The tables cannot bound what they discarded

A linear repair for a bilinear loss found that a colour computed from a lamp’s table and a sample’s table, each blurred through its own slit, is wrong by a covariance: inside every slit window, how the lamp and the sample vary together. No linear correction applied to either table puts that term back, because it depends on both factors at once. A second slit buys a quarter found that no choice of slit widths does either.

That essay ended on a hope. A covariance is bounded by the two variances it is built from — the Cauchy–Schwarz inequality — and a blurred table might still carry how much structure it had. If both variances could be estimated from the tables, a colour engine could compute an upper bound on its own error for any pair of tables, with no fine data and no reference. The question was whether that bound tracks the error within a factor of two or three, close enough to act on, or says four when the answer is 0.02.

It does neither. The bound that can be computed from the tables is not a bound, and the bounds that always hold cannot be computed from the tables or are too loose to use.

A valid bound is loose, and the computable one is not valid #

Over sixty-eight notched samples under a fluorescent tube, Cauchy–Schwarz with the true variances inside each slit window never falls below the actual error, and sits a median 14.7 times above it. With the variances estimated from the blurred tables it falls below the error on 6 notches, to as little as 0.45 of it. Under a three-laser projector the estimate falls below on 22 of 68, to 0.15.

• The estimate fails where the error is largest. Under the tube all six failures are the two narrowest notches within six nanometres of the mercury line, including the worst notch in the census: 2.58 colour differences against an estimate of 1.17.
• Two lamps can have nearly the same table and very different errors. Widening the tube’s lines from 0.6 to 3.6 nanometres at the same power moves its blurred table by under 7 per cent of its peak and the estimate by a tenth. The true bound falls by a factor of 3.5 and the error by a third.
• The only bound computable from the tables that always holds needs the lamp’s peak and the sample’s reflectance range declared, and sits a median 196 times above the error under the tube.
• Even with perfect knowledge the inequality costs a factor of three, on top of a factor of four from the covariance’s sign changing between wavelengths.
• As a ranking rather than a bound, the table estimate is almost as good as the truth under the tube, at a rank correlation of 0.72 with the error against 0.76. It cannot say how large the error is.

Four candidates, from least to most computable #

The error has an exact form. A colour’s tristimulus values are sums over the grid of lamp times sample times a matching function. Blurring the product through a slit gives the product of the two blurs plus, at each grid point, the covariance of lamp and sample inside the slit window. The separately blurred tables keep the first term and discard the second, so the error in each tristimulus value is the discarded covariance weighted by the matching function and summed.

Four quantities bound that sum, and they differ in what they need:

• The size of the covariance at every grid point, summed without letting positive and negative terms cancel. It is a bound, it is the tightest of the four, and it needs the covariance itself, which is what is missing.
• Cauchy–Schwarz with the true variances: at each grid point, the covariance is at most the square root of the lamp’s variance times the sample’s inside the window. It needs the fine spectra to compute the variances.
• Cauchy–Schwarz with estimated variances: the variance a smooth function would need inside the window to produce each table entry and its neighbours, from their slope and curvature. It needs only the tables.
• Bhatia–Davis: a function confined between a floor and a ceiling cannot vary inside a window by more than its mean’s distance from the floor times its distance from the ceiling. It needs only the tables and two declared numbers per factor — here, the lamp’s highest spectral value and the sample’s lowest and highest reflectance.

Each tristimulus bound becomes a colour difference as the largest CIEDE2000 at the corners of the box it allows around the computed colour. The census is the one the slit comparison used: notches 5, 8, 12 and 20 nanometres wide, walked in two-nanometre steps across sixteen nanometres either side of the 546.1-nanometre mercury line, through a five-nanometre slit on each table.

The figure at the top of the page places every notch by its actual error and by each bound. The diagonal is where a bound equals the error. The true-variance points are all above it. The table-estimate points straddle it, and the declared-range points sit two orders of magnitude up.

How loose each bound is #

Summing the covariance’s size without cancellation gives a median of 4.4 times the error, ranging from 2.4 to 68. That factor is the price of knowing sizes but not signs. The covariance can change sign from one grid point to the next, and a box that lets X, Y and Z move independently includes colour shifts the real error, which moves them together, cannot make. Nothing computed from magnitudes avoids it.

Cauchy–Schwarz with the true variances gives 14.7, from 5.4 to 261. The further factor of about three is the inequality’s own. Cauchy–Schwarz is tight only when the two functions inside the window are exactly linearly related, and a lamp’s line and a sample’s notch never are.

