Field

Difference and uniformity

How far apart two colours are, whether the space you measured in was uniform, and how MacAdam's ellipses settle the question by measurement.

53 essays. Read in order: Metric.

Three colour-difference formulae, disagreeing. ΔE76, ΔE94 and ΔE2000 for the same 9 pairs of colours. The largest disagreement between ΔE76 and ΔE2000 here is 26.6 units — larger than the threshold usually quoted for a just-noticeable difference, so the choice of formula can decide whether two colours count as matching.

How far apart are two colours

MacAdam's discrimination ellipses, drawn 10 times actual size. Twenty-five ellipses of colours indistinguishable from their centres. They are drawn at 10× because at true scale most are thinner than a line. Their areas vary by a factor of 74, which is the whole result: a step of the same size in xy means very different things in different places.

MacAdam measured it

The midpoint is not half

Threshold and suprathreshold contours, normalised to the same size. At five of MacAdam's centres: the measured just-noticeable-difference ellipse in grey and the ΔE2000 = 1 contour in gold, each scaled to the same mean radius so that only shape and orientation are being compared. A scale change preserves orientation exactly, so any rotation between the pair settles the question. They differ by 24° on average and by 70° at worst, and the ratio between their sizes varies 4.8-fold across the diagram — so no single factor turns one into the other.

A threshold is not a unit

A box tolerance and a ΔE tolerance around the same colour. A slice through CIELAB at L* 50, with the ΔE2000 = 1 contour traced point by point and a ±1 component box drawn over it. The contour is 2.1 times longer in one direction than the other, and the box is square. Of every sample either rule accepts, the two disagree about 74% — accepted by one specification and rejected by the other, on the same measurement.

A tolerance is a shape

Distinguishable colours in sRGB, counted under two difference formulae. The gamut volume divided by the volume of a ΔE = 1 ellipsoid, integrated over the solid because that ellipsoid changes size and orientation from place to place. Under the 1976 formula the answer is 195,720; under ΔE2000 it is 41,819 — 4.68 times fewer, from the same solid and the same lattice. Both assume perfect packing, which nothing achieves, so each is an upper bound rather than a count of anything. The gap between them is the result: "how many colours are there" is a question about a metric before it is a question about vision.

How many colours are there

The ΔE = 1 contour at points across the L* = 55 plane, magnified 14×. Each closed curve is the set of colours exactly one unit of difference from the dot at its centre, traced by bisection along 30 directions and drawn 14 times life size. Under ΔE2000 the contours run from 0.68 to 5.07 CIELAB units across this slice — a ratio of 7.44 — and they are not circles and not aligned with each other. A formula whose contours were circles of one radius everywhere would be claiming that CIELAB is uniform, which is what the 1976 formula claims and what the measurements refuse.

Where the formula is not smooth

A difference is not a distance

A chromatic line screen at 10.0 cycles per degree. The upper strip is the pattern as delivered and the lower one is the same pattern after each opponent channel has been low-passed at its own cutoff. Against a flat field of the same mean, the delivered pattern differs by up to ΔE00 14.22 and the filtered one by 7.77 — a ratio of 1.8. Every number quoted elsewhere for a difference of this kind is the first one.

A difference has no size

How often two colour-difference formulae disagree about which pair is worse. Pairs of colours sampled in CIELAB, compared two at a time. A rank inversion is a case where one formula calls pair A worse and the other calls pair B worse; no monotone rescaling of either can remove one. The left bar of each group is the rate over the whole space and the right bar is the rate among pairs sitting near a tolerance of ΔE 1, where the decision is actually made — and it is between 35 and 44 per cent, against a coin flip at fifty.

Which of two is worse

One colour difference, at four places in the visual field. The same pair of colours — ΔE00 22.0 at the fovea — with each of its three components divided by that channel's own threshold scaling at the stated eccentricity. What is left at 20° is 4.4, and its hue has turned by 26 degrees, because the red–green part is divided by more than the blue–yellow part. The swatches are the predicted colours, drawn where a reader will look straight at them; the figure states a prediction it cannot stage.

