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The thread: Identical, and checked — page 2

"These two patches are the same colour" is the most repeated and least checkable sentence in perception, because the whole point is that it does not look true. Every instance here is checked by computation.

Nine eigenvalues, six of which exist. Nine points on a logarithmic vertical axis: the eigenvalues of the Hessian of the adaptation residual at its own optimum, largest to smallest. The first six run from 6.8×10² down to 7.6×10⁻¹, a condition number of 890. Then the axis drops: the seventh is 1.9×10⁻⁴, and the last three are separated from the sixth by a factor of 4.0×10³. Those three are not small curvatures. They are the finite-difference truncation error on directions along which the objective is exactly constant, and a shaded band marks them as the numbers the objective does not have.

What the eye does

The rank is the invariance

A von Kries gain cannot see the scale of a row of its basis. That is an identity, proved in a line, and it can be measured instead — as the rank of a second-derivative matrix. Both objectives this collection minimises over the observer's nine free numbers have a Hessian of rank exactly six, and the three directions they cannot see are the three scalings, to a hundredth of a degree.

Room is not safety: two orderings of the same three claims. Three pairs of bars, one pair per published statement about the confusion points. The upper bar in each pair is the margin — how far the measured number is from the threshold that makes the statement true, as a ratio. The lower bar is the headroom — the factor by which one declared width of the population model would have to be wrong for the statement to fail. Both start at one, which is the line. Ordered by margin the three read the protan margin, the tritan margin, the deutan margin; ordered by headroom they read the protan margin, the deutan margin, the tritan margin, and the middle two change places. Every one of the three is inside a factor of two of failing, which the margins do not say.

Where the model breaks

What would have to be wrong

A great many statements here have thresholds written into them, which turns out to make an audit possible — for each one, the smallest change in a declared input that would stop it holding. Most are unreachable. One is inside a factor of one and a third.

No one surface carries the answer, and the set is smaller than it looks. A falling bar chart of the 125 surfaces in the test set, ordered by how much each contributes to the published mean for daylight to tungsten. The tallest bar is 1.50 per cent of the total, so the mean is not a few awkward objects with a crowd behind them and a leave-one-out would move it by well under a per cent. The tail is the other half of the story: 5 surfaces contribute essentially nothing, because a flat grey is a surface an adaptation gain handles exactly. Counting the set by how evenly it contributes rather than by how many members it has gives 108.5 effective surfaces out of 125, which is what "a mean over a hundred and twenty-five surfaces" is really worth.

What the eye does

The surfaces that answer nothing

Five of the hundred and twenty-five test surfaces contribute exactly zero to every number the adaptation census reports — not approximately, exactly — and the reason is the one fact about von Kries adaptation that makes it worth having at all. Counting the set by how much it contributes gives about a hundred members rather than a hundred and twenty-five.

A confusion point is about the pigments that remain. Three groups of three bars. Each group is one dichromat's confusion point; each bar is how far that point moves in chromaticity when one of the three cone pigments has its absorption peak shifted by eight nanometres. In every group the bar for the pigment that dichromat is missing has length zero — exactly zero, to machine precision, not merely small. The protanope's point does not move when the L pigment moves, the deuteranope's does not move when the M pigment moves, and the tritanope's does not move when the S pigment moves. The reason is algebraic rather than physiological: a confusion point is the direction that excites only the missing cone, which is the null space of the other two receptors' rows, and rescaling a row does not move where the other two are zero. So the point at which a protanope's confusion lines meet is not a fact about the pigment a protanope lacks, which is why the claim about it restates in nanometres of the M pigment.

What the eye does

A point about the pigments that remain

The chromaticity at which a protanope's confusion lines meet does not move at all when the long-wave pigment moves — not slightly, exactly not at all. It moves a great deal when the medium-wave pigment does. A dichromat's confusion point is a fact about the two receptors they have rather than about the one they lack.

The secondary band, dialled from nothing to twice what Govardovskii published. Govardovskii's template has a second, smaller absorption band below 400 nm, published at 0.26 of the α-band's peak. It is the one coefficient in the whole template whose contribution is somewhere else in the spectrum than the peak, and it is the part of the template a reader is least likely to have heard of. Sweeping it from nothing to twice the published value moves the median observer's distance from the 1931 standard from 0.9354 to 0.9790 ΔE₀₀, monotonically upwards. That is a small effect — about a fiftieth of the residual — and its being monotone is the interesting part: there is no interior optimum, so nothing here recommends the published value over any other, and a coefficient whose measured value is not the one that best fits an unrelated agreement is a coefficient to leave where the measurement put it.

What the eye does

The band below four hundred

The pigment template every observer here is built from carries a second, smaller absorption band in the ultraviolet, published at 0.26 of the main one. Dialling it from nothing to twice that moves the median observer monotonically away from the standard one, with no interior optimum — which is what a physical constant looks like when nothing downstream is pulling it anywhere.

