An implicit radial basis function based reconstruction approach to electromagnetic shape tomography

Abstract

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.

Radial-basis-function level-set-based regularized Gauss–Newton-filter reconstruction scheme for dynamic shape tomography

“…In an explicit parametrization, this boundary has been described in terms of a spline basis [9,12,14] in two dimensions or with spherical harmonics [10] in three dimensions. In implicit formulations [6,[17][18][19], typically the shape unknowns are the values of the function sr on the reconstruction grid. In [17] with the objective of retaining an implicit representation coupled with significant search-space-dimensionality reduction (as in explicit schemes), we represent sr as a RBF via a Hermite interpolation scheme to fit a few on-curve points (called centers of the RBF, and denoted by r c 1 …r c m ) and the normal unit vectors at those points (denoted by n 1 …n m , where n i ≡ cos θ c i ; sin θ c i for some θ c i ).…”

“…While the first (explicit-representation) class of schemes (as in [9][10][11][12][13][14]) has the advantage of fewer unknowns, which is useful in potential threedimensional reconstructions, the second (implicitrepresentation) class [6,15,16] is better suited to handle topological changes in the evolving shape of the boundary. Radial basis function (RBF)-based implicit-representation reconstruction schemes were first suggested in [17] followed by recent works such as [18,19], extending the capability of the approaches in [10,14] by allowing for topological changes, while retaining their advantage over conventional implicitrepresentation schemes of having few unknowns. A detailed literature survey of these various classes of schemes is given in [10,13,14,17,18].…”

“…Radial basis function (RBF)-based implicit-representation reconstruction schemes were first suggested in [17] followed by recent works such as [18,19], extending the capability of the approaches in [10,14] by allowing for topological changes, while retaining their advantage over conventional implicitrepresentation schemes of having few unknowns. A detailed literature survey of these various classes of schemes is given in [10,13,14,17,18].…”

“…In recent works [18,19], a compactly supported RBF-based parameterized level-set representation with centers in the interior of the domain (the centers in [17] are placed on the boundary) is used to solve single and multi-objective reconstruction problems in static nonlinear tomographic settings.…”

3D shape based reconstruction of experimental data in Diffuse Optical Tomography

“…Based on the assumption that the unknowns in such cases could be defined by the subdomain boundaries, a field of shape based reconstruction techniques has emerged in recent years. There have been many results in literature of either level-set techniques [18,19,20,21,22,23,24], where the boundaries of the subdomains are implicitly modeled by the zero level of level-set functions, or parametric methods [25,26,27,28] where an explicit parameterisation of the boundaries is used. In the case of the shape based reconstructions that use an explicit parameterisation of the boundaries, many different ways to describe the shapes have been used such as spherical harmonics, ellipsoids and spheres in 3D or Fourier curves, and Hermite polynomials, in 2D.…”