Two slits are not one slit

The slit is what makes it legal found that a spectrum integrated through a five-nanometre slit before it is tabulated can be summed on a five-nanometre grid honestly, even a laser’s, because the blur puts the line’s power where the grid can see it. The line can be in the sample then found that a notched sample puts a calculation in trouble under any light, and ended on the obvious repair: a slit for the sample too. Both results are true, and put together they do not give the result they appear to.

Legal tables, illegal product #

A slit on the sample’s table and a slit on the lamp’s table each make their own table honest, and the colour computed from the two is still wrong wherever a narrow feature in one meets a narrow feature in the other — while one slit on the light the sample actually reflects is right everywhere.

• Under a 6500 K thermal source a two-nanometre notch costs 2.14 colour differences point-sampled and 0.005 with a slit on the sample. Slitting the lamp as well changes nothing, because the lamp has no features.
• Under a fluorescent tube a twelve-nanometre notch on the 546.1 nm line costs 2.54 point-sampled, 0.73 with a slit on each table, and 0.02 with one slit on the reflected light.
• The two separate slits’ error is not the grid’s: summed at a tenth of a nanometre the same two tables still leave 0.75.
• For some notches two slits are worse than none: a five-nanometre notch near the line costs 0.20 point-sampled and 0.95 with two slits.

What a slit does, and to what #

A slit replaces each value of a spectrum by a weighted average of its neighbourhood — here a triangle five nanometres wide at half its height. For a lamp with lines narrower than the grid, that average spreads each line across enough of the grid for a five-nanometre sum to see its whole power, which is why the slit turned a laser projector’s thirty-seven colour differences into a fifth of one.

A spectrophotometer measures a sample the same way. Its reflectance is a ratio of light returned to light supplied, each read through the instrument’s slit, and the table it writes is the sample’s reflectance blurred by that slit. So a colour computed from measured data is the product of two blurred tables: the lamp blurred by whatever instrument measured the lamp and the sample blurred by whatever measured the sample.

The colour itself is a sum over the product of the lamp and the sample at each wavelength, and a grid is not a resolution found that it is the product’s narrowest feature, not either factor’s, that decides whether a coarse sum is safe. The slit that would make that sum honest on a coarse grid is a slit on the product — the blur of the light the sample returns under that lamp. The blur of a product is not the product of two blurs, and the question is how different they are.

The lamp’s slit was measured on a smooth sample #

The earlier result about slits was right, and the reason it was right is visible in what it measured.

That measurement slit the light and summed it against a sample it called an interference filter, a Gaussian notch forty-eight nanometres wide at 545 nm. Across any five-nanometre window that sample is very nearly a straight line, so whatever the slit did to the lamp, the sample had nothing inside the window to pair wrongly with it. Point-sampling a mercury line there cost 1.26 colour differences and a five-nanometre slit on the lamp cost 0.014: a factor of ninety, measured correctly.

The two essays that followed changed the other factor. The line can be in the sample narrowed the notch until it had structure inside a cell, and the cost of a steep notch repeats every step gave it steep edges. Neither changed the lamp’s slit, and neither asked what happens when both factors have structure inside the same window — which is the only case in which a slit on each factor and a slit on the product disagree.

When they are exactly the same #

There is an exact answer. Averaging a product over a window gives the product of the two averages plus a correction, and the correction is the covariance of the two factors inside the window: how much they rise and fall together across the slit’s five nanometres.

If either factor is flat across the window, the covariance is zero and the two blurs are identical — the same reason a neutral has no grid: a factor with no structure has nothing for any operation on the other to disagree with. A smooth lamp is flat across five nanometres everywhere, so under a smooth lamp a slit on the sample is a slit on the product, whatever the sample does.

Point-sampled, the two-nanometre notch costs 2.14 colour differences under the thermal source; with a slit on the sample it costs 0.005, and with slits on both tables, or one on the product, the same. A slit on the lamp alone does nothing — 2.14 — because the lamp was never the problem. Under a phosphor white LED, whose narrowest feature is twenty-two nanometres wide, the same notch goes from 2.40 to 0.012. The repair the line can be in the sample proposed works exactly where the lamp is smooth.

A notch on a mercury line #

A fluorescent tube is not flat across five nanometres — a lamp is not a blackbody, and a tube puts a large share of its light into its lines. Its mercury lines are 1.2 nm wide, and a notch whose edge sits on one has both factors changing steeply inside the same slit window.

That is where the covariance lives. Across the window, the lamp is a sharp peak and the sample is a steep slope, and the product of their averages puts the line’s power at the notch’s average depth, while the average of their product puts it at the depth the notch actually has under the line. The separately blurred tables record the right amount of line and the right amount of notch, and pair them wrongly.

