The appearance model has no slot for it

An appearance model takes a stimulus and returns what it looks like. Hand it two objects that are unmistakably different to look at and whose stimuli are identical, and it returns one answer twice.

The claim #

CIECAM16 returns identical lightness, chroma and hue for a translucent object and an opaque one with the same reflectance, and the identity is exact.

• The two slabs’ correlates agree to six decimals: J 65.962397, C 6.486745, h 58.065908, twice.
• That is not a defect in the model. Its input is a stimulus and the two stimuli are the same, so any other answer would be inconsistent.
• Translucency is a recognised appearance attribute — the CIE has a technical committee on it — and no appearance model has a correlate for it.
• The reason is structural: the model has no space in it. Its inputs are a stimulus, a background and a surround, and all three are points.
• So it cannot distinguish a translucent object from an opaque one painted with the object’s own gradient, which is the sharpest form of the gap.

What an appearance model is a function of #

CIECAM16 takes the tristimulus values of a stimulus, the tristimulus values of the adapting white, a background luminance factor, an adapting luminance, and a surround. Out of those it computes lightness, brightness, colourfulness, chroma, saturation, hue and hue quadrature.

That is a great deal of context by the standards of colorimetry — matching is not appearance, and the whole point of the model is that a colour’s appearance depends on the situation. The surround alone moves the lightness exponent by thirty-one per cent.

But look at the shape of the inputs. Every one of them is a number or a triple. There is no image, no neighbourhood, no gradient, no distance. The model describes what one patch looks like in a stated context, and the context is described by four numbers.

A translucent object’s appearance is not a property of one patch. It is carried by how the light falls off across an edge, by whether a shadow on it is soft, by what happens when it is held up to a lamp. All of those are spatial, and the model has no place to put them.

The demonstration #

The pair is the one constructed for the aperture: two slabs whose bulk reflectance agrees at every wavelength to fifteen figures, built by inverting the kernel’s total band by band under a different scattering coefficient.

Their tristimulus values are therefore identical under every illuminant and every observer, so any function of tristimulus values returns the same answer for both. Through CIECAM16 under an average surround at 100 cd/m², both give J = 65.962397, C = 6.486745, h = 58.065908.

The identity is exact rather than close, and that exactness is the point: no tightening of the model, no better parameters, no extra correlate computed from the same inputs could separate them. The information that distinguishes the two objects is not in the model’s arguments.

The sharper form #

The stimulus can be made spatial without helping, and this is the version that shows the gap is structural.

Take a translucent slab beside a shadow edge, and compute what leaves it at each position: a soft transition several millimetres wide. Now take an opaque slab and paint it with exactly that gradient, and light it uniformly. The two emit identical light from every point.

An appearance model applied point by point gives identical correlates at every point, because the stimuli are identical at every point. A spatial vision model applied to the whole field gives identical answers too, for the same reason.

And the two do not look the same. One is a translucent object beside a shadow; the other is a painted panel. A viewer distinguishes them easily, by moving, by seeing the shadow’s own edge, by the way the transition changes when the light moves.

So the difference is not in the light arriving at all. It is in what the visual system infers about the cause of the light — which is scene interpretation rather than photometry, and is the part of vision no colorimetric model has ever tried to hold.

What the model says about the instrument’s readings #

There is a second, more practical thing the model does faithfully, and it is worth separating from the gap above.

The same stone read through a series of apertures gives a series of different tristimulus values, and CIECAM16 reports them correctly: lightness J = 51.69 at a two-millimetre radius, 62.51 at four, 70.31 at eight, 77.74 at forty, and 79.62 with no aperture at all. The chroma falls from 7.78 to 1.69 and then rises again to 6.09, with the hue quadrant changing on the way.

Every one of those is what the model should say about the stimulus it was given. None of them is what a viewer sees, because a viewer is not looking through a four-millimetre hole. A person looking at a piece of marble under a window satisfies the wide-illumination condition and sees the bulk colour, which is the last row.

That is a useful separation to be able to make. The model’s answers are right about the stimuli and the stimuli are wrong about the object, and no amount of work on the model repairs the second problem.

What was computed, and how #

The pair is constructed by bisection on the absorption coefficient, band by band, against the closed-form total of the dipole model — the same construction the matching essay uses, so the two agree by sharing code rather than by both being right.

