## 2 Three-Dimensional Imaging Theory
### Measurement Principle  
The two-dimensional signal of the target image, with a resolution of \(n\times n\), can be connected in columns or rows, and converted to a one-dimensional discrete signal \(x\). There are \(N=n^{2}\) elements in \(x\), of which there are \(K\) non-zero values. After obtaining the linear measurement values \(y\in R^{M}\) under the measurement matrix \(\Phi\in R^{M\times N}(M<N)\), the compressed sensing sampling process can be described as follows [15]:  
\[y=\Phi x. \tag{1}\]  
Equation (1) is solved by the reconstruction algorithm; a one-dimensional signal \(x\) can be reconstructed from \(y\), and then restored to a two-dimensional signal by \(x\). According to the above-mentioned compressed sensing imaging theory, \(M\) measurements must be performed by the system used in this study. That is, the DMD needs to be flipped \(M\) times. Each row element (1 or 0) in the pre-selected measurement matrix \(\Phi\) (\(M\times N\)) corresponds to the switching state of the micro-mirror at each pixel of the DMD in a single measurement. The resolution of the DMD micro-mirror array is \(n\times n\). Let the horizontal and vertical coordinates of the DMD pixel be \(i\) and \(j\), respectively. In the \(k\)th measurement of these \(M\) measurements, the state of a single micro-mirror in (\(i\), \(j\)) in DMD is  
\[\Phi_{ij}^{k}=\left\{ \begin{array}{c}1\\ 0\end{array} \right.. \tag{2}\]  
\(\Phi_{ij}^{k}=1\) means that the micro-mirror is on (+ 12\({}^{\circ}\) flip), and \(\Phi_{ij}^{k}=0\) means that the micro-mirror is off (-12\({}^{\circ}\) flip). The transmitted laser signal after sine wave modulation is  
\[X_{T}^{k}(t)=M_{T}^{k}\cos(wt+\varphi_{0}), \tag{3}\]  
where \(M_{T}^{k}\) is the amplitude of the transmitted signal, \(w\) is the modulation angular frequency of the transmitted signal, and \(\varphi_{0}\) is the initial phase of the transmitted signal. The optical signal after DMD modulation can be expressed as  
\[X_{T}^{k}(t)=\Phi_{ij}^{k}M_{T}^{k}\cos(wt+\varphi_{0}). \tag{4}\]  
Because the distance from each point of the target surface to the transmitting end is different, the transmitted signal will have different phase delays at each point. After the emitted light is reflected by the target, the reflected signal at the corresponding pixel point (\(i,j\)) is  
\[X_{R_{ij}}^{k}(t)=\Phi_{ij}^{k}M_{R_{ij}}^{k}\cos(wt+\varphi_{0}+\Delta\varphi _{ij}), \tag{5}\]  
where \(M_{R_{ij}}\) is the amplitude of the reflected signal at (\(i,j\)) due to the attenuation of light intensity, and \(\Delta\varphi_{ij}\) is the delay phase of the reflected signal at (\(i,j\)) relative to the transmitted signal. The total reflected light received at the \(k\)th time step can be expressed as the sum of the reflected light at each point:  
\[ \begin{array}{l}X_{R}^{k}(t)=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\Phi_{ ij}^{k}.X_{R_{ij}}(t)\\ =\cos(wt+\varphi_{0})\cdot\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}(\Phi_{ ij}^{k}.M_{R_{ij}}\cos\Delta\varphi_{ij})\\ -\sin(wt+\varphi_{0})\cdot\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}(\Phi_{ ij}^{k}.M_{R_{ij}}\sin\Delta\varphi_{ij})\end{array}  \tag{6}\]  
The total reflected light can be received by a single-pixel photo-detector and measured. The total reflected signal amplitude is \(M_{R}^{k}\), and the delay phase relative to the transmitted signal is \(\Delta\Phi^{k}\). An alternative expression of the total reflected light is  
\[ \begin{array}{l}X_{R}^{k}(t)=M_{R}^{k}\cos(wt+\varphi_{0}+\Delta\Phi^{k})\\ =M_{R}^{k}\Big{[}\cos(wt+\varphi_{0})\cos\Delta\Phi^{k}-\sin(wt+\varphi_{0}) \sin\Delta\Phi^{k}\Big{]}\end{array}  \tag{7}\]  
Comparing Equations (6) and (7), the following can be obtained:  
\[\left\{ \begin{array}{l}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\Phi_{ij}^ {k}.M_{R_{ij}}\cos\Delta\varphi_{ij}=M_{R}^{k}\cos\Delta\Phi^{k}=d^{k}\\ \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\Phi_{ij}^{k}.M_{R_{ij}}\sin\Delta \varphi_{ij}=M_{R}^{k}\sin\Delta\Phi^{k}=b^{k}\end{array}