The table estimate’s median is 4.1, which looks tighter than the true bound and is a warning rather than an achievement. An estimate that sits closer to the error than any valid bound can manage gets there by guessing, and on six notches the guess lands below the error. Its range, 0.45 to 941, is wider than any valid bound’s.

Bhatia–Davis with declared ranges gives 196, from 13 to 67,000. It never fails, because it assumes the worst arrangement of the lamp inside every window its table allows. It holds on every notch and is too loose to act on for any of them.

So the hope asked for a factor of two or three. With the variances handed over from the fine spectra, the inequality delivers fifteen. The best a colour engine can compute honestly is two hundred.

Where the estimate fails #

A bound that fails on six notches in sixty-eight might still be useful if those six were harmless. They are the ones where the error is worst.

At five positions of seventeen the estimate is below the error: 540.1, 542.1, 546.1, 548.1 and 552.1 nanometres. With the notch on the line the error is 2.58 colour differences and the estimate says at most 1.17. Walk the notch twelve nanometres away, to 558.1, and the error falls to 0.023 while the estimate still says 0.52. The estimate is too small where the error is large and too large where it is small, which is the opposite of what a flag needs to do.

The sixth failure is the eight-nanometre notch centred on the line. The twelve- and twenty-nanometre notches never fail under the tube, because a wide notch has little covariance with a narrow line and the smooth estimate over-covers it.

Under the laser projector the same pattern is stronger. The five-nanometre notch fails at every position from 526 to 540 nanometres, the eight-nanometre notch at seven of the nine positions around the 532-nanometre line, and even the twenty-nanometre notch fails at four. The worst is 25.1 colour differences against an estimate of 3.7.

The table sees a bump where the window holds a line #

The mechanism shows at the level of individual grid points.

At 545 nanometres the discarded covariance is 0.098, the true bound 0.149 and the estimate 0.006. At 550 the covariance is 0.059 and the estimate 0.009. At 540 and 555, where the slit window holds almost none of the line and the covariance is nearly nothing, the estimate is 0.011 and 0.001, the largest of the three.

A 1.2-nanometre line blurred through a five-nanometre triangular slit and tabulated every five nanometres becomes two or three raised entries. The estimator reads those entries the only way three neighbouring numbers can be read: as the slope and curvature of something smooth. A smooth bump of that height has a small variance inside a five-nanometre window. The line, with all its power inside 1.2 nanometres, has a variance many times larger. The estimator has not lost the line’s structure; it has spread it out, onto neighbouring grid points where it does no harm, and taken it away from the grid point where it does.

Nearly the same table, different covariances #

If the estimate’s failure is that a table cannot tell a narrow line from a broad one, then two lamps that differ only in line width should show it directly.

Widening the lines sixfold at the same power moves the lamp’s blurred table by at most 6.5 per cent of its peak. The actual error falls from 2.71 to 1.81 colour differences, because a broader line covaries less with a narrow notch. The true bound falls from 19.6 to 5.7. The estimate from the tables moves from 1.17 to 1.05, and it is below the error at every width.

This is the argument in its shortest form. The covariance depends on a width the blurred table does not carry, so no function of the table can bound it for every lamp that could have produced that table. A bound computed from the tables alone must either assume the narrowest line physically possible, which is what Bhatia–Davis does and why it costs two hundred, or assume something about the width and fail on lamps narrower than the assumption, which is what the smooth estimate does.

A lamp is two audits at once and which end to buy both came down to one length compared against another: a lamp’s narrowest feature against the grid’s step. The same length decides this. A table that recorded its lamp’s narrowest feature beside its values would carry the one number the bound needs, and a table that does not record it carries none of what matters here.

Under a laser projector #

Under the laser the true-variance bound again always holds, at a median of 15.4 times the error. The table estimate falls below on 22 of 68 notches, as low as 0.15 of the error. Bhatia–Davis holds on all of them at a median of 30. The declared bound is tighter here than under the tube only because the laser has almost no light between its lines, so most windows have a lamp mean near zero and contribute nothing.

The laser’s lines are 0.2 nanometres wide, a sixth of the tube’s, and the failures are proportionally worse. A fifth of a nanometre blurred through a five-nanometre slit is indistinguishable in the table from a line ten times wider. Every notch that overlaps a laser line inside the slit window is a notch where the table understates the covariance, and the estimate’s rank correlation with the error falls to 0.39.

