ACTA TECHNICA CORVINIENSIS - Bulletin of Engineering

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ACTA TECHNICA CORVINIENSIS – BULLETIN of ENGINEERINGwhere the Green’s function can be given by( x, y;x , y )j 2G ′ ′ = - H0⎜⎛ 22k (x x′ ) + (y y′) ⎟⎞(3)4 ⎝⎠Here ( x , y ) and ( x', y') are the observation andsource points respectively.The solution of the forward are carried out bymoment method [7] using pulse-basis functionand point matching technique.INVERSE PROBLEMThe aim of the inverse problem is to find a stable*solution for permittivity distribution ε whichminimizes the squared error output at thereceivers i.e.2E(ε )-e(4)2nwhere e∈ C , the n electric fields we measure atmnreceiver points, E : C ⎯→C , a functionmapping the complex permittivity distributionwith m degrees of freedom into a set of napproximate electric field observations, and alsomε ∈ C , the complex permittivity distributionswith m degrees of freedom..The details of the Levenberg-Marquardtregularization technique has already beendiscussed in our earlier work [6]. The Levenberg-Marquardt regularization technique for theminimization of the (4) leads to an iterativesolutionε i+1 = εi+ Δ εi(5)where ε k+1 is the permittivity distribution at thek+1 th iteration.Δ ε can be written asΔ ε = ( E′ ( ε ) † E′ ( ε)+ λ I) -1 E′ ( ε)†( E(ε ) - e ) (6)where E′ is the Jacobian matrix, † denotes theconjugate transpose, λ is a monotonicallydecreasing regularization parameter, I is theidentity matrix, E(ε)is the calculated electricfields at the receivers.Figure1. Numerical modelThe target is illuminated with TE fields radiatingfrom an open ended dielectric filled wave guidehaving sinusoidal aperture field distribution. Thetransmitter is moved along four mutuallyorthogonal directions. For each of thetransmitter positions along a particulartransmitting plane, the received fields ateighteen locations in the other three orthogonalplanes were measured theoretically at afrequency of 1 GHzIf the same meshes are used both in thesimulation of the measured data and in thesolution of inverse problem, we may commit aso-called inverse crime, where numerical errorsmay be cancelled out inadvertently. To avoidinverse crime, different meshes are used asshown in figure 2. The finer mesh is used in theforward problem (Figure 2(a)) whereas theinverse solver uses the coarse mesh (Figure 2(b)).a)NUMERICAL MODELThe theoretical model used to test our algorithmis shown in Figure 1.It is a high contrast square biological object 9.6cm × 9.6 cm consisting of muscle and bonehaving complex dielectric constants 50-j23 and8-j1.2 respectively at a frequency of 1 GHz. Theobject is kept immersed in saline water havingcomplex dielectric constant 76-j40.b)Figure 2. Meshes used to overcome the inverse crime(a). Mesh used in forward problem (b). Mesh used ininverse problem1022010/Fascicule 2/AprilJune /Tome III
ACTA TECHNICA CORVINIENSIS – BULLETIN of ENGINEERINGIn case of forward problem, the rectangularmodel is divided into 1024 square cells ofdimension 0.3cm X 0.3cm and the saline waterregion is divided into 32 cells of dimension0.6cm X 0.6cm. During the inverse problem, therectangular model together with saline waterregion is divided into 324 equal square cells 0.6cm × 0.6 cm. The measurement set contains 288independent data [3].During the iterative reconstruction, the complexpermittivity values of the cells filled up withsaline water were assumed to be known, thusrendering the problem of estimating thecomplex dielectric constants of the remaining256 cells.RESULTS AND DISCUSSIONSa) b)Figure 4. Reconstructed model with λ= 02 (a) Realpart (b) Imaginary partAfter applying the different image Arithmeticfilters and image enhancement filters to theabove reconstructed model, the quality of thereconstructed model have been improved interms of noise as shown in the Figure 5, Figure 6and Figure 7.To apply the reconstruction algorithm, it wasinitially assumed that the biological medium isfilled up with muscle only. The received fields atdifferent receiver locations were computed foreach transmitter position.In our earlier works, the regularizing parameterwas monotonically decreasing by a factor of 10.In this paper when the same decrement factor isused, the reconstructed model was notsatisfactory as shown in the figure 3. Instead,when the decrement factor is chosen as 2, thereconstructed model is quite acceptable. So, thedecrement factor for the regularizing parameteris chosen as 2 here.a) b)Figure 5. Reconstructed model after the imagearithmetic filter being used (using imageaddition/subtraction technique) (a) Real part (b)Imaginary parta) b)Figure. 3. Reconstructed model with λ=10 (a). Realpart (b). Imaginary partAs usual, the only priori information we haveused in our algorithm is that the real part of thecomplex dielectric constant cannot be negativeand the imaginary part cannot be positive.Figure 4 (a) and Figure 4(b) shows the real partand imaginary part of the reconstructed modelwith our present algorithm with λ =2.a) b)Figure 6. Reconstructed model after the imageenhancement filter being used (using imageadjustment technique) (a) Real part (b) Imaginary parta) b)Figure 7. Reconstructed model after the imageenhancement filter being used (using histogramequalization technique). (a) Real part (b) Imaginary part2010/Fascicule 2/AprilJune /Tome III 103
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ACTA TECHNICA CORVINIENSIS - Bulletin of Engineering

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