Two blurs before the eye's own

This collection has modelled the eye’s own blur for several rounds — its optics, its mosaic, its contrast sensitivity. It has never modelled the blur that happens before the light leaves the object.

The claim #

A translucent object’s own edge blur is coarser than the eye’s resolution limit at ordinary viewing distances, on every material tested.

• The eye’s finest resolvable detail is about 1.2 arcminutes, from an achromatic cutoff near 50 cycles per degree.
• Coated paper’s own edge blur subtends 2.53 arcminutes at 300 millimetres — twice the limit, and it is the sharpest material in the table.
• Skin subtends 12.8 arcminutes at the same distance; marble 67; wax 121.
• So the softness is the object’s, and no amount of optical quality in the eye or in a camera removes it.
• And the distance at which it becomes invisible is about 2,900 times the blur width, which is 63 centimetres for paper and seventeen metres for marble.

Two convolutions in series #

Light from a scene reaches the retina through a chain of blurs, and this collection has measured most of them. The cornea and lens have a point spread; the pupil diffracts; longitudinal chromatic aberration puts one wavelength in focus and the others not; the cone mosaic samples; and the visual system’s own contrast sensitivity attenuates high frequencies, chromatic ones far more than achromatic.

Every one of those happens after the light has left the object. Before any of them, a translucent object applies a convolution of its own: light entering at one place leaves at another, so what the object emits is already the illumination convolved with the material’s kernel.

The two compose in the ordinary way — a convolution followed by a convolution — and whichever is wider dominates. The question is therefore quantitative, and it has an answer that depends on distance, because the object’s blur is fixed in millimetres and the eye’s is fixed in angle.

The comparison, in arcminutes #

The object’s blur width is the distance over which its edge response runs from a tenth to nine tenths of the far-field value: 0.22 millimetres on coated paper, 0.51 on a wall emulsion, 0.96 on a pigmented plastic, 1.11 on skin, 5.83 on marble, 10.52 on wax.

Converted to angle at three ordinary distances:

The eye’s achromatic cutoff of about 50 cycles per degree corresponds to a finest resolvable period of 1.2 arcminutes. Every entry in the first column is above it, and the sharpest material in the table is above it by a factor of two.

So at reading distance the object’s blur is what limits the sharpness of a translucent edge, not the eye’s.

The chromatic answer is different #

The eye has three spatial channels and they do not share a cutoff. The achromatic one runs to about 50 cycles per degree; the red–green channel to 12; the blue–yellow to 8, which is a finest period of 7.5 arcminutes.

That changes the comparison materially. A coated paper’s edge at 300 millimetres, at 2.53 arcminutes, is comfortably resolvable achromatically and far below the chromatic channels’ limits — so its lightness edge is visible and any colour fringe at that edge is not.

For skin at the same distance, at 12.8 arcminutes, both channels resolve it. And the object’s blur is spectrally selective — longer wavelengths travel further — so a translucent edge is not a grey blur but a coloured one, red-shifted on the dark side.

That is exactly why a hand held over a torch has a red rim, and why the shadow edge on a face is warmer than the lit skin beside it. The chromatic part of the object’s blur is real, is larger than the eye’s chromatic resolution on most translucent materials, and has no counterpart in an opaque one.

The distance at which it stops mattering #

Every material has a distance beyond which its own blur is finer than the eye can resolve, and it is a simple ratio.

An edge width w subtends 1.2 arcminutes — the eye’s finest resolvable period — at a distance of about 2,900 times w. So coated paper’s blur vanishes beyond 63 centimetres, a wall emulsion’s beyond 1.5 metres, a pigmented plastic’s beyond 2.8, skin’s beyond 3.2, marble’s beyond 17 and candle wax’s beyond 30.

Those numbers explain a set of ordinary observations without needing anything else. A printed page held at reading distance has a visibly soft edge and the same page across a room does not. A face is soft at conversational distance and sharp in a crowd. A marble surface never looks sharp at any distance a person stands from a wall, and a candle never looks sharp at all.

They also say something about reproduction. A photograph of a face displayed at a size and distance that put the subject’s blur below the observer’s limit has thrown the cue away — the image is sharper than the material, which is one of the several reasons a large print of a portrait reads differently from the sitter.

What a rendering has to get right #

The composition also decides what a renderer is obliged to compute.

If the object’s blur is finer than a pixel at the rendered size, it can be ignored: the display’s own sampling has already thrown it away. If it is coarser — which for skin at any reasonable rendered size it is — then leaving it out produces exactly the failure computer graphics spent the 1990s with. Rendered faces looked like plastic because plastic is what an opaque surface with skin’s reflectance looks like, and the reflectance was right.

