Near-optical-coaxial phase measuring deflectometry for dual-sided specular components based on a prior parametric model

Opposing working surfaces of a dual-sided specular component cannot be observed in a common field of view, which complicates simultaneous measurement of their surface forms and spatial relationship in a unified frame. To address this issue, a near-optical-coaxial phase measuring deflectometry (PMD) method based on a prior parametric model is presented, combining a dual-branch shared-display geometry with common-reference calibration. One physical display forms two equivalent virtual screens, and a dual-mirror circular-pattern plate establishes the rigid transformation between branches. Each branch reconstructs a relative surface from its measured phases, selects a model family according to the specimen's nominal surface type, and fits its initial parameters to the FC relative surface. Full reflection reprojection then estimates the active surface and pose parameters relative to the local reference plane, with the optimized parameter set determined by the selected surface family. The calibrated inter-branch transformation maps the two parametric absolute surfaces into a common coordinate system. Experiments with a plane mirror, stepped surface, and dual-sided specular components demonstrate datum-referenced axial localization with a maximum deviation of 0.1506 mm over 10-50 mm offsets, improved in-frame fringe coverage on the stepped specimen, and common-frame representation of opposing surfaces without a common field of view.