Analytical eccentricity correction for spherical projection based on tangent cone symmetry

Spherical targets are extensively employed in high-precision 3D vision measurement for system calibration and industrial positioning due to their omnidirectional rotational invariance. However, under the perspective projection model, the oblique intersection between the camera tangent cone and the image plane forms a projected conic contour, which appears as a quasi-elliptical contour in practical imaging. This projection mechanism induces an inherent systematic eccentricity error between the detected contour center and the true perspective projection of the sphere center. Existing nonlinear optimization-based compensation methods are limited by initial value dependency, high computational latency, and susceptibility to local optima. To address these challenges, we propose an analytical eccentricity correction algorithm for sphere-center projection based on spatial geometric constraints. We integrate sub-pixel edge extraction with point-level distortion correction in preprocessing, and derive a closed-form non-iterative solution for unbiased sphere center localization via tangent cone spatial symmetry. Experiments and simulations demonstrate that optimized multi-directional illumination suppresses imaging artifacts and that the proposed model corrects perspective geometric bias while significantly reducing radius measurement error, thereby providing a robust and efficient analytical solution for precision vision metrology.