For a twelve-nanometre notch centred on the 546.1 nm line, point sampling costs 2.54: the grid cannot hold the line. A slit on the lamp alone costs 0.38, because the line is now spread for the grid. A slit on the sample alone costs 2.72. Slits on both cost 0.73. One slit on the product costs 0.019. And the two separately blurred tables summed on a grid a tenth of a nanometre fine cost 0.75: once the tables are blurred separately, no grid gets the colour back, because the information about how the line and the notch overlapped was averaged away before anything was tabulated.

Across the line #

Walking the notch across the line shows where the pairing goes wrong.

With slits on both tables, the cost is 0.05 with the notch centred at 530 nm, rises to 0.73 at 547 nm, falls to 0.07 at 552 and rises again to 0.34 at 556. Point-sampled it runs from 0.37 to 2.52, and with one slit on the product it never leaves the range 0.016 to 0.022. The two-slit cost is largest where the notch sits over the line and where each of its edges crosses it, and smallest where the line falls in the notch’s flat floor or wholly outside it — where one of the two factors is flat across the window and the covariance vanishes again.

That is the structure of the exact correction appearing in the data. Two slits fail exactly where the lamp and the sample both change inside the same five nanometres, and succeed exactly where one of them does not.

Worse than no slit #

The obvious expectation is that each slit can only help, so two slits help at least as much as one. The width sweep says otherwise.

With the notch centred at 552.3 nm, six nanometres from the line, two separate slits cost 0.95 at five nanometres wide where point sampling costs 0.20, and 0.58 at eight nanometres where point sampling costs 0.46. At two nanometres they cost 0.43 against 1.83, and at twelve 0.04 against 1.20 — so they are sometimes a large improvement and sometimes a large deterioration, depending on whether the notch’s edge reaches the line inside the slit’s window.

Point sampling at five nanometres has its own luck: its error depends on where the grid lands, as where the grid starts found for a line lamp on its own, and at some widths the grid lands kindly. The two slits have no luck to fall back on. Their error is a bias written into both tables before any grid existed, and a notch five nanometres wide, one slit-width from a line, is the geometry that maximises it. One slit on the product costs between 0.003 and 0.031 at every width.

Centred on the line #

Centring the notch on the line makes the case as hard as it gets.

A one-nanometre notch on the line costs 6.18 point-sampled and 4.15 with two slits; a two-nanometre notch, 4.96 and 4.46; a five-nanometre notch, 3.63 and 2.60. Only once the notch is several slit-widths wide, so that the line sits in a flat floor for most of the window, do the two slits approach the truth: 0.28 at twenty nanometres and 0.07 at forty-eight. One slit on the product costs under 0.033 at every width, several hundred times less than two slits for a narrow notch.

A narrow notch filter on a line is not an exotic object. It is what a notch filter is for: blocking a laser or a lamp line while passing its neighbours, and its purpose is to sit exactly where the covariance is largest.

The check that cannot see it #

The line can be in the sample recommended one test a calculation can run on itself without a reference: slide the grid’s origin a nanometre at a time and watch whether the answer moves. For point-sampled notches under a line lamp it works well.

Point-sampled, the twelve-nanometre notch on the line costs anything from 0.80 to 8.94 colour differences depending on where the grid starts, a spread of 8.1 that no calculation could miss. The same notch tabulated through two separate slits costs 0.54 at its best origin and 0.90 at its worst: a spread of 0.36 around an error that never falls below half a colour difference at any origin.

The two-slit error is not the grid’s luck. It was written into the two tables before any grid was chosen, so every origin inherits it, and a check that looks for disagreement between origins finds a modest disagreement and no hint that all of them are wrong together. For the five-nanometre notch six nanometres from the line the pattern is starker: point-sampled it costs 0.12 to 0.79 across origins, and through two slits 0.66 to 1.19 — worse than point sampling at every origin, with a spread that looks like reassurance.

So the self-check separates two kinds of error, and it can only see one of them. Sampling error moves with the grid and shows up as a spread. A pairing error made before tabulation does not move with the grid, and only a reference — or a calculation on the product, which is the same thing — can see it. The three numbers at the end carry no trace of which kind they suffered, for the reason three numbers cannot see a line: the tabulation’s history is not in the answer.

Which instruments measure which #

The distinction between the two tabulations is a distinction between two kinds of measurement, and both are made every day.

A spectrophotometer measures a sample’s reflectance with its own lamp, through its own slit, and a lamp’s spectrum is measured separately by a spectroradiometer through another. A colour computed from the two for a scene under that lamp is the two-slit tabulation. It is right for every sample under a smooth lamp and for every smooth sample under any lamp, which is most of the colorimetry anybody does, and it is wrong for a sample with structure narrower than the slit under a lamp with structure narrower than the slit.

