A pixel has an aperture too

A camera and a spectrophotometer measure the same surface through the same two apertures. They differ in which one is wide, and the difference decides where their errors go.

The claim #

A camera’s flat-field colour on a translucent surface is exact, and all of its aperture error is at edges.

• Every measurement has two discs, the lit one and the looked-at one, and widening either recovers the whole kernel.
• A camera’s lit disc is the scene. A face under a window is illuminated over its whole area, so the light-side factor of the lateral departure is zero.
• So magnification does not matter in the middle of a uniform region. A pixel covering a tenth of a millimetre and one covering ten give the same colour.
• The error is entirely in the edges, where the illumination stops being uniform — a shadow boundary, an object’s silhouette, a printed mark on a translucent card.
• And the width of that error is the sample’s, not the lens’s. It is a fraction of a millimetre on paper and five millimetres on marble, which at ordinary viewing distances is larger than the eye’s own blur.

The two arrangements, side by side #

A spectrophotometer lights a disc a few millimetres across and looks at a disc a few millimetres across, and both are small compared with a translucent sample’s kernel. That is the worst case: the shared-area weight falls quickly with separation, and the reading comes out low and desaturated.

A camera lights nothing at all — the scene’s own illumination does that — and looks at a disc the size of a pixel’s footprint, which may be a hundredth of a millimetre. Naively that sounds far worse, and it is exactly the opposite, because what matters is the pair. With the lit region effectively infinite, every point of the measured region receives the whole plane’s worth of returning light, and the reading is the kernel’s total whatever the pixel’s size.

The arithmetic is the symmetry from the aperture essay used in the other direction. Both discs at two millimetres recovers 51.75 per cent of a translucent slab’s reflectance; a lit disc of four hundred millimetres with a two-millimetre measured one recovers 100.00 per cent; and shrinking the measured disc further changes nothing, because it was never the binding constraint.

A camera therefore gets the colour of a flat translucent surface right, and gets it right for a reason that has nothing to do with its own quality.

Where the error goes instead #

Uniform illumination is a property of a region, not of a scene. Every scene has boundaries — a shadow, an edge, a mark — and at a boundary the light-side factor stops being zero.

What happens there is the kernel convolved with a step rather than with a disc. Inside the lit region near the edge, the reading is low, because some of the neighbourhood that would have supplied returning light is dark. Outside it, the reading is above zero, because light is arriving from under the lit side. The transition runs over a distance that is a property of the material: 0.22 millimetres on coated paper, 0.96 on a pigmented plastic, 1.11 on skin, 5.83 on marble, 10.52 on wax.

So a photograph of a translucent object has a blur that is not the lens’s. It is in the object, it is there before any light reaches the camera, and no amount of optical quality removes it. Stopping down, focusing better, or using a larger sensor changes the lens’s contribution and leaves this one exactly where it was.

That is a different statement from the ordinary one about a camera’s own point spread function, and the two compose: the sensor has a footprint and the object has a kernel, and what reaches a pixel is both convolutions applied in series.

What that means for a printed translucent thing #

The practical case is anything printed on material light passes through: a laminated card, a plastic label, a thin paper, a photographic print on a translucent base.

A dark mark on such a substrate has ink over it and clear substrate around it. Light entering the substrate beside the mark travels sideways and leaves under it, so the mark is lighter than its ink would suggest — and the effect is largest for the smallest marks, because a small mark has more perimeter for its area.

This collection has met that arithmetic before under a different name. Optical dot gain is exactly this: light entering the paper beside a halftone dot and emerging under it, making the dot print darker than its area. The Yule–Nielsen exponent that the printing industry fits is a one-parameter summary of the same kernel this round computes from coefficients.

So the two are the same physics with the sign of the mark reversed — a dark dot on light paper gains, a light mark on dark ink loses — and the exponent is what a whole industry uses in place of a diffusion length. A fitted exponent and a computed kernel are the same object at different resolutions, and the exponent’s well-known dependence on ruling, paper and ink is what a one-parameter summary of a two-parameter material looks like.

The two errors are one number #

An instrument’s aperture departure and a camera’s edge width are the same material fact seen through two arrangements, and the correspondence can be checked rather than asserted.

