De Zaeytijd, Jürgen

Franchois, Ann

Undetermined

Ghent University Academic Bibliography

3D Quantitative Microwave Imaging ofInhomogeneous Dielectric ObjectsJu¨rgen De ZaeytijdSupervisor(s): Ann FranchoisAbstract—This paper deals with 3D quantitative microwave imaging ormicrowave tomography (MT), i.e. the reconstruction of inhomogeneous di-electric objects from measurements of the scattered field for different il-luminations of those objects with microwaves. In contrast to qualitativemicrowave imaging methods, no approximations are made to the scatteringmodel, thus MT tries to solve the full-vectorial 3D inverse scattering prob-lem exactly. This is accomplished by casting the inverse problem into anoptimization problem where a permittivity profile is sought that minimizesa properly chosen cost function, which measures the distance between themeasured data and the simulated scattering from a given permittivity pro-file. In this work, a regularized Gauss-Newton method is employed to thisend.Keywords—Microwave Imaging, Inverse Problem, Optimization, Gauss-Newton, RegularizationI. INTRODUCTIONTHE aim of microwave imaging is the reconstruction of anobject – or, more precisely, some characterizing physicalparameter of an object – from measurements of the scatteredelectromagnetic fields that occur when that object is illuminatedby a number of known incident microwaves. Apart from appar-ent applications in non-destructive testing and geophysical ex-ploration, microwave imaging is a promising alternative to theexisting imaging methods in the biomedical world. An interest-ing example is the detection of breast cancer tumors, for whichmicrowaves are especially suited, because they easily penetratethe body without the risk that is attached to the ionizing X-raysthat are commonly used for mammography these days.Microwave imaging methods can be devided in two majorclasses. On one hand you have the qualitative reconstructionmethods that provide approximate solutions, but linearized andtherefore easy-to-solve problems, and on the other hand thereis microwave tomography (MT) that attempts to solve the full-wave 3D inverse scattering problem in an exact manner. Despiteimproved image qualities, the latter class is known, however,for its non-linearity and its members are invariably burdened bycomputationally heavy iterative algorithms.In this work, the 3D inverse scattering problem is solved usinga regularized Gauss-Newton optimization algorithm [1]. Manyefforts have been made to overcome computational burdens,which results in a robust and effective algorithm for quantita-tive microwave tomography.II. THE FORWARD PROBLEMThe label “forward problem” in our field of research refersto the calculation of the electric field Escat(r) that is scatteredJ. De Zaeytijd is with the Department of Information Technology, Ghent Uni-versity (UGent), Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium. E-mail: jur-gen.dezaeytijd@UGent.be .from a known complex permittivity distribution ²(r) in an in-vestigation domain D when a known incident field E inc(r) im-pinges on this domain (Figure 1).                                                                      PSfrag replacementsEincEscatD²(r)²bFig. 1. The 3D scattering configuration.A convenient way to solve the forward problem is by usingthe Volume Integral Equation (VIE) [2]. In this formulation,the scattered field is the solution to Maxwell’s equations for asource current density J scat which is the product of the contrastin permittivity and the total electric field inside D, i.e.Jscat(r) = (²(r)− ²b)E(r), (1)where ²b is the permittivity of the background medium. Sincethe scattered field Escat can be expressed in terms of the con-trast current J scat through an integral formulation and since thetotal field is the sum of the incident and the scattered fields, anintegral equation for the total field can be obtained. Once thisequation is solved, the total field – and consequently the currentJscat– is known and the scattered field can be calculated.The VIE is solved numerically using the Method of Moments(MoM). This means that the electric field inside the domain Dis expanded in basis functions defined on a grid discretizationof D and that a linear system for the expansion coefficients isobtained after testing the VIE with the same set of basis func-tions. Because of the large dimensions of this linear system, it issolved iteratively and an FFT technique [3] is used to speed upthe matrix-vector multiplications that are needed by the iterativesolver. In addition, an extrapolation procedure has been devel-oped [4] to determine initial guesses that are already close to thesolution vector, as a result of which the number of iterations isreduced.III. THE INVERSE PROBLEMSince the contrast