Counting is a measurement

An area, out of one photograph recovers an area through a rectifying homography and finds that it degrades under two different laws at once. This essay takes the other route, which most floors offer and almost nobody uses: count the tiles.

The two routes have nothing in common except the answer, and comparing their error laws is the point.

The measurement #

A paved court, a tiled floor, a field of identical stones, a brick wall. Any repeated unit whose size is known gives an area directly: count the units, multiply by the unit’s area, and stop.

No rectifier is built. No four marks are read. No homography is fitted, so no conditioning number applies, and the answer does not depend on where the camera was or how the floor is oriented in the frame. The only thing supplied from outside the picture is the size of one tile — which is a closure, in exactly the sense the field uses the word, and it is one that a reader can usually get by standing on the floor with a ruler.

Exact, and then absent #

The count is an integer. There is no such thing as counting sixty-four and a half tiles, so wherever the count can be made at all it is exact, and the recovered area is exact to whatever the tile’s own size is known to.

That is a different error law from anything else in this field. Every other measurement here degrades smoothly: a length read at four tenths of a pixel is a little wrong, and reading it at four pixels makes it ten times as wrong. A count read at four tenths of a pixel is right, and read at four pixels it is right, until the tiles stop being separable — at which point it is not slightly wrong but simply unavailable.

Twenty-one of thirty-six, and the fifteen that are missing contribute nothing rather than contributing an approximation. That step is the whole character of the method.

Where the counting stops, and why it is nearer than expected #

The distance at which a tiled floor stops being countable is set by whichever of the tile’s two edges vanishes first, and the two edges do not vanish at the same rate.

The across edge — the one running left to right in the picture — is foreshortened only by distance, so its image is $t f / d$ pixels for a tile of side $t$. At two pixels of separability, a 62-centimetre tile with a focal length of 700 pixels lasts to 217 metres.

The depth edge — the one running away from the camera — is foreshortened by distance and by the grazing angle at which the ray meets the floor. Its image is $t f h / d^2$, with $h$ the camera’s height. The same tile from a camera 1.62 metres up lasts to 18.7 metres.

An order of magnitude apart, and it is the smaller number that governs. A reader estimating how far a tiled floor can be counted from the size of the tiles in the picture will be wrong by that factor, because what looks like a tile at a hundred metres is a horizontal band whose depth extent has already collapsed.

This is the same grazing factor that makes the previous essay’s two error laws differ by the depth over the camera’s height. It is one fact wearing three costumes in three consecutive essays, and the third is next.

The near limit is a geometric mean #

The two ranges are worth writing beside each other, because the second is the first in disguise and the relation between them says how to buy more of it.

Setting each edge’s image to the separability threshold $s$ pixels gives

$$d_{\text{across}} = \frac{tf}{s}, \qquad d_{\text{depth}} = \sqrt{\frac{tfh}{s}} \;=\; \sqrt{d_{\text{across}}\,h}.$$

The distance a floor can be counted to is the geometric mean of the distance its tiles stay wide enough and the height of the camera. For the 62 cm tile at 700 pixels and two pixels of separability, $\sqrt{217 \times 1.62} = 18.75$ metres, which is the number the sweep returns.

A geometric mean is a brutal operator on a small number, and that is the whole reason the limit lands where nobody expects. It also decides what improving anything is worth. Every one of the three quantities a reader might change — the tile, the lens, the height — enters under a square root, so

doubling the focal length buys 41% more range, not double. Doubling the camera’s height buys the same 41%. Counting out to a hundred metres with that tile and that lens needs the camera 46 metres up.

That is the exact opposite of the previous rung’s advice, and the contrast is the useful part. For an area through a rectifier the error is proportional to $1/h$ over most of the range, so height pays back linearly and pays back a factor of seventeen on the arrangement measured there. For a count, height pays back as $\sqrt{h}$ and there is no error to reduce — only a boundary to push, at half the rate.