A difference has no place

One tolerance decision, at four spectral distances. Every column is a pair of samples ΔE00 1.0 apart for the observer a colorimeter models — solved to that value by bisection, so the instrument would report the same number for all four. What differs is how far apart the two spectra are, which is achieved by adding a metameric black the reference observer cannot see. The bands are what two hundred people report: from 1.13 at the ninety-fifth percentile when the spectra are the same shape to 2.32 when they are not, and the worst case reaches 3.7. No specification records the quantity on the horizontal axis.

A tolerance is a probability

The three constants a specification does not quote. ΔE2000 is defined with three parametric factors in it — kL, kC and kH — which the CIE leaves to the industry using the formula rather than fixing. Graphic arts uses ones throughout; the textile standard weighs lightness at half, which is kL = 2. Applied to 4000 pairs at a tolerance of 1, the two settings accept 46 and 75 per cent, and 29 per cent of pairs change verdict — a larger disagreement than any between the formulae themselves. The reference conditions the ones assume are diffuse illumination at 1000 lux, a mid-grey surround, samples abutting, subtending over four degrees, differing by under five units, with no visible texture.

Three constants nobody quotes

Which of two reproductions is better depends on which statistic is asked. A specification for a proof or a print run is written against a set and has to reduce the set to one number. Here are two candidates: the one with the lower mean has the higher ninety-fifth percentile, so the mean prefers A and the tail prefers B. Across 2000 pairs of candidates generated the same way, the two statistics disagree about the winner 30 per cent of the time. Both numbers are honest; only one of them is what somebody notices.

A mean is not a difference

A halftone tint away from the centre of gaze, two ways. The upper curve raises the threshold by the E2 rule and leaves the filter alone, which is the multiplication the last phase guessed at. The lower one also moves the cutoff, because eccentricity magnifies the whole spatial scale — implemented as the substitution that makes it exact, a screen of ruling r seen through a filter whose cutoff has been divided by s being a screen of ruling r·s seen at the fovea. The horizontal line is threshold. The screen is visible where you are looking and gone by 3°, while the product prediction has it visible across the whole page.

A tint at the edge of a page

One tolerance decision, about pairs that agree less and less about the spectrum. Every point is a pair of samples that the reference observer reports as exactly ΔE00 1.0 apart — the same number, the same decision, the same line in the same specification. Along the axis is how far apart their two reflectances are. Up the side is the 95th percentile of what two hundred other eyes report. It runs from 1.13 to 2.32. The document records the horizontal line and not the axis it is plotted against.

A tolerance needs a second number

How fine a screen each channel can see. A halftone screen at 35 per cent coverage, 45 degrees, seen by each of the eye's three spatial channels. The bar is the ruling at which it drops below that channel's own threshold. The luminance channel is still seeing it at 55 cycles per degree; the two chromatic ones have lost it by 15 and 10. The dashed line is an ordinary press ruling at reading distance, and only one channel is above it. A halftone is a luminance object, which is why the ink whose dots are nearly the lightness of the paper is the one nobody worries about.

A halftone is a luminance object

A tolerance of one unit, re-measured under every light in the census. Every one of the 23 pairs behind this figure is at exactly ΔE00 1.000 under D65 by construction. Each bar is what those same pairs measure under another light, after the observer has adapted to it: the line is the median and the bar spans the pairs. A tolerance is written as a property of a pair and it is not one — the light multiplies both members, and the difference between two products is not the product of the difference. The widest row is a lens at twenty against a lens at seventy, spanning 0.81 to 1.57.

One unit in another room

How much of a whiteness figure is the sheet, and how much is the lamp. Each bar is how many points of CIE whiteness a stock has above an unbrightened sheet of the same base, measured with an ultraviolet-included instrument. The dark part is what survives when the ultraviolet is removed — the part that is a property of the paper. On average 82 per cent of the scale is the pale part, which is a property of the instrument's lamp. A whiteness figure without a measurement condition beside it is therefore not a measurement of a sheet; it is a measurement of a sheet and a lamp, quoted as though it were the first.

