# FDBP-InSAR: An Efficient Algorithm for InSAR Imaging via Frequency Domain Back Projection
## 1 Introduction
Making use of the trajectory positions from auxiliary measuring equipment, such as the inertial measurement unit (IMU) or global positioning system (GPS), TDBPA can compensate for the phase delay of the scene directly and avoid range migration correction in the frequency-domain-based algorithm. Furthermore, the imaging plane of TDBPA can be selected as a surface, which greatly reduced the geometric distortion and the overlaying caused by complex rugged topography. Therefore, even in the condition of non-ideal motion trajectory and complex terrain, TDBPA still performances well for high-precision focusing and interferometric phase extraction of a large scene. However, huge complexity and low efficiency is one obvious shortcoming of TDBPA, suffer from due to the point-by-point integration, which restricts its practical application [30; 31; 32; 33]. Compared with the computational complexity of the frequency-domain algorithms \(\mathcal{O}\big{(}N^{2}\log N\big{)}\) for an \(N\ \times\ N\) image, the complexity of TDBPA grows as \(\mathcal{O}\big{(}N^{3}\big{)}\), which makes it difficult to be real-time imaging in InSAR large scene observation. To improve imaging efficiency of TDBPA, some efficient modified TDBP methods based on sub-aperture decomposition and scene segmentation, such as the fast factorized back-projection algorithm (FFBPA) [34; 35; 36; 37; 38; 39], have been proposed for SAR imaging. Compared with the conventional TDBPA, FFBPA can reduce the computational complexity, but its imaging accuracy may decrease if the number of decomposed sub-aperture increased. Hence, FFBPA may not be suitable for InSAR high-resolution imaging with a complex trajectory. To further overcome the drawback of TDBP, an efficient method via transform domain focusing, named as the frequency-domain back-projection algorithm (FDBPA) [40; 41; 42], is proposed for SAR efficient imaging. In Reference [40], the presented FDBPA transfers the point-to-point coherent integration of conventional TDBPA to the wavenumber domain. It samples the spectrum of the SAR echo data after range compression and projects the spectrum data into the wavenumber domain. Eachfrequency grid in the wavenumber domain contains the information of the whole image scene. Then the focused image can be obtained by taking inverse Fourier transform of the wavenumber transform data. Owing to the index of each frequency grid is uniform, the step of index value calculation can be performed only once. Compared to TDBPA, FDBPA has improved imaging efficiency and kept similar imaging precision. In Reference [41], the authors introduce the application of FDBPA in stepped-frequency SAR, which improves the computation efficiency and realizes the automatic spatial spectra cutting. In Reference [42], an efficient three-dimensional imaging method based on FDBPA is proposed for SAR imaging, the results also demonstrate the great improvement of imaging efficiency. However, only the efficiency and aggregation degree are analyzed in these FDBPA methods, without considering the phase retention, the topographic relief effect, and the phase derivation relationship of rugged topography. These FDBPA methods cannot be used for InSAR imaging directly. Moreover, the theoretical proof of these FDBPA is only based on the simulation data, without the support of the experiment results of a large scene. Exploiting the advantages of FDBPA is still a challenging issue in InSAR imaging. In this paper, inspired by the FDBPA [40], we propose an efficient imaging method to achieve high-resolution focusing and accurate phase-preserving of InSAR imaging. The proposed method adopts a non-uniform Fourier transform instead of coherent integration and uses the accurate distance between antennas and imaging spots instead of reference distance to compensate for the delayed phase. In the scheme, the antenna position information is introduced in the azimuth non-uniform Fourier transform and the fully utilized in compensating the delayed phase of the echo signal.