This research involves the development of a set of computational tools for addressing problems of image formation arising in fields ranging from medical imaging and non-destructive evaluation of the manufactured components to subsurface environmental remediation and civil infrastructure monitoring. Common to these as well as many other problems is the need to characterize the internal structure of a region of space from data collected along the periphery. While common examples such as X-ray Computed Tomography or Magnetic Resonance Imaging yield high resolution images, the problems of interest in this effort are far more challenging due to complications associated with the physics of the underlying sensing technology as well as restrictions concerning where and how data can be acquired.
To address these challenges, the work in this project exploits the fact that in many cases, the processing objective is the identification of specific regions of interest containing flaws or other anomalies. Through a multidisciplinary effort in engineering, mathematics, and computation, the investigators develop and study a powerful new class of innovative shaped based computational tools which combine variational Active Surface models, Level Set Methods, Boundary Element Methods, and Partial Differential Equations to detect/reconstruct geometric structures of interest directly from raw tomographic data in a general class of PDE-based tomographic modalities. This study has the potential to yield a substantial paradigm shift in extracting geometric knowledge from tomographic measurements for a diverse collection of societally important applications characterized by challenging physics and restrictive sensing scenarios. Of specific interest here are problems of geotechnical surface wave inversion, non-destructive testing, and impedance tomography for environmental monitoring.
