The tilted-wave interferometer is a promising technique for the development of a reference measurement system for the highly accurate form measurement of aspheres and freeform surfaces. The technique combines interferometric measurements, acquired with a special setup, and sophisticated mathematical evaluation procedures. To determine the form of the surface under test, a computational model is required that closely mimics the measurement process of the physical measurement instruments. The parameters of the computational model, comprising the surface under test sought, are then tuned by solving an inverse problem. Due to this embedded structure of the real experiment and computational model and the overall complexity, a thorough uncertainty evaluation is challenging. In this work, a Bayesian approach is proposed to tackle the inverse problem, based on a statistical model derived from the computational model of the tilted-wave interferometer. Such a procedure naturally allows for uncertainty quantification to be made. We present an approximate inference scheme to efficiently sample quantities of the posterior using Monte Carlo sampling involving the statistical model. In particular, the methodology derived is applied to the tilted-wave interferometer to obtain an estimate and corresponding uncertainty of the pixel-by-pixel form of the surface under test for two typical surfaces taking into account a number of key influencing factors. A statistical analysis using experimental design is employed to identify main influencing factors and a subsequent analysis confirms the efficacy of the method derived.

翻译：倾斜波干涉仪是一种有望发展为非球面及自由曲面高精度形貌测量参考测量系统的技术。该技术结合了特殊装置采集的干涉测量数据与复杂的数学评估流程。为确定待测表面形貌，需建立能精确模拟物理测量仪器测量过程的计算模型。该模型参数（包含待求的待测表面形貌）通过求解反问题进行调整。由于真实实验与计算模型的嵌套结构及整体复杂性，全面评估其不确定性充满挑战。本文基于倾斜波干涉仪计算模型导出的统计模型，提出采用贝叶斯方法求解该反问题。该方法天然支持不确定性量化。我们提出一种近似推断方案，通过结合统计模型的蒙特卡洛采样高效获取后验分布样本。特别地，将该方法应用于倾斜波干涉仪，在考虑若干关键影响因素的前提下，对两种典型表面的逐像素形貌进行估计并给出相应不确定性。采用实验设计的统计分析识别主要影响因素，后续分析证实了所提方法的有效性。