1558 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 5, MAY 2008 A Nonlinear Iterative Reconstruction and Analysis Approach to Shape-Based Approximate Electromagnetic Tomography Naren Naik, Jerry Eriksson, Pieter de Groen, and Hichem Sahli Abstract—A nonlinear Helmholtz-equation-modeled electro- magnetic tomographic reconstruction problem is solved for the object boundary and inhomogeneity parameters in a damped Tikhonov-regularized Gauss–Newton (DTRGN) solution frame- work. 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 pa- rameters 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. Index Terms—Boundary reconstruction, Gauss–Newton method, local analysis, nonlinear tomography, subsurface imaging, Tikhonov regularization. I. I NTRODUCTION T HE NEED for nonlinear reconstruction approaches oc- curs in many branches of electromagnetic tomographic imaging. Some examples are wave-equation-based diffraction tomography, electrical impedance tomography, and diffusion optical tomography, to name just a few. The area of tomo- Manuscript received May 30, 2006; revised September 30, 2007. This work was supported in part by the Vrije Universiteit Brussel Research Council under Concerted Action Project GOA-20 on “Numerical issues in tomographic shallow subsurface imaging.” N. Naik is with the Department of Mechanical Engineering and the College of Engineering, University of Canterbury, Christchurch 8140, New Zealand (e-mail: naren.naik@canterbury.ac.nz; nnaikt@yahoo.com). J. Eriksson is with the Department of Computing Science, Umeå University, 901 87 Umeå, Sweden (e-mail: jerry@cs.umu.se). P. de Groen is with the Department of Mathematics, Vrije Universiteit Brussel, 1050 Brussels, Belgium (e-mail: pdegroen@vub.ac.be). H. Sahli is with the Department of Electronics and Informatics (ETRO), Vrije Universiteit Brussel, 1050 Brussels, Belgium, and also with the Interuniver- sitair Micro-Elektronica Centrum, B-3001 Leuven, Belgium (e-mail: hsahli@ etro.vub.ac.be). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2008.916077 graphic subsurface imaging is of interest in important applica- tions such as geophysical prospecting and landmine detection, to name only a couple. Typically, the reconstruction of the desired parameters is achieved in the following possible ways: 1) by a minimization of an objective functional comprising the residual of the measured and modeled data when either the full scattering operator [1]–[3] or an approximate operator such as a first- or a second-order Born approximation [4], [5] is considered and 2) via analytical methods, which are based on approximations of the forward operator [6]. The minimization class of problems, being ill posed in na- ture, needs additional a priori information (such as support of the object) to aid the convergence of the solution scheme to the desired correct solution. The motivations of using boundary information in a tomographic reconstruction are typically to lend stability to the iterative reconstruction scheme (by better demarcation of object support constraints and also by the pos- sible reduction of the number of parameters characterizing the object) or to solve an “approximate” inverse scattering problem wherein the object shape, location, and an approximate (as against a more exact) estimate of the object’s interior physi- cal parameter values are reconstructed. 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), whereas, 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 3-D reconstructions, the second (implicit represen- tation) class is better suited to handle topological changes in the evolving shape of the boundary. In this paper, we have formulated a scheme for the solution and convergence analysis of the nonlinear 2-D approximate reconstruction problem in the first class of schemes, in a framework for arbitrary parameter- izations, that is conceptually generalizable to three dimensions as well. We use a level-set representation for the shapes in order to calculate the first- and second-order shape and elec- tromagnetic parameter derivatives of the objective functional in a parameterized representation and not in the more customary implicit level-set representation of the boundaries. Miller et al. [7] solved the approximate inverse scattering problem using a partially nonlinear Born iterative scheme with a spline representation of the image boundary and a global basis representation for the object and background inhomogeneities. They use a greedy-search-type technique to find the best change 0196-2892/$25.00 © 2008 IEEE