A spectroradiometer pointed at the sample in the scene measures the product through one slit — the light the sample returns under the real lamp, blurred once. What the instrument reports is, in that case, the right quantity for the colour, and its five-nanometre table sums honestly because the blur acted on the product. It is also a measurement of one sample under one lamp, which cannot be reused under another.

So the two measurements trade generality for exactness in a specific way. Separate tables can be combined for any lamp and any sample and are exact whenever one of the two is smooth; a measurement of the product is exact for the pair it measured and nothing else. For a notch filter under a line lamp, only the second is right, and the first is the one a colour calculation almost always uses.

What a colour engine could do #

The repair follows from the exact correction and costs nothing new to measure.

Keep the fine tables when both factors have fine structure. A filter maker’s table at a nanometre or finer and a lamp maker’s line list are enough to form the product at a tenth of a nanometre and blur it once, which reproduces the one-slit answer. The instrument’s slit belongs to the product, and a calculation can apply it there if it has not been applied to the factors first.

Flag the pairs that need it. The covariance can only be large where both factors vary within a slit’s width, so a colour engine can check each factor’s steepness and do the fine calculation only for pairs where both are steep — a filter and a discharge lamp, a laser and an interference coating — and use its coarse tables everywhere else.

Do not slit a smooth factor. A slit on the smooth factor of a pair buys nothing and, as the lamp-only case shows, costs nothing either; a slit on the structured factor alone is the right repair when the other factor is smooth, and the wrong one — sometimes worse than nothing — when it is not.

How the tabulations were computed #

The sample is a constant reflectance of 0.72 with a Gaussian notch 0.66 deep of stated width and centre. The lights are closed-form: a 6500 K thermal source, a phosphor white LED, a three-emitter LED and a fluorescent tube with mercury lines 1.2 nm wide on a phosphor bed. The observer is given in closed form.

A slit is a triangle with a full width at half maximum of five nanometres, integrated at a tenth of a nanometre. The point tabulation evaluates the light and the sample at the grid’s points; the lamp and sample tabulations replace one factor by its blurred version; the two-slit tabulation replaces both; the product tabulation blurs the product of the two and divides it by the blurred lamp, so that its white is the blurred lamp’s. Each colour is summed on a five-nanometre grid from 380 to 780 nm with its own white and compared in ΔE₀₀ with the unblurred product summed at a tenth of a nanometre.

What the measurement leaves out #

The slit is triangular and the same width for both instruments. Real instruments have slits of different shapes and widths, and a lamp measured at two nanometres and a sample measured at ten pair their errors differently; the correction is still their covariance inside the combined window.

The notch is Gaussian. The cost of a steep notch repeats every step found that a flat-bottomed notch’s edges behave differently from a Gaussian’s on a grid, and a slit softens those edges before they are tabulated, which would change the point-sampled column more than the others.

And the deconvolution that the slit essay measured — the three-term correction that undoes a triangular slit of one interval — is not applied. It sharpens each table back towards its unblurred form, and whether sharpening two tables separately recovers a product that blurring them separately lost is a question of its own.

The habit: an operation that commutes with sums may not commute with products #

Blurring, averaging and integrating are linear, and linear operations commute with sums: the blur of two spectra added is the sum of their blurs. They do not commute with products, and a colour is an integral of a product.

The move is to apply a smoothing operation to the quantity that is actually integrated, and to ask, whenever it has been applied to factors instead, whether both factors vary inside the smoothing window. If either is flat, nothing is lost.

The failure mode is to treat each table’s honesty as the calculation’s. Two tables each fit to be summed on a coarse grid are not a product fit to be summed on one, and the pair that breaks the rule — a steep filter under a line lamp — is the pair a filter was designed to be used in.

Where the identity comes from #

That the average of a product is the product of the averages plus a covariance is elementary statistics, and that spectrometer slit functions should be applied to the measured quantity rather than to its factors is a standard caution in spectroradiometry, where the difference between measuring reflectance and measuring radiance is part of every calibration.

How large the separately blurred error is for a notch under a mercury line, that it survives any refinement of the grid, and that two slits can be worse than none, are computed here on closed-form spectra.

Still open: whether sharpening each table restores the product #

The Stearns correction sharpens a table blurred by a triangular slit of one interval, and it is widely applied to spectrophotometer data before colour is computed. Applied to both a sample’s table and a lamp’s, it would move each towards its unblurred form; whether the product of two sharpened tables approaches the one-slit answer, overshoots it, or leaves the covariance where it was, is the same calculation with the correction inserted, and it would say whether corrected data can be trusted where uncorrected data cannot.