The distance at which two lamps part

How many lamps a drawing has settles the method: cut the bundle of drawn lines into pencils, and take the smallest count whose residual is inside what the reader’s own clicking produces.

Run it on two lamps five centimetres apart and it returns one.

That is not a failure of the method. It is the drawing’s own answer, and this essay is about how to say so precisely — because between “these are two lamps” and “this drawing does not contain two lamps” there is a distance, and the distance is a number.

Not a threshold #

The first thing the measurement shows is that the question has no yes-or-no answer at the interesting separations.

Twelve drawings of the same two lamps, each with its own clicking, at each of seven separations. At twenty centimetres and beyond, all twelve are counted as two. At five centimetres, one of twelve is. At ten centimetres, eleven of twelve — the same two lamps, the same posts, the same camera, and different answers from different drawings.

So the honest report is a rate rather than a threshold, and the number worth quoting is where the rate crosses a half.

This is the shape acrossSeeds exists to enforce and it is worth restating: one run is an anecdote. A limit quoted from a single drawing is a statement about which pixel that reader happened to click, and the collection has enough single-run results in its history to be wary of them.

Twelve drawings, and what the rate is worth #

A rate estimated from twelve trials is not a precise number, and the essay should say what it is before quoting a crossing to two figures.

Twelve draws put a standard error of about fourteen percentage points on a rate near a half, so the crossing itself is uncertain by roughly the width over which the rate climbs through that band — on this sweep, a little under a factor of two in separation. Quoting 7.2 centimetres therefore means about seven, and the second digit is there to make the proportionality below readable rather than because it is known.

That is enough for the question being asked, and it is worth being explicit about why. The finding is not a threshold to be engineered against; it is the shape of the dependence on the noise, and a shape survives an uncertainty on each point that a threshold would not. Doubling the trials would halve the error bars and would not change a single sentence here.

The bisection that finds the crossing is run on the rate rather than on a single drawing’s verdict for the same reason: a bisection on a noisy predicate converges to wherever it happened to step, and a bisection on a rate estimated at each step converges to the crossing.

The number, and the law #

The crossing sits at 7.2 centimetres for a picture clicked to a pixel, on the arrangement here: five posts scattered across a view, lamps about three metres away, a 690-pixel frame.

Sweep the clicking and the crossing moves in exact proportion. Half a pixel gives 3.6 centimetres; two pixels give 14.5; four give 28. The fitted exponent is 1.00.

A proportionality rather than a floor is a strong statement and it is worth reading carefully, because the collection has an essay about exactly this distinction. An error law with two terms — one that falls with effort and one that does not — has a floor, and past a certain point care buys nothing. This one has no second term at all over the range measured: nothing stands between a reader and any separation except how carefully they click.

That is a rare shape here. A camera recovery’s floor is the lens; a panorama’s vertical miss does not respond to more frames; an anamorph’s height ambiguity is not reduced by looking harder. A lamp count, over this range, is limited by nothing but the measurement.

Why proportional, and where it would stop #

The proportionality has a mechanism, and stating it says where the law would break.

Two lamps at separation $s$ produce two pencils whose centres are apart in the picture by roughly $s$ times the projective scale — a factor set by the camera and the geometry, not by the lamps. A drawing clicked to $\sigma$ pixels puts each centre in doubt by about $5.5\sigma$, which is the leverage the previous rung measures. The two are distinguishable when the separation of the centres exceeds their uncertainty, which is one number proportional to $s$ against another proportional to $\sigma$. Hence a straight line through the origin.

