## 3 The Dynamic PMP Principle Based on the Tricolor Binary Fringe
The phase \(\Psi(x,y)\), which s modulated by the height of the measured object, is the difference \(\Phi(x,y)\) and \(\Phi_{0}(x,y)\). \(\Psi(x,y)\) can be expressed as  
\[\Psi(x,y)=\Phi(x,y)-\Phi_{0}(x,y) \tag{14}\]  
Figure 5: The schematic diagram of the proposed method. Figure 6: The phase measuring profilometry (PMP) principle of the proposed method: (**a**) Captured fringe patterns from R, G, and B channels and corresponding extracted sinusoidal fringe patterns and (**b**) cutaway views of the three extracted sinusoidal fringe patterns. So the 3D shape of the measured object is reconstructed by phase-to-height mapping relationship [35] as  
\[\frac{1}{h(x,y)}=a(x,y)+b(x,y)\frac{1}{\Psi(x,y)}+c(x,y)\frac{1}{\Psi^{2}(x,y)} \tag{15}\]  
where the system constants \(a(x,y)\), \(b(x,y)\), and \(c(x,y)\) can be calibrated by several planes with known heights. Furthermore, due to the phase of the binary fringe will be slightly changed during the FFT process, it may introduce a phase error. However, this phase error is the system phase error, it can be eliminated by the three-step PMP algorithm used in the proposed method. The corresponding reconstruction process of the proposed method is shown in Figure 7. The captured three deformed patterns, \(I_{DR}(x,y)\), \(I_{DG}(x,y)\), and \(I_{DB}(x,y)\), from R, G, and B channels were processed as outlined in the above-mentioned process. By FFT, their Fourier spectrums (R spectrum, G spectrum, and B spectrum) are shown in Figure 7. After proper filtering, their zero frequency components and the fundamental frequency components as shown in the dotted rectangle regions are extracted respectively. By IFT, the corresponding three nearly unbroken sinusoidal deformed patterns with an equivalent shifting-phase of \(2\pi/3\) one another can be retrieved successfully. The wrapped phase modulated by the measured object can be calculated with the three deformed patterns combined three-step PMP and the 3D shape of the measured object can be reconstructed successfully as above-mentioned method. Due to the binary feature of the encoded binary fringe, the projection fringe refresh rate of the HDLP can be improved greatly, so the proposed method can be used to reconstruct the 3D shapes of the real-time changing or dynamic object. The specialized TDM timing sequence is designed as shown in Figure 8. It can be generated by a microcontroller signal circuit which can actively control the HDLP and the high frame rate CMOS monochrome camera synchronously [36] in our research group. The frame rate of the HDLP for a 1-bit image can be 4225 fps while that of the high frame rate CMOS monochrome camera can be reach 337 fps. When the synchronized microcontroller signal is at the rising edge of the projector trigger signal, the binary fringe data \(I_{R}(x,y)\) saved in the memory of the HDLP is output and completely projected onto the real-time changing or dynamic object by the HDLP at \(1/500\)s with its monochrome light projection mode, the corresponding high frame rate CMOS monochrome camera synchronized with the HDLP starts to integrate at the rising edge of the camera trigger signal until it completes the process of capturing the current deformed pattern from R channel and the corresponding deformed pattern was \(I_{DR}(x,y)\) is effectively saved into the personal computer (PC) by the universal serial bus in one period (1/234s). Then the HDLP refreshes its digital micromirror device (DMD) and starts projecting the next two binary fringes \(I_{G}(x,y)\) and \(I_{B}(x,y)\) in the next two periods, respectively,  
Figure 7: The reconstruction process of the proposed method. with the above projection process, the corresponding two deformed patterns (\(I_{DG}(x,y)\) and \(I_{DB}(x,y)\)) from G and B channel can be captured and saved into the PC.