An even spend of the panel is an uneven picture

Keystone correction spends the panel unevenly turned a projector fifteen degrees, corrected its picture to an upright rectangle on the wall, and filled the rectangle with a source image at the panel’s own resolution. The correction used 82.2 per cent of the panel and spent it unevenly: at the edge of the picture nearest the projector each source pixel got 0.99 of a panel pixel, at the far edge 0.76. The unevenness came from filling a picture with a uniform grid of source pixels when the panel’s pixels land on the wall small at one edge and large at the other.

The essay ended by proposing the obvious repair. Nothing requires the source’s samples to be uniform. Lay them out across the picture in proportion to the panel pixels available at each place — densely where the panel lands densely, sparsely where it lands sparsely — and no panel pixel is spent on detail the source does not have there, and none is missing where it does. It asked what that density would be, and what the uneven source would cost.

The even spend exists and is simple. What it buys is less than the question hoped, and the reason is a small piece of arithmetic that changes what the question should have been.

The panel’s own grid #

The layout that gives every place the same number of panel pixels per sample has an obvious form: one sample per panel pixel, at the panel pixel’s own position. Rendering the picture straight into the panel pixels that land inside the corrected rectangle — computing each one’s colour from the scene or the source at the place on the wall it lights — is that layout, and it needs no resampling at all.

On the wall the panel’s grid is what it always was: a projective image of a rectangle, its cells small and crowded at the near edge and large and spread at the far. Every sample now has exactly one panel pixel, so in the sense the earlier essay meant the spend is perfectly even. It needs 82.2 per cent of a full frame’s samples, because that is how many panel pixels reach the corrected picture; the other 17.8 per cent landed outside the rectangle and were never used. Drag the projector’s turn and both the crowding and the waste grow together: at 5 degrees 94.1 per cent of the panel reaches the picture and the grid is nearly square, at 30 degrees only 63.8 per cent does and the far cells open visibly wider than the near. The render is distorted on purpose found a headset’s renderer drawing into a warped grid to spend its pixels where a lens will put them; a projector that renders into its own panel grid is the same idea with a projective warp instead of a radial one.

The cost to the source is as the earlier essay predicted: the source is now sampled unevenly, finely at the near edge and coarsely at the far. In units of a full-resolution source, a cell is 1.01 source pixels wide at the near edge and 1.32 at the far. The source is rendered with less detail at the far edge than a uniform frame would have given it.

What reaches the wall is the same #

That sounds like a loss at the far edge, and it is not one, for a reason that follows from the earlier essay’s own numbers.

The detail a corrected picture delivers at any place is the lesser of two densities there: the source’s samples and the panel’s pixels. A uniform source at the panel’s resolution has one sample per unit everywhere; the panel has 0.986 pixels per unit at the near edge and 0.758 at the far, and less than one everywhere between. So the panel is the coarser of the two at every place, and the panel sets the delivered detail everywhere: 0.986 at the near edge, 0.758 at the far. The uniform source’s extra samples — the ones above the panel’s density — are averaged away in the resampling and never reach the wall.

The even layout renders exactly the panel’s density, no more, and so it delivers exactly the same picture, to the last digit, on 82.2 per cent of the samples. That is the whole of what it buys: the rendering of detail the panel was always going to discard, and the resampling that discarded it. A projector is a camera run backwards is the right way to see it: a projector’s panel samples the wall as a camera’s sensor samples the scene, and a keystone correction resamples a source into that sampling. Rendering directly into the panel’s own grid skips a step without changing what the grid can hold.

The unevenness of the corrected picture was never in the correction. It was in where the panel’s light lands, which is fixed by the turn, and no layout of the source can change it.

On fewer samples, the even layout goes the wrong way #

The comparison changes when the source has fewer samples than the panel — a renderer with a budget, a video decoded at a lower resolution, a stream that must be light.

