A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets

Abstract

A t wo-step shape reconstruction method for electromagnetic (EM) tomography is presented which uses adjoint elds and level sets. The inhomogeneous background permittivity distribution and the values of the permittivities in some penetrable obstacles are assumed to be known, and the number, sizes, shapes, and locations of these obstacles have to be reconstructed given noisy limited-view EM data. The main application we address in the paper is the imaging and monitoring of pollutant plumes in environmental cleanup sites based on cross-borehole EM data. The rst step of the reconstruction scheme makes use of an inverse scattering solver which rst recovers equivalent scattering sources for a number of experiments, and then calculates from these an approximation for the permittivity distribution in the medium. The second step uses this result as an initial guess for solving the shape reconstruction problem. A key point in this second step is the fusion of the 'level set technique' for representing the shapes of the reconstructed obstacles, and an 'adjoint eld technique' for solving the nonlinear inverse problem. In each step, a forward and an adjoint Helmholtz problem are solved based on the permittivity d i s t r ibution which corresponds to the latest best guess for the representing level set function. A correction for this level set function is then calculated directly by combining the results of these two r u n s. Numerical experiments are presented which s h o w that the derived method is able to recover one or more objects with nontrivial shapes given noisy cross-borehole EM data.

From Sinograms to Surfaces: A Direct Approach to the Segmentation of Tomographic Data

“…Santosa [13] lays out this basic strategy and demonstrates the idea on some small problems such as de-blurring 2D, binary images. Dorn et al [16], [17] use this strategy to solve for permittivity using a sequence of electromagnetic measurements, and show examples for simulated data in a 2D domain. This paper makes several important contributions; first we describe a specific formulation of the direct surface estimation problem for tomographic data, second we present a level-set implementation of this formulation that is practical on large data sets, and third we demonstrate, using real data sets, the effectiveness of direct surface estimation for a specific class of tomographic problems that are subject to noise, limited-angle sinograms, and misregistration.…”

GPR Estimation of the Geometrical Features of Buried Metallic Targets in Testing Conditions

“…A first class of approaches are based on the solution of a nonlinear inverse scattering problem via an iterative minimization scheme [32,33]. However, due to the nonlinearity of the inverse problem, the solution scheme may suffer of the presence of local minima ('false' solutions) affecting the overall reliability of the reconstruction procedure, and suitable regularization strategies have to be adopted to assure the consistency of the results [34].…”

Level-set-based reconstruction algorithm for EIT lung images: first clinical results

“…The level set approach has become popular because of its ability to track propagating interfaces [8,9], and more recently it has been applied in variety of applications in inverse problems and in image processing [10,11,12,13]. Level set based reconstruction method (LSRM) is a nonlinear inversion scheme using an optimization approach to iteratively reduce a given cost functional, which is the norm of the difference between the simulated and measured data.…”

“…The level set method for shape based reconstruction is well studied in electrical and electromagnetic imaging for simulated data [10,11,12,15,16,17,18,19]; however, it has been seldom shown to be used for human data. This study is the first implementation of LSRM for EIT breathing real data demonstrating the results of applying a difference solver.…”

“…LEVEL SET METHOD One effective method to allow the reconstruction of sharp images is the Level Set method [12]. The classic formulation of this method assumes that the reconstructed image can take only two conductivity values: one for background with value σ b and another one for inclusions with value σ i .…”