Whiteness is mostly the lamp

The best possible 3×3, and the patches it makes worse. Each row is one patch printed on a brightened sheet, measured under both conditions. The pale bar is how far apart the two measurements are; the dark bar is what is left after the best least-squares 3×3 over the whole set has been applied. It leaves 23 per cent of the mean, and — the part a mean hides — it makes 4 patches worse than doing nothing. The solids are the ones it damages: the ink blocks the ultraviolet, so a solid barely disagrees between the two conditions and the correction has no business touching it. A matrix has no way to apply itself only where the paper is showing.

A tolerance cannot cross a condition

MacAdam's ellipses, drawn on one diagram. The twenty-five measured discrimination ellipses at 10× actual size, on CIE xy (1931). Mean axis ratio 2.95 — one would mean every contour is a circle — and a size spread of 10.42 between the largest and the smallest. Both numbers depend on the plane, which is why the 1976 revision existed; neither can be taken to one, which is why the revision did not finish the job.

No diagram makes them circles

The same formula, applied after six different changes of basis. CIELAB's arithmetic — divide by a white, take a cube root, difference the results — run on six of the bases the matching data leave free. A linear change of basis leaves every match alone; a cube root does not commute with one, so the space, and therefore every colour difference computed in it, depends on which basis was in place before the nonlinearity. CIELAB's own choice gives an axis ratio of 3.44 and the best row here is LMS (confusion points) at 2.60.

A difference needs a basis too

Five answers to how far the ellipses are from circles. Five mean axis ratios on the same twenty-five measured ellipses, measured the same way in every row: the boundary points carried through, the longest radius over the shortest, averaged. What differs is which class of map is allowed. The first two rows are chromaticity diagrams, which divide by a sum; CIE xy as printed leaves 2.95 and the best diagram there is leaves 2.02. The last three are lightness–chroma spaces, which divide by a white point; CIELAB as specified leaves 3.44, the best space with no compression leaves 2.33, and the best space with a cube root in it leaves 1.61. Neither family contains the other, and only the last one gets below two.

A compression goes below the floor

The floor as a function of the exponent, and the fixed basis beside it. Two curves against the compression exponent on a logarithmic axis from 1 to 10. The lower curve is the best mean ellipse axis ratio any basis can reach with that exponent applied after it, and it falls from 2.33 at no compression to 1.66 at a square root and 1.61 at a cube root, then hardly moves — 1.57 at a tenth root. The upper curve is CIELAB's own basis at the same exponents and gets steadily worse, from 3.57 to 3.77. Almost everything a compression buys arrives with the first step away from linearity, and after that the exponent is choosing between 1.66 and 1.61 while the basis is choosing between 1.61 and 3.44.

The exponent was never the argument

The minimum sits in a notch the width of the answer's reciprocal. The distance from the centre to the boundary, all the way round one MacAdam ellipse mapped into a lightness–chroma space built on the CIE RGB primaries. The curve has two broad maxima and two very narrow minima: the dip is about 2.5 degrees wide at a third above its floor, because the width of the minimum of an ellipse's radius is the reciprocal of its axis ratio, and this ratio is 41. Forty-eight sample points, marked, are spaced 7.5 degrees apart, so none of them lands in either notch and the smallest one found is 2.2 times the true minimum. The ratio comes out 18.59 where it is 40.76.

An ellipse is not a ring of points

Three numbers for one set of ellipses, and which of them is which. Two curves and a horizontal line, against the size the ellipses are drawn at. The line is the analytic axis ratio — the ratio of the singular values of the map's own derivative, which is what "does this space make discrimination contours circles" means. The upper curve is a very finely sampled ring, which sits 0.6 per cent above the line at full size and converges onto it as the ellipse shrinks, because the gap between them is the second-order distortion of the map across a real ellipse rather than an error. The lower curve is the forty-eight-point sample used for this until now: it does not converge onto anything, because its error is set by the sample and not by the size.

Three numbers for one ellipse

How wrong would the data have to be

Twenty-five ellipses is a sample, and the score has an error bar. One row per colour space this collection ranks: the mean axis ratio its ellipses come out at, with the standard error of that mean over the twenty-five ellipses it was computed from. No literature is quoted — a mean of twenty-five numbers has a standard error those twenty-five numbers determine. The bars are far from equal: the best space carries ± 0.07 and the worst ± 1.56, because a space that makes the ellipses nearly circular makes all of them nearly circular and one that does not is dominated by whichever ellipse it handles worst.

