Stable and Convergent Computational Algorithms for Current Density Imaging

将医学和地球物理成像中的两个不同的物理场结合在一起，能够提高成像方式的空间分辨率和对比度分辨率。该项目涉及这样一种模式-电流密度成像(CDI)，它将磁共振成像(MRI)中使用的磁场和射频电磁场与电阻抗断层成像(EIT)中使用的电场相结合。这种耦合产生的电导率图像比通过常规医学成像模式获得的图像具有显著更高的质量和精度。这个项目的总体目标是为CDI开发计算高效和健壮的算法。这项研究是在耦合物理(混合)反问题领域进行的，其显著特点是利用新成像模式中可用的内部数据来重建待研究对象的材料参数。谈到这种反问题的可能应用，应特别强调医学诊断学和海洋控制源电磁测深，因为它们对于癌症的早期发现、最大限度地减少误诊率、更有效地治疗疾病等以及对于陆架碳氢化合物矿藏的地球物理勘探具有重要意义。该项目的创新之处在于利用加权平均曲率流动方程的初边值问题发展了CDI的计算工具。为了实现这一总体目标，主要的研究人员将：(1)利用与Dirichlet问题相关的加权平均曲率分析运动的水平集形式，作为CDI的数学模型；(2)开发稳定和收敛的计算算法；(3)进行数值收敛研究，以在数值实验中验证计算算法的有效性和重建的质量。在这个项目中开发的方法有望促进对CDI几何特征的理解。一名本科生将参与该项目，并接受逆问题数值方法的培训。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Coupling together two distinct physical fields in medical and geophysical imaging is capable of enhancing both the space and contrast resolution of imaging modalities. This project concerns one of such a modality - Current Density Imaging (CDI) that couples the magnetic and radio-frequency electromagnetic fields used in Magnetic Resonance Imaging (MRI) with the electric field used in Electrical Impedance Tomography (EIT). Such a coupling is resulted in images of the electrical conductivity with significantly higher quality and accuracy than those obtained by the routine medical imaging modalities. The general aim of this project is to develop the computationally efficient and robust algorithms for CDI. The proposed research is in the field of coupled physics (hybrid) inverse problems whose distinctive feature is utilizing the interior data available in new imaging modalities in order to reconstruct material parameters of an object to be investigated. Speaking about the possible applications of such inverse problems, it should be particularly emphasized medical diagnostics and marine controlled source electromagnetic sounding due to their importance for early detection of cancer, minimizing the rate of false diagnoses, more efficient treatment of diseases, etc., and for geophysical exploration of hydrocarbon deposits on a shelf. The innovation of this project is that it is concerned with the development of computational tools for CDI by utilizing an initial boundary value problem for the weighted mean curvature flow equation. In order to achieve the general aim, the principal investigator will: (1) analyze the level set formulation of motion by the weighted mean curvature associated with the Dirichlet problem for the weighted 1-Laplacian, which is considered as a mathematical model of CDI, (2) develop the stable and convergent computational algorithms, and (3) conduct the numerical convergence study in order to demonstrate in numerical experiments the effectiveness of computational algorithms and quality of reconstruction. The approach developed in this project is expected to advance understanding of the geometrical features of CDI. One undergraduate student will be involved in the project and trained in numerical methods for the inverse problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.