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物理场耦合在一起，能够提高成像模式的空间和对比度分辨率。该项目涉及其中一种模式——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扫描（EIT）中使用的电场耦合在一起。这种耦合导致电导率图像的质量和准确性明显高于常规医学成像方式获得的图像。本课题的总体目标是开发计算效率高、鲁棒性强的CDI算法。本文的研究方向是耦合物理（混合）反问题，其特点是利用新的成像模式中的内部数据来重建待研究对象的材料参数。谈到这类逆问题的可能应用，应特别强调医学诊断和海洋可控源电磁测深，因为它们对于早期发现癌症、尽量减少误诊率、更有效地治疗疾病等，以及对大陆架上碳氢化合物矿床的地球物理勘探具有重要意义。该项目的创新之处在于，它通过利用加权平均曲率流动方程的初边值问题来关注CDI计算工具的发展。为了达到总体目标，首席研究员将：(1)分析作为CDI数学模型的加权1- laplacian的Dirichlet问题与加权平均曲率相关的运动水平集公式；(2)开发稳定收敛的计算算法；(3)进行数值收敛性研究，以便在数值实验中证明计算算法的有效性和重建的质量。本项目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，并接受反问题数值方法的培训。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.