4.5. 压缩感知成像¶

4.5.1. 压缩感知SAR成像¶

越来越多的基于压缩感知的SAR成像方法被提出 [3][4][5] .

匹配追踪用于量子模拟 [1]

SAR imaging process can be formulated as

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

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

If \(\bm g\) is not sparse enough, assume that exist a basis \({\bm D} = ({\bm d}_1, {\bm d}_2, \cdots, {\bm d}_n)\) that satisfies \({\bm g} = {\bm D}{\bm x}\) , where, \(\bm x\) is a \(K\) sparse \(n\times 1\) vector, and \(\bm D\) is the so called dictionary matrix of size \(n\times n\) .

Our goal is minimize

\[\mathop {\rm min}\limits_{\bm x}\|{\bm x}\|_p, \ \ s.t. \ {\bm s} = {\bm A}{\bm D}{\bm x} + {\bm n}, \]

i.e.

\[\mathop {\rm min}\limits_{\bm{x}} = \|{\bm s} - {\bm A}{\bm D}{\bm x}\|_2 + \lambda \|{\bm x}\|_p, \]

where, \(\lambda\) is the balance factor, and \(|\cdot|_p, (0<p<1)\) is the \(\ell_p\) norm.

Let \({\bm \Phi} = {\bm A}{\bm D}\) , then we have

\[\mathop {\rm min}\limits_{\bm{x}} = \|{\bm s} - {\bm \Phi}{\bm x}\|_2 + \lambda \|{\bm x}\|_p. \]

Note that, if \({\bm s, \Phi, x} \in {\mathbb C}\) , the problem changes to

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

so we have:

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

提示

对于非稀疏场景, 无需先对场景进行稀疏表示, 再进行观测; 然而在重构信号时, 由于信号非稀疏, 需要假设其在某一字典下稀疏.

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