Across the six materials the edge width and the departure a four-millimetre aperture costs rank together at exactly 1.0000. Coated paper is smallest on both, wax is largest on both, and every material between holds its place. The relation is close to proportional as well: fitted between the extremes the exponent is 0.905, and the ratio of departure to width sits between 1.67 and 2.41 on five of the six.

Skin is the sixth, at 5.24, and the exception is the informative part. Its edge is about a millimetre wide, comparable with opal plastic’s, and its aperture departure is three times the plastic’s. The difference is not how far the light wanders but what happens to it while it wanders: skin absorbs with strong structure in the band haemoglobin works in, so a given amount of lateral transport costs far more colour than the same transport through a spectrally flat scatterer.

So the correspondence between the two instruments is a good rule with one exception, and the exception separates the two things a kernel does. How wide the blur is, is transport. How much colour it costs, is transport multiplied by absorption — and a material can be ordinary on the first and extreme on the second.

That is worth carrying because it says which measurement to make. An edge in a photograph is easy to measure and reports the transport directly; the departure an instrument makes is what a specification gets written against, and needs the spectral half as well. On five materials in six the first predicts the second to within a factor of one and a half. On the sixth — the one anybody photographing people would care about — it under-predicts by a factor of three.

What was computed, and how #

The edge profile is the same kernel used everywhere in this round, convolved with a half-plane rather than with a disc. The convolution collapses to one dimension exactly: at radius r from the sample point, the fraction of the ring lying inside the lit half-plane is an arc-cosine of the distance divided by the radius, so the two-dimensional integral becomes a single radial one against that share.

The width is defined as the distance from a tenth of the far-field value to nine tenths, found by bisection on each side. That is a stated definition rather than a natural one — the profile has no edges, so any width is a convention — and it is chosen to match how a modulation-transfer measurement would quote it.

Two checks are worth naming. Far inside the lit region the profile has to return the kernel’s total, which is asserted to within one per cent; and far outside it has to return zero. Both are trivial and both would fail loudly if the arc-cosine share were wrong by a factor of two, which is the mistake that construction invites.

The three marginalisations a camera makes #

The lateral one is the interesting case because a camera gets it right. The other two are worth listing, because a camera makes all three and gets a different answer to each.

Over wavelength, a camera integrates against three broad filters rather than three colour-matching functions, so it is a fourth observer rather than a copy of the standard one, and its metamers are not the eye’s. That is the largest of its three departures and the one every camera profile exists to patch.

Over direction, a camera collects light over the solid angle its lens subtends, which is a few degrees — a much narrower window than a sphere’s hemisphere and a much wider one than a laser’s. So a photograph of a glossy object records a directional reflectance, and moving the camera changes it. That is not an error; it is what a photograph is, and it is why a rendered scene needs the whole bidirectional function rather than a colour.

Over place, it does what this essay describes, and it is the one of the three where the camera is the more faithful instrument.

Three marginalisations, three different outcomes: worse than an instrument, differently from an instrument, and better than an instrument. That is a useful corrective to the idea that a camera is a poor spectrophotometer — on one axis of three it is the better of the two, for reasons of geometry rather than of quality.

Where the model stops #

The camera’s exactness rests on a real assumption and it fails in a case worth naming.

The illumination has to be uniform over several diffusion lengths. A face under a window is; a face lit by a fibre-optic ring light a centimetre across is not, and the reading there would be the spectrophotometer’s problem rather than the camera’s. Macro photography of translucent subjects with small sources is exactly the arrangement that reproduces the instrument’s error in a camera.

And a pixel’s footprint is not a disc. It is a rectangle, convolved with the lens’s own point spread and with whatever anti-aliasing filter sits over the sensor. None of that changes the argument, because the argument does not depend on the measured region’s shape — only on the lit region being large.

The third limit is the substrate. Everything here is a semi-infinite slab; a thin translucent sheet held up to a window is being transilluminated, which is a different geometry with its own arithmetic and a much larger effect.

A two-millimetre aperture is the small end of what an instrument offers, and it is where the material matters most.

What a lens contributes, and why it is the smaller term #

It is worth putting a number beside the claim that the object’s blur dominates.

A good lens at a working aperture has a point spread function a few microns wide at the sensor, which for a subject at ordinary magnifications corresponds to a fraction of a millimetre on the subject itself. The sensor’s pixel pitch and its anti-aliasing filter add a comparable amount.

