Lens relative illumination and cos4 roll-off explained

Machine vision images often fade toward the corners, and the cause is rarely one thing. Relative illumination (RI) is the datasheet number that predicts how much of the center's brightness your sensor corners will receive, and cos4 roll-off is the part of that fade no iris setting can remove. What follows is the definition, the cos4 law with the arithmetic worked through, the split between the two losses — vignetting and chief-ray angle — and the decision path: match the image circle first, stop down for the vignetting term, then go image-space telecentric or rebalance the lighting when the angle itself is the problem.

What relative illumination measures

Relative illumination is the brightness at a point on the sensor as a percentage of the brightest point in the field, which is almost always the center. A curve reading 100% at image height zero and 57% at the corner of a 2/3" sensor means that corner receives 57% of the center's illumination.

Datasheets print this as the RI curve: image height on the x-axis, relative brightness in percent on the y-axis, one curve per f/#, with markers at the diagonals of common sensor formats. It sits on the same sheet as MTF and distortion; read all three together as how to read lens performance curves shows.

RI is relative, not absolute. Closing to f/8 dims the whole image; the percentage shape is a separate property with its own physics.

Two losses: vignetting and cos4 roll-off

Vignetting is mechanical. Rays from the outer field are clipped by element edges or mechanical stops and never reach the sensor. It bites hardest at open iris, at short focal lengths and in cost-optimized designs, and designers sometimes build it in deliberately to suppress stray light, hold off-axis resolution or cut cost. Stopping down shrinks the ray cone until it clears every stop, and the vignetting term disappears.

Roll-off is radiometric. Even with zero vignetting, irradiance at image height y falls with the fourth power of the cosine of the image-space chief ray angle θ. It follows the angle of the rays, not their number, so no f/# moves it.

Doubling the corner angle from 15° to 30° costs 31 points (87.1 − 56.3). Steep image-space angles come from wide fields, large sensors and short focal lengths, which is why ultra-wide designs live permanently at the steep end — the geometry in fisheye lens selection for M12 board cameras is the extreme case.

Worked example: two focal lengths, two formats

Sensor diagonals are fixed by format: about 8 mm for 1/2", 11 mm for 2/3", 16 mm for 1" and 17.6 mm for 1.1". Roll-off is set at the half-diagonal, the field corner, so each pairing computes like this:

The LS-FFL-1582 is an 8 mm FA lens for 2/3" sensors with a 57.6° field of view — exactly the Pair A geometry.

The LS-FFL-0567 pairs a 16 mm focal length with a 1.1" format, an F2.8-22 iris and under 0.3% distortion, built for 20 MP sensors.

Angle, not sensor size alone, sets the floor: the 1.1" diagonal is 60% larger (17.6 / 11 = 1.6), yet doubling the focal length lifts the floor from 46% to 59%. Both floors still sit far from flat, which is the cue to change geometry rather than exposure. Datasheet curves show the same structure — a typical f/1.4 curve reads 72% at the 1/2" corner and 57% at the 2/3" corner, and both readings rise as you stop down until they stop moving. Those resting levels are the roll-off floors.

Run your own pairing through the focal length calculator:

Match the image circle before anything else

The fix that costs nothing happens at selection. Keep the sensor diagonal inside the image circle with a 5–10% margin: an 11 mm diagonal calls for roughly a 12 mm circle, so a lens rated exactly for 2/3" is a borderline fit and a 1" rating is comfortable.

A sensor larger than its lens's image circle puts the corners outside the illuminated field, where brightness falls steeply toward black. Stopping down does not enlarge the image circle by a usable amount: a lens designed for one format will not serve a larger sensor at any iris setting.

CCTV lenses migrate onto vision cameras because the mounts fit, so check the quoted format first — many CCTV varifocal designs serve diagonals well below 2/3". The CCTV lenses range carries its own format filters for the match.

When roll-off hurts, and when you can ignore it

Roll-off hurts when the brightness level itself carries information: global thresholding without shading correction, gauging where edges cross a fixed gray level, low-light runs where corners need extra gain, and line scan, where uniformity along the whole line is the product. Wide-angle, short-focal-length designs on large formats — Pair A's 46% floor — live with it constantly.

