Keystone correction spends the panel unevenly

A projector is a camera run backwards turned a projector fifteen degrees from square and watched it throw a trapezium. Keystone correction cannot add light outside the trapezium, so it shrinks the picture until an upright rectangle fits inside, and the panel pixels that land outside that rectangle are thrown away: a sixth of the panel at fifteen degrees. That essay measured the loss where it is paid, on the panel, and found it grows fast with the angle.

It also noted, in passing, two things correction cannot buy back. One was focus, which is optics. The other was geometry: the far edge of the trapezium is further from the projector than the near edge, so each panel pixel lands larger there, and the corrected picture — a uniform rectangle on the wall — is drawn from panel pixels of varying size. The essay put the difference at about a third and left it there.

That difference is the larger half of what keystone correction costs, and it is worth measuring properly, because the number a specification quotes — the share of the panel used — describes an average, and a picture is judged at its worst.

A density across the corrected picture #

The corrected picture is an upright 16:9 rectangle on the wall, filled with a source image at the panel’s own resolution, 1920 by 1080. At every point of it, some number of panel pixels carries each source pixel. The figure shades that number across the picture for a projector turned fifteen degrees, taking the worse direction at each point, since a picture is only as sharp as its coarsest axis.

At the edge of the picture nearest the projector the panel’s pixels land small, and each source pixel gets 0.98 of a panel pixel in its worse direction: very nearly one to one. At the far edge the panel’s pixels land large and spread, and each source pixel gets 0.76 of one. The shading runs smoothly between. The correction uses 82.2 per cent of the panel, and it does not spread what it uses evenly.

The reason is the homography itself. Straightening does not move the eye described a keystone correction as a projective map — exact, reversible, and unable to invent information — and a projective map between a tilted plane and an upright one stretches one end more than the other, by the ratio of the two ends’ distances from the centre of projection. Here the centre is the projector’s lens and the tilted plane is the panel as the wall sees it. The map from the corrected picture back onto the panel compresses the far side, where the wall was further away, into fewer panel pixels than the near side.

From near edge to far #

The profile across the picture shows the gradient directly, at three angles.

At five degrees the density falls from 0.997 at the near edge to 0.915 at the far: a gentle slope, hardly worth remarking. At fifteen it falls from 0.986 to 0.758. At thirty it falls from 0.932 to 0.549, so the far edge of the corrected picture is drawn with a little over half the source’s linear resolution while the near edge is drawn with nearly all of it.

The near edge stays close to one at every angle, and that is the important detail. Keystone correction shrinks the picture until it fits the trapezium’s narrow end, and it is the narrow end — the near one, where the projector is closest — that decides how large the picture can be. So the near edge is drawn at nearly full resolution, and all the loss is concentrated toward the far edge, where the trapezium was wide and the picture had to be pulled in to stay rectangular.

Three prices, and the one that matters #

The figure below sets the usual price of keystone correction beside two others, against the projector’s angle.

The share of panel pixels used — the price quoted on the specification — is 94 per cent at five degrees, 82 at fifteen, 64 at thirty. The far edge’s linear resolution is lower at every angle: 92, 76 and 55 per cent. And the detail per area the far edge can carry, the square of its linear resolution, is lower again: 84, 58 and 30 per cent.

The gap between the first and third curves is the whole of the argument. At fifteen degrees the correction spends 82 per cent of the panel, which sounds like a modest loss; where the picture is thinnest, it delivers 58 per cent of the source’s detail. The average and the worst differ because the spend is uneven: the near edge gets more panel than it needs to reproduce the source, the far edge less, and panel pixels spent on the near edge cannot be moved to the far one. A source image with fine detail near its far edge — text, a line of a spreadsheet, a face — loses more than two-fifths of it to a fifteen-degree turn that the specification describes as costing under a fifth.

Tipped costs less than turned #

The earlier essay measured a projector turned about the vertical. Projectors are more often tipped — mounted on a ceiling or a table and aimed up or down at the screen — and on a wide panel the difference matters.

