## 3 The Dynamic PMP Principle Based on the Tricolor Binary Fringe  
The schematic diagram of the dynamic PMP principle based on the proposed tricolor binary fringe is shown in Figure 5. Before measuring, the proposed tricolor binary fringe is encoded and saved into the flash memory of the HDLP in advance. Under the control of a special time-division multiplexing (TMD) timing sequence, the HDLP projects the encoded tricolor binary fringe saved in the flash memory with its R, G, and B channels (\(I_{R}(x,y)\), \(I_{G}(x,y)\), and \(I_{B}(x,y)\)) onto the reference plane separately and sequentially. At the same time, the projected light source mode was set as monochrome mode which means that all the RGB LEDs remain lighting mode, so that every frame binary fringe could be projected as a grayscale fringe, effectively avoiding the grayscale imbalance problem caused by the monochrome camera's different sensitivity to the R, G, and B light. Meanwhile, a high frame rate monochrome CMOS camera synchronized with the HDLP was used to capture the corresponding three fringe patterns \(I_{Ref}(x,y)\), \(I_{Gref}(x,y)\), and \(I_{Bref}(x,y)\) from R, G, and B channels, as shown in Figure 6a. The filtering operation is as discussed in Section 2, the corresponding extracted three nearly unbroken sinusoidal fringe patterns--\(I_{FRef}(x,y)\), \(I_{FGF}(x,y)\), and \(I_{FBFf}(x,y)\)--can be obtained as shown in Figure 6a. Their cutaway views of the three extracted sinusoidal fringe patterns are shown in Figure 6b, it can be seen that the three extracted sinusoidal fringe patterns have an equivalent shifting-phase of \(2\pi/3\) one another. According to the calculation process of Equations (2)-(6), they can be expressed as  
\[\left\{ \begin{array}{l}I_{FRFP}(x,y)=A(x,y)+B(x,y)\cos[\frac{2\pi y}{T}+\varphi _{0}(x,y)]\\ I_{FGref}(x,y)=A(x,y)+B(x,y)\cos[\frac{2\pi y}{T}+\varphi_{0}(x,y)+\frac{2\pi}{ 3}]\\ I_{FBFf}(x,y)=A(x,y)+B(x,y)\cos[\frac{2\pi y}{T}+\varphi_{0}(x,y)+\frac{4\pi}{ 3}]\end{array} \right. \tag{10}\]  
where \(T\) denotes the period of the captured fringe pattern. \(\varphi_{0}(x,y)\) denotes the phase caused by the reference plane, it can be expressed as  
\[\varphi_{0}(x,y)=\arctan[\frac{\sqrt{3}(I_{FRef}(x,y)-I_{FBFf}(x,y))}{2I_{FGref }(x,y)-I_{FRFf}(x,y)-I_{FBFf}(x,y)}] \tag{11}\]  
As \(\varphi_{0}(x,y)\) is wrapped in \((-\pi,\pi]\) due to the arctan function it should be unwrapped to be \(\Phi_{0}(x,y)\) by a phase unwrapping algorithm [34] and saved in the computer in advance. Figure 4: The feature of the encoded tricolor binary fringe: (**a**) Encoded tricolor binary fringe and (**b–d**) cutaway views of its R, G, and B components. While measuring, under the same condition above-mentioned, the HDLP projects the encoded tricolor binary fringe in R, G, and B channels (\(I_{R}(x,y)\), \(I_{G}(x,y)\), and \(I_{B}(x,y)\)) onto the measured object separately and sequentially at three times of dynamic frame rate, the corresponding three deformed patterns (\(I_{DR}(x,y)\), \(I_{DG}(x,y)\), and \(I_{DB}(x,y)\)) from R, G, and B channels can be captured by the high frame rate CMOS monochrome camera synchronized with the HDLP as shown in Figure 5. In the same way as discussed in Section 2, the captured three deformed patterns--\(I_{DR}(x,y)\), \(I_{DG}(x,y)\), and \(I_{DB}(x,y)\)--which were processed the corresponding extracted nearly unbroken sinusoidal deformed patterns \(I_{FDR}(x,y)\), \(I_{FDG}(x,y)\), and \(I_{FDB}(x,y)\) can be expressed as  
\[\left\{ \begin{array}{l}I_{FDR}(x,y)=A(x,y)+B(x,y)\cos[\frac{2\pi y}{T}+\varphi (x,y)]\\ I_{FDG}(x,y)=A(x,y)+B(x,y)\cos[\frac{2\pi y}{T}+\varphi(x,y)+\frac{2\pi}{3}]\\ I_{FDB}(x,y)=A(x,y)+B(x,y)\cos[\frac{2\pi y}{T}+\varphi(x,y)+\frac{4\pi}{3}]\\ \end{array} \right. \tag{12}\]  
where \(\varphi(x,y)\) is the phase modulated by the measured object located the reference plane; it can also be expressed as  
\[\varphi(x,y)=\arctan[\frac{\sqrt{3}(I_{FDR}(x,y)-I_{FDB}(x,y))}{2I_{FDG}(x,y)- I_{FDR}(x,y)-I_{FDB}(x,y)}] \tag{13}\]  
It also should be unwrapped to \(\Phi(x,y)\).