# Interpolation Methods with Phase Control for Backprojection of Complex-Valued SAR Data +
## 3 Interpolation Methods
### Extended Cubic-Spline Interpolation  
Cubic spline function for complex-valued data interpolation \(p_{\rm c}(\tau)\) is a piecewise cubic polynomial function on the interval \([\tau_{0},\tau_{2}]\), \(\tau\in\mathbb{R}\), and where the samples \(\tau_{i-1}<\tau_{i}\) for \(i=1,2\) are the knot points. The cubic spline \(p_{\rm c}\) can be uniquely defined by  
\[p_{\rm c,i}(\tau)=a_{i}+b_{i}(\tau-\tau_{i-1})+c_{i}(\tau-\tau_{i-1})^{2}+d_{i} (\tau-\tau_{i-1})^{3},\qquad\tau_{i-1}\leq\tau\leq\tau_{i}, \tag{14}\]with corresponding first  
\[p^{\prime}_{c,i}(\tau)=b_{i}+2c_{i}(\tau-\tau_{i-1})+3d_{i}(\tau-\tau_{i-1})^{2},\qquad\tau_{i-1}\leq\tau\leq\tau_{i}, \tag{15}\]  
second  
\[p^{\prime\prime}_{c,i}(\tau)=2c_{i}+6d_{i}(\tau-\tau_{i-1}),\qquad\tau_{i-1} \leq\tau\leq\tau_{i}, \tag{16}\]  
and third derivatives  
\[p^{\prime\prime\prime}_{c,i}(\tau)=6d_{i},\qquad\tau_{i-1}\leq\tau\leq\tau_{i}. \tag{17}\]  
Here, \(a_{i},b_{i},c_{i},d_{i}\in\mathbb{C}\) for \(i=1,2\) are the polynomial coefficients. Furthermore, the spline function satisfies the following conditions  
\[\left\{ \begin{array}{l}p_{c,i}(\tau_{i-1})=\tilde{y}_{i-1},\\ p_{c,i}(\tau_{i})=\tilde{y}_{i},\\ p^{\prime}_{c,i}(\tau_{i})=p^{\prime}_{c,i+1}(\tau_{i}),\\ p^{\prime\prime}_{c,i}(\tau_{i})=p^{\prime\prime}_{c,i+1}(\tau_{i})\end{array} \right. \tag{18}\]  
for \(i=1,2\), and where the last two conditions are not applicable at the edge knot points \(\tau_{0}\) and \(\tau_{2}\). Here, \(\tilde{y}\) denotes the reference data values, the phase of which is controlled under the procedure introduced in (6). It should be noted that controlled phase value may vary with a factor of \(\pi\) due to corresponding location of data values in the complex plane, i.e., either in the right \(\mathbb{C}_{+}\) or in the left complex half-plane \(\mathbb{C}_{-}\). Assume that the range time \(\tau_{\rm p}\) defined in (4) satisfies the condition \(\tau_{0}<\tau_{\rm p}<\tau_{1}\). Let \(p^{\prime\prime}_{c,i}(\tau_{i-1})=k_{i-1}\), from which the unknown polynomial coefficient can be expressed as \(c_{i}=k_{i-1}/2\). Furthermore, let \(\delta_{i-1}=\tau_{i}-\tau_{i-1}\). By employing spline conditions (18), the unknown polynomial coefficients can be determined as  
\[\left\{ \begin{array}{l}a_{i}=\tilde{y}_{i-1},\\ b_{i}=\frac{\tilde{y}_{i}-\tilde{y}_{i-1}}{\delta_{i-1}}-\frac{k_{i}+2k_{i-1} }{6}\delta_{i-1},\\ c_{i}=\frac{k_{i-1}}{2},\\ d_{i}=\frac{k_{i}-k_{i-1}}{6\delta_{i-1}}\end{array} \right. \tag{19}\]  
for \(i=1,2\). Substituting (15) and (19) into the third spline condition in (18) and applying the natural boundary conditions [26; 27], i.e., \(p^{\prime\prime}_{c,1}(\tau_{0})=p^{\prime\prime}_{c,2}(\tau_{2})=0\), one can construct a system of linear equations  
\[\mathbf{Ak}=\mathbf{b}, \tag{20}\]  
which provides a unique solution to unknown parameters \(k_{i}\), \(i=0,1,2\). Here,  
\[\mathbf{A}=\left[ \begin{array}{ccc}1&0&0\\ \delta_{0}&2(\delta_{0}+\delta_{1})&\delta_{1}\\ 0&0&1\end{array} \right], \tag{21}\]  
\[\mathbf{b}=6\left[ \begin{array}{c}\frac{\partial}{\delta_{0}}-\tilde{y}_{1} \frac{\delta_{0}+\delta_{1}}{\delta_{0}\delta_{1}}+\frac{\tilde{y}_{2}}{\delta _{1}}\\ 0\end{array} \right], \tag{22}\]  
and \(\mathbf{k}=[k_{0}\ k_{1}\ k_{2}]^{\rm T}\). Note that since the interpolation of complex SAR data is of the local type, where only three data samples are used, we assume that signal is compactly supported on \([\tau_{0},\tau_{2}]\) and employ the natural boundary conditions.