## 2 Principles
### The Measuring Limitation of TFTP  
The optical geometry of TFTP is shown in Figure 5, where the optical axis \(\overline{P_{1}P_{2}}\) of the projector crosses that of the camera \(\overline{I_{1}I_{2}}\) at point \(O\) on the reference plane, which is perpendicular to \(\overline{I_{1}I_{2}}\). \(d\) expresses the distance between \(P_{2}\) and \(I_{2}\), and \(L_{0}\) is the working distance between \(I_{2}\) and \(O\). Along the temporal axes, \(\overline{AC}\) is the distance between points \(A\), \(C\) on the reference plane, \(D\) expresses a tested point, \(C\) and \(D\) correspond to the same pixel of the camera at different moments. Assuming the TFTP system is perfect and the system structural parameters \(d\), \(L_{0}\) are exactly known. The height distribution can be calculated by  
\[h(u,v,t)=\frac{\overline{AC}L_{0}}{d+\overline{AC}}=\frac{L_{0}\Delta\phi(u,v, t)}{2\pi f_{t}d+\Delta\phi(u,v,t)}, \tag{9}\]  
As the Fourier spectra along the temporal axis usually have multiple components that can be expressed as  
\[f_{\mathrm{n}}=nf_{\mathrm{t}}+\frac{n}{2\pi}\frac{\partial\phi(u,v,t)}{ \partial t}, \tag{10}\]  
In order to obtain the height information accurately, the fundamental component must be separated from all other spectra components, it is necessary that  
\[\left(f_{1}\right)_{\mathrm{min}}>f_{b}, \tag{11}\]  
\[\left(f_{1}\right)_{\mathrm{max}}<\left(f_{n}\right)_{\mathrm{min}},n>1, \tag{12}\]  
where \(f_{b}\), \((f_{1})_{\mathrm{max}}\) represent the maximum frequency of the zero frequency component and the fundamental frequency component respectively. \((f_{1})_{\mathrm{min}}\) and \((f_{n})_{\mathrm{min}}\) sequentially represent the minimum frequency of the fundamental frequency component and \(n\)-order spectral component.  
Therefore, the phase variation caused by height modulation should be limited in  
\[\left|\frac{\partial\phi(u,v,t)}{\partial t}\right|_{\mathrm{max}}<\frac{2\pi f _{t}}{3}, \tag{13}\]  
As \(L_{0}\) is much larger than \(h(u,v,t)\), Equation (9) can be rewritten as  
\[\phi(u,v,t)\approx\Delta\phi(u,v,t)=\frac{2\pi f_{t}d}{L_{0}}h(u,v,t), \tag{14}\]  
Then the height variation should be limited in  
\[\left|\frac{\partial h(u,v,t)}{\partial t}\right|_{\mathrm{max}}<\frac{L_{0}} {3d}, \tag{15}\]  
Figure 5: Optical geometry.  
During the measurement process, if the height variation along the temporal axis exceeds the limitation, the fundamental components would overlap the zero component and other components in their Fourier spectrum domain. If this occurs, the tested object's shape at that moment cannot be restored correctly.  
In Reference [6], because of the discrete nature of discrete Fourier transform (DFT), [PERSON] et al. proposed the relationship between the sampling frequency \(m\) and the frequency spectra orders \(n\) of the adjacent frequency period.  
\[m\geq\frac{4}{3}(n+1),n\geq 2, \tag{16}\]  
Consequently, in case of sinusoidal fringe pattern projection, at least four sampling points were required to avoid spectral overlapping.  
For the convenience of data processing, a serial of sinusoidal periodic signals--five sampling points per period (T = 5)--was simulated. Considering the length of time that the dynamic object lasted, the appropriate sampling rate of the employed high-speed camera was selected to capture as many frames of deformed fringe patterns as possible. To ensure temporal resolution and measuring accuracy, multiple periodic signals were constructed to analyze the phase variation along the temporal axis with TFTP. During the following computer simulation and dynamic measurement process, 201 and 630 frames of images were captured along the temporal axis respectively. This suggests that the presented TFTP method is especially suitable for multi-frame 3D shape measurement of a complex dynamic object which cannot be successfully digitalized by traditional FTP in 2D spatial fringe analysis.