# Reconstruction of Isolated Moving Objects by Motion-Induced Phase Shift Based on PSP
## 1 Introduction  
Phase shifting profilometry (PSP) is one of the most popular technologies for 3D reconstruction [1, 2, 3]. PSP projects multiple sinusoidal fringe patterns with phase shift by a projector and then captures the reflected fringe patterns from the object surface by the camera. The phase information existing in the sinusoidal fringe pattern is retrieved by eliminating the influence of the background light and object surface reflectivity, leading to the advantages of high accuracy and robustness [4, 5]. However, the measured object must keep static when implementing the reconstruction. The motion of the object not only introduces position mismatch, but unknown phase shift is also generated, resulting in errors in the reconstruction [6]. Recently, much work has focused on the issues caused by the motion in PSP [7, 8, 9]. [PERSON] et al. [10] proposed compensating the motion errors for rigid moving object reconstruction. The motion speed is limited to be constant, and its influence is removed by using the statistical nature of the fringe patterns. However, it cannot reconstruct the object with rotation movement, as the motion-induced phase shifts are different for different points on the object surface. [PERSON] et al. [11] proposed reconstructing the dynamic object based on a modified three-wavelength phase unwrapping algorithm and phase error compensation method. The sequence of the projected fringe pattern is adjusted and less fringe patterns are required. Then, the Hilbert transform is employed to remove the motion-induced errors. [PERSON] et al. [12] reconstructed a moving object based on feature correspondence. One fringe pattern is projected, and five fringe patterns of the object are captured. Based on the oriented fast and rotated brief feature algorithm, the correspondence of the object fringe pattern can be defined. Then, the phase value is retrieved by the traditional PSP. The method requires the equivalent phase variation, which means the object should have onedirection of constant speed movement. [PERSON] et al. [13] address the motion-induced error by separating the fringe patterns of four-step PSP into two groups. The first group is the first three fringe images, and the other is the last three images. As the frequency of the error caused by motion is twice of the fringe pattern frequency, the error is compensated by the average phase of the two groups. [PERSON] et al. [14] estimate the motion from the object images in the beginning and end state. Based on the phase map and the obtained initial displacement data, the displacement for each point is retrieved. Then, the phase jump caused by the object motion is compensated. However, the method limits the object's uniform motion. [PERSON] et al. [15] proposed solving the motion-induced error issue by using an adaptive reference. Two planes are put in the up and down sides, and the object is put in the middle. Their phase maps are used to compensate the phase error. Our group [16] reconstructed multiple objects by motion tracking. Through object identification, the area of interest for each object is defined. Then, the object motion is tracked with the kernel correlation filter algorithm. The mathematical descriptions of movement for all objects are calculated. At last, the 3D information of the multiple objects is obtained. Recently, our group reconstructed the object by projecting a single fringe pattern [17]. Multiple fringe patterns with motion-induced phase variation were captured and the phase information was retrieved. However, the method can only reconstruct the object with a smooth surface, as the spatial phase unwrapping method is employed. Based on [17], this paper focuses on the reconstruction of multiple objects with two-dimensional movement by using motion-induced phase shift based on PSP. The motion information is not only used to generate the motion-induced phase variation, but it is also employed to compensate the phase variations between the two frequencies.