Tipped fifteen degrees, the same projector keeps 83 per cent of its linear resolution at the far edge and uses 88 per cent of its panel, against 76 and 82 when turned by the same angle. The trapezium’s taper runs across the dimension the tilt is about: a turn tapers the picture’s height from one side to the other, a tip tapers its width from top to bottom. The corrected rectangle has to fit inside the taper, and on a panel sixteen units wide by nine high, trimming the short side to keep a rectangle costs less of the panel than trimming the long side.

That favours the common installation. A projector on a table aimed up at a screen, or hung from a ceiling and aimed down, pays less for its keystone than one placed off to the side at the same angle. The ordering reverses for a portrait panel, and would vanish for a square one.

A lens shift places the picture for nothing #

Keystone correction is one of two ways to put a projector’s picture somewhere its lens is not pointing. The other is a lens shift: the lens is moved across the panel, or the panel across the lens, so the picture lands off the projector’s axis while the panel stays square to the wall.

With the panel square to the wall, the whole panel lands as an upright rectangle — there is no trapezium to correct — and every source pixel is exactly one panel pixel, across the whole picture. To place the picture’s centre fifteen degrees off the projector’s line needs a shift of 0.80 of the picture’s half-width at a throw ratio of 1.5; thirty degrees needs 1.73. The centre of the picture is not the centre of the paper is the same principle for a camera: a shifted lens moves the picture without turning the picture plane, so parallel lines stay parallel and nothing is resampled.

The shift has a price of its own, but it is paid in glass, not pixels: the lens must cover an image circle large enough for the shifted panel, and a large shift needs a large circle. What it never costs is resolution. Between a lens that can shift the picture where it is needed and a keystone correction that resamples it there, the shift delivers the whole panel evenly and the correction delivers, at fifteen degrees, three-quarters of it at the far edge.

What the earlier measurement showed #

The earlier essay’s figure is the trapezium this whole argument lives inside.

The inscribed rectangle is the corrected picture, and the quadrilateral around it is the panel as it lands on the wall. Everything in the density figures is a statement about how the panel’s grid, drawn inside that quadrilateral, is carved up by the rectangle: small, dense cells at the near side, where the quadrilateral is narrow and the rectangle nearly fills it; large, sparse cells at the far side, where the quadrilateral flares and the rectangle pulls away from its edges. A projector in the viewer’s eye found that on a flat wall a projector’s picture is a homography from every seat, so a keystone correction can cancel the whole chain; it cancels the shape exactly and leaves the density exactly as uneven as the figure draws it.

Where the viewer sits changes nothing #

A keystone-corrected picture is an upright rectangle on a flat wall, and so, as a projector in the viewer’s eye found, every viewer in the room sees it as a correct flat picture from wherever they sit. The density gradient does not depend on the viewer either. It is a property of how the panel was mapped onto the wall, fixed at the moment of projection, and a viewer at the far side of the room sees the same softened far edge as one at the near side — larger or smaller on their retina, sharper or softer by their own distance, but carrying exactly the source detail the panel delivered there and no more.

What does change with the seat is which edge is nearer. A viewer sitting near the far edge of the picture — the edge the projector was turned away from — sees that edge largest, at the smallest distance, and is the viewer most likely to notice its loss. A projector placed to one side of a room, turned toward a screen at the front, softens the side of the picture nearest the far wall, and the audience seated along that wall is closest to the softest part. The geometry is indifferent to who is looking; the eyes that notice are not.

The loss per degree #

The three price curves fall at different rates, and the rates say which measure a designer should budget with.

The share of the panel used falls by almost exactly six points for every five degrees of turn: 94.1, 88.2, 82.2, 76.2, 70.1, 63.8 and 57.5 per cent from five degrees to thirty-five. The far edge’s linear resolution falls a little faster at first and a little slower later — 91.5, 83.5, 75.9, 68.6, 61.7, 55.0 and 48.6 — about a point and a half a degree throughout. Its square, the detail per area, falls fastest: 83.8, 69.7, 57.6, 47.1, 38.1, 30.3 and 23.6. By thirty-five degrees the panel is still more than half used and the far edge carries less than a quarter of the source’s detail.