The constant can be written out too, and it turns out to be two factors this collection already publishes separately. The image separation of the two centres is $f s/D$ for lamps $s$ apart at distance $D$ from a camera of focal length $f$; the uncertainty in each centre is the leverage $\lambda$ times the clicking error. Setting the first above the second,

$$s^{*} \;=\; \lambda\,\sigma\,\frac{D}{f}.$$

The resolvable lamp separation is the leverage times what one pixel is worth in metres at the lamps’ distance — and $D/f$ per pixel is exactly the quantity the metrology field reports as the cost of a pixel on the ground. The measured 7.2 centimetres per pixel of clicking is $\lambda = 5.5$ against a pixel worth about 1.3 centimetres at the lamps’ distance, which is an ordinary camera in an ordinary room.

Both factors are computable before any clicking is done, which is the useful part: a reader can say in advance what separation their photograph could resolve, from the focal length, the lamps’ distance and the height ratio of whatever objects they intend to use. And both are things they can change — a longer lens, a nearer camera, taller objects — which is why the law has no floor to run into.

It would stop in three places, none of which is reached here.

When the lamps are far enough that the pencils are nearly parallel, the projective scale collapses and the limit stops being proportional to anything simple — which is a light far enough away, where a lamp becomes the sun and the pencil becomes a family of parallels.

When the drawing runs out of lines, the leverage argument fails: with two posts each pencil is exactly determined and has no residual, so nothing can be tested at all.

And when the noise stops being small, the first-order propagation the leverage rests on is no longer the whole story. At four pixels of clicking on a 690-pixel frame the fit is still linear; at forty it would not be.

What “one lamp” means when there are two #

The strongest thing in this measurement is the case that reads as a failure.

Below the crossing the method returns one lamp, and the single centre it returns is not either lamp: it is a least-squares compromise sitting between the two images. A reader taking that answer to the room and computing a lamp position gets a lamp that is not there.

And the method is right to have returned it. Every line in the drawing is within the reader’s own clicking of that single pencil, so the drawing is consistent with one lamp — there is a one-lamp scene that would have produced marks indistinguishable from these. Reporting two would be reporting a distinction the evidence does not carry.

That is not a comfortable answer, and it is the same one the one-post recovery gives: a residual of zero and an answer that is a whole one-parameter family, which the field calls its sharpest counter-example to the idea that a small residual means a right answer. Here the residual is small and the answer is a single point and it is wrong, which is a step worse.

The defence is the same as it is everywhere in this collection: report the ambiguity, not the point. A drawing below the crossing determines a single centre and a radius of confusion around it, and both should be quoted.

Two ways to buy a smaller limit that do not work #

The law says care buys the limit outright. The obvious question is what else does, and the two obvious answers were both written into this essay before they were measured, and both are wrong.

More posts do not help. Three posts give a limit of 6.5 centimetres, five give 7.2, and nine give 7.9 — if anything slightly worse, and certainly not the factor of $\sqrt{3}$ that averaging three times as many lines would suggest.

A wider scatter of posts does not help either. Spreading them over eight metres instead of one and a half moves the limit from 6.4 centimetres to 7.8, again in the wrong direction and again by very little.

The reason is the same for both, and it is a fact about the criterion rather than about the geometry. The test compares a residual against an expectation, and both of them are per-line quantities computed from the same lines: adding lines does not reduce the residual a two-lamp bundle leaves at one centre, because that residual is structural — it is roughly half the separation of the two pencils, whatever the count — and it does not reduce the expectation either, because the expectation is a root-mean-square over the same set. The number of lines cancels out of both sides.

That is a real limitation and it is worth naming precisely rather than apologising for. A test that asked how confident is this rather than is this residual consistent with noise would exploit the extra lines and would buy the root: with nine posts the two centres are individually better determined even though the one-centre residual is not smaller. The criterion used here throws that away, in exchange for needing nothing but a noise level.

So the honest summary is: this test’s resolution is set by the noise and by the arrangement, and not by how much drawing there is. A reader who wants the extra factor has to go to a likelihood, and has to state a noise model rather than a noise level.

Which direction the lamps are apart in #

What does change the answer is the direction, and by a factor the projection makes inevitable.

