Full-Pipeline Differentiable Ray-Tracing Modeling for Bi-prism Phase Measuring Deflectometry

Bi-prism phase measurement deflectometry (BPMD) is able to transform a monocular camera into a virtual binocular with the help of a single bi-prism element. Therefore, the simultaneous capture of fringe patterns from two perspectives can be achieved. Three-dimensional (3D) reconstruction can be performed using binocular phase measurement deflectometry (PMD) technology. However, the used bi-prism introduces a complex imaging model that requires intricate formula characterization and calibration. To solve this issue, a comprehensive BPMD model utilizing an accurate and differentiable ray-tracing (DRT) technique that characterizes freeform surfaces by B-spline is proposed and simulated within a GPU-based Python framework. Forward tracing accordingly constructs a fringe signal propagation model. Combined with backward tracing, it obtains the gradient of the measured specular surface through automatic differentiation. DRT can be used to perform a differentiable optimization on parameters, e.g., the pose and the angle of the bi-prism. Combining phase analysis and surface reconstruction algorithms, it optimizes and suppresses those errors to improve the accuracy of 3D reconstruction. The simulation results have verified the effectiveness of the proposed method based on reflection-typed BPMD.