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.