What the estimate is good for #

One number in the census survives: the table estimate ranks the tube’s notches nearly as well as the true bound does, at a rank correlation of 0.72 against 0.76. Across a large set of pairs it would put most of the bad ones near the top.

That is a triage tool, not a bound, and a mean is not a worst case is the reason the distinction matters. It cannot say that a pair is safe, because its lowest readings include the pair with the largest error. It can say which pairs deserve the fine measurement that would settle them. Under a laser it cannot even do that reliably.

Two slits are not one slit found the arrangement that does not need a bound, and the slit is what makes it legal found why a slit belongs there at all: one slit on the reflected light, which keeps the covariance in what it measures. Where that arrangement is available the bound is unnecessary. Where it is not, the honest statement from two blurred tables is the colour and the lamp’s narrowest feature, not the colour and a number calling itself its error.

How the bounds were computed #

The tube and the laser are the standard models these essays use: the tube as broad phosphor bands with mercury lines of 1.2 nanometres at 404.7, 435.8, 546.1 and 578.0, and the laser as three lines of 0.2 nanometres at 465, 532 and 638. The line-width variants of the tube scale each line’s height inversely with its width, so each line’s power is held. A notched sample is a gaussian dip 0.66 deep in a flat reflectance of 0.72, of a stated full width at half maximum.

At each five-nanometre grid point from 380 to 780, the lamp and the sample are integrated through a triangular window of five nanometres half-width at a tenth of a nanometre, giving the two table entries, both variances, the covariance and the lamp’s maximum. The estimated variance uses the table’s central slope and second difference with the triangle’s second and fourth moments, which is exact for a quadratic. Bhatia–Davis uses the lamp’s highest value anywhere in the visible range as its ceiling and zero as its floor, and the notch’s deepest and flat reflectance as the sample’s range.

Each bound is a tristimulus box around the colour from the separate tables, and its colour difference is the largest CIEDE2000 from that colour to any of the box’s eight corners, with the white taken from the blurred lamp. The actual error is the CIEDE2000 between the colours from the separate tables and from the blurred product.

What this leaves out #

The corner of a box is not a strict bound on a colour difference. CIEDE2000 is not convex, so a colour inside the box can in principle sit further from the centre than any corner. For boxes this small the difference is negligible, and the tristimulus bounds themselves are strict.

Other estimators exist. A table could be deconvolved rather than read for slope and curvature, or a variance could be taken from a model of the lamp’s type. Any estimator that uses the table alone meets the same wall: two lamps with nearly the same table and different line widths need different bounds, so it must either assume the worst width or fail on narrower ones.

The notches are gaussian. A sample with a steep-sided or periodic structure has larger variances inside a window, which makes the true bound looser and the smooth estimate worse. A finer table is a worse table is the extreme case.

And the census asks about one slit width, five nanometres. A narrower slit reduces every covariance and every variance, and the ratios of bound to error need not stay where they are.

Still open: whether a declared line width makes the bound usable #

The census found the number the bound needs, and it is a number a lamp’s measurement could record: the width of its narrowest feature. What it did not measure is how tight a bound using it would be.

The calculation is a bound that takes the declared width as an argument: the largest variance a feature of that width and of the power the table implies can have inside the window, used in Cauchy–Schwarz with the sample’s own estimate. It would be valid for every lamp whose features are no narrower than declared. The prediction is that it lands between the true bound’s factor of fifteen and Bhatia–Davis’s two hundred, closer to the first for a tube and further for a laser, because the laser’s declared width would be so small that its worst-case variance approaches the peak’s.

If the answer is a factor of twenty or so, a colour engine with declared widths has a usable warning. If it is a hundred, the inequality itself is the limit, and the only useful statement is the arrangement of the measurement rather than an error bar attached to its result.

An error bar must bound the thing it was built from #

The habit is about error estimates computed from the data they describe.

It is natural to estimate an error from the output of a process when the input is unavailable, and it often works, because the output carries most of what the input had. It fails in one specific way: when the error comes from exactly what the process removed. A blurred table has lost its narrow structure, the covariance error lives in that structure, and any estimate built from the table inherits the loss.

The move is to find two inputs that give the same output and different errors. If they exist, no function of the output bounds the error, however clever. Here it took four line widths and one notch, and the estimate stayed flat while the error fell by a third.

The failure mode is a median that looks tight. The table estimate’s median ratio was 4.1, tighter than the true bound’s 14.7, and that apparent tightness came from being too small exactly where the error is largest.