## 2 The Tricolor Binary Fringe Encoding Principle
Furthermore, the effects of the traditional binary fringe, and the introduced binary fringe on their Fourier spectrums, are analyzed. At the same time, the amplitudes of the Fourier spectrums in binary fringes are also analyzed, from Equation (4), the amplitude \(A_{\max}(f_{0},P_{dc})\) of the fundamental frequency spectrum in binary fringe can be expressed as  
Figure 1.0  
The traditional 1/2 duty cycle binary fringe and the 1/3 duty cycle binary fringe as shown in Figure 3a,b, respectively, are taken as an example. In this example, their image sizes are all \(300\;\mathrm{pixels}\times 300\;\mathrm{pixels}\). By FFT for them, their corresponding Fourier spectrums are shown in Figure 3c,d, respectively; it can be seen that the second order harmonic frequency spectrum vanishes in the traditional binary fringe but exists in the 1/3 duty cycle binary fringe. Although the second order harmonic frequency spectrum exists in the 1/3 duty cycle binary fringe, its proportion is small enough. It means that the shape reconstruction error caused by the second order harmonic frequency spectrum is small enough. According to Equation (7), it can also be seen that the amplitude of the fundamental frequency spectrum in traditional binary fringe is 4700 and that of the 1/3 duty cycle binary fringe is 4000 from Figure 3c,d. But if the zero frequency spectrums and the fundamental frequency spectrums in Figure 3c,d are filtered out respectively by introducing a proper rectangular window low-pass filter as shown in the rectangle box in Figure 3c,d with the above-mentioned process, it can be seen that these retained frequency spectrum components' distributions contained the sinusoidal fringe information are similar. Then they aligned the same amplitude of the fundamental frequency spectrum by simply multiplying the latter spectrum with 4700/4000, it is found that the aligned frequency spectrums are very similar to those shown in Figure 3e. It reveals that the 1/3 duty cycle binary fringe has the same effect as the traditional binary fringe. By IFT for the aligned frequency spectrums in Figure 3e respectively, the corresponding nearly unbroken sinusoidal fringe patterns can be efficiently extracted from the traditional binary fringe and the 1/3 duty cycle binary fringe respectively. Figure 3f show the cutaway views of the extracted sinusoidal fringe patterns in one columns, it can be seen that the transmittance function of the sinusoidal fringe pattern extracted from the 1/3 duty cycle binary fringe is much close to that of the traditional binary fringe. Furthermore, it also reveals that the reconstructed  
Figure 1.1 Sinusoidal fringe pattern extracting process: (**a**) Captured fringe pattern, (**b**) Fourier spectrum, (**c**) remained spectrum, and (**d**) extracted fringe pattern. phase information of the measured object by using the 1/3 duty cycle binary fringe can be close to that of traditional binary fringe. In order to realize dynamic 3D shape measurement with the introduced binary fringe, a tricolor binary fringe is designed as shown in Figure 4a in which its R, G, and B components are encoded by three monochromatic binary fringes with the same duty cycle of 1/3 but shifted 1/3 period one by one. It can be seen that R, G and B components of the encoded tricolor binary fringe share the same fringe width of 1/3 periods and are independent without any color overlapping in the non-zero regions one by one as shown in Figure 4b-d, so the encoded tricolor binary fringe can avoid the color overlapping problem in the traditional composite color sinusoidal fringe. The mathematic model of the tricolor binary fringe \(I_{C}(x,y)\) can be expressed as  
Figure 1.2  
where \(I_{R}(x,y)\), \(I_{G}(x,y)\), and \(I_{B}(x,y)\) denote the grayscale distributions of the R, G, and B components in the tricolor binary fringe:  
\[\left\{