Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August

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1 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August A Noel Technique for Localizing the Scatterer in Inerse Profiling of Two Dimensional Circularly Symmetric Dielectric Scatterers Using Degree of Symmetry and Neural Networks Vinu Thomas, C. Gopakumar, Jaimon Yohannan, Anil Lonappan, G. Bindu A. V. Praeen Kumar, V. Hamsakutty, and K. T. Mathew Cochin Uniersity of Science and Technology, India Abstract A noel technique for localizing the scatterer in microwae imaging of two-dimensional circularly symmetric dielectric scatterers using degree of symmetry and neural networks is presented. The degree of symmetry for a transmitter position is computed as a function of the difference between the first half and the spatially reflected second half of the measured scattered field ector. A Probabilistic Neural Network (PNN) classifier is trained with the degree of symmetry ectors for the different object configurations. It classifies the degree of symmetry ector of the unknown circularly symmetric scatterer presented to it into one of the classes that indicate the radius and location of the centre of the scatterer. Thus the scatterer is localized in the imaging domain. This not only reduces the degrees of freedom in the inersion for the unknown object, thereby aiding the global conergence of the solution, but also results in a reduction in computation time. The technique has been tested on synthetic data and the results are promising. Introduction Microwae imaging is of great interest in many applications, ranging from medical imaging to non-destructie ealuation of buried pipelines, and inoles the solution of an electromagnetic inerse problem. The measured scattered field and the contrast function of the unknown object whose dielectric profile is to be estimated are nonlinearly related because of multiple scattering [1]. The inerse problem of microwae imaging is ill posed in the Hadamard sense [2]. Both deterministic and stochastic methods hae been deeloped to sole this inerse scattering problem. Some examples of the deterministic techniques are the Born iteratie method [3], distorted Born iteratie method [4], Newton Kantoroich method [5, 6] etc. Due to the nonlinearity of the inerse scattering problem, there is a risk of the solution from a deterministic method, getting trapped in a local minimum. Neural Networks hae been employed for the solution of electromagnetic inerse problems [7-9]. Neural networks offer an adantage in that it can work as a black box system capable of soling the physical problem without directly considering the functional relation between the input and output alues of the problem itself [10]. In this communication the a priori knowledge about the cylindrical geometry of the 2-D dielectric scatterer is exploited, with the assumption that the complex permittiity distribution is symmetric with respect to the centre of the cross section. By a 2-D dielectric scatterer, one refers to the fact that permittiity does not ary along the axis. This a priori is alid for dielectric pipelines. A custom defined degree of symmetry is computed for each transmitter position. A Probabilistic Neural Network is trained with the degree of symmetry ectors computed for a training set of object configurations. PNN offers the adantage of rapid training and easy re-trainability. It is also guaranteed to conerge to the Bayesian classifier, the optimal classifier, with enough training data [11]. It classifies the degree of symmetry ector of the unknown circularly symmetric scatterer presented to it into one of the classes that indicate the geometry of the scatterer localizing the scatterer in the imaging domain. Thus the degrees of freedom in the inersion for the unknown object are reduced, which aids the global conergence of the solution. It also results in a reduction in computation time. The Newton Kantoroich procedure is employed to image the scatterer that has been localized. The details of the formulation of the problem and numerical simulations and discussions are presented in this paper. Formulation of the Problem A 2-D dielectric scatterer of circular cross section and complex permittiity distribution symmetric with respect to the centre of the cross section is located in an imaging domain I, which is usually a square or a

