The calibration object
From the IR marker to the blob, the white threshold, partial occlusion, resolution and fps, the right diameter, recognising the T from anonymous points, the terna and the ideal object.
Transcript
1. From marker to blob
A marker is a retro-reflective sphere. The ring of infrared LEDs around the lens lights it and the sphere sends the light back: in the monochrome image it is a bright disc on a dark background.
Here the disc is drawn pixel by pixel, as the sensor sees it. The yellow cross is the true centre of the sphere.
The program applies a white threshold: it keeps the pixels brighter than the threshold (blue outline) and computes their centroid, weighting each pixel by how much it exceeds the threshold. That is the red cross.
With a dark IR background the error is a few hundredths of a pixel, even for a 4-pixel marker. We see it in HD, Full HD and 4K: in 4K the same marker is twice as big.
2. The white threshold
How high should the threshold be? The chart shows the centroid error versus threshold, for 4, 8 and 16 pixel markers.
Between 15% and 45% the error is minimal: about 0.01 pixel. Above 75% the threshold cuts the rim of the disc, which carries information, and the error rises.
We choose 40%: high enough to cut weak infrared reflections, low enough to use the whole disc.
3. Partial occlusion
The real enemy is not noise. When an arm, the T’s rod or another marker hides part of the sphere, the centroid moves towards the visible part.
With a third of the disc hidden the error is 0.4 pixel for a 6-pixel marker and 0.8 pixel for a 12-pixel one: it grows with the diameter, and it is 30–100 times the noise.
In millimetres that error is proportional to the physical size of the sphere. That is why bigger markers are not better, and why deformed blobs must be discarded.
4. Resolution and fps
Now full calibrations: 8 cameras on the walls of a 6 by 6 metre room, a person moving the T for 20 seconds, the bundle adjustment.
With 120 degree lenses a 30 mm marker at 3.6 metres measures about 4.6 pixels in Full HD, 3 in HD and 9 in 4K.
Resolution matters: p95 error of about 0.29 mm in 4K against 0.43–0.44 in HD and Full HD at 60 fps. Frame rate matters little for calibration: the T moves slowly and blur is small even at 30 fps.
5. The right diameter
How big should the markers be? Error versus diameter, for the three resolutions.
Below about 3 pixels at the median distance far markers disappear and the calibration fails: in HD below 30 mm, in Full HD below 14–20 mm.
Above that limit the error grows again with the diameter, because of partial occlusion. Best: 30 mm in HD, 20 mm in Full HD, 10 mm in 4K.
6. Recognising the T from anonymous points
The camera does not know which blob is which marker. The program tries every assignment and for each one looks for the affine map that takes the T drawing onto the blobs.
White numbers: the right labelling. Red numbers: the best wrong labelling. The margin is the difference between their residuals: the bigger, the safer the recognition.
Asymmetric 4-marker T: margin about 0.12. Symmetric T: zero margin, left and right always swap. A 5-marker T: good margin, but it becomes ambiguous when a reflection appears.
With 3 markers the margin is always zero: an affine map has 6 degrees of freedom and 3 points leave nothing to check.
7. Optimising the shape
Margin of the 4-marker T versus the junction offset and the stem length, with a 75 cm bar.
The margin grows with the asymmetry and the stem, then flattens. The optimum is at a 14 cm offset and a 45 cm stem: margin 0.152, 28% more than the T of the preliminary study.
In calibration the 4-marker T leaves 0.05% of wrong observations; the 5 and 6 marker variants 0.25%, the L 0.7%.
8. The static terna
The terna standing at the origin fixes the reference frame. A 3D terna beats a planar L: 0.63 against 0.74 mm with 75 cm arms.
But what matters most is how precisely it is built: with 0.25 mm of tolerance the error rises from 0.68 to 1.6 mm, with 1 mm to 4.4 mm.
About half of this error is a rigid shift of the whole volume: it weighs on absolute measurements more than on joint angles. The terna must be built or measured within 0.1–0.2 mm.
9. The ideal object
In short, the ideal object for IR markers and monochrome cameras.
A T with 4 spheres, 75–100 cm bar, junction about 14 cm off-centre, 45 cm stem. Small markers: about 3 pixels at the median distance. Threshold at 40%.
And a precise 3D terna. In the next video: how many cameras are needed and where to put them.