Recent work in shape-based methods for diffusive inverse problems

We discuss recent research into a specific class of inverse medium problems: those associated with diffuse wavefields. Diffusive imaging problems are encountered in geophysical problems in environmental monitoring and remediation using electrical resistance tomographic (ERT) methods where one is interested in detecting and tracking in time plumes of chemical contaminants moving through the earth. We recover the Poisson equation that describes ERT. For such application areas often one is primarily interested in extracting information from the data concerning the structure of anomalies. We consider geometric inversion methods designed to directly determine information concerning the size, shape, location, and perhaps number of anomalies in a region of interest. First, a parametric approach to the problem is considered and demonstrated using a DOT sensing example. An adjoint-field method in conjunction with a steepest descent algorithm is employed to estimate the size, location, and contrast of a spherical anomaly in the absorption properties of the medium under investigation. A second approach to identify boundaries of an unknown number of objects is based on the idea of curve evolution. This approach functions by mathematically shrink-wrapping a deformable surface in 3D or curve in 2D to the boundary of one or more objects.