Efficient geometric modeling and joint calibration for biaxial MEMS-based fringe projection profilometry

Fringe projection profilometry (FPP) enables high-precision 3D shape measurement and is increasingly deployed with biaxial micro-electro-mechanical system (MEMS) laser-scanning projectors for large-range projection. However, the fast and slow axes of a biaxial MEMS mirror exhibit coupled and nonlinear motion, causing the actual scanning trajectory to deviate from an ideal separable sweep. Consequently, points sharing the same phase no longer lie on straight lines or planar isophase surfaces, but instead follow curved spatial loci, making accurate calibration challenging. This work introduces a phase-angle-ray model (PARM) that establishes a bivariate polynomial mapping between the phase and MEMS deflection angles, from which projected rays are analytically derived. Two calibration frameworks based on pinhole and ray-based imaging models are developed by integrating geometric constraints from a standard plane and multiple standard spheres. Experiments demonstrate that the proposed method achieves high calibration accuracy, improved metric consistency, and precise 3D reconstruction.