The disc and the streak

A frame is an interval established that an exposure is an integral of projections, so a moving point draws a streak, that the streak is straight because the image of a straight path is straight, and that its length goes as one over the depth.

The centre has an area established that a pupil with a radius turns a point into a disc, and that the disc’s centre is exactly the pinhole’s mark.

A real frame does both at once, and this essay asks whether they factorise.

The patch a moving point makes #

The set of sensor positions reached by any ray, from any instant of the exposure, through any part of the pupil. Not a description of the blur — the actual set, sampled at forty-one instants times a forty-eight-point pupil rim.

For a point that holds its depth while it moves, the answer is a stadium: the streak of the disc’s centres, with one disc slid along it. That is a Minkowski sum, which is the shape “the two integrals commute” takes when the integrals are drawn rather than written.

Measured against exactly that construction, the sampled patch agrees to 1.8 × 10⁻⁵ of its own width — which is the sampling’s floor rather than the geometry’s, since both shapes are point clouds and their support functions differ by the chord a rim sample cuts off.

Why they commute at one depth #

Because neither integral’s kernel depends on the other’s variable.

The pupil integral’s kernel is a disc whose radius is R|1 − v₀/v(Z)|, a function of depth alone. The exposure integral’s kernel is a segment whose direction and length come from the world velocity projected at that depth. Hold Z fixed and the disc is the same at every instant, so the whole patch is the segment convolved with one disc.

That is a real and useful statement rather than a triviality: it says the two ways a photographer loses sharpness are separable for a subject at constant depth, so the familiar rule that a fast shutter and a small aperture are independent choices is exactly right for a runner crossing the frame and exactly wrong for one running toward the camera.

Where they stop commuting #

Let the point recede from 1.3 metres to nine over the same exposure, focused at four.

The disc’s radius at the two ends of the streak is 0.958 and 0.256 millimetres — a factor of 3.7 — so the patch is a taper rather than a stadium, and it departs from the best single-kernel construction by 4.1 per cent of its own width, which is 16.4 pixels.

Sixteen pixels is not a subtlety. It is the difference between a smear that ends abruptly and one that fades, which is exactly the visual signature of a subject moving in depth, and it is why a car approaching the camera does not blur like a car crossing it — the same distinction stepping closer is not zooming makes about a camera rather than a subject.

The taper is a bow-tie, and its waist is computable #

The arrangement above deserves one more reading, because its own numbers say something the word taper does not.

The point recedes from 1.3 m to 9 m while the lens is focused at 4 — so it passes through the focus distance during the exposure. The disc’s radius is $K|1/Z - 1/Z_{f}|$, which is zero there exactly. The patch is therefore not a wedge that narrows; it is a bow-tie, with a genuine point at the middle and two straight-sided halves.

Both the radius and the image position are affine in $1/Z$ — the radius by the inverse-depth form, the position because a receding point’s image goes as $fX/Z$ — so the patch’s outline is two straight lines meeting at a vertex. Not approximately: for uniform motion in depth the boundary has no curvature at all, and a curved boundary would mean the motion was not uniform.

The vertex’s position is a pair of numbers, and they disagree, which is the interesting part.

Along the streak it sits at 79%. The fraction is $(1/Z_{1} - 1/Z_{f})/(1/Z_{1} - 1/Z_{2}) = 0.519/0.658$, so the waist is nearly four fifths of the way toward the far end in the picture.

In time it sits at 35%. The fraction is $(Z_{f}-Z_{1})/(Z_{2}-Z_{1}) = 2.7/7.7$, so the subject reaches focus barely a third of the way through the exposure.

The two disagree because one axis is depth and the other is its reciprocal, which is the same mismatch every quantity in this row turns on. And the ratio of the two end radii, 0.958 to 0.256, is $|1/Z_{1}-1/Z_{f}|$ against $|1/Z_{2}-1/Z_{f}|$ — 3.74 against the measured 3.74.

