# FDBP-InSAR: An Efficient Algorithm for InSAR Imaging via Frequency Domain Back Projection
## 3 InSAR Imaging via FDBPA
### Algorithm Flow  
According to the theoretical derivation above, there are mainly four steps for InSAR imaging: The first step is range compression, the second step is calculating the index, the third step is back projection, and the last one is inverse Fourier transform. The main flow of InSAR imaging based on is demonstrated as follows (Algorithm 1):```
Input: raw echo data \(\mathbf{S}_{m}\left(n,\mathbf{\tau}\right)\) and \(\mathbf{S}_{s}\left(n,\mathbf{\tau}\right)\), antennas information \(l_{m}\left(n\right)\) and \(l_{s}\left(n\right).\) Output: focused image data \(\mathbf{f}_{m}\left(x,y\right)\) and \(\mathbf{f}_{s}\left(x,y\right).\)
1: Range Compressed \(\mathbf{s}_{m}\left(n,r\right)=\text{IFFT}\left\{\mathbf{S}_{m}\left(n,f_{r}\right)*H \left(f_{r}\right)\right\},\) where \(\mathbf{S}_{m}\left(n,f_{r}\right)\) is the raw echo data in the range frequency domain, \(H\left(f_{r}\right)\) is the range reference function and \(H\left(f_{r}\right)=\exp\left(j\pi{f_{r}}^{2}/K_{r}\right).\)
2: Calculation of Index Value a: grid division \(dkr=\frac{2\pi}{W_{r}},d\text{$y$}=\frac{2\pi}{\text{$\eta$}y}\) \(Nr=\left|\frac{W_{r}}{d\
u}\right|,Ny=\left|\frac{\text{$\eta$}y}{d\
u}\right|\) where \(\left|\cdot\right|\) is the sign of cell; \(Wr\times Wy\) is the size of the target scene; and \(d_{r}\) and \(dy\) are range and azimuth spatial resolution, respectively. b: the calculation of index value \(ID=\left(\sqrt{k_{r}^{2}+k_{r}^{2}}/2-k_{t}\right)/dk+1,\) where \(k,\) and \(k_{r}\) are the range and azimuth frequency scaled by \(2\pi/C\), respectively; \(dk\) and \(k_{t}\) are the frequency space and starting frequency scaled by \(2\pi/C\), respectively; and \(dk=2\pi f_{r}/N/C\), \(k_{t}=4\pi/\lambda-dk\cdot N/2.\)
3: Back Projection a: 1D Fourier transform \(\mathbf{s}_{m}\left(n,\omega\right)=FFT\left\{\mathbf{s}_{m}\left(n,r\right)\right\}\) b: ID extraction by Sinc-interpolation \(\mathbf{s}_{m}\left(n,k_{r}\right)=\sin c\left(\mathbf{s}_{m}\left(n,\omega\right)\right)\) c: non-uniform Fourier Translation \(\mathbf{S}_{m}\left(k_{r},k_{r}\right)=NFFT\left\{\mathbf{s}_{m}\left(n,k_{r}\right)\right\}\) where \(NFFT\left\{\cdot\right\}\) denotes the non-uniform Fourier translation. d: Phase compensation \(\hat{\mathbf{S}}_{m}\left(k_{r},k_{r}\right)=\mathbf{S}_{m}\left(k_{r},k_{r}\right) \exp\left(-j\left(\sqrt{k_{r}^{2}+k_{r}^{2}}-k_{r}\right)R_{r,r}\right),\) where \(R_{r,r}\) is the distance from the center of the target point \(\left(y,r\right)\) to the antenna. 4: Two-Dimensional Inverse Fourier Transform \(\mathbf{f}_{m}\left(x,y\right)=IFFT\left\{IFFT\left\{\mathbf{S}_{m}\left(k,k_{r}\right) \right\}\right\}\) The processing step of \(\mathbf{S}_{s}\left(n,\mathbf{\tau}\right)\) is the same as \(\mathbf{S}_{m}\left(n,\mathbf{\tau}\right).\) The interferometric phase can be obtained by the images data of \(\mathbf{f}_{m}\left(x,y\right)\) and \(\mathbf{f}_{s}\left(x,y\right)\) after registration. ```

**Algorithm 1**InSAR Imaging via FDBPPA

### Phase Stability

Under the influence of an unstable imaging environment, the trajectory of the antenna platform is not an ideal linear motion. We assume the actual coordinates of radar antennas are \(\mathbf{P}_{m}\) and \(\mathbf{P}_{s}\)respectively; and \(\mathbf{P}_{w}\) is the actual coordinates of imaging points. The error-free antenna signal of scattering point after two-dimensional focused imaging can be expressed as follows:

\[\mathbf{S}_{r}(\mathbf{P}_{w})=\mathbf{S}_{m}(\mathbf{P}_{w})\exp(-j4\pi\|\mathbf{P}_{m}-\mathbf{P}_{w}\| _{2}/\lambda) \tag{13}\]

where \(\mathbf{S}_{m}(\mathbf{P}_{w})\) denotes the range-azimuth ambiguity function of the master antenna corresponding to the target point \(\mathbf{P}_{w}\). Taking the RD algorithm as an example of the traditional frequency-domain algorithm, it firstly compresses the original echo signal, then corrects the distance migration in the two-dimensional frequency domain, and finally compresses the corrected echo data.