4.4. 正则化成像方法¶

4.4.1. 正则化成像¶

SAR imaging process can be formulated as

\[{\bm s} = {\bm A}{\bm g}, \]

where, ${\bm s}$ is the $MN\times 1$ recieved SAR raw data vector in phase history domain, $\bm g$ is the $HW \times 1$ reflection vector of scene, and $\bm A$ represents the the mapping from scene to SAR raw data.

Given $\bm s, \bm A$ , regularization methods try to reconstruct $\bm g$ by minmizing

\[\mathop {\rm min}\limits_{\bm{g}} {\left\| {{\bm{Ag}} - {\bm{s}}} \right\|_2} + \lambda |{\bm{g}}{|_p}, \]

where, $\lambda$ is the balance factor, and $|\cdot|_p$ is the $\ell_p$ norm.

Note that, if ${\bm s, A, g} \in {\mathbb C}$ , the problem changes to

\[{\mathop{\rm Re}\nolimits} ({\bm{s}}) + j{\rm Im}({\bm{s}}) = {\rm Re}({\bm{Ag}}) + j{\mathop{\rm Im}\nolimits} ({\bm{Ag}}) \]

so we have:

\[\left[ {\begin{array}{ccc} {{\mathop{\rm Re}\nolimits} ({\bm{s}})}\\ {{\mathop{\rm Im}\nolimits} ({\bm{s}})} \end{array}} \right] = \left[ {\begin{array}{ccc} {{\mathop{\rm Re}\nolimits} ({\bm{A}})}&{ - {\mathop{\rm Im}\nolimits} ({\bm{A}})}\\ {{\rm Im}({\bm{A}})}&{{\mathop{\rm Re}\nolimits} ({\bm{A}})} \end{array}} \right]\left[ {\begin{array}{ccc} {{\mathop{\rm Re}\nolimits} ({\bm{g}})}\\ {{\mathop{\rm Im}\nolimits} ({\bm{g}})} \end{array}} \right] \]