## 2 Methods
### 2D Compressive Sensing Spectral Domain Optical Coherence Tomography  
In this section, we detail the basic principle and procedures of 2D CS-SDOCT, and simulate the reconstruction of the 2D image of the two mirror surfaces \(p_{1}\) and \(p_{2}\) for a more vivid explanation. The simulation results are shown in Figure 2e-h, and Figure 2a-d shows the recovery process of the spectral pattern. The processes represented by the green arrows in Figure 2i illustrate the reconstruction steps of 2D CS-SDOCT. Figure 2: Schematic demonstration of the compressive sensing reconstruction strategy. (**a**)–(**d**) The patterns of the data matrixes, (**e**)–(**h**) the simulation results of two mirror surfaces, (**i**) processing steps of the method combining compressive sensing spectral domain optical coherence tomography (CS-SDOCT) and ISAM. In (**i**), \(\beta=-2k_{z}(\overrightarrow{Q_{//}}/2)\). According to CS theory, CS reconstructs signal \(\mathbf{a}\) by solving the following constrained optimization problem [6]:  
\[\underset{\mathbf{a}}{\min}\|\psi\mathbf{a}\|_{1}\qquad\mathrm{s.t.}\|\mathbf{Ta }-\mathbf{y}\|_{2}^{2}\leq\varepsilon \tag{5}\]  
where \(\mathbf{T}\) is the measurement matrix, \(\mathbf{y}\) is the measured data, \(\psi\) is the sparse operator that transforms \(\mathbf{a}\) to a sparse representation, and \(\varepsilon\) is the parameter that controls the fidelity of the signal in the measurement domain. \(\|\bullet\|_{1}\) and \(\|\bullet\|_{2}\) are the \(l_{1}\) norm and \(l_{2}\) norm, respectively. In SD-OCT, after removing the DC term and the self-coherent signal of the sample, the 2D under-sampled spectral data can be formulated as a digital signal as follows [10]:  
\[\mathbf{S}(x_{c},k_{c})=\mathrm{Re}(\mathbf{M}_{x}\mathbf{M}_{k}\mathbf{Fa}( x,z^{\prime})), \tag{6}\]  
where \(\mathbf{a}(x,z^{\prime})\) represents the complete sparse structural signals in the spatial domain; \(\mathbf{F}\) represents the 1D discrete Fourier transform (\(DFT\)); \(\mathbf{M}_{k}\) and \(\mathbf{M}_{x}\) are the sampling masks in wavenumber \(k\) domain and lateral spatial domain, respectively; and a parameter with the subscript '\(c\)' indicates that it is compressed. According to Equation (3), we know \(z^{\prime}=-z/2\). The pattern of \(\mathbf{S}(x_{c},k_{c})\) is shown in Figure 2b, and the corresponding structural image shown in Figure 2f was obtained by performing inverse discrete Fourier transform (\(IDFT\)) on the \(k\)-domain of \(\mathbf{S}(x_{c},k_{c})\). The 2D CS reconstruction is to restore Figure 2f to Figure 2e dimension-by-dimension. For CS reconstruction of the axial direction, i.e., each A-scan, measurement matrix \(\mathbf{T}\) is \(\mathbf{M}_{k}\)\(\mathbf{F}\). Because the structural signal \(\mathbf{a}\) to be recovered is considered sparse in SD-OCT [24], \(\psi\) is chosen as the identity matrix [13]. Therefore, based on Equation (5), we reconstruct A-scans data by solving the following constrained optimization problem:  
\[\underset{\mathbf{a}}{\min}\|\mathbf{a}(x_{c},z^{\prime})\|_{1}\qquad\mathrm{ s.t.}\|\mathbf{M}_{k}\mathbf{Fa}(x_{c},z^{\prime})-\mathbf{S}(x_{c},k_{c})\|_{2}^{2} \leq\varepsilon. \tag{7}\]  
After solving Equation (7), structural signal \(\mathbf{a}(x_{c},z^{\prime})\) and data matrix \(\mathbf{S}(x_{c},k)\) are obtained, which are shown in Figure 2g and Figure 2c, respectively. This process corresponds to step I shown in Figure 2i. For CS reconstruction of the lateral direction, i.e., B-scan, sparse machine \(\psi\) cannot be identity matrix since the measurement domain cannot be the same as the sparse domain [13]. In this paper, we select the sparse machine \(\psi\) as \(\mathbf{F}\)[13], then Equation (5) becomes  
\[\underset{\mathbf{a}}{\min}\|\mathbf{a}(Q_{//},z^{\prime})\|_{1}\qquad \mathrm{s.t.}\|\mathbf{M}_{x}\mathbf{F}^{\mathbf{H}}\mathbf{a}(Q_{//},z^{ \prime})-\mathbf{S}(x_{c},z^{\prime})\|_{2}^{2}\leq\varepsilon, \tag{8}\]  
in which \(\mathbf{a}(Q_{//},z^{\prime})=\mathbf{Fa}(x,z^{\prime})\).