Machine vision lens distortion: pincushion, barrel and calibration

Every lens bends straight lines. Distortion is the geometric error left when everything else is perfect: points land slightly off the place a rectilinear projection would put them, by a percentage that grows toward the corner of the image. It never blurs anything, so a system can look tack-sharp and still measure wrong. Below is the definition, the formula that turns a distortion percentage into pixels and then into millimetres of measurement error, and a decision path: when a standard lens plus a calibration grid is enough, when a low-distortion fixed lens is worth it, and when only a telecentric lens at its rated working distance holds the tolerance. A worked example converts a real 0.5 mm field error into a distortion number and a fix.

What distortion is, in one number

Distortion is the percentage difference between the distance of an image point from the image center and that same distance in an undistorted image. Read it as the deviation of an object's imaged size from its true size. Positive radial distortion is pincushion; negative is barrel.

A point that lands 198 px from center where the ideal image would put it 200 px away carries (198 − 200) / 200 = −0.01 = −1%: barrel, and the point sits 2 px closer in than it should. Two properties follow. Distortion varies with radial position and can change sign; barrel mid-field with pincushion at the edge is common enough to have its own name, mustache. And it is smooth: as a transform from the real world into the virtual image space the lens creates, it stays close to a second- or third-order polynomial in radius, which is exactly why a calibration target can map it out.

Radial distortion displaces points along the radius, so it is symmetric about the center. Tangential distortion, from decentered or tilted elements, displaces them sideways and breaks that symmetry. The straight-line reference all of this is measured against is rectilinear projection, where scene lines stay straight.

The human eye compensates up to 1–2% distortion. Photographs look fine at levels that would wreck a measurement.

How much distortion each lens type carries

Ordinary optical designs run from a few percent to tens of percent, worst in wide angles. A quality telecentric lens holds within 0.1%, and even that, on a high-resolution camera, approaches a pixel of measurement error. That is why most measurement applications calibrate distortion in software instead of chasing it in glass.

The corner of a 200 × 150 mm field sits 125 mm from center, worked out below, so a spec translates to millimetres directly: 1% means 1.25 mm there.

The last row's millimetres use the pixel scale from the worked example. A fixed lens with a published spec is the usual starting point for the middle row:

The LS-FFL-0567 is a 16 mm, 1.1", 20 MP C-mount machine vision lens rated at <0.3% distortion.

From percent to pixels: the deciding conversion

Take a 2448 × 2048 px sensor, a common 5 MP resolution used here as an example input, and substitute your own. The corner sits sqrt(1224² + 1024²) = sqrt(1,498,176 + 1,048,576) = sqrt(2,546,752) ≈ 1596 px from center. A 0.1% spec therefore moves the corner by 0.001 × 1596 ≈ 1.6 px: the best figure in the table still costs the better part of two pixels, raw. Any metrology working to sub-pixel accuracy calibrates, telecentric or not.

The other half of the conversion is the pixel scale: object size per pixel equals field of view divided by pixel count. That is the arithmetic the camera resolution calculator runs when sizing a sensor for a smallest feature.

Worked example: 0.5 mm error on a 200 × 150 mm field

A positioning cell reports picks that are consistent but off target by about 0.5 mm at the worst point. The field is 200 × 150 mm and the camera is the 2448 × 2048 px sensor used above. The question: is 0.5 mm consistent with ordinary lens distortion, and what fixes it?

Inputs and rule:

• Worst placement error: 0.5 mm, measured on the part.
• Field of view: 200 × 150 mm, so the corner is the worst point.
• Rule: distortion = error ÷ distance from the image center; the corner sits at half the field diagonal.

Arithmetic:

1. Corner radius = sqrt((200/2)² + (150/2)²) = sqrt(100² + 75²) = sqrt(10,000 + 5,625) = sqrt(15,625) = 125 mm.
2. Distortion needed to produce 0.5 mm there = 0.5 ÷ 125 = 0.004 = 0.4%.
3. Pixel scale = 200 ÷ 2448 = 0.0817 mm/px.
4. Error in pixels = 0.5 ÷ 0.0817 ≈ 6.1 px. Cross-check: 6.1 ÷ 1596, the corner radius in pixels from above, = 0.38% ≈ 0.4%. Same answer both ways.

Decision: 0.4% is unremarkable for a standard lens, so the optics explain the error and focus is not the culprit. A full-field calibration driving residuals to 0.1–0.2 px leaves 0.008–0.016 mm, roughly 30× inside the 0.5 mm tolerance. Keep the lens, add the grid. If the tolerance were ±0.05 mm instead, a ≤ 0.1% telecentric, which gives 0.125 mm raw at this corner, plus the same grid is the pair that holds it.

