Comparative study of variant shifting-phase coding method based on noise error model

Abstract

Phase shifting algorithm, which is well-known for its high-accuracy and high-resolution, plays a crucial role in fringe projection profilometry. A class of variant shifting-phase coding method (VSPCM) has been proposed for high-speed 3D measurement. The three kinds of VSPCM include the inner shifting-phase method (ISPM), the intensity-coded method (ICM) and the modified Carré algorithm (MCA). These methods project only four patterns avoiding the need for extra images, yet achieve accuracy comparable to the classical three-frequency heterodyne method. In VSPCM, the shifting-phase, as a key code-word for phase unwrapping, is encoded as a variable value. In practice, the noise and the encoding of shifting-phase have varying degrees of impact on the performance of these methods. But the coded ranges of the shifting-phase were previouly empirical values. Besides, there are no studies related to these factors, and also a lack of comparative researches on such methods. This paper compares three kinds of VSPCM for the first time. We derive the noise error models for shifting-phase and wrapped phase in three methods and examine the impact of variant shifting-phase on fringe orders and wrapped phase in these methods. We find that the abnormal shifting-phase errors predominantly cause fringe orders errors. Fortunately, median filtering can effectively handle shifting-phase errors to ensure successful phase unwrapping. To improve their accuracy, we also deduce that the optimal shifting-phase encoding range is [-77°, 77°] for ISPM, and is [-61°, 61°] for ICM, and is [34°, 74°] for MCA. For wrapped phase errors, the distributions of ICM and MCA exhibit a U-shaped distribution, while ISPM demonstrates a uniform distribution. Based on theoretical analysis and experiment results, this paper gives recommendations for better use of these methods to boost their accuracy and offers valuable guidance for selecting an appropriate variant shifting-phase coding method under the specific measurement requirements.