Current Density Impedance Imaging from Minimal Interior Data

该项目关注的问题，从最小的边界和内部数据的知识电导率成像。数学公式导致的非线性偏微分方程的1-拉普拉斯变系数。这是一个奇异的退化椭圆方程，其解可以定义在粘性意义下。虽然不是所有的粘度解决方案是感兴趣的逆电导率问题，它原来的解决方案的利益是加权最小梯度。研究的一个方向是考虑加权最小梯度函数（在氡措施的意义上）与规定的边界数据的研究。一个主要的目标是证明极小化问题在较大的有界变差函数空间中的唯一性和稳定性。微妙的问题有关的规律性的重量，以及会发生什么时，系数消失的开放子集进行调查。研究的另一个方向涉及的类型的边界数据，将先验排除奇点的理解。这样的数据的存在，然后将问题简化为一个哈密尔顿-雅可比系统，其中时间的作用是由一个空间变量。更一般地说，一些正则化技术来自图像处理的算法方面将在加权最小梯度问题的背景下进行分析。在成像的第二个方面，PI将研究表征在衰减X射线变换中获得的数据的范围条件与A解析映射理论之间的关系，以及当低计数辐射不保证传输模型中假设的大数定律时在X射线层析成像中产生的随机模型。成像科学中的一个热门话题，使用耦合物理来确定体内的材料属性。PI旨在寻找和分析新的数学方法，这些方法可以定量恢复电导率（对电磁激励的特征编码响应），其精度和分辨率比目前可能的要高得多。一种定量显示，不仅可以显示物体的内部结构，还可以区分同一部分的状态。应用范围从材料科学的无损检测到医学成像和诊断。在医疗诊断应用中，使用无害辐射以高分辨率成像将降低目前通过CAT扫描等方法的误诊率，并将在发育的早期阶段检测到有缺陷的组织，从而增加治愈的可能性。至少有两名学生将参加并接受培训，在这项调查。

This project concerns the problem of conductivity imaging from minimal knowledge of boundary and interior data. The mathematical formulation leads to the non-linear partial differential equation of 1-Laplacian with a variable coefficient. This is a singular and degenerate elliptic equation for which solutions can be defined in the viscosity sense. While not all viscosity solutions are of interest in the inverse conductivity problem, it turns out that solutions of interest are of weighted least gradient. One direction of research considers the study of weighted least gradient functions (in the sense of Radon measures) with prescribed boundary data. A major goals is to prove uniqueness and stability for the minimization problem in the larger space of functions of bounded variations. Delicate questions concerning the regularity of the weight, as well as what happens when the coefficient vanishes on open subsets are to be investigated. Another direction of research concerns the understanding of types of boundary data which would a priori exclude singularities. Existence of such data would then reduce the problem to a Hamilton-Jacobi system, in which the role of time is played by one spatial variable. More generally, some regularization techniques coming from the algorithmic side of Image Processing will be analyzed in the context of the weighted minimum gradient problem. On an second facet of imaging, the PI will investigate the relation between the range conditions that characterize the data obtained in the attenuated X-ray transform, and the theory of A-analytic maps, as well as a stochastic model arising in X-ray tomography when low count radiation does not warrant the law of large numbers assumed in the transport model.The proposed research is in the area of Inverse Problems of hybrid type, a hot topic in imaging sciences that uses coupled Physics to determine a material property inside a body. The PI aims to find and analyze new mathematical methods which quantitatively recover the electrical conductivity (a characteristic encoding responses to electromagnetic excitations) with significantly higher accuracy and resolution than currently possible. A quantitative display that not only reveals the inner structure of the object, but also allows for discriminating in the status of a same part. Applications range from nondestructive testing in Material Sciences to Medical Imaging and Diagnostic. Of special interest in medical diagnostic applications, the use of harmless radiation to image with high resolution would decrease the current rate of misdiagnosis by methods such as CAT-scans, and would detect defective tissue at an earlier stage of development, with the benefit of increasing the odds of a cure. At least two students will participate and be trained in this investigation.