# FDBP-InSAR: An Efficient Algorithm for InSAR Imaging via Frequency Domain Back Projection
## 3 InSAR Imaging via FDBPA  
### Theory  
Inspired by the basic focusing theory of both frequency-domain-based algorithms and time-domain-based algorithms, FDBP adopts the idea of back projection to eliminate the azimuth and range coupling. As mentioned above, the equation of TDBPA can be described as follows:  
\[\mathbf{s}(l)=\int\limits_{t}\mathbf{s}(t,l)\cdot\exp(j4\pi l(t)/\lambda)dt \tag{5}\]  
where \(l\) and \(t\) are the distance in range direction and azimuthal time in the time domain; and \(\lambda\) denotes the wavelength. Converting (5) to the two-dimensional wavenumber domain, the formula will be converted to the following:  
\[\mathbf{S}(\mathbf{K})=\int\limits_{\Omega}\int\limits_{t}\mathbf{s}(t,l)\cdot\exp(j4\pi l( t)/\lambda)dt\exp(-j\mathbf{KL})d\mathbf{L} \tag{6}\]  
where \(\mathbf{K}\) is the wavenumber vector and \(\mathbf{K}=[k_{y},k_{r}]\) for 2D imaging; \(k_{y}\) and \(k_{r}\) are the wavenumber in azimuth and range direction, respectively; \(\mathbf{L}\) is the space vector, and \(\mathbf{L}=[l_{y},l_{r}]\); \(l_{y}\) and \(l_{r}\) are the distance in azimuth and range direction, respectively; and \(\Omega\) is the selected imaging space. Equation (5) is the processing result of the conventional time-domain back-projection algorithm, (6) is the results of the wavenumber-domain back-projection algorithm. Obviously, the representations of them are very similar. Therefore, we can apply the coherent accumulation in the wavenumber domain as the equivalent of focusing imaging in the time domain. After some rearrangements, (6) can be written as follows:  
\[\mathbf{S}\big{(}k_{y},\omega\big{)}=\int\limits_{l_{r}}\int\limits_{t}\mathbf{s}(l) \exp\big{(}-jk_{y}vt\big{)}dt\cdot\exp(-j\omega l_{r})dl_{r} \tag{7}\]where \(l\) is the distance between platform and imaging scene and \(l=\|\mathbf{P}-\mathbf{P}_{m}\|_{2}\), \(v\) is the azimuthal velocity of the platform, \(\omega\) is the angular frequency of the echo signal. According to the wave theory, \(k_{y}\) and \(\omega\) are not orthogonal. Their relationship can be expressed as \(\omega=\frac{\zeta}{2}\sqrt{{k_{y}}^{2}+{k_{r}}^{2}-\omega_{0}}\), where \(\omega_{0}\) is the carrier angular frequency, \(\mathrm{k}_{y}=2\pi f_{d}/v\) and \(f_{d}\) is the Doppler frequency. The central wavenumber in the azimuth direction is \(\mathrm{k}_{yc}\) and \(\mathrm{k}_{yc}=4\pi\sin(\beta)/\lambda\), \(v\) is the velocity of the radar platform, \(\beta\) is the squint angle. It shows the coupling in azimuth and range direction. FDBPA uses the idea of back projection to realize decoupling. Taking the back projection, (7) can be converted to the following:  
\[\mathbf{S}\big{(}k_{y},k_{r}\big{)}=\int\limits_{t}\mathbf{s}(t,k_{r})\exp\big{(}-jk_{ y}vt\big{)}dt \tag{8}\]  
where \(\mathbf{s}(t,k_{r})=\mathbf{B}\mathbf{P}[\mathbf{s}(t,\omega)]\), denoting back-projection processing on \(s(t,\omega)\). Owing to the fact that \(k_{y}\) and \(t\) are the uniform variables in the frequency domain and time domain, respectively, the coherent accumulation in (8) can be considered as the azimuthal non-uniform Fourier transform of \(s(t,k_{r})\). To reflect the phase factor term in (8), the formula above also can be described as follows:  
\[\mathbf{F}\big{(}k_{y},k_{r}\big{)}=\mathbf{S}\big{(}k_{y},k_{r}\big{)}\exp\left(-j \big{(}\sqrt{{k_{r}}^{2}+{k_{y}}^{2}}-k_{r}\big{)}R\right) \tag{9}\]  
where \(\mathbf{S}\big{(}k_{y},k_{r}\big{)}=\int\limits_{\Omega}\int\limits_{\Omega}s(t,l) \exp(-j\mathbf{K}\mathbf{L})dL\), \(R=\|\mathbf{p}_{0}-\mathbf{p}_{m}(t_{0})\|_{2}\), \(\mathbf{p}_{0}\) is the coordinates of central point of the target scene, \(\mathbf{p}_{m}(t_{0})\) is the coordinates of the main antenna at \(t_{0}-th\) slow time, in which \(t_{0}\) is the slow time when the distance from platform to target scene is smallest, \(\mathbf{F}\big{(}k_{y},k_{r}\big{)}\) denotes the wavenumber-domain-focused image data, imaging result \(\mathbf{f}(x,y)\) can be obtained from the 2D IFFT of \(\mathbf{F}\big{(}k_{y},k_{r}\big{)}\).