The appearance correlates use this collection’s own CIECAM16 implementation, which is checked against the published test vector to four significant figures in all six correlates and round-trips through its own inverse to four parts in ten thousand million million.

The identity of the two correlate sets is not asserted as a discovery — it follows from the inputs being identical — and it is computed rather than argued because the exactness is the content. A pair matched to a tolerance would give correlates differing by something, and somebody would then ask whether the difference was meaningful.

The aperture series uses an average surround at an adapting luminance of 100 cd/m², which is a viewing booth. Nothing about the conclusion depends on that choice; a dim surround changes every J and leaves every equality.

Where the model stops #

This is an essay about where a model stops, so its own stopping points matter more than usual.

Nothing here measures what the two objects look like. The claim that a translucent slab and a painted panel look different is an appeal to ordinary experience, not a measurement, and this collection’s standing rule is that a claim about appearance needs a model or an experiment. There is no experiment here.

And the gap is not evidence that appearance models are badly built. They are built to answer a question — what does this stimulus look like in this context — and they answer it well. Asking them what an object looks like is a category change, and the honest description is that the question has never been posed in a form a model could take.

The model is not the blind part #

The two slabs give identical correlates because they give identical stimuli, and it is worth showing that the model separates them the moment they do not.

Hand it the two slabs as an instrument with a four-millimetre aperture reads them, rather than as their bulk reflectances, and the correlates come apart at once: lightness by 7.41, chroma by 3.69 and hue by 9.05 degrees, for a colour difference of 6.25. At one millimetre the lightness gap is 18.66, and at four tenths of a millimetre it is 23.53, with the two slabs 23.68 ΔE00 apart.

So the model has no difficulty at all with this pair. Nothing in CIECAM16 resists the distinction, no correlate saturates, and no assumption is strained. Every one of its inputs does exactly what it was designed to do, on two stimuli that genuinely differ.

That places the gap precisely. It is not that an appearance model cannot represent translucency. It is that by the time an appearance model is called, the object has already been reduced to a stimulus, and the reduction is where the information went. An instrument with a four-millimetre aperture makes that reduction one way and gets a difference of six units; an instrument with a wide one makes it another way and gets zero. The model sits downstream of both and reports whichever it is handed.

Which is why a translucency correlate cannot simply be added. A correlate is a function of the inputs; the inputs are four numbers; and those four numbers are identical for both slabs whenever the reduction was made at a wide enough aperture. Adding a fifth input is not adding a correlate — it is changing what the model is a function of, and what that fifth input would have to be is a kernel, which is a function rather than a number.

Why this is not the surround argument again #

The model does have context arguments, and it is worth being clear that the gap here is not the one they fill.

The surround is three constants describing how bright the field around the stimulus is, and it moves the lightness exponent by about a third between a cinema and a lit room — which is a large effect and is the model’s own subject. The background is a luminance factor and produces simultaneous contrast. The adapting luminance produces the Hunt effect.

Every one of those is a scalar summary of the surroundings, and each was added because a scalar summary turned out to predict something. The pattern suggests an obvious remedy for translucency — find the right scalar and add it — and the perceptual literature’s answer is that no such scalar has been found. Perceived translucency tracks image cues that are not summarisable in that way: how soft a shadow’s edge is, how bright a thin part is, where the highlights sit.

So this is not a missing parameter of the same kind as the surround. It is a missing argument of a different type, and adding a fourth number to a list of four numbers does not supply an image.

What a translucency correlate would have to be a function of #

Setting out the requirements makes the difficulty concrete.

It would have to be a function of an image region rather than of a patch, because the evidence is spatial. It would have to know the illumination geometry, because the same object under a diffuse sky and a point source gives different cues and looks differently translucent. And it would have to survive a change of viewing distance, because the object’s own blur is fixed in millimetres and the eye’s resolution is fixed in angle.

None of those is impossible and all three are outside what an appearance model’s signature admits. What could be built inside the existing signature is a discriminant rather than a correlate — a number saying whether two images differ enough in their edge structure to be told apart — and that is a question this collection’s spatial machinery could answer.

The generalisation #

The transferable point is about what a model’s argument list forecloses.