Two slabs with one reflectance, and two colours through an aperture. Two constructed media whose bulk reflectance agrees at every wavelength to fifteen figures, and whose diffusion lengths differ by a factor of four. The upper curve is that shared reflectance — both slabs lie on it exactly. The two patches on the right are what a 4 millimetre radius returns from each, and they are 6.3 ΔE₀₀ apart. This is a metamerism with no observer in it: the two samples are the same colour to anybody under any light, and the instrument separates them because it is measuring a kernel through a hole rather than measuring a reflectance.

Matching and measuring

A pair the aperture separates

Two constructed slabs with the same reflectance at every wavelength, to fifteen figures — the same colour to any observer under any light — and 6.25 ΔE₀₀ apart when measured through a four-millimetre aperture. It is a metamerism with no observer in it, no illuminant in it, and no spectral difference to construct it from.

The two tabulation choices over forty-two surfaces, under a tungsten lamp at 2856 K. Each column is one choice, measured over a family of forty-two analytic reflectances rather than on a single example: an absorption band of stated centre, width and depth. The four marks are the smallest, the median, the ninety-fifth percentile and the largest cost in ΔE₀₀, logarithmically. Under a smooth light the range is worth 6.3 times the step at the median, so a collection wanting one repair should widen its range rather than refine its step — and under a fluorescent tube the ranking reverses outright.

Difference and uniformity

A neutral has no grid

A perfectly flat reflectance computes to exactly the same colour on every wavelength grid, through every slit, at every origin, and for every observer — not nearly, but to the last bit of a floating-point number. The condition is an identity rather than a limit, and what makes it one is the white point.

How far this collection's analytic observer is from the tabulated one. The construction every figure in this family uses is three pigment absorptances through one fitted 3×3, and this is the residual of that fit against the CIE's 1931 functions: root-mean-square error as a percentage of each curve's peak, and the colour difference it produces over forty-two surfaces. The short-wavelength function is the worst at 16.4 per cent, which is where a pigment template is weakest and where the ocular media are doing most of the work. The median colour difference is 1.42 ΔE₀₀, so this is an observer of the right shape rather than a copy of the table — and every departure in this family should be read beside that number rather than against zero.

Where the model breaks

The reference had to be built

A five-nanometre error cannot be measured with five-nanometre data. Interpolating the tables and integrating finely measures the interpolator, not the grid — so the audit of this collection's index had to be run against an observer made of formulae, and the price of that is a residual of 1.42 ΔE₀₀ that every number in the section is read beside.

The conditions under which an observer's departure is exactly zero. A departure of the observer is the pairing of something belonging to the observer with something belonging to the stimulus, so emptying either factor empties the product. The axis is logarithmic in what is left when the condition is imposed. Six rows empty the stimulus's factor — a perfectly neutral sample is the same colour for every observer, at any age and any field size — and two empty the observer's, since a gain on each cone and a change of basis are both absorbed exactly. All eight are identities rather than small numbers. The last two are the same two conditions imposed in a published cone space rather than in the observer's own, and they are worth eight and thirteen units: the identity is about the eye, and the arithmetic everybody uses is in somebody else's coordinates.

What the eye does

A neutral is everyone's colour

Two eyes differing by fifty years of lens yellowing, by a factor of three in macular pigment and by six nanometres of long-wavelength peak agree about a grey card to four parts in ten thousand billion. The agreement is an identity rather than a coincidence, and it says exactly what an observer disagreement is a disagreement about.

The three cone absorptances at two settings of the cone optical density. Solid and dashed are the same construction at the two ends of two standard deviations of the reported spread. The curves are built from one pigment template through its ocular media, which is the same model its population of two hundred eyes is drawn from. The largest difference between the two sets is 18.8 per cent of the peak, and where it sits along the wavelength axis is what decides which stimuli the two observers disagree about — a departure concentrated in the blue is invisible on a sample with no blue in it.

What the eye does

A gain is not an observer

Multiply one eye's three cone sensitivities by 1.6, 0.7 and 2.4 and it is not a different eye. The white-point division is that multiplication's inverse, so the two agree exactly — and most of what a cone optical density change does is that multiplication, which is why the largest number in the table of individual variation is the one that matters least.

The arguments a standard observer does not have. Seven choices inside a set of colour-matching functions, each with the shape it takes and what it is worth in ΔE₀₀ on a red pigment under a 6500 K radiator. Six are measurements: a field size, an age, a macular density, a cone optical density, three peak wavelengths and a rod contribution. The seventh is not — a change of basis is a change of curves and not a change of observer, and its entry is exactly zero because the space an experiment measures is what an observer is. Printing that zero beside the others is the clearest statement of what the other six are measurements of.

What the eye does

Three curves for one space

Rotate a set of colour-matching functions by an arbitrary invertible matrix, undo the rotation at the end, and the computed colour is identical to eight parts in a thousand million million. An observer is a three-dimensional subspace, not a set of curves, and the literature keeps reopening a question that is a theorem.

A departure against how far the sample sits from the light. The sample is mixed with a flat reflectance, from the flat one at the left to its own at the right, and two observers differing in the macular pigment look at each mixture. The straight line is the distance between their relative cone excitations, and it is straight to 0.0 per cent: the departure is a pairing, and scaling one factor scales the product. The curved line is the same sequence in ΔE₀₀, which is not a linear function of the excitations and cannot be — it has cube roots in it and a chroma weighting underneath. The identity is about the eye; the curvature belongs to the unit.