Twenty-five is a sample of the diagram

Every census row under five constructions of the same test set. A slope chart with 5 columns — lattice, coarse, fine, uniform, natural — and one line per change of light in the census, each line joining that row's mean residual under each construction. Four of the five columns describe the same region of surfaces walked at different densities or against different measures; the last is the clamped, realistic family, which is not linear in its parameters and is therefore answering a slightly different question. The levels move: between the coarse and fine lattices every row shifts by seven to nine per cent, in the same direction, which is a common-mode factor no published residual here has ever carried. The order almost survives. Inside the region exactly one pair crosses, and it is the pair the standard error had already flagged; under the clamped set two more cross, including one the error separates by nearly nine standard errors. The crossing lines are drawn heavy.

A lattice is a quadrature rule

The error on a gap is not the two rows' errors added. Two bars for each of the 13 adjacent pairs in the census ranking. The upper, shorter bar is the standard error of the gap taken as a paired difference — the same 125 surfaces score both rows, so a surface that is awkward under one change of light is usually awkward under the other and the difference is quieter than either. The lower bar is the two rows' own errors added in quadrature, which is what comparing error bars by eye amounts to. Pairing is worth a factor of 1.78 on average and 3.36 on the pair it helps most, and it is the difference between 6 adjacencies unordered and 4. The gain is largest where the two rows are two daylights or two tungstens, because then the surfaces they find awkward are nearly the same surfaces.

The error on a gap is not the errors at its ends

How far each census row moves when the test set's own description does. A grid of bars, one row per change of light in the census and one bar in each row per number that describes the region the test surfaces are drawn from: how saturated they are, how bright, and how far the two modulations may go together. A bar's length is the elasticity — the proportional change in the published residual for a proportional change in that number. Saturation runs from 0.49 to 0.91 and brightness averages 0.104, so a test set's chroma range is nearly everything and its lightness range is nearly nothing. For scale, the largest elasticity found anywhere among this collection's five declared population widths is about a half — and those at least have declared ranges, while these three numbers have never been quoted with one.

Saturation is nearly everything

part 11 $How far each unit is from being a rescaling of the one this collection publishes in. One row per unit on the menu. The bar is the root-mean-square scatter about that unit's own best rescaling of ΔE2000, over 374 pairs of surfaces differing by a fraction of a unit to about ten. A bar of zero would mean the unit is ΔE2000 in different money — every printed number would change and no conclusion would. ΔE2000's own row is zero by construction and is the check that the table is computed the right way round. The two units that divide a chroma difference by the chroma it was measured at, ΔE94 at 15 per cent and CAM16-UCS at 24, are closer to it than the three that do not, which run from 28 to 35. The split is by weighting and not by whether the unit is a matching difference or an appearance one.$

The weighting is the disagreement

The census along the line from ΔE76 to ΔE94, and past it. ΔE94 is ΔE76 with two weighting constants in it, and at zero those constants make every weight exactly one, so the two formulae are joined by a line rather than separated by a choice. The horizontal axis is how much of the published weighting is applied: 0 is exactly ΔE76, 1 is exactly ΔE94, and 3 is three times more weighting than anybody has proposed. The falling curve is the census's mean elasticity to how saturated its test set is, which drops from 1.11 to 0.74 — most of the fall happening before the published value is reached. The other curve is Kendall's τ against ΔE2000's ranking, and it peaks at w = 0.5, not at 1: the weighting that best reproduces the published ordering is about half the published weighting. There is no value of this dial that reaches ΔE2000, whose rotation term is not on this line at all.

A dial through a discrete menu

The disagreement is at the near end

An aperture and a gloss lobe, apart and together. Six materials, each measured through a four-millimetre radius and each given a gloss lobe, alone and at the same time. The pale bar is what the two cost added together as if they were independent; the dark one is what they cost when both are present. Every material comes out below the sum, by between 0.8 and 3.6 ΔE₀₀. The two departures partly cancel: the aperture removes light that went into the material and came back out too far away, and the interface returns light that never went in at all. Measuring either one alone therefore overstates what both together do, which is the opposite of the way interacting errors are usually assumed to behave.

