neighbours’ light rather than taking the largest would make the halo broader and smoother. Either would change the numbers and not the finding, which depends only on the halo’s floors being continuous between the clusters.

The pictures are three, and stated. A picture with many small highlights — a city at night — would carry many haloes and a larger halo share; one with large smooth gradients would have no clusters at all, which the earlier census’s even spread already covers.

Pixel-level light is ignored. A zone’s floor is taken as uniform across it; in fact light from the lit LEDs falls off across a zone as well as between zones, and a floor declared per zone is exact only to that extent.

Still open: whether the halo is where a viewer looks #

Declaring the clusters exactly gives up the halo, and the halo sits round the brightest content. Whether that matters depends on whether grey steps in the halo are ones a viewer inspects — the dark sky next to the moon, the window frame next to the window — or ones nobody sees, because the highlight beside them has taken the eye’s adaptation.

The calculation is this census with each zone’s steps weighted by their visibility in the presence of the neighbouring highlight — the veiling glare inside the eye adding a floor of its own, which falls off from the highlight much as the diffuser’s halo does. The prediction is that the eye’s own glare is larger than the panel’s halo floor in the zones next to a full-brightness highlight, so that the halo’s misreading is hidden there, and that the cluster declaration’s real failures are confined to the halo’s outer edge, where the panel’s floor has fallen less than the eye’s glare has. If that holds, the cluster pair is the right summary after all, for a reason the display alone could not supply.

A share of the screen is not a share of the error #

The habit is about which statistic a declaration is judged by.

The proposal’s reasoning was sound as far as it went: most of a picture’s zones sit at one of two levels, so two numbers should describe most of the picture. They do. But a tolerance is not a description of most of the picture; it is a statement about every part of it, and the part two numbers miss is continuous, spanning the whole range between them. A small share of the screen with the full range of errors carries the worst error in full.

The failure mode is to size a summary by how much of the data it fits, when it will be judged by the part it fits worst. Two floors fit ninety per cent of a dark picture and misread the other ten per cent as badly as one floor misread all of it, because the other ten per cent is not near either of them.