That makes the case a sharper counter-example than a taper would be. A tapering patch might be mistaken for a stadium blurred unevenly; a patch with an exact zero in the middle of it cannot be produced by convolving anything with anything, because a convolution of two things with support cannot have a hole. So the arrangement this essay already measures rules out the single-kernel model by a topological argument as well as by a 4.1 per cent departure — and the 4.1 per cent, which is a comparison against the best single kernel, is the mild way of saying it.

What the failure is and is not #

It is worth separating this from the previous rung’s failure, because both are called “blur is not a convolution” and they are different statements.

This one is a shift-variance. The kernel varies along the streak, so the patch is not a convolution of the sharp image with any single kernel — but it is a superposition of kernels, each one valid at its own instant, and a model that integrates over the exposure with a depth-dependent kernel reproduces it exactly. The information is all present; the model was too small.

The previous one is an impossibility. A pupil sees around an edge shows two scenes with identical pinhole pictures and 42 per cent different pupil pictures, so no model of any size that takes the pinhole picture as its input can succeed. The information is absent.

So this essay’s finding is a warning about model class and the previous one’s is a proof about information, and only the second is irreparable.

The exposure and the aperture are the same trade twice #

A frame is an interval names this and does not compute it, so it is worth computing.

Both integrals are over a set of viewpoints: the exposure integrates over where the camera and the subject were during an interval, and the aperture over where on the pupil a ray passed. Both produce a patch rather than a point. Both are traded against light: a shorter exposure and a smaller aperture each cost the same factor in exposure and each buy a smaller patch, which is the trade the sharp band is a decision spends on one side and this one on both.

Where the analogy stops is the one place worth knowing. The aperture’s patch is centred on the pinhole mark exactly, for any pupil symmetric about the axis. The exposure’s patch is centred on the mark at the middle of the exposure only if the motion is uniform; accelerate, and the streak’s density is uneven and its centroid moves off the midpoint’s mark.

So of the two, only the aperture comes with the exactness guarantee, and a measurement made on a streak of an accelerating subject carries a bias that a measurement made on a defocused disc does not.

What a renderer does with this #

A real-time renderer computes motion blur and depth of field as two separate post-process passes, each reading a per-pixel buffer, and composites them.

The stadium result says that is exactly right when nothing moves in depth: two separable convolutions compose into the correct patch, and the order does not matter. The taper result says it is wrong by a computable amount when things do, and the amount grows with the depth excursion during one frame — which for a fast subject at close range is large.

And the previous rung’s result says both passes fail at every occluding edge for a reason neither is equipped to fix, which is the subject of one depth per sample is not enough.

The correct method is the one that evaluates the integral: sample the scene at many instants and many pupil positions and average. That is distributed ray tracing, it is exact in the limit, and its cost is the reason nobody does it in real time — the information a single depth buffer does not have is exactly the information those samples are gathering.

What it means for reading a photograph backwards #

A blurred photograph carries information about the motion and about the depths, and the two integrals decide how much.

A stadium says one thing. Its length gives the world displacement over the exposure divided by the depth, and its width gives the defocus at that depth. Two numbers from one patch, and — as a frame is an interval notes about speed and time — neither separates into its factors from a single frame.

A taper says more. The two end radii give the defocus at the beginning and the end of the exposure, so the depth excursion is readable off a single blurred blob, which is a genuinely surprising amount of information from one frame. The ratio here is 3.7, and converting it back through the disc formula gives the two depths up to the usual scale ambiguity — the one the one thing a single view cannot give is about, arriving here as well.

That is a small recovery and it is real, and it is the kind of thing this collection likes: a quantity that looks like an artefact turning out to be a measurement, once the geometry that produced it is written down.

Which order the two integrals are taken in #

A question that sounds like a technicality and is not, because it decides whether a two-pass renderer can be correct in principle.

Both integrals are linear and their variables are independent, so for a fixed scene they commute in the strict sense: integrating over the pupil first and then over time gives the same answer as the reverse. That is why the stadium comes out however it is computed, and it is the licence a two-pass renderer is implicitly claiming.

What breaks is not the order but the factorisation. The joint kernel is not a product of a time kernel and a pupil kernel unless the depth is constant, so no sequence of two independent one-dimensional passes reproduces the taper, in either order. Composing two correct passes gives a correct answer only when the thing being composed is a product, and here it is not.

