The shutter samples the lamp

A person under a lamp flickering at a hundred hertz sees one steady colour, and the reason is Talbot’s law: above the fusion frequency the eye reports the time average of what arrived, exactly. This site asserts that exactness to a part in ten thousand billion, and it has a corollary that the fused colour is a colour nothing emits.

A camera does not integrate over the same interval. Its shutter is open for a thousandth of a second, or a two-thousandth, and a hundred-hertz cycle is ten milliseconds long — so an exposure samples a part of a cycle, and which part depends on when the shutter happened to open.

The claim #

A camera records what nobody can see and misses what everybody does.

• Two frames of the same scene differ by 2.37 stops at a thousandth of a second under a hundred-hertz lamp, from nothing but when the shutter opened.
• They differ in colour too, by ΔE00 24.8 from the colour the eye fuses to, because a hybrid-dimmed lamp is switched between two drive currents and two drive currents are two spectra.
• A shutter long against the period averages the flicker away: at a twenty-fifth of a second the range is 0.000 stops, which is Talbot’s law arriving in a camera.
• And a rolling shutter turns the same flicker into bands whose count is a camera setting. The lightest band and the darkest are 2.37 stops apart, and ΔE00 5.4 apart in colour at matched luminance.

The lamp #

The lamp is a phosphor-converted white LED under hybrid dimming: switched between two drive currents rather than between on and off. That is a real technique and it exists because the two obvious alternatives each have a defect.

Pulse-width modulation between on and off preserves colour exactly — the spectrum during the on phase is the full-current spectrum, and the average of a spectrum with itself is itself — at the cost of a hundred per cent modulation depth. Reducing the current instead preserves the waveform and changes the colour, because a diode’s peak moves with current and its junction cools as it dissipates less.

Switching between two currents is a compromise, and its consequence is the subject here: the lamp genuinely emits two different spectra, alternating, above fusion. A person sees the mixture. A camera sees one or the other.

What a short exposure does #

The integral is elementary. The sensor receives whatever arrived while the shutter was open, so a shutter of duration T opening at phase φ receives the integral of the waveform over that window.

When T is long against the period the window contains many whole cycles and the phase does not matter: every frame gets the mean. When T is short the window sits inside one phase or straddles a transition, and the answer runs from the high level to the low one.

At a thousandth of a second under a hundred-hertz lamp the window is a tenth of a cycle, so most openings land entirely inside one phase or the other. The result is a bimodal distribution rather than a smooth one, and the range is 2.37 stops — a factor of five in exposure between two frames a photographer would describe as identical.

The colour goes with it. The high phase’s spectrum and the low phase’s are ΔE00 24.8 apart from the fused colour at the extremes, most of which is a lightness difference. Held at matched luminance the two are still 5.4 units apart, which is the part an exposure correction cannot fix.

The range is flat, then it falls off a cliff #

The observation that a shorter shutter does not average anything is right and is a special case of something sharper, which the square wave makes exact.

For any shutter shorter than half the lamp’s period, the range is the full ratio between the two levels and does not depend on the shutter at all. A window under five milliseconds can sit entirely inside the high phase or entirely inside the low one, so the extremes it can sample are the levels themselves, and shortening it further changes nothing about what it can catch. That is why a thousandth of a second and a five-thousandth report the same 2.37 stops.

Past half a period the window must contain some of both phases, and the range starts to close. It reaches exactly zero when the window is a whole number of periods, which is the camera setting everybody knows:

The collapse happens in the last ten per cent. The range is still above a tenth of a stop at 1/110 of a second and only drops below it past 1/105 — so of the whole interval between the flat region and the first null, nine tenths of it is spent above a tenth of a stop.

That is a sharper rule than use a long shutter. There is no partially safe shutter speed. Every setting faster than 1/200 is equally bad, everything between 1/200 and about 1/105 is bad in decreasing degree, and the safe settings are a discrete set — the reciprocals of whole multiples of the lamp’s frequency: 1/100, 1/50, 1/33, 1/25, 1/20 under a hundred-hertz ripple.

