Nonlinear Inverse Scattering Methods for Three Dimensional Objects

在之前的五年PYI资助下，加上随后的一年和三年的资助，使研究者在逆散射和正向散射理论方面取得了重大进展。为了考虑散射对象内部的多重散射效应，提出了一种新的非线性逆散射方法。提出了born -迭代法（BIM）和畸变born -迭代法（DBIM），对EZ （TM）极化波的反演对比度高达10:1，并表现出0.1波长量级的超分辨率。另一种被称为局部形状函数（LSP）方法的新技术被开发出来，用于对金属散射体，HZ （TE）极化波和变密度声波进行逆散射，其中以前的方法由于极强的非线性和格林函数奇点而无法产生收敛解。最近，已经发展了跳频方法，其中可以重建具有高对比度的大散射体。此外，还获得了初步的三维逆散射结果。对大散射物体进行反演的需要刺激了快进散射求解器的研究。这些快速算法准确地解决了散射问题，并利用散射计算中的固有冗余来节省计算量。首先是递归算法（RATMA， RTMA），它通过利用波现象的平移特性来解决0（N2）操作中所有入射源的散射问题。新算法（尼泊尔）使用表面等效原理，产生可并行处理的0 （N1.5）算法。该算法已推广到三维空间。最近，各种快速迭代求解器被开发出来，它们提供了0（N log N）次运算和使用0(N)存储器的一个激励的解。它们涉及到快速多极子方法的使用，以及对平移算子进行对角化的各种方法。目前的建议是继续发展新的非线性逆散射理论，特别强调三维散射体。研究人员计划摆脱波恩类型的方法，并开发新的参数化散射对象的方法，以实现最佳的重建。他们将利用他们在快进散射解算器上的进展，作为加快反演和求解更大的物体和三维物体的基础。其他解决大型物体的方法，特别是埋藏在缓慢变化的非均匀背景中的物体，也将被研究。最后，他们希望开发一种新的实验装置，可以实现高保真度的测量，以便在实验中验证超分辨率现象。＊＊＊

9906651ChewThe previous support period under the five-year PYI grant plus a subsequent one-year and then three-year grant enabled the investigator to make significant progress in both inverse and forward scattering theory. New nonlinear inverse scattering methods were developed to account for multiple scattering effects within the scattering object. The Born-iterative method (BIM) and distorted-Born-iterative method (DBIM) were developed that demonstrated the inversion of objects with contrasts as great as 10:1 and exhibited super-resolution on the order of 0.1 wavelengths for EZ (TM) polarized wave. Another new technique known as the local-shape-function (LSP) method was developed to perform inverse scattering on metallic scatterers, for HZ (TE) polarized waves, and variable density acoustic waves, where previous methods failed to produce a convergent solution due to extremely strong nonlinearities and Green's function singularity. More recently, the frequency hopping approach where large scatterers with high contrasts can be reconstructed, have been developed. Furthermore, preliminary three-dimensional inverse scattering result has been obtained. The need to invert very large scattering objects stimulated research in fast forward scattering solvers. These fast algorithms solve the scattering problem exactly and achieve computational savings by exploiting inherent redundancies in the scattering calculations. The first of these are recursive algorithms (RATMA, RTMA) that solve the scattering problem valid for all incidence sources in 0(N2) operations by exploiting translational properties of wave phenomenon. The new algorithm (NEPAL) instead uses the surface equivalence principle resulting in an 0 (N1.5) algorithm that is amenable to parallel pro-cessing. This algorithm has been generalized to three dimensions. Recently, various fast iterative solvers have been developed that furnish the solution for one excitation in 0(N log N) operations and using 0(N) memory. They involve the use of fast multipole method, and various methods to diagonalize the translation operator.The current proposal is to continue developing new nonlinear inverse scattering theo-ries with special emphasis on three-dimensional scatterers. The investigators plan to move away from the Born-type methods and exploit new ways of parameterizing the scattering object in order to achieve the best possible reconstruction. They will use their progress in fast forward scattering solvers as building blocks to speed up the inversion and solve much larger objects and three-dimension-objects. Other means of solving large objects, particularly objects buried in a slowly varying inhomogeneous background would be investigated as well. Finally, they wish to develop new experimental apparatus that can achieve high-fidelity measurements so that the super-resolution phenomenon can be verified experimentally. ***