Two lamps separated across the view part at 7.2 centimetres. Separated in height they part at 13.3. Separated in depth — one behind the other along the line of sight — they part at 32.0, four and a half times the lateral figure.

The mechanism is the one every essay in this collection about depth eventually reaches. A separation across the view is a separation in the picture, at the full projective scale. A separation along the line of sight is projected almost entirely away: two lamps a third of a metre apart in depth put their images a few pixels apart, because the direction they differ in is the direction the camera collapses.

So a photograph of a room with two lamps in it resolves them well if they are side by side on a wall and badly if one is behind the other — and the ratio is the same one that makes depth a reciprocal and makes a stereo pair blind past a range. It is the projection’s own anisotropy, arriving in a measurement that has nothing to do with stereo.

The vertical case sits between the two for a reason worth a sentence: a vertical separation is in the picture, so it is not collapsed, but the posts’ own geometry gives less leverage in that direction — the drawn lines run mostly along the shadows, so they pin the horizontal position of a centre better than its height.

The comparison worth making #

Seven centimetres at three metres is an angular separation of about a fortieth of a radian, which is a little over a degree.

Set that beside the collection’s other resolution limits and it is a good one. A silhouette’s smallest recoverable feature is bounded by a concavity nothing reaches at any effort; a curvature fitted to a shadow has a detection floor set by the pixel size and reports a number for a floor that has none; a stereo pair’s range ends where the disparity falls under a pixel.

The lamp count is unusual in this company because its limit is set by a quantity the reader controls. That makes it a rare kind of measurement in this collection: one where the honest advice is measure more carefully rather than measure something else.

What a reader with a photograph should do #

The measurement is worth turning into instructions, because a reader with a real photograph is in a slightly better position than the sweep above assumes.

Estimate the clicking, honestly. Not the resolution — how far apart two attempts at the same shadow tip land. A soft shadow edge in a real photograph is worth several pixels even at high resolution, which is why the penumbra rung matters here: the width of the edge is a lower bound on the clicking, and it is often the dominant one.

Look at the direction the candidate lamps are apart in. Two lamps side by side on a ceiling are the good case; a lamp and its reflection in a window behind it are the bad one, because the reflected source is displaced almost entirely in depth.

And quote the answer with the limit attached. “One lamp” from a drawing whose limit is thirty centimetres means one lamp, or several within thirty centimetres of each other — which for a ceiling fitting with three bulbs in it is a completely reasonable and completely different description of the same room.

That last point is the one that decides whether the method is useful in a real setting. Most rooms that contain more than one light contain lights that are metres apart, and a metre is fourteen times the limit at a pixel of clicking. The method’s weak case is not two lamps across a room; it is one lamp with a reflector, or a fitting with several bulbs, or a lamp and a bright window — all of which are physically several sources within centimetres or in the wrong direction, and all of which the drawing will honestly report as one.

The other end of the range #

The sweep stops at six metres and it is worth saying what is beyond it.

As two lamps move very far apart, their pencils separate completely and the count is trivially two — but a second thing happens that the count does not report: each pencil is then fitted from lines that no longer span the picture, and the positions of the two centres get worse even as the count gets easier. A lamp whose posts are all on one side of it is badly determined along one direction, which is the degenerate arrangement in miniature.

So the useful regime has two ends. Below the limit the count fails; far above it the count is easy and the positions are poorly conditioned. The comfortable middle is where the two pencils are separated by many times the clicking and each still has lines crossing the picture at several angles — which, on the arrangement here, is roughly from a fifth of a metre to a few metres.

The short version #

Two lamps become two when they are further apart than the reader’s clicking makes them, and that distance is proportional to the clicking with no floor: 3.6 centimetres at half a pixel, 7.2 at one, 28 at four.

Below it the drawing genuinely contains one lamp, in the strong sense that a one-lamp scene would have produced the same marks — and the single centre the method returns is a compromise between two images rather than either of them, which is a thing to report with a radius rather than as a point.