With the full frame’s samples, or the panel’s 82.2 per cent laid out evenly, the far edge gets the most it can: 0.758 of the source’s detail, which is the panel’s limit there. Cut the budget below that and the two layouts part. On half a frame’s samples, a uniform source is 0.707 of the panel’s resolution along each axis, which is coarser than the panel everywhere in the corrected picture — including at the far edge, where the panel has 0.758 — so the uniform source sets the detail everywhere and delivers 0.707 evenly across the whole picture. The even layout on the same half frame puts its samples where the panel is dense, and so delivers 0.769 at the near edge and 0.591 at the far.

On a fifth of a frame the pattern repeats: uniform 0.447 at both edges, even 0.486 near and 0.374 far. The even layout follows the panel, and the panel is densest where the picture was already best; with a short budget it spends more of it at the near edge and less at the far, and makes the corrected picture’s worst place worse than a plain uniform source would.

So the earlier essay’s proposal inverts at exactly the budgets where a renderer would want it. With samples to spare it is harmless and saves a sixth of them; with samples short it takes detail from the far edge, where the corrected picture was weakest, and gives it to the near edge, where the panel could have carried it anyway.

The even picture #

The question that should have been asked is not how to spend the panel evenly but how to make the picture even — to deliver the same detail at every place on the wall. The arithmetic above answers that directly.

A uniform source coarser than the panel everywhere is already an even picture: it delivers its own density at every place, because the panel can carry that density at every place. The finest uniform source that is still coarser than the panel everywhere is the one whose density equals the panel’s at its worst place, the far edge — 0.758 along each axis, which is 57.6 per cent of a frame’s samples. That source delivers the far edge’s full detail at every place in the picture, gives up only the near edge’s surplus, and needs a little over half the samples of a full frame.

This is the even picture, and it is an uneven spend of the panel. At the near edge each source sample now has more than one panel pixel, and the panel’s extra resolution there carries nothing. That is the right trade for a picture that will be judged at its worst place — a line of text across the far edge, a face at the side of the screen away from the projector — and the wrong one for a picture whose important content sits near the projector’s side. The earlier essay’s lesson, that a keystone-corrected picture is judged at its far edge, is the lesson that says which.

How the savings move with the turn #

The two budgets — the even layout’s, which delivers a full frame’s detail exactly, and the even picture’s, which delivers the far edge’s detail everywhere — both fall as the projector turns further, and they fall at different rates.

The even layout needs the panel’s used share, which falls by about six points for every five degrees of turn: 94.1 per cent at five degrees, 82.2 at fifteen, 63.8 at thirty. The even picture needs the square of the far edge’s density, which falls faster: 83.7, 57.4 and 30.1 per cent. At thirty degrees a source of less than a third of a frame, uniform, delivers everything the far edge of the corrected picture can show at every place in it.

Both are savings in rendering, not in what reaches the wall, and they save different things. The first keeps the corrected picture exactly as uneven as the full frame made it, and is worth having whenever the renderer can write into the panel’s own grid. The second makes the picture evenly sharp at the far edge’s standard, and is worth having whenever the far edge is what matters. Neither changes the one thing straightening does not move the eye found no correction can change: a projective map cannot invent detail, and the detail at each place on the wall is set by how large the panel’s pixels land there.

Why the panel is the coarser everywhere #

The result that the even layout delivers exactly what a full frame does rests on one inequality: that inside the corrected rectangle the panel’s density never exceeds the source’s. It is worth seeing why that holds, because it is not an accident of fifteen degrees.

Keystone correction shrinks the picture until the largest upright rectangle of the right shape fits inside the trapezium the turned panel throws. The trapezium’s narrow end is the near edge, and it is the near edge that decides how large the rectangle can be: the rectangle’s near side is pulled in until it just fits between the trapezium’s near corners. At that side the panel’s pixels land at very nearly the size the rectangle’s source pixels would have if the whole panel had been used square on, which is why the density there is 0.986 rather than one. Everywhere else in the rectangle the trapezium has flared outward and the panel’s pixels have grown, so the density only falls. A source at the panel’s own resolution is therefore at least as fine as the panel at every place, and exactly as fine nowhere.

