Stereo calibration accuracy: the measurement volume decides

A stereo pair promises micron-level measurement, and the first question in every design review is: how accurate will it be? The honest answer is a fraction of the volume you calibrate, not a figure on a camera data sheet. Here you get the planning rule, the triangulation formula behind it, a worked example that moves the same two cameras between two volumes, and the target and verification choices that decide whether a rig lands at 1:3,000 or 1:10,000 of its volume. It is written for anyone specifying a stereo rig for dimensional inspection or robot guidance.

Plan on 1:10,000 of the depth you must cover

Published industrial stereo systems give the scale to plan against. One rig that swaps optics to span volumes of interest from 70 × 50 × 20 mm³ to 390 × 290 × 200 mm³ reports 2 µm to 20 µm of measurement precision — each figure is 1:10,000 of that volume's depth extent (20 mm gives 2 µm, 200 mm gives 20 µm). Treat 1:10,000 of the working distance as the system budget for a disciplined calibration. That means reprojection residuals at or below 0.1-0.2 px, targets filling the volume and temperature held. The triangulation physics alone is better than that; distortion residuals, target fit and thermal drift consume the difference.

The decision follows. Take the tightest depth tolerance t and require the budget to be t/10 for metrology work or t/4 for pass/fail gauging — the same thinking as the 10 to 1 accuracy and repeatability rule. With a 1:10,000 budget that caps the working distance at about 1,000 × t: a 50 µm tolerance means a 50 mm working distance at 10:1, or about 125 mm at 4:1. That line decides whether stereo fits the task before any hardware is specified.

The formula: distance squared up, baseline and focal length down

The terms, all in millimetres except the pixel quantities:

• d — disparity in pixels between the rectified images
• f_px — focal length in pixels, f ÷ p
• B — baseline, the camera separation
• Z — depth, the working distance to the feature
• p — pixel pitch
• σ_d — disparity precision per feature: 0.1 px on marked targets or projected patterns, 0.3-1 px on natural texture
• σ_Z — depth error, a first-order estimate
• σ_xy — lateral error per pixel of feature location

Read the scaling: halve the working distance and depth error falls fourfold; double the baseline or the focal length and it halves, at the cost of overlap or field of view. Lateral error is the object-space pixel size, and marked features locate to about a tenth of a pixel, so Z is the weak axis at stand-off. Disparity precision is where lighting earns its keep: a 0.3 px texture fit is three times the error of a 0.1 px marker fit at identical geometry.

One geometric cap: disparity at the nearest point of the volume must stay below roughly 70% of the sensor width, or the two views stop overlapping. That cap is why small-volume rigs shorten the baseline as they shorten the distance.

Pairing two different sensor formats is workable, but the rectified images must share one scale; rescaling versus recalibrating is its own decision — see Stereo calibration resolution mismatch: rescale or recalibrate?.

Worked example: the same cameras, two volumes

Both configurations use two 12 MP area scan cameras, 4096 × 3000 px with 3.45 µm pixels and 25 mm lenses on a rigid beam. Config A inspects a 300 mm welded assembly at 600 mm working distance with a 200 mm baseline. Config B measures small machined parts at 120 mm with the baseline cut to 50 mm.

Config A, step by step:

1. f_px = 25 mm ÷ 0.00345 mm = 7,246 px
2. Field of view = 14.13 mm × 600 ÷ 25 = 339 mm × 248 mm
3. d = 7,246 × 200 ÷ 600 = 2,415 px (59% of the 4,096 px width)
4. σ_Z = 600 mm × 0.1 ÷ 2,415 = 0.025 mm ≈ 25 µm, a 1:24,000 relative figure
5. Lateral = 0.083 mm per pixel, about 8 µm at 0.1 px feature precision

Config B:

1. Field of view = 14.13 mm × 120 ÷ 25 = 68 mm × 51 mm
2. d = 7,246 × 50 ÷ 120 = 3,019 px (74% — no room left to move closer)
3. σ_Z = 120 mm × 0.1 ÷ 3,019 = 0.004 mm ≈ 4 µm, a 1:30,000 relative figure
4. Lateral = 0.017 mm per pixel, about 2 µm

Scaling the whole rig down kept relative error near 1:30,000 while the absolute figure fell with the volume — that is the measurement volume doing the work. The decisions: assembly-level features with a ±0.3 mm tolerance sit in Config A with a 5:1 margin (300 µm against a 60 µm budget). A 100 µm tolerance needs 10 µm at 10:1; Config B's 4 µm triangulation figure says a good calibration reaches it, the 12 µm budget says only just — measure an artifact before committing. A 20 µm tolerance defeats both configurations and calls for the shorter-standoff techniques at the end.

Calibration targets: board, body or projected grid

Fill roughly half to two-thirds of each camera's field of view in every pose; published evaluations of large-FOV stereo calibration find that small central boards leave lens distortion unconstrained at the edges and bias the scale there. Spread the poses through the depth of the volume too: depth is only constrained where calibration points exist. The capture procedure, pose by pose, is in Stereo camera calibration from paired checkerboard images; target size versus distance is in Checkerboard calibration fails at distance: depth camera target size.

Verify: residual, artifact, temperature

Reprojection RMS at or below 0.1-0.2 px on marked targets is the entry ticket, not the proof: camera extrinsic calibration accuracy beyond reprojection error shows how small residuals coexist with wrong scale. Measure a certified step gauge or ball bar at several positions through the volume — near, far, corners. The mean deviation from the certified length is the bias; the spread over ten repeats is the repeatability. Inside tolerance divided by ten, the rig is an instrument; outside, an indicator. The same gate applies when the camera rides a robot, per hand-eye calibration accuracy verification.

Temperature moves the scale through the baseline. A 200 mm aluminium beam grows 4.6 µm per kelvin at 23 ppm/K. A 10 K swing adds 46 µm to the baseline; at 600 mm depth that shifts measurements by 600 × 46 ÷ 200 = 138 µm — more than twice the Config A budget. Steel at 12 ppm/K cuts it to about 70 µm. Hold the cell within a few kelvin, or recalibrate on schedule.

When stereo is the wrong tool

If the tolerance demands better than 1:10,000 of the available stand-off, or the task needs full-surface maps rather than feature positions, change technique instead of chasing cameras. Laser profilers put micron-class depth on one profile per scan at short stand-off; the MV-DP4180-03P is a 3D laser profile sensor of that class. Structured-light stereo covers a surface in one shot and ships factory-calibrated: in the 3D sensors category, the MV-DLS300P-04 is a laser galvanometer stereo camera, and the MV-DPS200P-02 is a projected structured-light 3D camera.

Building the pair yourself starts in the area scan cameras category — two matching sensors on a rigid beam — with the 25 mm class drawn from the lenses category. Size the rig with the camera finder and the lens finder, or lay out the whole chain in the camera configurator. If a volume and tolerance pair looks marginal on these numbers, send the drawings through the contact page before ordering hardware.