A model that takes a triple cannot distinguish two objects with the same triple, however different they are. That is not a limitation to be improved away; it is what the argument list means. The interesting move is to notice the foreclosure early, because it says what class of question the model can never answer, and that class is usually larger than it looks.

Colour has several. A model taking one stimulus cannot see simultaneous contrast without a background argument, which is why the background is in the list. A model taking a background but no image cannot see filling-in or an illusory contour. A model taking an image but no time cannot see an afterimage, which is why the model has no clock.

Translucency needs the argument that comes after all of those: not the light, and not the light’s arrangement, but the cause the arrangement implies. Material appearance — gloss, translucency, roughness, texture — is the collective name for what depends on that, and it is the current frontier of the subject rather than an oversight in it.

The instrument produces those stimuli by the same mechanism, and it is worth seeing the two halves of the object side by side before the model is asked about either.

What the model is not being told about has a shape and a scale, and both are measurable without leaving this generator.

Three millimetres is the aperture a bench instrument most often has, and it is where the shortfall is large enough to matter and small enough to be missed.

What this does not say about the model #

Two things could be read into this essay that are not in it, and both are worth closing.

It does not say the model is wrong. Every number CIECAM16 produced here is the correct answer to the question it was asked. Handed two identical stimuli it returned identical correlates, which is the only consistent thing to do. A model that returned different answers for identical inputs would be broken in a much more serious way.

And it does not say a correlate is impossible. It says that no correlate computable from the model’s current arguments can exist, which is a statement about the signature rather than about the world. A model with an image argument might well have one. Whether it would be a colour appearance model at that point is a question about naming rather than about arithmetic.

What the essay does say is that the gap is not closable by the usual route. Every previous extension of an appearance model has added a scalar to the context — a background, a surround, an adapting luminance — and every one of those was justified by a measurable effect on a patch. This one has no patch-level effect at all: the two objects’ stimuli are identical, so no experiment on patches could ever produce the data such an extension would be fitted to.

Who found it, and when #

Material appearance became a research programme in its own right in the 2000s, with Hunter’s older four attributes — colour, gloss, translucency, texture — turning from a list into a measurement problem. The CIE has a technical committee on translucency and a series of reports on the difficulty of measuring it, and the difficulty is exactly the one this round has been computing: the quantity depends on the geometry of the illumination and the size of the region.

The perceptual side has its own literature, and its central finding is that translucency judgements are driven by image cues — the softness of shadow boundaries, the brightness of thin parts, the way highlights sit — rather than by any physical parameter that could be read off a spectrum. Fleming and Bülthoff’s work in the mid-2000s is the standard reference and its conclusion is negative in a useful way: no single physical quantity predicts perceived translucency, so a correlate in the CIECAM sense may not exist to be found.

That is a stronger statement than nobody has computed it yet, and it is the reason this essay ends without proposing one.

The reduction happens earlier than the model #

It is worth locating exactly where the information about translucency is lost, because it is not lost in the appearance model at all.

A physical sample has a kernel. Taking its integral gives a reflectance, and that is the first reduction — the one this round is about. Multiplying by a light and integrating against three curves gives tristimulus values, and that is the second, the one metamerism is named for. The appearance model takes those three numbers and produces seven correlates, which adds nothing and loses nothing.

So the model inherits both reductions and performs neither. Blaming it for having no translucency correlate is like blaming a thermometer for not reporting pressure: it was handed three numbers and it used them.

The practical consequence is about where to intervene. Adding an argument to an appearance model would not help, because the argument would have to be filled from something upstream, and upstream is where the information was discarded. Anything that recovers translucency has to start before the reflectance, which means at a kernel, a geometry and an image — three things no colorimetric pipeline currently carries.

Where the ladder goes next #

The nearest reachable thing is not a correlate but a discriminant: given two rendered or photographed objects, compute whether their edge profiles differ by more than a visual threshold. That is a computation this collection could do, since it has both the kernel and a spatial filter with measured thresholds, and it would answer the question a reproduction actually asks — whether a printed or displayed version of a translucent object carries the cue at all.

The further thing is out of reach and worth naming as such. Whether an object looks translucent is a judgement about a cause, and this collection has no machinery for causes. It has spectra, observers, models and gates, and every one of them is a function of what arrived.