Theoretical Model and Experimental Investigation on Integrated Calibration in MFPP With a Bi-Telecentric Camera and a Scheimpflug Projector

This work establishes a microscopic fringe projection profilometry (MFPP) system consisting of a bi-telecentric camera and a pinhole Scheimpflug projector to maximize the common focus area between them. Currently, phase-height mapping (PHM) and stereoscopy (SV)-based calibration approaches are still two representative families. The former generally relies on a precise translation stage accompanied by two separate procedures including the camera parameters calibration and the PHM process, making the calibration task relatively complicated and expensive in terms of required operation and devices. The latter only via a calibration target is relatively simple and cheap to implement, but its accuracy fails to be guaranteed because it is difficult to undistort the projector by indirectly resorting to the camera. To cope with these issues, we propose a flexible integrated calibration (IC) solution for the specially designed 3-D system while maintaining a high measurement accuracy and a low requirement of operation and devices. It implements a pixel-wise lookup table (LUT) 3-D coordinate computation only via a low-cost calibration target. The phase artifacts of the dot target are addressed by phase optimization strategy. Furthermore, an optimization strategy is presented to exclude the measurement errors caused by the projector’s distortion and Scheimflug projection structure. The proposed framework effectively simplifies the calibration procedure and reserves the reconstruction accuracy. A theoretical model of the calibration solution is mathematically derived, and rich comparative experiments imply that it can obtain an average height error of $7~\mu \text{m}$ around within the volume of $28\times 23\times5$ mm, as well as a standard deviation (STD) of plane fitting error better than $5~\mu \text{m}$ .