Level set based explicit and implicit representation reconstruction schemes in electromagnetic shape-tomography

Level set based explicit and implicit representation reconstruction schemes in electromagnetic shape-tomography

Abstract

The area of tomographic subsurface imaging is of interest in important applications such as geophysical prospecting, biomedical imaging and landmine detection to name only a few. This talk is about a couple of methodologies to solve an ``approximate'' inverse scattering problem especially useful in limited data situations, wherein the object shape, location and an approximate (as against a more exact) estimate of the object's interior physical parameter values are reconstructed. A Helmholtz-equation modelled electromagnetic tomographic nonlinear reconstruction problem is solved for the object boundary and inhomogeneity parameters in a Tikhonov-regularizedGauss-Newton (DTRGN) solution framework. In the present work, the electromagnetic parameter is the normalized (w.r.t the squared ambient wave-number) difference of the squared wave-numbers between the object and the ambient half-space, and is represented in a suitable global basis, while the boundary is expressed as the zero level set of a signed-distance function. The iterative ``shape based'' approximate reconstruction schemes broadly fall into two categories.

The objective functional minimized in the first class has as unknowns the coefficients in an explicit parametric representation for the boundary curve(s), while in the latter class, the unknowns are the values of a set function representing the image, with the zero level set of that function representing the boundary. While the first (explicit representation) class of schemes has the advantage of fewer unknowns which is useful in potential three-dimensional reconstructions, the second (implicit representation) class is better suited to handle topological changes in the evolving shape of the boundary.We present two approaches to the solution of this shape based reconstruction problem, one each in the explicit and implicit classes of schemes. The objective functional w.r.t boundary and electromagnetic parameters is set up and required Frechet derivatives are calculated. Reconstructions using a Tikhonov regularized Gauss-Newton scheme for this almost rank-deficient problem are presented for 2D test cases of subsurface landmine-like dielectric objects under noisy data conditions.In an explicit B-spline boundary-representation based reconstruction scheme, we evaluate the Frechet derivatives of the scattered fields measured above the surface w.r.t the control points of the spline, and minimize the objective functional using a Tikhonov regularization method.

On the other hand, with the objective of having an implicit scheme with few unknowns, we use an implicit Hermite interpolation based radial basis function (RBF) representation of the boundary curve. An object's boundary is defined implicitly as the zero level set of an RBFfitted to boundary parameters comprising the locations of few points on the curve (the RBF centers) and the normal vectors at those points. The required Frechet derivatives are calculated.