Two departures that partly cancel

Four departures from the model equation, each at an ordinary strength. What each of the four assumptions inside a colour integral costs, in ΔE₀₀, on a stated sample under a stated light. The wavelength index is a coated printing paper measured with and without the ultraviolet of D50; the range is the same paper integrated from 300 nanometres and from 380; the place index is a pigmented plastic through a four-millimetre radius; the direction index is an eggshell paint beside a window. The spread is a factor of 7.0. This is a ranking of four examples rather than of four departures — each of them can be made larger by choosing a more extreme sample, and the marble in the same collection of materials reaches 12.7 on the index that comes third here.

The departures are larger than the tolerance

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.

A neutral has no grid

What the normaliser cancels, per light. Two bars per light, logarithmic. The upper is the colour error a 5-nanometre sum makes when the white it is divided by is computed finely; the lower is the same sum divided by the white computed on the same coarse grid, which is what every colorimetric calculation actually does. The ratio is between 1.3 and 4.1. The grid appears twice in a tristimulus value and the two errors are the same error, so most of it divides out — which is why five nanometres has been good enough for a century without anybody having to be careful about it.

The normaliser carries the error too

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.

A departure is straight in the excitations

Each departure over forty-two surfaces rather than one. The same six departures measured over a family of forty-two analytic reflectances — an absorption band of stated centre, width and depth — with the smallest, the median, the ninety-fifth percentile and the largest marked. Every one of them spans more than a factor of three, and the ranking between them is not stable across the family: what decides a departure's size is which sample it is asked about, because a departure is a pairing and the sample is one of the two factors. Quoting any single number for what an observer's age is worth is quoting a choice of example.

A tolerance with an observer in it

The length of one grey ramp, cut into more and more steps. A neutral ramp from L 1 to L 100 cut into between one and ten thousand equal steps, each step measured and the steps added, in four units, both axes logarithmic. ΔE*ab and the model's Euclidean J′a′b′ give the same length at every step count, 99 and 96. ΔE₀₀ settles at 74.6 once the steps are small. The power-corrected ΔE′ does not settle: 25 in one step, 137 in a hundred, 755 in ten thousand, growing as the number of steps to the power 0.37.

A distance raised to a power has no length

The angle between the lens and the macular pigment, before and after adaptation. On each of 120 surfaces the angle, in the local metric, between what an older lens does to the reading and what a denser macular pigment does, binned in ten-degree steps. Read without adaptation, where both filters yellow the observer's white along with everything else, the two point nearly the same way: a median of 8 degrees. Read after each observer has adapted to its own white, the median is 156, and the two together cost less than the larger alone on 115 of the 120.

Two yellow filters cancel on a slope

The angle between the two filters against how completely the eye has adapted. The angle, in the local metric, between what an older lens does to a reading and what a denser macular pigment does, on 120 smooth reflectances, as the degree of adaptation runs from nought to one. The median angle is 8 degrees unadapted and 156 at complete adaptation, and almost all of the turn happens in the last tenth: it passes a right angle at a degree of 0.928. The marks are the degrees CIECAM16 gives five rooms — an overcast sky 1.00, an office 0.94, a lit living room 0.86, a dim room 0.75, a cinema 0.66 — so only the outdoor one is at the end of the dial.

Two filters cancel only in a bright enough room

The same pairs, held at one colour difference, read in a unit that knows the room. 23 pairs of reflectances built to sit at exactly ΔE₀₀ 1.000 under D65, read in CAM16-UCS as the adapting luminance runs from a third of a candela a square metre to ten thousand, in an average surround. ΔE₀₀ has no argument for the room, so in that formula every pair stays at 1.000 all the way across — the flat line. In the model's unit the same pairs rise from a median of 0.76 to 1.30, and they do not rise together: at the bright end they run from 1.11 to 1.65.

A tolerance has no light level

What is left of one colour difference when the two colours alternate. 23 pairs built at exactly ΔE₀₀ 1.000, alternated at a rate, with each part of the difference scaled by its own temporal channel and the formula then applied unchanged. At rest every pair is the flat line at one. By 7 hertz the median is 1.11 and the pairs run from 0.57 to 3.04 — a factor of 5.3 between pairs the formula calls identical. By sixty hertz the largest of them is 0.15.

