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Unbiased Estimator for Distorted Conics in Camera Calibration
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Derives a closed-form, unbiased moment-based estimator for the centroid of circular patterns under radial polynomial lens distortion and uses it for camera calibration, outperforming point, conic, numerical and checkerboard methods.
Problem
Circular calibration targets yield sub-pixel centers, but projection and nonlinear radial lens distortion destroy the conic property, so the projected centroid is biased and conic features are underused in camera calibration.
Approach
Shows that the centroid (first moment) of a distorted conic under polynomial radial distortion is a linear combination of moments of the undistorted conic, gives closed-form moment computation via rotation and translation equivariance plus binomial coefficients, and plugs this unbiased estimator into reprojection-error calibration.
Key results
- On synthetic images with high distortion (d1=-0.4), the proposed estimator recovers fx=599.9±0.09 and d1=-0.40±0.001 versus biased point-based fx=603.1±0.82 and conic-based fx=606.9±0.80, at 2.0 s runtime.
- Numerical unbiased method needs 75-87 s (1600 iterations) to reach similar accuracy, while the proposed method takes 2.0 s.
- On real TIR camera 6D pose evaluation, rotation error is 0.45° for the proposed method versus 1.63° for checkerboard; on RGB, rotation error is 0.24° versus 0.32° for checkerboard.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the moment-based unbiased estimator instead of the checkerboard method | Camera calibration · TIR camera | translation error: 16.5 mm -> 3.1 mm | Helped |
| Use the moment-based unbiased estimator instead of the checkerboard method | Camera calibration · RGB camera | rotation error: 0.32◦ -> 0.24◦ | Helped |
| Use the moment-based unbiased estimator instead of the checkerboard method | Camera calibration · TIR camera | rotation error: 1.63◦ -> 0.45◦ | Helped |
| Use the moment-based unbiased estimator instead of the numerical estimator with 1600 iterations | Camera calibration · RGB camera | rotation error: 0.26◦ -> 0.24◦ | Helped |
| Use the moment-based unbiased estimator instead of the numerical estimator with 1600 iterations | Camera calibration · TIR camera | translation error: 3.3 mm -> 3.1 mm | Helped |
| Use the moment-based unbiased estimator instead of the numerical estimator with 1600 iterations | Camera calibration · Synthetic images, low distortion (d1 = −0.2) | fx deviation from ground truth: 600.1±0.07 -> 600.0± 0.06 | Helped |
Surprising findings
- The checkerboard method, despite being theoretically unbiased, yields much higher reprojection error and pose error on TIR images than the circular-pattern estimator, due to unreliable corner detection under blur and noise.
Limitations
- Only radial distortion is modeled; tangential distortion is not considered.
- Assumes the distortion function is polynomial and invertible within the field of view; high-distortion synthetic setting required slight adjustment of d2.
- Requires circular targets; reliance on centroid of blob pixels with weights set to one in real data.
Experimental setup
Datasets: synthetic images (100 generated, 30 random subsets per trial) real RGB images (Trition 5.4 MP) real TIR images (FLIR A65)
Benchmarks: synthetic calibration images real RGB and TIR calibration sets
Metrics: reprojection error intrinsic parameter error (mean and std) runtime (s) 6D pose rotation and translation error
Baselines: checkerboard point-based estimator conic-based estimator numerical unbiased estimator (Kannala and Brandt)
Builds on: Zhang's calibration method Kannala and Brandt numerical unbiased estimator Heikkila circular pattern calibration
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- Unsupervised Vanishing Point Detection and Camera Calibration From a Single Manhattan Image With Radial Distortion (CVPR 2017 · similarity 0.84)
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