A Point Cloud Registration Method Based on Point-to-Triangulation Estimation for Optical Window Free-Form Surfaces Testing by Coordinate Measuring Machine

Optical window freeform surfaces have emerged as a critical research focus in advanced optical engineering owing to their extensive surface degrees of freedom. These surfaces enable the simultaneous correction of on-axis and off-axis aberrations while satisfying stringent requirements for high-performance, lightweight, and compact optical systems. In the initial metrological characterization of these surfaces, coordinate measuring machines (CMMs) are conventionally employed for target point cloud acquisition. However, the achievable measurement accuracy (>2 μm) inherently constrained by CMM precision imposes fundamental limitations for subsequent optical inspections requiring sub-micron to nanometer-level resolution. Meanwhile, although optical measurement methods can result in higher measurement accuracy, they also lead to an increase in costs and testing difficulties. To overcome these limitations, we propose an accelerated point cloud registration methodology based on point-to-triangulation distance estimation. In simulation, using optimal coordinate transformation enabled good capabilities for exceptional surface characterization: peak-to-valley (PV) surface error of 10−6 nm, residual error of 5 nm, and registration accuracy of log10 (mm/°). Further, in the experiment, the PV surface error was reduced from 27.3 μm to 6.9 μm, equivalent to a reduction of 3.95 times. These results confirm that the point-to-triangulation distance approximation maintains sufficient fidelity to the nominal point-to-surface distance, thereby empirically validating the efficacy of our proposed methodology. Notably, compared with conventional 3D alignment methods, our novel 2D estimation registration approach with point-to-triangulation surface normal vectors demonstrates significant advantages in computational complexity, which achieved a 78% reduction from O(n3) to O(n) while maintaining sub-millisecond alignment times. We believe that the method has potential for use as a low-cost optical precision measurement in manufacturing technology.