Difference and uniformity

A departure is straight in the excitations

Walk a sample a quarter of the way from the light towards its own reflectance and exactly a quarter of the observer disagreement remains — in cone excitations, to two parts in a hundred. In ΔE₀₀ the same quarter leaves 0.347 where proportionality wants 0.428, and the discrepancy belongs entirely to the unit.

What this collection's grid does to its own observer audit. Two bars per light: the mean departure of the observer computed on this collection's five-nanometre grid, and the same computation on a quarter-nanometre one. For five of the six lights the two agree to two decimal places, which is what a well-sampled spectrum looks like. For the laser projector the coarse answer is exactly zero and the fine one is 1.80 — the largest in the table. On a five-nanometre grid a three-line spectrum with lines at 465, 532 and 638 nanometres is a one-line spectrum, and a single wavelength is a stimulus every observer agrees about to the last bit. The two departures do not compound here; the first conceals the second.

Matching and measuring

One wavelength is everyone's colour

A stimulus with a single wavelength in it produces the same relative cone excitations for every observer, exactly, whatever their age or field size. A display made of three such stimuli is where observers disagree most. Both statements are consequences of the same algebra, and the second is why laser projection has an observer problem.

The directional solver reduces to the radiosity solver exactly. A solver with a new unknown in it is worth nothing until it reproduces the one it replaces. Setting every wall's bidirectional distribution to ρ/π collapses all thirty ordered-pair radiances onto their patch's radiosity divided by π, and the answer agrees with this collection's existing radiosity solution to 9.8e-16 relative — the floating-point floor. That is the check that makes every other number in this family a statement about lobes rather than about a new piece of arithmetic, and it is the reason the reduction is drawn rather than mentioned.

What a scene does

Thirty unknowns instead of six

A directional transport solver is worth nothing until it reproduces the one it replaces. Setting every wall's bidirectional distribution to ρ over π collapses thirty ordered-pair radiances onto six radiosities and reproduces this collection's existing answer to 9.8 × 10⁻¹⁶ relative — which is the only reason anything else it says can be believed.

Where a point-sampled patch stops resolving a lobe. The horizontal axis is the wall's roughness, logarithmic; the vertical is how far the answer moves when the cone quadrature is refined from eight directions to sixteen, also logarithmic. A patch in a cube subtends a cone of angular radius 25.8° at the face opposite, and a microfacet lobe of roughness a is about a radians wide, so a lobe below about 0.15 is narrower than the quadrature that samples it. The movement at 0.05 is 21.2 ΔE₀₀ and at 0.3 it is 0.033. This is where the method stops, not where the paint does: matt, eggshell and satin finishes are inside it and a high-gloss varnish is not.

Where the model breaks

Where a patch stops being a point

A patch in a cube subtends a cone of 25.8° at the face opposite. A microfacet lobe of roughness 0.05 is about three degrees wide. Point-sampling the second inside the first returned a chroma of thirty-four thousand and a negative lightness, and the boundary between working and not working is measured rather than declared.

What the eye does

Two observers and one metamer

An observer departure is invisible on a single sample compared with nothing. It becomes a disagreement the moment two spectra are being asked to match, because a match is an identity between three integrals and a different observer takes different integrals. Everything in this round is a statement about pairs wearing a single sample's clothes.

The pairing against the direct computation, for each departure that admits both. Each departure can be computed twice: directly, by taking the difference between the fuller model and the integral one, and as a pairing — an inner product of the sample's deviation with the light's. The bar is how far apart the two answers are, relative to the answer, on a logarithmic axis. The three directional rows agree to a part in a thousand billion, which is the arithmetic of one shared quadrature. The lateral row agrees to three parts in a hundred thousand, and the gap there is the radial quadrature rather than the identity: the two integrals are taken over different grids. The pairing is not an approximation to the departure. It is the departure, written so that its two factors are separate.

Where the model breaks

Three audits and one shape

The tabulation, the observer and the scene solver have nothing in common as subjects. Each turned out to hold a departure that is a pairing of two deviations, vanishes exactly when either is empty, and had been invisible because its parameter had no call site. Three subjects, one shape, and the shape is the round's result.

Where the model breaks

The conditions are the result

The round measured about twenty departures and established ten conditions. The departures are numbers that depend on a sample, a light and a construction; the conditions are exact, they hold under every substitution tried, and they are what a reader can act on. A size is a measurement and a condition is a mechanism.

What a camera does

Four places to clamp are two pipelines

The essay on clipped noise ended on a procedure: a black frame and a dim grey card at a high amplification would place each raw converter's clamp. A procedure is a claim that a measurement identifies something, and this one identifies less than it looks. A clamp at zero commutes with the white balance and with the tone curve and not with the colour matrix, so the four positions are two pipelines — and the measurement separates them, on a card at half a per cent of white rather than on the black frame.

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.

What a scene does

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.

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