So a designer who budgets keystone correction by the panel used underestimates the loss at the worst place in the picture by a factor that grows with the angle: at five degrees the two measures differ by ten points, at fifteen by twenty-five, at thirty-five by thirty-four. The linear resolution is the one to budget with for text, where legibility is set by how many pixels span a stroke’s width; the per-area detail for images, where what matters is how much of the source’s texture survives.

The lens shift’s own cost grows with the angle too, but in glass rather than pixels: 2 × throw ratio × tan of the angle, in half-widths of the picture, so a longer-throw lens — placed further back for the same picture size — needs proportionally more shift to reach the same angle, and a larger image circle to allow it.

The same fact, from the camera’s side #

The unevenness is not peculiar to projectors. It is what every rectification of an oblique view does, and it has been met before from the other direction.

Flattening a facade rectifies a photograph of a wall taken at an angle, recovering its true shape, and it cannot recover the resolution the oblique view failed to record: the far end of a rectified facade is genuinely softer, because the camera’s pixels spread across more wall there. A keystone-corrected projector is the same map run the other way. The camera’s pixels sample the wall unevenly and rectification cannot even them out; the projector’s pixels paint the wall unevenly and correction cannot even them out either. The render is distorted on purpose found a headset’s pre-warp spending its render’s pixels unevenly by a radial map; a keystone correction does it by a projective one, and the question in both cases is where the scarce pixels end up.

A panel pixel has a footprint #

The density measured here counts panel pixels as if each were a point, and one more consequence follows from their being areas. A pixel is not a point found that where a sample sits inside a pixel is a convention with a measurable cost; a projector’s pixel raises the companion question, how large a patch it lights. A panel pixel landing on the wall at the far edge of a keystone-corrected picture lights a larger patch than one at the near edge — larger by the same ratio the density figure draws, about a third at fifteen degrees. The corrected picture’s far edge is therefore not only sampled more coarsely; each sample is also blurred over more wall.

The two effects compound in the direction a viewer notices. A coarser grid of larger patches at the far edge makes both fine detail and sharp edges softer there, and neither the scaler nor the correction can undo it, since both are set by where the panel’s light lands. What the correction controls is only which source pixels are sent to which panel pixels; the light’s own geometry is fixed by the turn.

What was assumed #

The source is at the panel’s resolution. A source image smaller than the panel is upsampled everywhere, and a density below one at the far edge then means the source’s own pixels are enlarged there rather than lost. A source larger than the panel is downsampled everywhere, and the far edge loses proportionally more of it.

The resampling is ideal. Every figure here counts panel pixels per source pixel; how well a projector’s scaler interpolates between them is a further loss, largest where the density departs furthest from one — which is the far edge.

The corrected picture is centred in the trapezium. A correction that places its rectangle elsewhere — against the near edge, say, to keep the picture as large as possible — moves the gradient but cannot remove it, since the rectangle’s two ends are still at different distances from the lens.

Still open: whether a correction that spends the panel evenly exists #

The unevenness comes from filling an upright rectangle on the wall with a uniform grid of source pixels. Nothing requires the source grid to be uniform. A correction could distribute the source’s pixels across the rectangle in proportion to the panel pixels available at each place — denser where the panel is dense, sparser where it is sparse — and then no panel pixel would be spent on detail the source does not have at that place, and none would be missing where it does.

The measurement that settles what that buys takes a turned projector, finds the distribution of source samples across the corrected rectangle that makes the panel-to-source density equal everywhere, and asks two things: what that density is — whether it approaches the share of the panel used, 82 per cent at fifteen degrees, as it must if the panel is spent evenly — and what it costs the source, which is now itself sampled unevenly, more finely at the near edge than the far. If the even spend is worth having, it is because a source can be rendered to match: a projector that knows its own angle could ask the renderer for more detail where it will reach the wall and less where it will not, which is exactly what a headset’s foveated renderer does for a lens.