# Motion-Induced Phase Error Compensation Using Three-Stream Neural Networks
## 1 Introduction  
Three-dimensional (3D) measurement methods based on fringe projection profilometry (FPP) [1; 2; 3] has been widely used in computer vision [4], industrial inspection [5], and other fields [6] due to its advantages of non-contact, low cost, and high accuracy. A typical FPP system consists of a camera and a projector. By projecting specially designed fringe patterns onto a tested object, the depth information of the object's surface is modulated in the distorted fringe patterns. Then, the phase distribution can be retrieved through fringe analysis algorithms and converted to 3D geometry based on triangulation. At present, there are two main approaches to extract phase value in FPP: Fourier transform profilometry (FTP) [7; 8] and phase shifting profilometry (PSP) [9; 10]. In FTP, only a single-frame high-frequency fringe pattern is required, and a suitable designed bandpass filter is applied to separate fundamental components in the frequency domain. However, with the limitation of filter operation, FTP is sensitive to surface variation and non-uniform reflection. In PSP, multi-frame (usually at least three frames) fringe patterns are utilized to calculate the phase map by a least square algorithm. At the expense of reducing speed, a more accurate and robust result can be obtained compared with FTP. For the measurement of dynamic scenes, the moving object will no longer satisfy the assumptions of PSP, which include fixed position and known phase shift. More specifically, the location of a moving object changes over time between frames, while an extra value is introduced into phase shift on the same point of the object's surface [11]. The motion-induced error will cause periodical fluctuations in phase distribution, leading to the decrease of measurement accuracy. To suppress the motion-induced error in PSP, researchers have developed various kinds of methods. One of the approaches is to combine the FTP with PSP. [PERSON] et al. [12]extracted phase map using FTP from single fringe pattern in three-step PSP, and performed phase subtraction to estimate the unknown phase shift. [PERSON] et al. [13] used four fringe patterns to obtain the absolute phase with the stereo phase unwrapping (SPU) method, then developed a pixel-wised motion detection strategy to fuse the results of PSP and FTP. [PERSON] at al. [14] proposed a dual-frequency composite grating method to identify the motion region using the phase of a virtual high frequency and replaced the phase map of PSP with FTP in corresponding region. However, due to the implicit drawbacks of FTP, these hybrid methods are unadaptable for dealing with complex scenes. To avoid this issue, [PERSON] et al. tracked the motion of an object by manually placed makers [15] or scale-invariant feature transform (SIFT) [16] and estimated the translation vector and rotation matrix to calculate phase map. [PERSON] et al. [17] estimated the phase-shift error by averaging the difference of three adjacent phase maps, which are calculated from eight consecutive images of four-step phase-shift fringe patterns. [PERSON] et al. [18] applied Hilbert transform to shift the phase of three-step phase-shift fringe patterns by \(\pi/2\), constructed opposite distributional phase to compensate motion error. [PERSON] et al. [19] divided four-step phase-shift fringe patterns into two groups to calculate two phase maps which have opposite phase error distribution, and the periodical motion-induced phase error can be compensated by averaging them. However, most of these methods may not be suitable if the object undergoes non-uniform motion [11]. Recently, plenty of studies have introduced deep learning technique into 3D measurement. [PERSON] et al. [20] proposed a deep learning-based modulation-enhancing method that transforms two low-modulation fringe patterns into a set of three-step phase-shift fringe patterns. [PERSON] et al.