It is harmless when features sit well inside the field or when flat-field correction runs with light to spare: interior defects at 70% of the radius still carry usable signal on Pair B's 59% floor.

Shading correction is not free of charge. Dividing corner pixels by RI multiplies them, and their noise, by 1/RI: ×2.2 at the 46% floor (1 ÷ 0.46 = 2.17), ×1.15 at an 87% corner. Set gain deliberately, as automatic iris, exposure and gain control in machine vision describes, instead of letting the correction amplify noise for you.

The fixes, in order of cost

1. Buy the right image circle. It costs nothing extra at selection; every later fix is damage control.
2. Stop down one or two steps to clear the vignetting term. f/1.4 → f/2.8 costs 4× the light ((2.8 / 1.4)²) and leaves the cos4 floor untouched.
3. Go image-space telecentric. Chief rays arrive perpendicular to the sensor, so θ = 0 and cos⁴θ = 1: the floor disappears and illumination goes flat. Bi-telecentric designs add object-side telecentricity, so magnification — and with it the object space pixel size — stops wandering with part position. The trade against ordinary lenses is covered in entocentric vs telecentric lenses: when to use each, and the gauging case in when you need a telecentric lens for gauging.

For measurement work that needs even illumination and constant magnification at once, bi-telecentric designs are the standard answer:

The LS-TCL-1422 runs 0.166x over a 1" sensor at a 258 mm working distance; the LS-TCL-2118 gives 1x on 2/3" at 108 mm; the LS-TCL-0885 is an M58 bi-telecentric at 0.287x and 288 mm for APS-H formats. The rest of the range sits in the telecentric lenses category.

4. Rebalance the lighting. Add lamps near the edges of the object or dim the center; this pre-compensates roll-off on the object side for the cost of fixtures, and detection-only applications rarely need more.
5. Fit an apodizing ND filter. Its transmission varies radially, densest at the center, pulling the center down toward the corner level and flattening RI at the cost of overall light. Placement changes what a filter does; telecentric lens filters: where to place them and what they change walks through it.

A bench test that separates the two losses

1. Shoot a matte white target filling the field, lit evenly, at your working f/#. Plot brightness along the horizontal center line and note the corner-to-center ratio.
2. Stop down two stops, f/1.4 → f/2.8, and raise exposure 4× ((2.8 / 1.4)²) so the overall level stays put.
3. Corners brightened relative to the center: vignetting dominated, and more stopping down still pays.
4. Ratio unchanged: you sit on the cos4 floor, and only telecentric optics, edge-weighted lighting or an apodizer moves it.
5. Apply shading correction and re-check corner noise against the inspection threshold before trusting the corrected image.

Choose by case

• Corners near black on a sensor bigger than the lens rating: wrong image circle; replace the lens, no iris setting helps.
• Corners dark only at open iris and recover when stopped down: vignetting; stop down and budget the light, or pick a design with more aperture headroom.
• Corner level stuck near the floor your geometry computes — 46% in Pair A — whatever the iris: roll-off; image-space telecentric when you measure, edge-weighted lighting when you detect.
• Gauging or dimensional measurement of any kind: telecentric outright, because flat RI and constant magnification arrive together; the specs that matter are in telecentric lens distortion and telecentricity explained.

Checklist before you order

Selection:

• Sensor diagonal fits the image circle with a 5–10% margin.
• RI read at your corner: half-diagonal on the x-axis, your working f/# curve.
• Image-space chief ray angle at or under 7° for small-pixel sensors (cos⁴ 7° ≈ 97%).
• Telecentric decided by task, not habit: bi-telecentric for gauging, lighting compensation for detection.

Bench:

• Flat-field profile on a matte white target at the working iris.
• Two-stop stop-down test to split vignetting from roll-off.
• Shading-corrected corner noise checked against the inspection threshold.

Next step: send the sensor diagonal, working distance and target field into the lens finder, start the configurator at the lens step, or browse the machine vision lenses category by format. The lenses topic page collects the guides that surround this decision.