Ray-tracing imaging model and calibration method for cameras with position unconstrained spherical mirror

To enable compact design and wide field-of-view measurement in narrow spaces, catadioptric vision systems with spherical mirrors offer advantages of low cost, wide coverage, and isotropy. However, their curvature and spherical aberration limit measurement precision and design flexibility, hindering applications in confined environments. To overcome these limitations, we propose a ray-tracing imaging model and a two-stage calibration method for position unconstrained spherical mirrored cameras. The imaging model describes the true light propagation path without constraining mirror position, establishing a forward–backward precise mapping between three-dimensional spatial and two-dimensional image points without spherical aberration. The calibration method includes an initial stage that estimates parameters using the extracted mirror profile, and an optimized stage that refines them via bundle adjustment. Simulations and experiments demonstrate that the imaging model conforms to physical principles, and the calibration method exhibits robustness, stability, and high precision, with forward reprojection and backward projection RMS errors of 0.032 pixels and 0.006 mm, respectively. This paper provides a foundation for precise imaging and robust calibration of spherical mirrored cameras that are free from spherical aberration and allow arbitrary mirror placement, supporting future development of hybrid catadioptric vision sensors.