Concept

Model complexity — where it appears

How many numbers a model needs before it can produce an answer. The count is only useful once its parts are separated: numbers settled once in advance behave quite differently from numbers that must be estimated afresh from every scene, and a model needing quantities nobody can obtain is not a worse model but a different kind of object.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

What an observer is left with, by how much it is allowed to know about the room. Six ways of discounting a change of light, averaged over the fourteen changes in the adaptation census and 125 test surfaces each. The bar is what each leaves behind, on a logarithmic axis because the models span two orders of magnitude. The second line under each name is the count that matters: how many numbers about this room the model has to be given. Doing nothing leaves 15.7 ΔE₀₀. A single gain read off the two whites' luminances leaves 15.3. A matrix fitted across half the census and then applied everywhere, knowing nothing about the room at all, leaves 12.5. The published von Kries gain, which is told the white and nothing else, leaves 1.312 — and bolting a fixed correction onto it, at no cost in scene information, leaves 1.368, which is very slightly worse. The exact matrix leaves nothing and is not on the chart: its nine numbers are the change of light, which is the quantity being discounted.

Three numbers the scene supplies

An adaptation model's parameters are not all the same kind of thing. Some are numbers an observer must estimate from the room it is standing in; others could have been settled once by evolution. Counting them separately turns the diagonal gain from a crude approximation into the only model of the set that gets a large answer from information the observer can actually have.

scene · Scene $How much of the residual a partial correction removes. Between the diagonal gain and the exact matrix there is a line: apply the correction that would make a row exact, but only a fraction of it. The horizontal axis is that fraction and the vertical is the share of the row's residual it removes, for all fourteen census rows. The straight diagonal is where a correction worth exactly its fraction would fall, and in the published unit every curve lies on it to within 2.2 percentage points. The lower band of curves is the same interpolation measured in CAM16-UCS, which departs by up to 17 points — because its distance is a power of the Euclidean one and a power is not homogeneous along a ray, where every ordinary norm is. The straight line is therefore a property of the ruler rather than of the correction, and the exception is what says so.$

A partial correction is worth its fraction

Between a diagonal gain and the exact matrix there is a line, and a bounded observer's natural hope is that the first part of it is worth a disproportionate share. It is not. On all fourteen changes of light, at every setting, the share of the residual removed matches the share of the correction applied to within 2.2 percentage points — which closes the last way the gap could have been cheap.

scene · Scene

A correction an observer could have been born with, fitted on half the census and tested on the other. The same six models, each scored twice: on the seven census rows the fixed matrices were fitted to, and on the seven they were not. The split alternates by position so both halves contain daylight changes and discharge lamps. The upper bar is in sample and the lower is out, on a logarithmic axis. For the four models with nothing fitted the two bars differ only because the halves are different questions. For the two fitted ones the gap is the finding, and it is largest where it matters least: bolting a fixed correction onto the von Kries gain takes it from 1.2724 to 1.2592 on the rows it was fitted to, and from 1.3511 to 1.3679 — worse — on the rows it was not. There is no correction to the diagonal that an observer could arrive with.

A model is a claim about what can be known

The exact answer to chromatic adaptation is nine numbers, and the nine numbers are the change of light itself. A model whose parameters are quantities the observer cannot obtain is not a worse model of the same thing — it is a model of something else, and counting parameters without asking where they come from hides the difference.

scene · Scene

The glossiest finish the solver can report, against what it costs to report it. Six quadratures, each with the roughness at which its answer stops being stable, on logarithmic axes. The line is a fit and its slope is -0.350: the reachable roughness falls as the cost to the power of about a third, so reaching a finish twice as glossy costs about 7 times the work. The solver used here sits at 108 directions and reports down to a roughness of 0.145, which is where its own note put the boundary by inspection.

The boundary belongs to the quadrature

The directional solver stops at a roughness of about 0.15, and below that its answers are not imprecise but unphysical. The boundary is where the lobe stops being resolved by the sampling, so it belongs to the discretisation rather than to the room — and moving it is a purchase. Measured across six quadratures the reachable roughness falls as the cost to the power of a third, so a finish twice as glossy costs seven times the work and a polished varnish costs two hundred and thirty-six times.

scene · Scene

Named alongside it

The objects these essays reach for when they reach for this one.

Chromatic adaptation Matrix Residual The von Kries transform Basis Held-out Illuminant estimation Test set Bidirectional reflectance Colour difference Constancy Convergence

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