The same question on a 700 mm-wide field corrected with nine points answers differently: 3 × 3 nodes across 700 mm leaves 700 ÷ 3 ≈ 233 mm between nodes, and the polynomial bends unmeasured between them. Corners keep drifting. Use a dense target, not more gain.

Working distance and squareness change the number

Distortion depends on the distance to the object as well as on the glass, so a telecentric lens must run at its rated working distance; the published figure is computed there. The LS-TCL-1972 is a 1x object-side telecentric lens with a 440 mm working distance for 2/3" sensors. The LS-TCL-1422 is a 0.166x bi-telecentric lens with a 258 mm working distance for 1" sensors. Move either off its standoff and the distortion spec no longer describes the image.

Mount the lens precisely square to the part. A tilt adds keystone, the thin-prism effect, which is asymmetric and hard to calibrate out because it lacks the radial symmetry the correction model exploits. The focus mechanism contributes too: mechanical play or a decentered moving group adds symmetric or asymmetric distortion terms, one reason to lock focus after calibrating and to prefer a rigid barrel, which machine vision lens anatomy covers along with the rest of the barrel.

Why telecentric magnification ignores depth is covered in telecentric lens distortion and telecentricity; the case against ordinary lenses for gauging is argued in entocentric vs telecentric lenses.

Two C-mount telecentrics from the telecentric lenses range show the spread:

All 2 share: Mount C-mount. Values as printed in each product's data sheet; open a part for its drawing and full specifications.

Calibrating distortion out with a target

A calibration plate does two jobs in one pass: it fixes the pixel equivalent, in millimetres per pixel, and it maps the distortion. That is why most measurement applications calibrate rather than pay for distortion-free glass.

The radial share is the easy half. Fit the second- or third-order polynomial and residuals land well under a pixel. The tangential and keystone shares are asymmetric and stubborn: square the camera first, then let the full model absorb what is left.

Target choice matters as much as fit quality. The pattern must span the field with margin, per how much margin a calibration target needs, and a checkerboard should be odd by even so orientation is never ambiguous, per the odd-by-even checkerboard rule.

After the fit, read average and maximum residuals and hold them to targets you set, as camera calibration residual values explains, then prove the correction on a gauge part measured at center, mid-radius and corner. The correction workflow itself, pixel-to-mm included, is in lens distortion correction for accurate pixel-to-mm measurements, and measuring barrel distortion correction accuracy is the acceptance test.

Targets live with the lens accessories. The LS-LXR-0029 is a 76 x 76 mm dot-grid calibration board with 0.5 mm dots.

The decision: which lens when

Compute the allowed distortion before opening a catalog: allowed distortion = tolerance ÷ corner radius. On the worked field, 0.5 ÷ 125 = 0.4%.

• Presence, counting, code reading: a couple of percent changes nothing these algorithms look at. Any covering lens from lenses will do.
• Allowed distortion 0.2–0.5%: a low-distortion fixed lens plus a one-time grid calibration. The cheapest accurate answer.
• Allowed distortion below about 0.1% of the corner radius, or several feature heights in one image: a telecentric lens at its rated working distance, where magnification stops changing with depth, plus the same grid for the residual. Telecentric lens selection: magnification and image circle walks that match.
• Fields of 0.5 m and up: calibration is the lever, not glass; the polynomial does work no affordable lens does.

Start in the lens finder, or let the lens step of the configurator carry the sensor, field and standoff with you.

When the usual answer is wrong

• The calibration goes stale the moment anything moves. Refocusing shifts an internal group, a changed iris moves best focus, and remounting re-seats the lens differently. Recalibrate after any of these.
• A distortion figure belongs to one working distance, one wavelength band and one sensor size. Compare specs at the same format; board-camera lenses and varifocals drift most when re-zoomed.
• Keystone masquerades as distortion. If a radial fit leaves an S-shaped residual, the signature a lens distortion vector map exposes, the camera is tilted. Shim it square instead of feeding the polynomial.
• Datasheets may quote TV distortion, which understates the geometric distortion you actually correct. Geometric vs TV distortion shows the gap; compare like with like.
• The best telecentric spec is not zero: the pixel arithmetic above still leaves error at the corner before the grid.

Checklist before you trust the measurement

• Allowed distortion = tolerance ÷ corner radius, written beside the lens spec.
• The spec is geometric distortion at your sensor size, not TV distortion.
• The sensor diagonal fits the image circle; corners are where distortion and coverage both bite.
• The lens sits square to the part, shimmed if needed; C-mount flange distance and shims covers the mechanics.
• Calibration ran at the running working distance; focus and iris are locked.
• A gauge part measures in-spec at center, mid-radius and corner, and again after any mechanical change.
• The target stays on the shelf: recalibration is maintenance, not failure.

Borderline arithmetic or an odd geometry? Send the field size, tolerance and standoff through contact and we will run these numbers against the catalog with you. More lens fundamentals sit on the lenses topic page.