# FDBP-InSAR: An Efficient Algorithm for InSAR Imaging via Frequency Domain Back Projection
## 3 InSAR Imaging via FDBPA
### Algorithm Flow
The phase of the frequency-domain focusing images can be expressed as follows:

\[\mathbf{S}_{RD}(\mathbf{P}_{w})=\mathbf{S}_{m}(\mathbf{P}_{w})\exp(-j4\pi\|\mathbf{P}_{m}{}^{\prime} -\mathbf{P}_{w}{}^{\prime}\|_{2}/\lambda) \tag{14}\]

where \(\mathbf{P}_{m}{}^{\prime}=Vt\), \(V\) is the average velocity of the radar platform, and \(\mathbf{P}_{w}{}^{\prime}\) denotes the reference point, which is the center of the imaging scene. In the proposed algorithm, the original echo data are compressed firstly. Then the data are projected into image wavenumber domain, and the non-uniform Fourier transform is implemented in the azimuth; finally, the delayed phase of the echo is compensated according to the measurement position information of antennas and scene, and the two-dimensional inverse Fourier transform is performed. Different from the traditional frequency-domain algorithm, the proposed algorithm introduces the precise antennas position information in the process of azimuth-to-Fourier transform. Secondly, the actual distance between the target points and the antennas is used instead of the reference distance when compensating for the delayed phase of the echo. Thus, the equation of the focusing scene by the proposed algorithm can be written as follows:

\[\mathbf{S}_{FDBP}(\mathbf{P}_{w})=\mathbf{S}_{m}(\mathbf{P}_{w})\exp(-j4\pi\|\mathbf{P}_{m}{}^{ \prime\prime}-\mathbf{P}_{w}{}^{\prime\prime}\|_{2}/\lambda) \tag{15}\]

where \(\mathbf{P}_{m}{}^{\prime\prime}\) denotes the coordinate of the InSAR platform obtained by the GPS and IMU of the radar system, \(\mathbf{P}_{m}{}^{\prime\prime}=\mathbf{V}t\), \(\mathbf{V}\) is the instantaneous velocity of radar platform, and \(\mathbf{V}=\left[v_{r},v_{y}\right]\) for 2D imaging; \(\mathbf{P}_{w}{}^{\prime\prime}\) is the coordinate corresponding to the target point \(\mathbf{P}_{w}\) on the imaging plane. When antennas vibrate and deviate from the ideal antenna trajectory, the azimuth phase error of the traditional frequency-domain algorithm and the proposed algorithm can be expressed as \(\Delta\phi_{RD}(\mathbf{P}_{w})=4\pi(\|\mathbf{P}_{m}{}^{\prime}-\mathbf{P}_{w}\|_{2}-\| \mathbf{P}_{m}-(\mathbf{P}_{w})\|_{2})/\lambda\) and \(\Delta\phi_{FDBP}(\mathbf{P}_{w})=4\pi(\|\mathbf{P}_{m}{}^{\prime\prime}-\mathbf{P}_{w}\| _{2}-\|\mathbf{P}_{m}-\mathbf{P}_{w}\|_{2})/\lambda\), respectively. The precise antenna position information obtained by high-precision positioning tools and IMU(inertial measurement units) makes the coordinate \(\mathbf{P}_{m}{}^{\prime\prime}\) close to \(\mathbf{P}_{m}\). Compared with \(\Delta\phi_{RD}\), \(\Delta\phi_{FDBP}\) can be viewed as \(\Delta\phi_{FDBP}\approx 0\). For complex terrain, the azimuth phase error of the traditional frequency-domain algorithm and the proposed algorithm can be expressed as \(\Delta\phi_{RD}(\mathbf{P}_{w}{}^{\prime})=4\pi(\|\mathbf{P}_{m}-\mathbf{P}_{w}{}^{\prime} \|_{2}-\|\mathbf{P}_{m}-\mathbf{P}_{w}{}^{\prime}\|_{2})/\lambda\) and \(\Delta\phi_{FDBP}(\mathbf{P}_{w}{}^{\prime\prime})=4\pi(\|\mathbf{P}_{m}-\mathbf{P}_{w}{}^{ \prime\prime}\|_{2}-\|\mathbf{P}_{m}-\mathbf{P}_{w}{}^{\prime\prime}\|_{2})/\lambda\), respectively. FDBP-based InSAR algorithm can make use of the distance calculated by the measured antenna trajectory information and the coordinates of target points in the imaging field to compensate for the delayed phase. For non-reference points, the phase error of the RD algorithm is much larger than that of the FDBP algorithm. Thus, compared with the traditional frequency-domain algorithm, the proposed algorithm can better adapt to the complex terrain. In conclusion, compared with the traditional frequency-domain InSAR imaging algorithm, the proposed algorithm makes full use of the position information of the antenna and imaging scene, which improves phase stability, and avoids the phase error and compensation caused by the non-ideal trajectory of the antenna. ### Computational Complexity

InSAR imaging algorithm based on TDBPA can be divided into two steps: the range compression and TDBPA.