The centre has an area

Every projection on this site goes through a point. That is the sentence lib/camera.js implements, the one recovering the camera gets back out of a picture, and the one every theorem quoted anywhere here follows from.

No instrument has one. A camera gathers light through a hole with a diameter, an eye through a pupil four millimetres across, a projector through a lens. The pinhole is what is left when that diameter goes to zero, and it never does.

The eye is a place, not a point established where the centre is and what rotating about anywhere else costs. This essay gives it a radius.

The model, stated #

A thin lens of focal length f, a circular pupil of radius R on the axis, a sensor at distance v₀ behind it. Everything in millimetres of a real camera, with the sensor format converting to the pixels the rest of this collection works in.

The f-number written on a lens barrel is f/2R, so the pupil’s radius is f/2N — for a 50 mm lens at f/2.8, 8.93 millimetres. Focused at three metres, the sensor sits 50.85 millimetres behind the lens.

One line does the whole of the geometry. A ray from an object point through the pupil at height a passes through that object’s own image point, so its height at the sensor is the linear interpolation between the two:

$$u(a) = a + (x_{\text{img}} - a)\,\frac{v_0}{v}$$

with v the distance at which the object actually focuses. When v = v₀ the a drops out, which is what in focus means written as an identity rather than as a description.

The claim, and the number #

Take the mean over the whole pupil of where the rays land, and compare it with the mark a pinhole at the pupil’s centre would have made.

Across three apertures, five distances and three field positions, the worst disagreement is 5.7 × 10⁻¹⁴ millimetres. That is the arithmetic floor, not a tolerance.

It is also easy to see why, once the interpolation above is written down: the mean over a symmetric pupil of a·(1 − v₀/v) is zero, so the centre is x_img·v₀/v, which works out to −X·v₀/Z — the pinhole’s mark exactly, with the object’s own focus distance cancelling out.

So the whole of this collection’s geometry is licensed. A vanishing point is where a pinhole would have put it. A cross-ratio survives. A recovered camera is the real camera. What a finite aperture spends is sharpness, and nothing else.

The control #

An identity that always held would be a fact about the algebra. So the same measurement is made with the pupil not centred on the axis — a shifted stop, or a vignetted corner where the pupil is clipped on one side — and the mark moves by 0.106 millimetres, two pixels on this collection’s own figures.

That is a real and useful consequence rather than a formality. It says a lens whose effective pupil is asymmetric across the field has a centre of projection that wanders with position, which is the mechanism behind several small calibration residuals people attribute to distortion. And it says the symmetry, not the algebra, is what the exactness rests on.

What the disc’s size is #

R|1 − v₀/v|, which is the second half of the closed form, and it is worth having in a shape a reader can use.

Substituting the thin-lens relation gives the diameter on the sensor as

$$c = \frac{f^2}{N}\cdot\frac{|Z - Z_f|}{Z\,(Z_f - f)}$$

with Z_f the focus distance. The important features are all visible in it: it is zero at Z = Z_f; it grows without bound as Z approaches the focal length; and as Z → ∞ it approaches a finite limit f²/(N·Z_f), which is why a far background is never infinitely blurred and why a hyperfocal distance exists at all.

Measured on the 50 mm at f/2.8 focused at three metres: a point at 0.9 metres images as a disc 13.5 pixels across, at 2.1 metres 2.5, at 4.4 metres 1.9, at 16 metres 4.7. The near side is steeper, which is the asymmetry the sharp band is a decision inherits.

The blur is linear in inverse depth #

The four measured diameters — 13.5, 2.5, 1.9 and 4.7 pixels at 0.9, 2.1, 4.4 and 16 metres — are the closed form evaluated four times, and rearranging it once puts the whole family in the coordinate the rest of this collection keeps arriving at.

Factor the expression the other way:

$$c = \frac{f^{2} Z_{f}}{N\,(Z_{f} - f)} \left| \frac{1}{Z} - \frac{1}{Z_{f}} \right|.$$

The disc’s diameter is linear in the difference of reciprocals, exactly — the same coordinate a stereo pair’s disparity is linear in, and the same one a scroll’s sag turns out to live in. Blur is a disparity in the aperture rather than across a baseline, and it obeys the same law.

Three consequences drop out of that form, and none of them is visible in the version written in depths.