The ratio of the two limits is $\sqrt{s h/tf}$, which says something about tilings that is easy to get backwards. A coarser floor is more anisotropic, not less. Two-centimetre mosaic tesserae from the same camera last to 7 m across and 3.4 m in depth — a factor of two apart. Metre-square slabs last to 350 m and 24 m — a factor of fifteen. The bigger the unit, the further the across-limit runs ahead of the depth-limit, so the gap between what a floor looks countable to and what it is countable to grows with the size of the paving. A reader glancing at a picture of a large paved square and judging the range from how big the slabs look will be wrong by a larger factor than they would be on a mosaic, which is not how estimation errors usually behave and is worth carrying as a warning rather than as a formula.

Counting has no conditioning #

The comparison with the homography route is worth setting out as a ledger, because they fail in opposite ways.

The rectifier needs four marks, a plane, and a closure with a length in it. Its error grows smoothly with depth, with two exponents, and it can be improved without limit by reading the marks more carefully. It works at any distance and gets steadily worse.

The count needs a repeated unit and a closure with an area in it. Its error is zero or infinite. It cannot be improved by reading more carefully once the tiles are separable, because there is nothing to read more carefully — the count is already an integer. And it stops at a definite distance.

So they are complementary rather than competing, and the useful move is to run both: the count where the tiles are separable, the rectifier beyond it, and their agreement in the overlap as a check on the closure. A disagreement in the overlap means the reference is wrong, which is the failure mode neither method detects on its own.

The pavement is already a measuring instrument #

Counting a floor is not a modern trick. A tiled pavement is the standard measuring instrument of the whole construction field, and it has been since the first Renaissance treatise.

Alberti draws a pavement, and chooses where the reader stands is the collection’s account of the construction; the bay repeated by a straightedge shows that the repetition can be continued indefinitely with joins and meets alone; and measuring a room off the page uses the tiles to read distances in a picture that is not a perspective at all. In every one of those the tile is doing what it does here: supplying a unit that the picture repeats.

What is different here is the direction of the reasoning. Those essays use the tiles to construct — to place a figure at a stated depth, or to continue a floor. This one uses them to measure, which needs no construction and works on a photograph.

Partial tiles, and what to do with them #

A real region does not end on a tile boundary, and the honest treatment of the remainder is where the method’s exactness gets qualified.

Counting whole tiles and ignoring partial ones underestimates. Counting every tile the region touches overestimates. Counting halves by eye reintroduces exactly the smooth error the method was avoiding, and does so at the boundary, where the tiles are most foreshortened and hardest to read.

The clean answer is to report two numbers — the count of tiles entirely inside and the count of tiles the region meets — which bracket the true area and whose difference is the boundary’s own contribution. That is a genuine interval rather than an estimate with an error bar, and for a compact region it narrows as the tiles get smaller relative to the region, in proportion to the perimeter over the area.

It is the same accounting a pixel count makes, and a pixel is not a point is where the pipeline field sets out why a sample has an area and what pretending otherwise costs.

What has to be true of the units #

Three conditions, and the third is the one that fails in practice.

The units must be identical, or at least identically sized. A floor of hand-cut stones is not a counting instrument; a floor of machine-made tiles is.

They must be coplanar with each other. A count over a floor that steps is a count over two floors, and the units on the step are foreshortened differently.

And they must be separable in the picture, which is the whole of the previous section — with the additional wrinkle that separability depends on the contrast between the tile and its grout as much as on the geometry. A dark floor with dark joints stops being countable a long way before the geometry says it should, and that is a fact about the photograph rather than about the projection.

The closure, in different units #

Counting needs a closure just as the rectifier does, and it is worth being exact about which one.

The rectifier is closed by a length: a reference of stated size somewhere in the plane, from which the whole map gets its metre. The count is closed by an area: one tile’s size. Those are not the same fact, and a photograph may offer one and not the other.

They also fail differently. A reference length one per cent wrong makes the rectifier’s areas two per cent wrong, because the scale enters twice. A tile size one per cent wrong makes the count’s area one per cent wrong, because the tile’s area enters once and the count is exact. So counting is less sensitive to its closure than rectifying is, by a factor of two, which is a second advantage on top of the exactness.

