## 2 The Tricolor Binary Fringe Encoding Principle  
In the traditional phase measuring profilometry (PMP) based on binary fringe projection [32], the traditional binary fringe is shown in Figure 1. The width for non-zero transmittance (\(w_{1}\)) of the binary fringe is always encoded the same as that for zero transmittance (\(w_{0}\)) in one period (\(T_{0}\)) as shown in Figure 1. The corresponding sinusoidal fringe pattern can be approximated by properly defocusing the binary fringe [29] or using the nonlinear error suppression of the large-step PMP based on binary fringe projection [30]. The duty cycle \(P_{dc}\) can be expressed as  
\[P_{dc}=\frac{w_{1}}{T_{0}} \tag{1}\]  
Until now, the duty cycle is always \(1/2\) used in PMP based on binary fringe projection. In the proposed method, this traditional barrier is broken through. A new duty cycle binary fringe is introduced that means \(P_{dc}\) may not be \(1/2\) but \(1/3\), meaning that the \(w_{1}\) can be smaller than \(w_{0}\) in one period. It is found that although the duty cycle is not \(1/2\) in the encoded binary fringe, its fundamental frequency components of Fourier spectrum contains the sinusoidal fringe pattern information. Just by filtering operation in spatial frequency domain, the nearly unbroken sinusoidal fringe pattern can be extracted effectively. The introduced binary fringe can be modeled as the convolution of the rectangular window function and the comb function. Its gray value \(g(x,y)\) can be expressed as  
\[g(x,y)=A_{0}rect(\frac{y}{P_{dc}T_{0}})*comb(\frac{y}{T_{0}}) \tag{2}\]  
where \(A_{0}\) is a constant represented as the non-zero grayscale of the introduced binary fringe, \({}^{*}\) denotes the convolution operation. Its Fourier spectrum \(G(f_{x},f_{y})\) can be expressed as  
\[G(f_{x},f_{y})=A_{0}P_{dc}T_{0}\sum_{j=-\infty}^{j=\infty}\sin c(P_{dc}T_{0}f_{ y})\delta(f_{y}-jf_{0}) \tag{3}\]  
where \(f_{0}=1/T_{0}\) denotes the fundamental frequency of the Fourier spectrum. By introducing a proper rectangular window low-pass filter, the zero frequency component, the positive frequency component, and the negative frequency component \(G^{\prime}(f_{x},f_{y})\) can be retained as  
\[G^{\prime}(f_{x},f_{y})=A_{0}P_{dc}T_{0}\big{\{}\delta(f_{y})+\sin c(P_{dc})[ \delta(f_{y}+f_{0})+\delta(f_{y}-f_{0})]\big{\}} \tag{4}\]  
By inverse Fourier transform for \(G^{\prime}(f_{x},f_{y})\), the nearly unbroken sinusoidal fringe pattern \(g^{\prime}(x,y)\) can be extracted as  
\[g^{\prime}(x,y)=A_{0}P_{dc}T_{0}+2A_{0}P_{dc}T_{0}\sin c(P_{dc})\cos(\frac{2 \pi y}{T_{0}}) \tag{5}\]  
It can be simplified as the mathematic model of PMP [33] and expressed as:  
\[g^{\prime}(x,y)=A(x,y)+B(x,y)\cos(\frac{2\pi y}{T_{0}}) \tag{6}\]  
Figure 1: Traditional binary fringe. where \(A_{0}P_{dc}T_{0}\), simplified as \(A(x,y)\), represents the background light intensity, and \(2A_{0}P_{dc}T_{0}\sin c(P_{dc})\), simplified as \(B(x,y)\), reflects the contrast of the fringe pattern. The corresponding sinusoidal fringe pattern extracting process is show in Figure 2. When the encoded binary fringe is projected onto the reference by the HDLP, the fringe pattern is captured by a high frame rate complementary metal oxide semiconductor (CMOS) monochrome camera, as shown in Figure 2a. By fast Fourier transform (FFT), its Fourier spectrum is shown in Figure 2b, it can be seen that although there many higher order harmonic frequencies existed in the frequency domain, the Fourier spectrum distribution is discrete. By introducing a proper rectangular window low-pass filter, the frequency components contained the sinusoidal fringe pattern information as shown in the dotted box in Figure 2b can be retained as shown in Figure 2c. By inverse Fourier transform (IFT), the nearly unbroken sinusoidal fringe pattern (see Figure 2d) can be extracted efficiently. In the same way, the nearly unbroken sinusoidal deformed pattern also can be extracted from the captured deformed pattern when the encoded binary fringe is projected onto the measured object.