Centers and centroids of the cone-beam projection of a ball
Bibliographic record
Abstract
In geometric calibration of cone-beam (CB) scanners, point-like marker objects such as small balls are imaged to obtain positioning information from which the unknown geometric parameters are extracted. The procedure is sensitive to errors in the positioning information, and one source of error is a small bias which can occur in estimating the detector locations of the CB projections of the centers of the balls. We call these detector locations the center projections. In general, the CB projection of a ball of uniform density onto a flat detector forms an ellipse. Inside the ellipse lie the center projection M, the ellipse center C and the centroid G of the intensity values inside the ellipse. The center projection is invariably estimated from C or G which are much easier to extract directly from the data. In this work, we quantify the errors incurred in using C or G to estimate M. We prove mathematically that the points C, G, M and O are always distinct and lie on the major axis of the ellipse, where O is the detector origin, defined as the orthogonal projection of the cone vertex onto the detector. (The ellipse can only degenerate to a circle if the ball is along the direct line of sight to O, and in this case all four points coincide.) The points always lie in the same order: O, M, G, C which establishes that the centroid has less geometric bias than the ellipse center for estimating M. However, our numerical studies indicate that the centroid bias is only 20% less than the ellipse center bias so the benefit in using centroid estimates is not substantial. For the purposes of quantifying the bias in practice, we show that the ellipse center bias ||CM|| can be conveniently estimated by eA/(π ƒ(≈) where A is the area of the elliptical projection, e is the eccentricity of the ellipse and ƒ(≈) is an estimate of the focal length of the system. Finally, we discuss how these results are affected by physical factors such as beam hardening, and indicate extensions to balls of non-uniform density.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.016 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".