基于隨機(jī)平均梯度下降和對比源反演的非線性逆散射算法研究

周輝林; 歐陽韜; 劉健

doi:10.11999/JEIT190566

基于隨機(jī)平均梯度下降和對比源反演的非線性逆散射算法研究

doi: 10.11999/JEIT190566 cstr: 32379.14.JEIT190566

南昌大學(xué)信息工程學(xué)院南昌 330031

基金項(xiàng)目: 國家自然科學(xué)基金(61561034, 61261010, 41505015)

詳細(xì)信息

作者簡介:
周輝林：男，1979年生，教授，研究方向?yàn)槌瑢拵Ю走_(dá)成像、雷達(dá)信號處理

歐陽韜：男，1996年生，碩士生，研究方向?yàn)槌瑢拵降乩走_(dá)成像，逆散射成像方法研究

劉?。耗?，1995年生，碩士生，研究方向?yàn)槌瑢拵Т走_(dá)成像，逆散射成像方法研究

通訊作者:
周輝林　zhouhuilin@ncu.edu.cn

中圖分類號: O451
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出版歷程
- 收稿日期: 2019-07-26
- 修回日期: 2020-02-22
- 網(wǎng)絡(luò)出版日期: 2020-03-23
- 刊出日期: 2020-08-18

Stochastic Average Gradient Descent Contrast Source Inversion Based Nonlinear Inverse Scattering Method for Complex Objects Reconstruction

Institute of Information engineering, Nanchang University, Nanchang 330031, China

Funds: The National Natural Science Foundation of China (61561034, 61261010, 41505015)

摘要

摘要:
采用非線性對比源反演(CSI)算法求解電磁逆散射問題時(shí)，在每次迭代過程中都涉及到求解散射場數(shù)據(jù)關(guān)于對比源和總場的微分，即Jacobi矩陣，該矩陣求解導(dǎo)致算法存在計(jì)算代價(jià)大和收斂速度慢等問題。該文在CSI框架下，采用一種基于隨機(jī)平均梯度下降的對比源反演算法(SAG-CSI)代替原來的全梯度交替共軛梯度算法來重構(gòu)介質(zhì)目標(biāo)介電常數(shù)的空間分布信息。該方法在每次迭代中只需計(jì)算隨機(jī)抽取的部分測量數(shù)據(jù)在目標(biāo)函數(shù)中的梯度信息，同時(shí)目標(biāo)函數(shù)對未抽中的測量數(shù)據(jù)的梯度信息保持不變，用以上兩部分梯度信息共同求解出目標(biāo)函數(shù)的最優(yōu)值。由模擬數(shù)據(jù)結(jié)果表明，該方法與傳統(tǒng)CSI方法在成像精度相比擬的情況下，降低了計(jì)算代價(jià)并提高算法收斂速度。
- 非線性電磁場逆散射 /
- 對比源反演 /
- 隨機(jī)平均梯度
Abstract:
When using the nonlinear Contrast Source Inversion (CSI) algorithm to solve the electromagnetic inverse scattering problem, each iteration involves finding the differential of the dissolution radiation field data about the contrast source and the total field, i.e., the Jacobi matrix. the solution of the matrix leads to the problem of large computational cost and slow convergence speed of the algorithm. in this paper, a Contrast Source Inversion algorithm based on Stochastic Average Gradient descent (SAG-CSI) is used instead of the original full gradient alternating Conjugate Gradient algorithm to reconstruct the spatial distribution information of the dielectric constant of the dielectric target under the CSI framework. the method only needs to calculate the gradient information of the randomly selected part of the measurement data in the objective function in each iteration, while the objective function keeps the gradient information of the unscented measurement data, and the optimal value of the objective function is solved together with the above two parts of the gradient information. The simulation results show that the proposed method reduces the computational cost and improves the convergence speed of the algorithm when compared with the traditional CSI method.
- Inverse scattering /
- Contrast Source Inversion (CSI) /
- Stochastic Average Gradient (SAG)

HTML全文

圖 1 2維電磁逆散射模型

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圖 2 Austria散射體

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圖 3 SNR為10 dB 時(shí)SAG-CSI的成像結(jié)果

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圖 5 SNR為10 dB 時(shí)CSI的成像結(jié)果

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圖 4 不同信噪比下兩種方法的目標(biāo)函數(shù)

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圖 6 SAG-CSI中不同散射場數(shù)據(jù)量的目標(biāo)函數(shù)

下載: 全尺寸圖片幻燈片

表 1 SAG-CSI算法

　輸入：對比源初值：${{{w}}_0}$，

　　　　散射場數(shù)據(jù)：${{{E}}_{{\rm{sca}}}}$，

　　　　測量數(shù)據(jù)的索引：$i \in \left( {1,2, ··· ,N} \right)$，

　　　　隨機(jī)抽取數(shù)據(jù)的索引：${i_k} \in \left( {1,2, ··· ,N} \right)$。

　輸出：當(dāng)目標(biāo)函數(shù)${{{F}}_n}$達(dá)到設(shè)定誤差值$\delta $或最大迭代次數(shù)${{{N}}_{{\rm{iter}}}}$時(shí)，輸出對比度函數(shù)：${{{\chi}} _n}$