Which is exactly what a camera’s anti-flicker mode does, and the essay mentions the mode without connecting it to the nulls. The mode is not choosing a shutter long enough to average; it is choosing a shutter that lands on a null, and the difference matters because the two prescriptions give different advice. Long enough to average suggests a monotone trade in which a slightly longer shutter is slightly better; the null structure says a shutter one part in twenty short of a null is nearly as bad as a thousandth of a second, and a shutter on the null is perfect.

And it says which direction to err in. A shutter slightly longer than a null is in the flat approach to the next one and recovers quickly; a shutter slightly shorter is on the steep side and is much worse. So a camera locking to 1/100 under a fifty-hertz mains should round up rather than down, and a photographer choosing 1/125 because it is a whole stop is choosing a setting that carries half a stop of banding while 1/100 carries none.

Two things this does not cover, and both would soften the cliff. A duty cycle other than a half moves the flat region’s edge — the range stays at its maximum for any shutter shorter than the shorter of the two phases, which for a deeply dimmed lamp is a much smaller window. And a waveform with any smoothing at all — a filament’s thermal mass, a phosphor’s persistence, a driver with a capacitor — rounds the corners of the square wave and rounds the corners of this curve with them, which is why the effect was a nuisance under fluorescent lighting and is a defect under a directly driven diode.

The bands #

A rolling shutter does not expose the whole sensor at once. It scans, exposing each row a little later than the one above, so a frame is not an instant — it is a diagonal slice through time, typically a fiftieth or a sixtieth of a second from the first row to the last.

Under a steady light nothing follows from that except motion skew. Under a flickering one it means each row is exposed at a different phase, and the frame carries the waveform written down its own height.

The shutter is one stage of a chain, and it is worth seeing where in that chain the sampling this essay is about actually happens.

The count is arithmetic: readout time times lamp frequency. A sixtieth-of-a-second readout under a hundred-hertz lamp gives 1.7 bands, which is why the effect usually looks like one broad dark stripe drifting rather than a stack of stripes. Change the frame rate and the count changes; change countries and it changes, because a fifty-hertz mains gives a hundred-hertz ripple and a sixty-hertz mains a hundred and twenty.

The contrast between the lightest row and the darkest is 2.37 stops, the same range as the single-frame case, because the rows are sampling the same distribution. And the two extreme bands differ in colour by ΔE00 5.4 at matched luminance — so the bands are not merely light and dark stripes, they are stripes of two different colours, and a photographer correcting the exposure of one band will not have corrected its hue.

Why the eye is not the arbiter #

The temporal machinery on this site says what a person gets: the modulation left after the eye’s own temporal filter, in multiples of its own threshold. At a hundred hertz the luminance channel is far past fusion and the residual modulation is below one, so the flicker is not there.

That is a strong statement and it is worth being precise about what it does not say. It does not say the flicker has no effect — a modulation below the fusion frequency of the detection task can still produce a stroboscopic effect on a moving object, and this site computes that separately. It says the field looks steady, and a steady field is what a photographer believes they are shooting under.

So the camera is not making a mistake. It is integrating over a different window than the eye, and the two windows disagree by a factor of ten to fifty in duration. That is the whole mechanism, and it is why the failure is invisible until it is in the picture.

The repair, and its cost #

The repair is a shutter long against the period, and it works completely: at a twenty-fifth of a second the exposure range is 0.000 stops to three decimal places, which is Talbot’s law arriving through a different aperture.

Its cost is what a long shutter always costs. Motion blur, a smaller working aperture or a lower sensitivity, and — for video — a frame rate that has to divide into the mains frequency. That last constraint is why cinema cameras have a shutter angle control and why shooting at twenty-four frames a second under fifty-hertz lighting is a known nuisance.

The other repair is at the lamp. A driver switching at ten kilohertz rather than a hundred puts the period at a tenth of a millisecond, so even a ten-thousandth-of-a-second exposure contains whole cycles. Drivers of that kind exist, cost slightly more, and are specified by a flicker figure of merit — which is a measurement made with a photodiode and a bandwidth, and therefore is a statement about a detector rather than about an eye.

Two detectors, one waveform #

The compact statement of the whole essay is that a lamp’s waveform is read by two detectors with different windows, and everything else follows from the ratio between them.

The eye’s window is not a shutter and the comparison is loose in that respect — the temporal filter is a smooth low-pass rather than a rectangle — but the ratio of durations is what decides the answer and the ratio is not loose at all.

