A Nonlinear Iterative Reconstruction and Analysis Approach to Shape-Based Approximate Electromagnetic Tomography

A nonlinear Helmholtz-equation-modeled electromagnetic tomographic reconstruction problem is solved for the object boundary and inhomogeneity parameters in a damped Tikhonov-regularized Gauss-Newton (DTRGN) solution framework. In this paper, the object is represented in a suitable global basis, whereas the boundary is expressed as the zero level set of a signed-distance function. For an explicit parameterized boundary-representation-based reconstruction scheme, analytical Jacobian and Hessian calculations are made to express the changes in scattered field values w.r.t. changes in the inhomogeneity parameters and the control points in a spline representation of the object boundary, via the use of a level-set representation of the object. Even though, in this paper, a homogeneous dielectric is considered and a spline representation has been used to represent the boundary, the formulation can be used for a general global basis representation of the inhomogeneity as well as arbitrary parameterizations of the boundary, and is generalizable to three dimensions. Reconstruction results are presented for test cases of landminelike dielectric objects embedded in the ground under noisy data conditions. To confirm convergence and, at times, to know which of the obtained iterates are closer to the actual unknown solution, using a perturbation theory framework, a local (Hessian-based) convergence analysis is applied to the DTRGN scheme for the reconstruction, yielding estimates of convergence rates in the residual and parameter spaces.