A Probabilistic Approach for Spatio-Temporal Phase Unwrapping in Multi-Frequency Phase-Shift Coding

Abstract

Multi-frequency techniques with temporally encoded pattern sequences are used in phase-measuring methods of 3D optical metrology to suppress phase noise but lead to ambiguities that can only be resolved by phase unwrapping. However, classical phase unwrapping methods do not use all the information to unwrap all measurements simultaneously and do not consider the periodicity of the phase, which can lead to errors. We present an approach that optimally reconstructs the phase on a pixel-by-pixel basis using a probabilistic modeling approach. The individual phase measurements are modeled using circular probability densities. Maximizing the compound density of all measurements yields the optimal decoding. Since the entire information of all phase measurements is simultaneously used and the wrapping of the phases is implicitly compensated, the reliability can be greatly increased. In addition, a spatio-temporal phase unwrapping is introduced by a probabilistic modeling of the local pixel neighborhoods. This leads to even higher robustness against noise than the conventional methods and thus to better measurement results.

Keywords

• Phase-shifting
• temporal phase unwrapping
• circular statistics
• von Mises distribution
• structured illumination
• deflectometry