current (1) is the product of the permittivityand the total field, which in turn depends on the permittivitythrough the integral equation, the scattered field is a non-linearfunction of the permittivity. As a result, the inverse scatteringproblem is solved iteratively. It is formulated as an optimizationproblem in which the optimization variables are the values of²(r) in the cells of a cubic grid that covers D and in which acost functionF (ε) =[1 + αFR(ε)]‖escat(ε)− emeas‖2, (2)is minimized. In (2), the vectors escat and emeas contain the sim-ulated and measured scattered field data respectively, ε is a vec-tor containing the optimization variables, α is a positive param-eter and FR is a smoothing function that penalizes strong localvariations in the permittivity. The presence of the smoothingfunction is required to put some constraints on the optimizationprocess. Indeed, it is known that the inverse scattering problemis very ill-posed, i.e. large fluctuations on the permittivity pro-file can correspond to only small deviations on the scattered fieldand therefore noise on the data can be amplified to undesiredlevels in the reconstructions when no precautions are taken.The optimization itself is carried out using a descent methodwith line search. The (k + 1)-th iterate in the optimization pro-cess is calculated asεk+1 = εk + βksk, (3)where the search direction sk is the solution of the system(JHkJk + λ2kΣ)sk = −(JHk[escatk− emeas]+ λ2kΩ∗k),(4)with λ2k= α‖escat(ε)−emeas‖2k/(1+α(FR)k) and where thesubscript k indicates quantities evaluated in εk. The matrix J isthe Jacobian matrix containing the first order derivatives of thesimulated scattered field and the vector Ω and the matrix Σ con-tain first and second order derivatives of the smoothing function,respectively. It can be shown [1, 5] that with this procedure theoptimization process will end in a (local) minimum of the costfunction.This basic structure of the algorithm is implemented withgreat care for computational efficiency. For instance, the up-date systems (4) are ill-conditioned and therefore a subspacepreconditioned LSQR algorithm [6] has been implemented tosolve them iteratively in a limited number of iterations. Also,to have more control over the optimization process, upper andlower bounds on the real and imaginary parts of the permittivitycan be imposed.IV. EXAMPLEAs an example, consider the test domain and antennaconfigu-ration of Figure 2. Three circular antenna-arrays, each with 30antennas are placed on a cilinder around a test domain D in awater background. Inside the test domain a simplified leg struc-ture is placed, consisting of a bone inside a cilinder of musclematerial. The real part of the relative permittivity in the xy-plane is depicted in Figure 3(a) and the reconstruction with ourmicrowave imaging algorithm is shown in Figure 3(b).PSfrag replacementsD3λb3λb3λbFig. 2. The investigation domain and the antennaconfiguration for the biomed-ical example. The arrows indicate possible antenna polarizations. Sizes areexpressed in background wavelengths λb.(a) (b)Fig. 3. Biomedical example: real part of the actual (a) and reconstructed (b)relative permittivity in the xy-plane.V. CONCLUSIONSThe Gauss-Newton minimization method, combined with aproper regularization strategy offers an effective means to solvethe inverse scattering problem and thus allows for quantitativemicrowave imaging, which is a promising technique in biomed-ical applications.ACKNOWLEDGMENTSThe authors would like to acknowledge the suggestions ofmany people.REFERENCES[1] J. De Zaeytijd, A. Franchois, C. Eyraud, J.M. Geffrin, “Full-wave three-dimensional microwave imaging with a regularized Gauss-Newton method– theory and experiment”, accepted for IEEE Trans. Antennas Propagat.[2] W.C. Chew, J. Jin, E. Michielssen, J. Song, Fast and Efficient Algorithms inComputational Electromagnetics, Artech House, Boston, 2001[3] P. Zwamborn, P.M. van den Berg, “The Three-Dimensional Weak Formof the Conjugate Gradient FFT Method for Solving Scattering Problems”,IEEE Trans. Microw. Theory Tech., Vol. 40, No. 9, pp. 1757-1766, 1992[4] A.G. Tijhuis, M.C. van Beurden, A.P.M. Zwamborn, “Iterative Solution ofField Problems with a Varying Physical Parameters”, Turk J Elec Engin,Vol. 10, No. 2, 2002[5] R. Fletcher, Practical Methods of Optimization, 2nd edition, John Wiley,New York, 1990[6] M. Jacobsen, P.C. Hansen, M.A. Saunders, “Subspace preconditionedLSQR for discrete ill-posed problems”, BIT Numerical Mathematics, Vol.43, No. 5, pp. 975-989, 2003   

3D quantitative microwave imaging of inhomogeneous dielectric objects

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3D quantitative microwave imaging of inhomogeneous dielectric objects

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