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In a reconstruction problem for subsurface tomography (modeled by the Helmholtz equation), we formulate a novel reconstruction scheme for shape and electromagnetic parameters from scattered field data, based upon 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 RBF fitted to boundary parameters comprising the locations of few points on the curve (the RBF centers) and the normal vectors at those points. The electromagnetic parameter reconstructed 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. The objective functional w.r.t. boundary and electromagnetic parameters is set up and required Frechet derivatives are calculated. Reconstructions using a damped Tikhonov regularized Gauss-Newton scheme for this almost rank-deficient problem are presented for 2D test cases of subsurface landmine-like dielectric single and double-phantom objects under noisy data conditions. The double phantom example demonstrates the capability of our present scheme to separate out the two objects starting from an initial single-object estimate. The present implicit-representation scheme thus enjoys the advantages (and conceptually overcomes the respective disadvantages) of current implicit and explicit representation approaches by allowing for topological changes of the boundary curve, while having few unknowns respectively. In addition, the Hermite interpolation based RBF representation is a powerful method to represent shapes in three dimensions, thus conceptually paving the way for the algorithm to be used in 3D.

Since the innovation of the ubiquitous Kalman filter more than five decades back it is well known that to obtain the best possible estimates the tuning of its statistics X 0 , P 0 , Θ, R and Q namely initial state and covariance, unknown parameters, and the measurement and state noise covariances is very crucial. The earlier tweaking and other systematic approaches are reviewed but none has reached a simple and easily implementable approach for any application. The present reference recursive recipe based on multiple filter passes through the data leads to a converged 'statistical equilibrium' solution. It utilizes the pre, post, and smoothed state estimates and their corresponding measurements and the actual measurements as well as their covariances to balance the state and measurement equations and form generalized cost functions. The filter covariance at the end of each pass is heuristically scaled up by the number of data points and further trimmed to provide the P 0 for subsequent passes. A simultaneous and proper choice for Q and R based on the filter sample statistics and certain other covariances leads to a stable filter operation providing the results after few iterations. When only R is present in the data by minimizing the 'innovation' cost function J using the non filter based Newton Raphson optimization results served as an anchor for matching and tuning the filter statistics. When both R and Q are present in the data the consistency between the injected noise sequences and their statistics provided a simple route and confidence in the present approach. A typical simulation study of a spring, mass, damper system with a weak non linear spring constant shows the present approach out performs earlier techniques. The Part-2 of the paper further consolidates the present approach based on an analysis of real airplane flight test data.

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