超声逆散射图象重建问题中截断奇异值分解正则化方法研究

The Study on Truncated Singular Value Decomposition Method in Ultrasound Inverse Scattering Image Reconstruction

为解决超声逆散射成像问题中的非线性性,人们需要反复地求解前向散射方程和逆散射方程,以达到对全场和未知函数的精确近似,从而根据这一未知函数的精确近似,较好地重建物体内部的断层图象.前向散射方程是一个适定的方程组,可以采用通常的方法进行求解;而逆散射方程则是一个不适定性的方程组,即使数据中存在一个微小的误差,都可能引起解的较大偏离,因此,对这个不适定方程组的求解问题是整个迭代算法成功的关键.而在不适定性问题的求解过程中,正则化参数的选取又是非常重要的.求解不适定性方程的传统方法是Tikhonov正则化方法,这一方法的实质是在传统最小二乘方法上加上一个小于1的滤波因子,对于超声逆散射成像问题来说,效果并不太好.本文将截断奇异值分解正则化方法应用于逆散射方程的求解问题中,并对正则化参数的选取方法进行修正.数值仿真结果表明,这一方法配合适当的正则化参数选取,可以更好地滤除噪声,提高重建图象的质量与可信度,同时还可以减小迭代过程中的计算量.

To process the nonlinear property of ultrasound inverse scattering image, one should alternately solves the well-posed forward scattering equations for an estimated total field and the ill-posed inverse scattering equations for the desired object property function. Forward scattering equations can be solved by common method while inverse scattering equations is ill-posed and should be regularized. For ill-posed inverse scattering equations, very little perturbation in data will cause great change in the solution. So the iterative procedure depends strongly on the precision of the solution of ill-posed inverse scattering equations. Previous work on the ill-posed inverse scattering equations commonly used Tikhonov regularization which by adding small filter factors to original least squares problem and can't filter noise efficiently. The method for choosing regular parameter is difficulty in Tikhonov regularization because the parameter is continuous. This paper adopts the truncated singular value decomposition (TSVD) method to solve the inverse scattering equations which can filter noise better than Tikhonov regularization. Since the regularization parameter is an integer in TSVD method, it can be revised by an appropriate method. Different 'images with different structure are simulated by truncated singular value decomposition method equipped with a revised parameter choosing strategy. Simulation results show that this method associated with a good approach for choosing regular parameter can efficiently filter noise, and hence the quality and reliability of the reconstruction image can be improved. At the same time, this method can decrease computations at the iterative procedure.