Comparison of linear and nonlinear calibration methods for phase-measuring profilometry

In phase-shifting-based fringe-projection surface-geometry measurement, phase unwrapping techniques produce a continuous phase distribution that contains the height information of the 3-D object surface. Mapping of the phase distribution to the height of the object has often involved complex derivations of the nonlinear relationship. In this paper, the phase-to-height mapping is formulated using both linear and nonlinear equations, the latter through a simple geometrical derivation. Furthermore, the measurement accuracies of the linear and nonlinear calibrations are compared using measurement simulations where noise is included at the calibration stage only, and where noise is introduced at both the calibration and measurement stages. Measurement accuracies for the linear and nonlinear calibration methods are also compared, based on real-system measurements. From the real-system measurements, the accuracy of the linear calibration was similar to the nonlinear calibration method at the lower range of depth. At the higher range of depth, however, the nonlinear calibration method had considerably higher accuracy. It seems that as the object approaches the projector and camera for the higher range of depth, the assumption of linearity based on small divergence of light from the projector becomes less valid.