That is also why the even layout’s budget is the panel’s used share and not something smaller: every panel pixel inside the rectangle carries at most one source pixel’s worth of detail, and it takes one sample to give it that. A layout with fewer samples than panel pixels must leave some panel pixel without a sample of its own, and the question the budget figure answers is only which ones.

A pixel is not a point adds the refinement the densities leave out. A panel pixel is an area that lights a patch of wall, larger at the far edge; the delivered detail there is limited both by how far apart the patches are and by how much each blurs. The even layout, one sample per pixel, respects both limits by construction — it never asks a pixel for more than it can hold — and the uniform source asks the far edge’s pixels for detail they then average away.

The same accounting from the camera’s side #

A projector’s keystone correction is the picture-making half of a map whose other half is familiar from photographs. Flattening a facade rectifies a photograph of a wall taken at an angle into a straight-on view, and the rectified image is conventionally resampled onto a uniform grid — the camera’s pixels, dense on the near part of the wall and sparse on the far, redistributed into equal cells.

The arithmetic here applies unchanged. The rectified image’s detail at each place is the lesser of its grid’s density and the camera’s there, and a uniform grid at the camera’s own near-end resolution carries nothing the camera did not record: the far end of a rectified facade is soft because the camera spread its pixels thin there, not because the rectification lost anything. Rectifying into a grid that matches the camera’s own sampling — denser near, sparser far — stores exactly what was recorded in fewer cells; rectifying onto a uniform grid at the far end’s density gives an evenly sharp picture of the whole wall at the far end’s standard, in fewer cells still. The camera and the projector are one map run in opposite directions, and so are their two savings.

A lens shift, again #

The earlier essay’s last comparison still stands and is sharper for this one. A lens shift places the picture off the projector’s axis with the panel square to the wall, so every panel pixel lands the same size, the whole panel is used, and every source pixel is one panel pixel across the whole picture. The centre of the picture is not the centre of the paper is the camera’s version: a shifted lens moves the picture without tilting the picture plane.

Against that, both of this essay’s layouts are ways of living with a tilted picture plane rather than removing it. The even layout avoids wasting rendering on detail the tilted panel cannot show; the even picture avoids showing more detail at one edge than the other. A shifted lens avoids the tilt. Where the projector can be placed square to the wall and shifted, neither layout is needed; where it cannot, the choice between them is a choice about which edge of the picture matters.

What was assumed #

Detail is counted in the worse direction at each place. A panel pixel landing on the far side of a turned projector’s picture is stretched more in one direction than the other, and the figures take the smaller density; a picture whose content is mostly lines in the other direction loses less.

Rendering and resampling are ideal. Every density here assumes a sample reproduces the source’s detail up to its own spacing; a real scaler’s interpolation softens the uniform source slightly further wherever it resamples, which favours the even layout, since it does not resample.

The source can be rendered at any layout. A game engine or a presentation renderer can draw into the panel’s grid; a video decoded at a fixed resolution cannot, and for it only the uniform layouts — at whatever resolution the stream carries — are available.

Still open: whether the even picture should be even in the viewer’s eye #

The even picture delivers the same detail per unit of wall at every place. A viewer does not look at units of wall. A projector in the viewer’s eye found that on a flat wall a corrected picture is a correct flat picture from every seat, which is a statement about its shape and not its detail. A viewer sitting to one side of the room sees the far edge of the picture nearer and larger than the near edge, or the reverse, and what matters to that viewer is the detail per degree of their own view, which varies across the picture differently from seat to seat.

The measurement that settles it takes the turned projector, a room of seats, and for each seat the detail per degree the corrected picture delivers at each place, under the uniform source, the even layout and the even picture; and asks which layout gives the worst seat its best worst place — whether the even picture on the wall is also the fair picture across the audience, or whether a projector placed to one side should render for the seats on the far side, where its picture is both softest and nearest.