That is a useful test of whether a material model is needed at all, and it has nothing to do with how important the material is. It is a comparison between a length in the object and a length on the display, and either one can make the other irrelevant.

What was computed, and how #

The edge response is the diffusion kernel convolved with a half-plane of light, which reduces exactly to a one-dimensional radial integral against an arc-cosine share.

The width is a stated convention — from a tenth to nine tenths — chosen because that is how a modulation transfer measurement is usually quoted. It is not a natural quantity: the profile has no edges and any width is a definition. What is not a convention is the ordering, which is the same under any sensible definition.

The angular conversion is elementary trigonometry, and the eye’s limits come from this collection’s own contrast sensitivity model rather than from a quoted number, which matters because the achromatic cutoff is a fitted parameter of that model and the chromatic ones are fitted separately.

The one thing not computed is what the two blurs look like in series, and that is deliberate. Composing them properly needs the object’s kernel expressed in angle at a stated distance, resampled onto the visual model’s own grid, and the resampling introduces a third blur that would have to be shown not to matter. It is an afternoon’s work and it is not done.

Where the model stops #

The comparison is between a blur width and a resolution limit, which is not the same as a visibility calculation. Whether an edge looks soft depends on its contrast as well as its width, and a very low-contrast edge that is wide can be less visible than a sharp one that is faint. The right calculation puts the object’s edge profile through the contrast sensitivity function and asks whether the difference from a step exceeds threshold.

The kernel is at one wavelength. The table’s widths are computed at 550 nanometres, and the spectral spread is what makes the blur coloured; a full treatment would give three widths per material, one per opponent channel — and the chromatic channels stop far earlier than the achromatic one, so the three would not be compared against one limit.

And the illumination is a step. Real scenes have gradients, and a gradient wider than the kernel is unaffected by it — which is why translucency is invisible on a uniformly-lit flat sample and obvious at an edge.

The generalisation #

The useful observation is that a perceptual system’s resolution limit is only interesting relative to what arrives at it, and what arrives has usually been through something first.

Vision science measures acuity with gratings on displays, where the stimulus is as sharp as the display allows and the eye is the only blur in the chain. That is the right experiment for measuring the eye. It is the wrong basis for concluding anything about what limits the sharpness of ordinary scenes, because ordinary scenes contain objects that blur themselves.

This collection has made the same observation in a different variable: every threshold in the literature was measured with a grating, and a threshold measured on one kind of stimulus is a statement about that stimulus as much as about the observer. The same caution applies to an ellipse measured on one kind of field and to a name measured in one room. Here the stimulus is a real object and the object is doing part of the filtering.

There is a design consequence. Anybody choosing how sharply to render, print or display a translucent subject is choosing a number that the subject has already chosen — and a rendering sharper than the material is not more faithful, it is wrong in a way that reads as plastic.

Two more materials say how much of this is a property of skin in particular, which is the question a photographer would ask first.

The instrument meets the same kernel from the other side, and what it recovers of a sample is the part of that curve its aperture happens to contain.

The one case where the eye wins #

There is a material in the table where the eye is the limiting blur rather than the object, and it is worth naming because it is the ordinary case.

Coated paper at a metre subtends 0.76 arcminutes, which is below the eye’s 1.2. So a printed page across a desk is limited by vision, and every intuition about print sharpness formed at that distance is an intuition about eyes.

Bring the same page to reading distance and the ordering reverses. That is not a large change in the world — half a metre — and it moves the limiting term from one side of the chain to the other. Any statement of the form this is as sharp as it can be is therefore a statement about a distance, and usually the distance is not given.

The same reversal happens with a halftone screen, where a ruling that is resolvable at one distance and not at another produces two different lightnesses from one sheet — and this collection measured that in a previous round. The two are the same arithmetic in different variables: a length in the object against an angle in the observer, with a distance deciding which wins.

What the numbers do not settle #

The comparison here is between two widths, and a width is not a visibility. Three things stand between the table above and a statement about what anybody sees.

Contrast. A wide low-contrast edge can be less visible than a narrow one with more contrast in it, and the object’s blur reduces contrast as it widens the transition. The right computation puts the edge profile through the contrast sensitivity function and asks whether the result differs from a step by more than threshold.

The tail. The object’s kernel falls exponentially and then more slowly; the eye’s falls faster in its core and has a wide veiling component. Two profiles with equal nominal widths can look quite different, and the composite is dominated by whichever tail is heavier.