This is the same shape as a finding the site has already made about spatial filtering: reading a filtered signal at a point and reading it as components can give answers a hundredfold apart, and the comparisons that move are the ones where the two readings have different bandwidths. A shutter and an eye are two readings of one signal with bandwidths a factor of ten apart, and this is what that costs in time rather than in space.

What was computed, and how #

The two spectra come from the current-dimming model, which is not a scaling: a diode driven at a fraction of its current has its peak moved by a stated nanometres per decade, and its junction is cooler because it is dissipating less, so all three thermal slopes run backwards. That is why the two phases are two colours rather than two brightnesses.

The exposure is a direct numerical integral of the waveform over the shutter window at five hundred steps, which is far finer than any structure in a square wave and is chosen for simplicity rather than for accuracy — the analytic answer for a rectangular window on a square wave is available and would say the same thing.

The colour comparison is made twice: once with the brightness left in, which gives the large number, and once with both spectra scaled to a common luminance, which isolates the hue difference. Quoting only the first would overstate the case, and assertTheBandsAreTheCamerasOwn asserts on the second.

What a photograph of this looks like #

The artefact has a signature that separates it from everything else a photograph can go wrong with, and the signature is worth writing down because it is diagnostic.

It is stationary in the frame and moves between frames. A band from a mains-driven lamp drifts slowly up or down the picture at the beat frequency between the readout and the ripple, which is why it is obvious in video and easy to mistake for a sensor fault in a single still.

It is independent of the subject. Nothing in the scene produces it, so it crosses object boundaries, sits over faces and sky alike, and does not follow anything’s edges.

And — the part that is new here — it carries a hue. A band under a hybrid-dimmed lamp is not merely darker; it is a few units warmer or cooler, so correcting the exposure of one band leaves a stripe of the wrong colour where a stripe of the wrong brightness used to be. That is a repair that makes the picture look stranger rather than better, and it is the reason the automatic corrections work on brightness only.

Where it stops #

The lamp is one lamp with one dimming scheme. A lamp dimmed by pulse width between full current and zero has one spectrum and two levels, so the bands are light and dark and not two colours; a filament has a thermal mass that filters the mains ripple almost entirely; a fluorescent tube’s phosphor persistence does the same more weakly. The colour half of this result belongs to the hybrid scheme.

The rolling shutter model is a ramp of exposure times down the frame with no readout noise, no gain changes and no per-row calibration, all of which real sensors have and all of which would modify the amplitude rather than the existence of the bands.

And nothing here is about the eye’s stroboscopic effects, which are a separate mechanism with a separate calculation on this site. A field can be fused and still produce a visible artefact on a moving object; a camera can record bands from a field that is fused and produces no such artefact. Those are three different questions about the same waveform and keeping them apart is most of the discipline.

Who found it, and when #

Talbot’s law is from the 1830s and is one of the oldest exact results in visual science. Rolling shutters arrived with CMOS sensors in the 1990s, and banding under discharge lighting was a known defect from the start — the mitigation, a shutter locked to a multiple of the mains period, is in every camera’s firmware and is offered to users as an anti-flicker mode.

What has changed is the lamp. A fluorescent tube’s phosphor smooths its ripple and a filament’s thermal mass smooths it further, so the modulation depth under twentieth-century lighting was tens of per cent rather than a hundred. A directly-driven diode has neither, and the depth is whatever the driver leaves.

So a defect that was a nuisance in fluorescent-lit factories became a general problem when the lamps changed, and the part of it that is a colour difference rather than a brightness one arrived with dimming schemes that switch current instead of switching on and off. That last step is recent enough that the artefact does not have a settled name.

Where the ladder goes next #

Two directions, and the first is a measurement nobody publishes. A lamp’s flicker is specified by a percentage and an index, both of them luminance quantities. Neither says anything about whether the two phases are the same colour, and a lamp could score identically on both while emitting two spectra ΔE00 five apart. A chromatic flicker index is one integral away and would separate lamps the current figures cannot.

The second is the eye again. A field can be above fusion for detection and still produce a chromatic afterimage or a stroboscopic artefact, and this site now has the temporal channels, the adaptation clock and the two spectra in the same place — which is what a computation of what an alternating pair of colours above fusion does to an adapted observer would need.