The blur is symmetric in $1/Z$ and not in $Z$. Two distances blur equally when $1/Z_{1} + 1/Z_{2} = 2/Z_{f}$ — the focus distance is the harmonic mean of any two equally blurred depths. Focused at three metres, a point at 2.1 m is exactly as blurred as one at 5.25 m, which is what the measured 2.5 pixels at 2.1 m pairs with. The apparent asymmetry of the near and far sides is the depth axis being the wrong coordinate to look at it in.

The far side is bounded and the near side is not. Inverse depth runs from $1/Z_{f}$ down to zero going outward, so the disc cannot exceed $f^{2}/(N Z_{f})$ — 5.69 px on this lens — however distant the object. Going inward it runs to infinity, so nothing bounds the near disc. The measured 4.7 px at sixteen metres is already 83% of everything the far half of the world can produce, and the remaining 17% is spread over the rest of the universe.

And the hyperfocal distance is that bound read backwards. Choosing an acceptable disc picks a value of $|1/Z - 1/Z_{f}|$, and focusing so that the acceptable interval reaches zero — that is, so that infinity is exactly at the edge of it — is the whole of the rule. Hyperfocal focusing is not a trick; it is the observation that the far end of the inverse-depth axis is a wall, and that a criterion which does not reach it is wasting the half of its interval that lies beyond.

Why this had to be checked #

It is the kind of assumption that is obviously true and has never been written down here, which on this site is exactly the condition under which things turn out to be false.

Two of this collection’s own findings are of that shape. The camera basis was inverted for a whole phase and every check passed, because a consistent flip is a valid projection of a mirrored scene. A shared tick routine returned an empty array for a descending domain, so two figures carried no gridlines at all, at full gates, because every check asks whether a label fits and none asks whether it exists.

“The blur is centred on the sharp position” is a sentence of the same kind: everybody believes it, nothing on this site had run it, and the consequences of its being false would have been invisible — a systematic bias in every recovered vanishing point, growing with defocus, indistinguishable from lens distortion.

Where it stops being true #

Three places, each of which is a real lens rather than a quibble.

Vignetting. Toward the corner of the frame the pupil is clipped by the barrel, and a clipped pupil is not symmetric about its own centre. The mark then moves, by the amount the control above measures. This is the cat’s-eye aperture visible in the corners of an out-of-focus photograph, and it is a genuine departure from the identity.

Pupil aberration. In a real lens the entrance pupil is the image of the stop through the front group, and that image is not perfect: it moves and changes shape with field angle. The centre of projection therefore depends slightly on where in the frame the point is, which is one of the things a distortion model absorbs — and a lens destroys the invariant is where this collection measures what that absorption costs.

And a non-circular stop. A polygonal aperture is still symmetric about its centre if it has an even number of blades or is regular, so the identity survives — but a stop that is asymmetric, or one blade of which sticks, does move the mark.

None of these is large. All of them are of the same size as the residuals a careful calibration is trying to explain, which is the reason for naming them.

The pinhole is not the limit anybody uses #

A remark on the other end of the same axis, because it is where the model’s honesty is easiest to check.

Shrinking the aperture does not improve a picture without bound: diffraction sets in and the disc grows again. That crossover is real optics rather than geometry and this collection does not compute it — but the crossing itself is geometry and has been met once already here, in a hole is not preserved’s own field, where a five-millimetre gap between leaves stops imaging itself and starts imaging the sun at half a metre.

That measurement is geometric and it is the same statement in a different costume: a hole is a pinhole camera when the source’s image through it is smaller than the hole, and an aperture when it is larger. The blur disc computed here is the second case, and where the two cross is a length that the geometry supplies.

The disc is the pupil’s own image #

A way of reading the result that makes the exactness obvious and is worth having, because it also predicts the exceptions.

The patch a defocused point makes on the sensor is a picture of the pupil, projected from that point’s own image. It is a disc because the pupil is a disc; it is an ellipse toward the corners because the pupil is seen obliquely there; it is a cat’s eye where the barrel clips it; and it is a heptagon on a lens with seven blades, which is why out-of-focus highlights have visible edges.

So “the centre of the patch is the pinhole’s mark” is the same statement as “the centre of the pupil’s image is the image of the centre of the pupil”, which is true because projection is linear and the pupil is symmetric. Everything that breaks the identity breaks that symmetry.

