一种处理分层有耗色散介质的时域逆散射方法

An Approach to the Time-Domain Inverse Scattering Problem for the Stratified Frequency-Dispersive Lossy Media

为了重建分层有耗色散介质的特征参数,我们应用泛函分析和变分法,提出一种时域逆散射新方法.该方法首先以最小二乘准则构造目标函数,将逆问题表示为约束最小化问题;接着应用罚函数法转化为无约束最小化问题;然后基于变分计算导出闭式的拉格朗日(Lagrange)函数关于特征参数的Férchet导数;最后借助梯度算法和时域有限差分(FDTD)法迭代求解.此外,为了对抗噪声污染和逆问题的病态特性,采用了一阶吉洪诺夫(Tikhonov)正则化方法.数值应用中,利用一组非线性共轭梯度(CG)法,在几种情形下对一维乳房模型进行了数值测试,测试结果显示了算法的鲁棒性和识别病灶的有效性.

We present a time-domain inverse scattering approach using functional analysis and variational method,which is applied to reconstruct dispersive dielectric properties of stratified lossy media.This approach firstly formulates a cost functional,according to least squares criterion,to turn the inverse problem into a constrained minimization problem.The resulting constrained minimization problem is then transformed into an unconstrained minimization problem by a penalty function technique.And then,the closed Fréchet derivatives of the Lagrange function with respect to the properties are derived based on the calculus of variations.Thus,the unconstrained minimization problem can be solved by using any gradient-based algorithm and the finite-difference time-domain（FDTD） method.Also,the first-order Tikhonov＇s regularization is adopted to cope with noise and the ill-posedness of the problem.In numerical examples,the presented algorithm is applied to a one-dimensional（1-D） breast model in several cases with the help of a set of nonlinear conjugate gradient（CG） methods,and the simulated results demonstrate its robustness and validity in the lesions identification.