## 2 Principles  
### Principle of TFTP  
As is shown in Figure 1, a typical experimental arrangement of TFTP system is similar to that of the FTP system, except for a multi-period phase-shifting projector. The system is mainly composed of a self-made projection device, an imaging device and a dynamic object to be measured. To improve the projection speed of the measuring system, a mechanical projector is employed. Multi-periodic fringe patterns with a fixed phase-shifting interval are generated by the rotating wheel of the mechanical projector and continuously projected onto the surface of the tested object. The fringe pattern is modulated and deformed by the 3D shape of the measured dynamic object. A high-speed camera located on another viewing angle is employed to sample the tested object in the temporal domain and works synchronously with the self-made projector. At the initial moment \(t_{0}\), the intensity distribution of fringe pattern can be expressed as  
\[I(x,y)=\mathrm{A}(x,y)+\mathrm{B}(x,y)\cos\phi(x,y), \tag{1}\]  
where (\(x\),\(y\)) is the pixel coordinate, \(\mathrm{A}(x\),\(y\)) is the average intensity, \(\mathrm{B}(x\),\(y)\) is the intensity modulation, and \(\phi(x\),\(y)\) is the fringe deformation determined by 3D shape of the tested object. The intensity distribution of the fringe pattern at the moment \(t\) can be written as  
\[I(x,y,\delta_{t})=\mathrm{A}(x,y)+\mathrm{B}(x,y)\cos[\phi(x,y)+\delta_{t}], \tag{2}\]  
where \(\delta_{t}\) is the phase modulation introduced by multiple phase shifts during \(t_{0}\)\(\sim\)\(t\). During the whole dynamic measurement process, \(n\) frames of modulated fringe patterns are obtained by sampling the measured object along the temporal axis. As illustrated in Figure 2, a 3D temporal-spatial coordinate system has two spatial coordinates (\(x\),\(y\)) and one temporal coordinate (\(t\)). It can be seen that the intensity distribution in the temporal domain is an approximate sinusoidal curve, which also carries height information of the measured object. At time \(t\), the intensity distribution along the temporal axis can be represented by  
\[g(x,y,t)=a(x,y)+b(x,y)\cos[2\pi f_{0}x+\phi(x,y,t)+2\pi f_{t}t], \tag{3}\]  
Figure 1: Schematic diagram of a typical temporal Fourier transform profilometry (TFTP) setup. where \(a(x,y)\) is the background intensity, \(b(x,y)\) is the intensity modulation, \(f_{0}\) is the spatial frequency, \(\phi(x,y,t)\) is the phase distribution modulated by the height information of the measured object; \(2\pi ft\) is the phase modulation caused by the multi-period phase-shifting and \(f_{t}\) is the temporal frequency of the phase-shifting. After the expansion of Equation (3), it can be rewritten as  
\[g(x,y,t)=a(x,y)+\frac{b(x,y)}{2}e^{i2\pi ftx}e^{i\phi(x,y,t)}e^{i2\pi ft}+\frac {b(x,y)}{2}e^{-i2\pi ftx}e^{-i\phi(x,y,t)}e^{-i2\pi ft}, \tag{4}\]  
As \(f_{0}\) is the spatial frequency, for any point \((x,y)\), the \(e^{i2\pi ftx}\) and \(e^{-i2\pi ftx}\) are constant in temporal domain, then Equation (4) can be expressed as  
\[g(x,y,t)=a(x,y)+\frac{c(x,y)}{2}e^{i\phi(x,y,t)}e^{i2\pi ftt}+\frac{c^{\prime} (x,y)}{2}e^{-i\phi(x,y,t)}e^{-i2\pi ftt}, \tag{5}\]  
It is quite clear that the last two terms of Equation (5) on the spectrum are located on both sides of the zero-frequency component. As is shown in Figure 3, there is no spectral aliasing and the fundamental frequency can be easily filtered out. Inverse Fourier transform operations can be carried out along the temporal axis to obtain the wrapped phase value. The wrapped phase value can be further expressed as  
\[\phi(x,y,t)=phase\bigg{\{}F^{-1}\bigg{\{}F\bigg{\{}\frac{c(x,y)}{2}e^{i\phi(x,y,t)}e^{i2\pi ftt}\bigg{\}}\bigg{\}}\bigg{\}}=\arctan\frac{\text{Im}[\hat{g}( x,y,t)]}{\text{Re}[\hat{g}(x,y,t)]}, \tag{6}\]  
where \(\hat{g}(x,y,t)=F^{-1}\bigg{\{}F\bigg{\{}\frac{c(x,y)}{2}e^{i\phi(x,y,t)}e^{i2 \pi ftt}\bigg{\}}\bigg{\}}\), \(F[\#]\) is the Fourier transform operator, \(F^{-1}\{\#\}\) is the inverse Fourier transform operator.