And the scene. A blur is only visible where there is an edge to blur, so a uniformly-lit flat sample shows none of this. Translucency is invisible on exactly the arrangement a laboratory measures and obvious on the arrangement a person looks at, which is the same inversion the aperture produces from the other direction.

Three distances, not one #

The distance at which it becomes invisible is about 2,900 times the blur width is exact — the constant is 2,865, being 3,437.75 arcminutes in a radian divided by the eye’s 1.2 — and the whole table of distances reproduces from it to the millimetre. What the single constant hides is that there are three of them, one per channel, and the essay’s own chromatic section needs the other two.

Dividing the same 3,437.75 by each channel’s finest resolvable period — 1.2 arcminutes achromatic, 5.0 red–green, 7.5 blue–yellow — gives three multipliers: 2,865, 688 and 458. So each material has three distances:

The colour goes before the softness, and the ratio is fixed: a translucent edge stops looking coloured at a quarter of the distance at which it stops looking soft in red–green, and at a sixth in blue–yellow. Those two numbers are 5.0/1.2 and 7.5/1.2, so they are properties of the eye and hold for every material at once.

That is the quantitative form of what the essay says qualitatively about a hand over a torch and a shadow edge on a face. Skin’s warm shadow rim is visible to about 0.76 metres and its softness to 3.2 — so at conversational distance a face is soft and not visibly warm at its shadow edges, and at arm’s length it is both. The transition happens between the two distances people habitually use.

Which materials clear which limit at reading distance #

The chromatic comparison is made for two materials and the table settles it for all six.

At 300 millimetres the red–green channel resolves anything above 5.0 arcminutes and the blue–yellow anything above 7.5. Coated paper at 2.52 clears neither, which is the essay’s own example: its edge is visible achromatically and carries no visible fringe. Wall emulsion at 5.84 clears red–green and not blue–yellow — a case the essay does not have, and the only one in the table where the two chromatic channels disagree. The other four clear both.

So at reading distance five of six materials show a coloured edge and one does not, and the boundary falls between coated paper and a wall emulsion — which is to say between the two materials nobody would call translucent. Everything an observer would describe as translucent is above both chromatic limits at arm’s length, and the interesting cases are the two that are not.

The constant is the whole content #

One consequence of the arithmetic being this simple is worth stating, because it is what makes the comparison portable.

Every number in this essay is 3,437.75 × w / d compared against one of three thresholds. There is no model of the eye in it beyond three cutoff frequencies, and no model of the material beyond one width. So the table can be recomputed for any material whose edge width is known and any distance somebody cares about, without any of the machinery either half was built with.

That is also the honest limit of it. A width against a threshold is a comparison of two numbers, and the essay’s own list of what stands between it and a visibility statement — contrast, the tail, the scene — is a list of everything the two numbers do not carry. The multipliers 2,865, 688 and 458 are exact and they are exact about a proxy.

Who found it, and when #

Painters have known it for a very long time and have a technique for it. The reason a face painted in opaque colour looks like a mask is that skin’s own blur has been left out, and the remedy — thin translucent layers over a lighter ground — reproduces the kernel physically rather than depicting it. Italian painters were doing that deliberately by the fifteenth century.

Computer graphics rediscovered it in the same way and much faster. Rendered faces before subsurface scattering looked like plastic and everybody knew it; Jensen and colleagues’ 2001 model is the point where they stopped, and the reception of that paper is a good demonstration that the effect is not subtle.

Vision science has the measurement side and does not usually connect it. The eye’s spatial cutoffs are well established and are measured on displays, where the object contributes nothing.

Why the chain’s order does not matter and its widths do #

Two convolutions applied in series commute, so it makes no difference to the result whether the object blurs first and the eye second. What matters is only the two widths, and specifically the larger of them.

That is a useful simplification and it hides one real subtlety: the two blurs are not the same kind of function. The object’s kernel has an exponential tail that falls slowly; the eye’s is closer to a Gaussian core with a wide veiling component. Two functions with the same nominal width but different tails compose differently, and the composite’s tail is dominated by whichever tail falls more slowly — which here is the object’s.

So the quantity that decides how a translucent edge looks is not the width at all but the tail, and a comparison of widths is a first approximation to a comparison this round has not made. The ordering it gives is right; the composite profile it implies is not.

Where the ladder goes next #

The composition is the obvious next step and it is a real computation rather than a gesture: put the object’s edge profile through this collection’s own spatial filter at a stated viewing distance, and report the visibility of the difference between a translucent edge and a sharp one.

That would answer the question this essay only frames — at what distance does a material stop looking translucent — and the answer would be a number per material, in metres, which is exactly the kind of quantity a specification for a rendered or printed reproduction would want.