Characterization of the interpolation bias in the analysis of deflectometry measurement data

摘要

Zonal integration techniques are commonly used for surface shape reconstruction from slope measurements for phase measuring deflectometry (PMD) systems. The stability of zonal surface reconstruction algorithms relies on the symmetry of numerical differential equations for uniformly sampled grids. Modified zonal integration methods for non-uniformly sampled slope data can yield accurate results for the measurement of a single camera yet fail to produce accurate results for stereo deflectometry. In this publication, we identify and quantify a large systematic bias in the deflectometry measurement that is introduced during resampling or interpolation of phase data onto a uniform sample grid. This interpolation is commonly done in all forms of deflectometry and cannot be avoided in stereo phase measuring deflectometry. This surface reconstruction bias is largely independent of the integration method and the accuracy of the integration algorithm under different grid assumptions. This bias is a significant contributor to the well-known micron-level low spatial-frequency errors in deflectometry that are commonly attributed to calibration errors. We show the dependence of the bias on different system parameters including surface shape, curvature, and the geometry of the system components, providing guidance on setup geometry for stereo PMD.