On the precision models of fringe projection profilometry: unification, simplification and connection

Fringe projection profilometry (FPP) is one of the most prominent non-interferometric high-precision 3D shape measurement techniques for measuring dimensions ranging from 1 mm to 1 m, with increasing applications in precision engineering. Thorough theoretical understanding of the precision of the prevailing FPP methods is critical when facing the challenges of higher precision demands. In this paper, through theoretical analysis of three reconstruction methods based on the projection of vertical fringes (Ver3), horizontal fringes (Hor3) and optimal-angle fringes (OptE3), we establish a general unified precision model, from which a general Pythagoras relationship is discovered, clarifying the optimal precision of OptE3. We then simplify the model to reveal the role of a system's geometry for easier configuration. First, a special yet not restrictive case where the direction of the optical axis of the projector is perpendicular to the baseline of the FPP is considered, resulting in an intermediate unified precision model. We show that the effective baselines of the three methods obey a special Pythagoras relationship. Second, the angle between the optical axes of the camera and the projector is interestingly delineated from the projector's extrinsic parameters, leading to our final and simplified unified precision model. This simplified model is geometrically tangible and thus eases the determination of the positions and orientations of the projector and camera in system configuration, given a precision budget, for which, an FPP-Planner software tool is prototyped. Going beyond, we theoretically equivalate FPP with stereo vision and laser triangulation regarding precision. Our findings are experimentally validated. These theoretical models are believed to potentially enhance or even change the way of FPP design and performance evaluation.