　(1) 計(jì)算格林函數(shù)${{{G}}_{\rm{D}}}$, ${{{G}}_{\rm{R}}}$

　(2) 對比度函數(shù)初值${{{\chi}} _0} = 0$，

　(3) 由前向模型計(jì)算得到對比源初值
　　　　　　　　　　　　　　　　　　　　　　　${ {{w} }_0} = \dfrac{ {\parallel { {{G} }^H_{\rm R} }{ {{E} }_{\rm sca} }{\parallel ^2} } }{ {\parallel { {{G} }_{\rm R} }{ {{G} }^H{\rm R} }{ {{E} }_{\rm sca} }{\parallel ^2} } }{ {{G} }^H_R}{ {{E} }_{\rm sca} }$ (*為共軛轉(zhuǎn)置) (16)

　(4) $n \leftarrow 0$(n為迭代次數(shù))

　(5) while ${{{F}}_n}\left( {{w_n}} \right) < {{\delta}} $或${{n}} < {N_{{\rm{iter}}}}$ do

　　　步驟 1 ?每次迭代隨機(jī)抽取測量數(shù)據(jù)的索引：${i_k} = {\rm{randperm}}\left( i \right)$

　　　步驟 2 將步驟1抽取的測量數(shù)據(jù)按照進(jìn)行梯度更新計(jì)算(其中$\rho _{n - 1}^{{i_k}}$, $\gamma _{n - 1}^{{i_k}}$和${\chi _{n - 1}}$分別表示隨機(jī)抽取散射場數(shù)據(jù)的數(shù)據(jù)方程誤差和場誤差以及上次迭代保存的對比度函數(shù))
　　　　　　　　　　　　　　　　　　　　${{g} }_n^{ {i_k} } = - \frac{ { { {\left( { {{G} }_{\rm{R} }^{ {i_k} } } \right)}^H}\rho _{n - 1}^{ {i_k} } } }{ {\displaystyle\sum\limits_k {\parallel {{E} }_{ {\rm{sca} } }^{ {i_k} }{\parallel ^2} } } } - \frac{ {\gamma _{n - 1}^{ {i_k} } - {{G} }^H_{\rm{D} } { { {{{\chi} } }^* _{n - 1} } } \gamma _{n - 1}^{ {i_k} })} }{ {\displaystyle\sum\limits_k {\parallel { {{\chi} } _{n - 1} }{{E} }_{ {\rm{inc} } }^{ {i_k} }{\parallel ^2} } } }$(*為復(fù)共軛) (17)

　　　步驟 3 ?對抽取的散射場數(shù)據(jù)在目標(biāo)函數(shù)中的梯度值進(jìn)行更新，其余散射場數(shù)據(jù)對應(yīng)的梯度不變(其中${\nabla _{{w_n}}}{{F}}_n^i$表示隨機(jī)抽取散射場數(shù)據(jù)在目標(biāo)函數(shù)中關(guān)于對比源的更新梯度值，${{g}}_{n - 1}^{{i_k}}$表示上次迭代該散射場數(shù)據(jù)對應(yīng)的梯度值，${\overline { { {{g} }_{n - 1} } } }$表示所有散射場數(shù)據(jù)在目標(biāo)函數(shù)中的梯度矩陣)

　　　　　　　　　　　　　　　　　　　　　　　　　　${ { { {{\overline{{g} } } }_n} } } ={ { { { {\overline{{g} } } }_{n - 1} } } } - {{g} }_{n - 1}^{ {i_k} } + { {\text{?} } _{ {w_n} } }{{F} }_n^i$ (18)

　將所有測量數(shù)據(jù)構(gòu)成的梯度矩陣$\overline{\overline {{{{g}}_n}}} $求和取平均得出當(dāng)前迭代的梯度${{{g}}_n}$,即搜索方向的反方向。

　　　步驟 4 ?由步驟3所得的搜索方向和式(15)所求得的搜索步長可計(jì)算更新的對比源(上次迭代保存的對比源${{{w}}_{n - 1}}$)：${{{w}}_n} \!=\! {{{w}}_{n - 1}} \!-\! {{{\alpha}} _n}{{{g}}_n}$

　　　步驟 5 ?更新總場的值：${{E}}_{{\rm{tot}}}^n = E_{{\rm{tot}}}^{n - 1} - {{{\alpha }}_n}{{{G}}_{\rm{D}}}{{{g}}_n}$

　　　步驟 6 ?由對比源作為輔助變量公式可知
　　　　　　　　　　　　　　　　　　　${ {{\chi} } _n} = \frac{ {\displaystyle\sum\limits_K { { {{w} }_n} ({ { {{{E} } } }_{ {\rm{tot} } }^n})^* } } }{ {\displaystyle\sum\limits_K { {{E} }_{ {\rm{tot} } }^n ({ {{{E} } }_{ {\rm{tot} } }^n})^* } } }$ (*為復(fù)共軛,K為抽取樣本的總個(gè)數(shù)) (19)

　　　步驟 7 ?${{n}} = {{n}} + 1$

　(6) end

　(7) ${{\chi}} = {{{\chi}} _n}$

下載: 導(dǎo)出CSV

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