That reading also says which measurements are safe on a defocused image. A centroid of a symmetric blob is safe. A centroid of a clipped blob — a highlight near the corner, half of which is cut off by vignetting — is not, and the bias is toward the frame’s centre.

What is licensed, precisely #

Worth stating as a list, because “the geometry survives” is a large claim and it is not unlimited.

Licensed: the position of the centroid of a point’s image; every construction built on centroids; vanishing points, horizons, cross-ratios, homographies, recovered focal lengths and recovered poses, at any aperture and any degree of defocus.

Not licensed: anything that reads an edge rather than a centroid. An occluding edge does not blur symmetrically, and its position in a defocused picture is not where a pinhole would have put it — which is a pupil sees around an edge, and it is the one thing in this row that does not survive.

And not licensed: any assumption that the blurred picture is a function of the sharp one. Two scenes with identical pinhole pictures can have different pupil pictures, which is the same essay’s stronger result.

So the licence covers point features and stops at silhouettes, which is a boundary a practitioner can use: a calibration target of dots is safe out of focus and a calibration target of checkers is not. That is a sharper statement of the same caution the eye is a place, not a point raises about where a camera’s centre actually is, and it is worth putting beside one depth per sample is not enough, where the same failure appears in a renderer.

Where the disc’s size becomes a decision #

The size is a number and nothing about it says whether a picture is sharp, which is the subject of the next rung and worth a sentence here because it is the commonest confusion about the whole topic.

There is no aperture at which a picture becomes sharp. There is a diameter of disc that a particular reader, at a particular viewing distance, looking at a particular print size, is prepared to ignore — and every quantity usually quoted about depth of field is that acceptance restated in metres. The sharp band is a decision measures the whole family of them on one lens at one focus setting, and the band goes from half a metre deep to unbounded without anything about the optics changing.

Quoting the criterion in pixels of the picture rather than in millimetres of sensor is this collection’s own choice and it is deliberate: a circle of confusion in millimetres presupposes a print size and a viewing distance that are almost never stated, and this is a site that computes viewing distances.

What a pupil costs a measurement, in practice #

One number, and it is smaller than the topic’s prominence suggests.

A mark located as the centroid of a blurred blob is located to roughly the blob’s diameter divided by the square root of the number of photons in it — which for an ordinary exposure is a fraction of a pixel even when the blob is thirteen pixels across. Defocus therefore loses less positional accuracy than intuition expects, and a calibration target photographed slightly out of focus is often located better than one photographed sharply, because a larger blob has more pixels to average.

What defocus destroys is not position but discrimination: two nearby marks merge, an edge stops being a step, texture disappears. So the practical failure of a defocused calibration is not that the marks move, but that fewer of them can be found and told apart.

That is worth stating because it is the reverse of the usual advice, and because it follows directly from the identity at the top of this essay. If the centre of the disc were biased, defocus would introduce a systematic error that no amount of averaging removes; because it is not, defocus introduces only a variance that averaging does remove.

The sampling behind the number #

The centre is computed as the mean of sixty-four landings around the pupil’s rim rather than from the chief ray, and the distinction is the whole reason the measurement is a measurement.

Taking the chief ray and calling it the centre would assume what is being shown. Taking the mean over the rim of a symmetric pupil gives the same answer for a reason — the rim is symmetric about the axis, and the map from pupil position to sensor position is affine at a fixed depth — and the agreement to 5.7 × 10⁻¹⁴ millimetres is that reason confirmed rather than restated.

The same sampling reports one more thing the closed form cannot: the patch’s roundness, the spread of its rim radii about their own mean, which comes back at 5.9 × 10⁻¹⁵. A patch that was slightly elliptical on the axis would have said the projection was not what it was thought to be, and nothing else would have noticed.

The short version #

Give the centre of projection an area and every world point images as a disc of radius R|1 − v₀/v|, whose centre is the pinhole’s mark to 5.7 × 10⁻¹⁴ millimetres at every aperture, distance and field position. The identity follows from the pupil’s symmetry rather than from the algebra, and moving the pupil off the axis moves the mark by 0.106 millimetres.

So a finite aperture spends sharpness and buys nothing and costs no geometry — for point features. The exception is an edge, and it is large.