That distinction is worth having because the usual objection to a two-pass approach is stated as an ordering problem, and fixing the order fixes nothing. The problem is the shape of the kernel, and the only remedies are to integrate jointly or to accept the departure.

The measurement’s own floor #

Stated because the first number in this essay is a comparison of two point clouds and could be misread as an exact identity.

Both shapes are sampled: forty-one instants along the streak, forty-eight points around each rim. Their support functions therefore differ by the chord a rim sample cuts off, which is a few parts in a hundred thousand of the patch’s width. That is the 1.8 × 10⁻⁵, and it is the sampling rather than the geometry.

The claim is made in the form that survives the floor: the taper’s departure is a hundred times larger than the stadium’s, so the difference between the two cases is not a difference in how carefully they were sampled. Quoting a tolerance without saying where it comes from is the thing this collection’s gates exist to prevent, and a support-function comparison of two clouds is exactly the place it would have been easy to do.

What a photographer chooses between #

Turning the result into advice takes one observation: the two integrals are traded against the same resource, and the patch they make is not the same shape.

Crossing motion. The streak is long and the disc is whatever the aperture makes it. Opening up shortens the exposure and lengthens nothing, so a wide aperture is straightforwardly the right choice: the stadium’s length falls and its width rises, and length is what a viewer reads as smear.

Motion in depth. The streak is short — a subject approaching the camera barely moves across the frame — and the defocus changes during the exposure, so the patch tapers even at a short exposure. Stopping down shrinks the taper as well as the disc, and opening up makes the far end of the streak sharp and the near end soft, which is the look of a subject arriving.

And a subject at the focus distance moving in depth through it. The patch has a waist: the radius falls to zero in the middle of the exposure and rises again. Nothing about a single kernel produces that, and it is the strongest visual signature of the failure this essay measures.

That last case is the one to look for in a photograph, because it is unmistakable once named and there is no other geometry that makes it.

Where the joint patch is drawn from #

A note on the figure, because what it shows is a set rather than an image and the two are easy to confuse.

The outline drawn is the convex hull of every sampled landing position — the patch’s support, which is where light arrives at all. The faint discs behind it are the individual instants’ patches, and the line through their centres is the streak of chief rays, which is the pinhole camera’s own answer.

What the figure does not show is the density: how much light lands where inside the support. That is a genuine part of the answer and it is not a shape, so a picture of it would be a picture of a brightness rather than of a geometry, and this collection draws geometry. The density is uniform along the streak for uniform motion and it is not uniform across the disc once the pupil is clipped, so a full treatment would need both the previous rung’s vignetting and a photometric model.

Naming the omission is the point. The support is a geometric object and it is computed exactly; the density is optics, and stating which half is drawn is the difference between a figure and a claim.

Why forty-one instants and forty-eight rim points #

Both counts are chosen against the shape being measured rather than picked, and saying so is the difference between a floor and a tolerance.

Forty-one instants put about two samples per pixel along the longest streak in the sweep, so the support’s outline is resolved rather than polygonised. Forty-eight rim points put the chord a sample cuts off below a part in a hundred thousand of the patch’s width, which is exactly the 1.8 × 10⁻⁵ the stadium result is quoted against.

Coarser sampling would make the stadium’s agreement look worse and leave the taper’s departure unchanged, which is the direction that matters: the claim is a ratio between the two, and it survives any sampling fine enough to resolve either.

The short version #

A frame is an integral over the pupil and over the exposure. For a point at constant depth the two factorise exactly: the patch is the streak of centres with one disc slid along it, agreeing to 1.8 × 10⁻⁵ of its own width, which is the sampling’s floor.

For a point that moves in depth they do not. The disc’s radius falls by 3.7 along the streak here, and the patch departs from any single kernel’s stadium by 16.4 pixels — a shift-variance rather than an impossibility, so a model that integrates properly reproduces it and a model with one kernel does not. The taper is also a measurement: two end radii give two depths from one blurred blob.