The object’s kernel is 0.22 millimetres on coated paper and 5.83 on marble, measured on the subject. So for anything more translucent than paper the object’s contribution is larger than the camera’s, and for marble it is larger by more than an order of magnitude.

That ordering has a practical consequence which photographers know as a rule of thumb without the arithmetic: a translucent subject cannot be made to look sharp, and trying makes it look wrong. Sharpening a photograph of skin or marble amplifies a blur that was in the object, and the result reads as a material that is not the material photographed.

The generalisation #

The useful shape is that two instruments measuring the same quantity can have complementary error structures, and knowing which is which is what says what each is good for.

An instrument that lights small and looks small is wrong everywhere by a bias. A camera that lights large and looks small is right everywhere except at boundaries. Those are not degrees of the same failure; they are different failures, and they call for different remedies — a second aperture for the first, and an edge-aware treatment for the second.

The same complementarity turns up wherever a measurement has a source and a detector that can be sized independently. It is worth asking, of any pair of instruments that disagree, whether they disagree uniformly or at features, because the answer names which factor of the pairing each of them has set to zero.

Skin is the material a camera photographs most often. The most translucent thing it is likely to meet is stone, and the same sweep on stone costs a great deal more.

What a camera pipeline assumes instead is a surface with no length in it at all — a reflectance per wavelength, applied at a point, with nothing arriving from anywhere else.

What it means for measuring a face #

Skin is the case where all of this arrives at once, and the practical conclusion is unusually clean.

A spectrophotometer on a cheek lights four millimetres and reads 5.83 ΔE₀₀ away from the skin’s own colour, in a direction that is darker and less chromatic — because haemoglobin’s absorption bands are narrow and melanin’s slope is smooth, so the diffusion length runs from a fifth of a millimetre in the blue to three and a half at the red end.

A camera photographing the same cheek under a window gets the flat colour right, and blurs the edges of everything on it — a mole, a vein, a line — by about a millimetre of the subject’s own optics.

So the two instruments disagree about a face in a way that has nothing to do with either one being badly calibrated, and the disagreement has a sign: the instrument reads darker and less saturated than the camera, always. Anybody comparing a measured skin tone with a photographed one is comparing two different quantities, and the difference is several times the tolerance a cosmetics or a medical application would work to.

What a studio light does to the argument #

The camera’s exactness depends on the illumination being wide compared with the subject’s kernel, and studio lighting is where that assumption is most often broken deliberately.

A softbox a metre across, a metre from a face, is enormous compared with skin’s millimetre of transport, so the condition holds comfortably. A snoot or a fibre-optic light a centimetre across is not, and a macro photograph of skin or of a translucent object under a small source reproduces the spectrophotometer’s problem in a camera — the lit region becomes comparable with the kernel, and the reading goes low and desaturated exactly as an instrument’s does.

That gives a rule with a number in it rather than a taste: the light’s apparent size at the subject should exceed the subject’s transport length by a wide margin, which for skin is any ordinary source and for marble is a window rather than a lamp. It is the same rule the standards give an instrument, arrived at from the other side, and it explains why small hard sources make translucent subjects look like something else.

Who found it, and when #

Optical dot gain was measured by Yule and Nielsen in 1951, and their exponent has been fitted rather than derived ever since. The tissue-optics literature arrived at the same kernel from the other direction in the 1980s, and computer graphics adopted it in 2001 because rendered faces looked like plastic without it.

The observation that a camera’s illumination geometry saves it from the instrument’s problem does not seem to be written down anywhere, probably because nobody has needed to say it: photographers do not compare their results with spectrophotometry, and colour scientists do not photograph their samples. The two communities have the same two discs and never put them on the same page.

What is written down, at length, is that a photograph is not a measurement — and this is a case where the photograph is the more faithful of the two, for one property, for a reason that has nothing to do with either instrument’s design intent.

Where the ladder goes next #

The obvious extension is quantitative: how large must a uniformly-lit region be, in diffusion lengths, before the middle of it is exact to a stated tolerance? The answer is a number a photographer could use — it is the size of the softbox, in millimetres, for a given subject.

The other open item is the one the printing industry has already half-solved. If a fitted exponent is a one-parameter summary of a two-parameter kernel, then the two parameters ought to be recoverable from a tone scale printed at two rulings — and if they are, an industry that has been fitting a fudge factor since 1951 could be measuring a material property instead.