And in one case counting needs no closure at all. A ratio of two areas on the same floor is a ratio of two counts — two integers, with no tile size anywhere in it — so a coverage fraction from a tiled floor is exact and requires nothing supplied from outside the picture whatsoever. That is the one measurement in this whole field with no closure attached.

What else can be counted #

The method is not about tiles. It is about any quantity a picture repeats, and once that is seen there are more instruments in an ordinary photograph than a reader expects.

Courses of brick give a vertical scale on a wall with no ruler: count the courses between two features and multiply by the course height. That is a length rather than an area, and it is the same closure — a repeated unit of stated size — applied one dimension down. The wall under the paint uses exactly that on a façade whose openings have been filled in.

Fence posts, sleepers, paving joints, window bays all repeat along a line, and counting them measures along that line without any construction at all. Where they run away from the camera the count is a projective measurement rather than a metric one — the tenth post is ten units away in the world however crowded it is in the picture — which is why the map a row of posts is treats a repeated row as a coordinate rather than as a scale.

People, at a stretch, and badly: a crowd’s count times an assumed area per person is the standard way of estimating an attendance, and its error is entirely in the assumption rather than in the count.

What all of these share is that the integer is read from the picture and the unit is supplied from outside, which is the division this whole field turns on. The unusual thing about counting is that the picture’s half of the division is exact.

The one place counting is worse #

Fairness requires the case where the method loses, and it is not the far distance.

A count has no way to represent a partial unit, so a region much smaller than one tile cannot be measured at all — the answer is zero or one, and neither is right. The rectifier measures it perfectly well, because a homography does not care how big the patch is.

More generally, the count’s precision is set by the unit rather than by the reading. A floor of two-metre slabs measures a courtyard to within a slab, which is a coarse instrument however good the photograph; a floor of five-centimetre mosaic measures it finely. So counting is exact in the sense that the count has no error, and coarse in the sense that the count has a granularity — and those are different words that a careless summary would run together.

The right statement is that counting quantises the answer rather than perturbing it. That is the same distinction the precision a depth buffer has left makes for a fixed-point depth: a quantised quantity is not a noisy one, its error is bounded rather than distributed, and the two behave differently when many of them are combined.

The boundary, stated #

A plane, identical units, and enough resolution. The first two have been covered; the third has a limit that is not the geometric one.

The 18.7 metres above is where two tile edges are two pixels apart in an ideal image. A real image has a point spread function, a demosaicing filter and a compression scheme between the scene and the pixels, all of which blur, and the practical limit is nearer. This collection computes geometry and states that limit as a geometric one, with the note that it is an upper bound.

There is also a case where the method is better than the geometry suggests. Tiles are periodic, and a periodic pattern can be counted from its frequency long after individual units stop being separable — the same principle a moiré pattern works on. Recovering a count that way is a signal-processing measurement rather than a geometric one and it is not attempted here, but it means the hard limit above is a limit on counting by eye rather than on counting.

What is measured here #

Three numbers and a control.

A tiled floor near the camera is counted exactly: the relative error is 0.0 × 10⁰, because the count is an integer. A floor straddling the limit gives twenty-one of thirty-six tiles, so the method reports a partial answer rather than a degraded one. Past the limit the count is wrong by the whole floor, which is what says the failure is a step rather than a slope. And the limit itself is 18.7 metres, set by the depth edge, against 217 metres for the across edge — a factor of 11.6, which is the grazing foreshortening.

The short version #

A repeated unit turns an area measurement into a count, which is exact wherever it can be made and unavailable where it cannot — a step rather than a slope, and the opposite failure mode from a rectifier’s smooth decay.

The distance at which it fails is set by the tile’s depth edge, which foreshortens as one over the depth squared rather than one over the depth, so it arrives an order of magnitude sooner than the across edge would suggest: 18.7 metres against 217 for a 62-centimetre tile from a camera at head height. And a ratio of two counts needs no closure at all, which makes it the one measurement in this field with nothing supplied from outside the picture.