A difference has no rate

Every pair of departures, before adaptation and after it. The fifteen pairs of the six audited observer departures. Each row runs from the angle between that pair's two deviations with no adaptation to the angle with complete adaptation; an angle past ninety degrees is a pair pointing apart, which is where a pair can cost less together than the larger of the two costs alone. 3 pairs gain that behaviour as the eye adapts, 3 keep it, 5 lose it and 4 never have it. The pair followed here — the lens against the macular pigment — is in the smallest group that is not empty, and every result quoted from it generalises in the wrong direction.

Adaptation turns more pairs off than on

Six departures, and three ways of adding them up. The six audited observer departures on 120 smooth reflectances, across the dial. Summing them assumes they all point the same way and is an overestimate everywhere; taking the largest alone assumes only one matters and is an underestimate everywhere. Quadrature — the usual way of combining contributions taken to be independent — assumes they are mutually perpendicular, and the measured combination crosses it at a degree of 0.8618. Below that the departures are on balance pointing together and quadrature is too small; above it they are on balance pointing apart and quadrature is too large. It is exactly right in one room.

Quadrature is exact in one room

Where a surface's band sits decides the room it needs. Each of 168 surfaces at its own crossing — the degree of adaptation at which the part adaptation has yet to remove falls to the size of the part it will leave — against where that surface's absorption band sits. The line joins the median at each band centre and the rooms are marked across. A surface absorbing at 470 nm crosses at 0.883 and one absorbing at 670 nm at 0.972. Both of the filters this is about absorb in the blue, so a surface with a blue band is where the two differ in shape and has a large residual, while a surface with a red band is nearly invisible to both and its whole deviation is the shared yellowing of the observer's white.

The room a surface needs is written in its band

Twenty-three pairs at one ΔE₀₀, read in two units that know the light level. Twenty-three pairs of surface colours, each exactly one ΔE₀₀ apart, on a display whose white runs from 1.5 to 10,000 cd/m² across, with a background at a fifth of the white. ΔE₀₀ has no argument for the light and stays at one. The median ΔEITP rises from 1.01 to 2.75 and flattens near the top, and the median CAM16-UCS distance from 0.76 to 1.20.

Two units with a light level disagree about lightness

How each unit balances lightness against chroma, as a display brightens. Twelve pairs built to differ in lightness alone and twelve built to differ in chroma alone, each at exactly one ΔE₀₀, read in both units at nine display levels. Up is the median lightness pair's reading divided by the median chroma pair's, so a falling curve means chroma differences becoming relatively more visible. Both fall. ΔEITP's median runs from 0.91 at a 1.5-candela white to 0.81 at 10,000, a fall of 11 per cent; CAM16-UCS's from 1.13 to 0.87, a fall of 23 per cent. Neither turns: both units say chroma differences gain on lightness differences as a display brightens, and CAM16-UCS says it twice as strongly.

The units part by hue, not by light level

What a lit room takes, and where it takes it. On a display whose white is 1,000 cd/m², how much of the ΔEITP a one-unit lightness step is given survives a veiling luminance reflected off the screen, against where on the lightness scale the step sits. At L 2 — a deep shadow — one candela of reflected light removes 22 per cent of the difference and three candelas remove 46. At L 90 the same veils remove nothing measurable. The shadows the unit counts most are the ones a room removes first.

The shadows a unit counts are the ones a room removes

A lit room brings the units' medians together

How wrong a declared veil makes the unit, for four true veils. On a 100 cd/m² display, the worst error over grey steps from L 2 to L 90 — the size of the natural logarithm of the declared reading over the true one — against the veil declared, for rooms putting 0.1, 0.3, 1 and 3 cd/m² on the screen. Each curve reaches nought at its own true veil and rises on both sides. The flat stretch at the left is declaring almost nothing, which is declaring none: 0.22 for a true veil of 0.1, 0.54 for a true veil of 0.3, 1.14 for a true veil of 1, 1.89 for a true veil of 3.

A guessed veil halves the error