# 4.5. 压缩感知成像¶

## 4.5.1. 压缩感知SAR成像¶

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:

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

ABC

## 4.5.2. 实验与分析¶

### 仿真数据¶

#### 实验说明¶

• 仿真场景大小: $$32 \times 32$$

• 回波矩阵大小: $$32 \times 32$$

• 稀疏表示字典: 无, DCT , DWT

• 优化方法: Lasso , OMP

#### 实验结果¶

1. OMP优化, 不采用字典进行稀疏表示, 和采用DCT字典进行稀疏表示的结果如下:

2. Lasso优化, 不采用字典进行稀疏表示, 和采用DCT字典进行稀疏表示的结果如下:

• 场景的稀疏性决定了重构的性能, 越稀疏重构越精确

• 字典的选择很重要