2 116 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August rectangle. Howeer, since the location of the object to be imaged is not known, the imaging region has to be chosen sufficiently large. The background and the object are non magnetic. To simplify the implementation, TM polarization of the incident field is considered. A circular geometry for the imaging system is employed, with M line sources equispaced on a circle, one of which will be transmitting and the others receiing at a gien time as shown in figure 1. Figure 1: The measurement setup. The total field satisfies the scalar electric field integral equation for the iew, where e inc (r) is the incident field and e scat e scat (r) = I e r = e inc (r) + escat (r) (1) (r) is the scattered field gien by k 2 m c(r )e (r )G(r, r )dr, = 1, 2,..., M (2) with G being the two-dimensional Greens function, k m the propagation constant of the medium and c the object contrast. The integral equations are discretized with pulse basis functions and point matching. The imaging region is discretized into N cells. The degree of symmetry for a transmitter position is defined as (assuming een number of receiers M 1 per iew). S real () = S imag () = (M 1)/2 k=1 (M 1)/2 k=1 Re(e scat (k) e scat (M k)) 2 Im(e scat (k) e scat (M k)) 2 (3) When circular geometry of measurement employed, if the dielectric cylinder is circularly symmetric and off centred, the measured scattered field ector e scat exhibits symmetry with respect to its centre, only for the iews 1 and 2 that are diametrically opposite as shown in figure 1. The symmetry plots also exhibit two maxima, a 1 and a 2, which correspond to the transmitter positions where the symmetry of the measured scattered field ector is minimum. The plots of S real and S imag with respect to, exhibits two significant minima, at 1 and 2, and two significant maxima at a 1 and a 2, the maximum asymmetric iews. It is seen that the location of the centre of the scatterer and its radius decide the alues and features of the degree of symmetry ector. This indicates that the object maybe localized in the imaging domain by employing the degree of symmetry data for the scatterer [12]. For this purpose, a Probabilistic Neural Network, trained with degree of symmetry ectors for different circularly symmetric scatterers is employed. The PNN has been chosen because of its rapid training and easy re-trainability. It is also guaranteed to conerge to the Bayesian classifier, the optimal classifier, with enough training data. The PNN has three layers: The input layer is presented with the training set of degree of symmetry ectors for different scatterer configurations, haing arying radii, offset from the centre and permittiity profiles. The weight matrix of the second layer, the pattern layer, is set to the transpose of the matrix formed from the training set of degree of symmetry ectors. The pattern layer units compute the

3 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August distance between the input test ector and the units weight ector. The actiation function of the pattern units is the radial basis function. When a test degree of symmetry ector is presented to the PNN, each neuron in the radial basis layer outputs a alue that indicates how close the input ector is to each neurons weight ector. The pattern layer outputs are selectiely connected to units in the summation layer depending on the class of patterns they represent. Since each summation unit collects inputs from the pattern units of the same class, their output is an estimate of the probability density function of that particular class. The output layer is a competitie layer that produces a binary output, with only one of the output neurons firing, indicating that the test degree of symmetry ector belongs to that particular class. Numerical Simulations and Discussions The coupled equations 1 and 2 are employed to generate synthetic scattered field data in the measurement domain for the inersion, for known object profiles. For the simulations, 49 transceiers hae been used, that is 48 receiers per iew. The degree of symmetry for each transmitter position is computed as per equation 3, for arious radii and distances of the cylinder centres from the origin. For the forward simulations, imaging domain is a square discretized into pixels, with a sampling interal of 0.05 waelengths. The inerse problem is soled at a coarser resolution of 0.1 waelengths. The cross sectional profiles considered for the cylinders are a homogenous profile and a profile consisting of two concentric circles, which are usually the practical cases for a dielectric pipeline. The training set is minimized by considering only diagonal displacements along the leading diagonal of the centres of the circularly symmetric scatterer from the centre of the imaging domain. The training data set is generated with the following parameters: The outer radius is aried in the range between 6 and 11 sampling interals, and the inner radius in the range 4 and 8 sampling interals. The relatie permittiity of the unknown cylinder is aried in the range [1:16]. The offset from the centre is aried in steps of one sampling interal. Figure 3a: The actual profile of the object. Figure 3b: The degree of symmetry ector for the scatterer. Figure 3c: The circularly shifted degree of symmetry ector. The degree of symmetry ector is now aligned with the diagonal iews.

4 118 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Each output target class indicates the offset from the centre and the range of radii the unknown circularly symmetric scatterer belongs to. The weight matrix of the summation layer is deried from the target classes. An example is shown in figure 3. The synthetic data is generated from the profile shown in figure 3a. The degree of symmetry for the scatterer is shown in figure 3b. Since the training data consists of scatterers that are displaced only along the main diagonal, the test degree of symmetry ector is circularly shifted in the anticlockwise direction by n elements, so that the two minima of the degree of symmetry are aligned with the iews 1 and 2 of the main diagonal, to that in figure 3c. The PNN classifies the test degree of symmetry ector into one of the target classes that indicates the offset and range of radii the circularly symmetric scatterer belongs to. The Newton Kantoroich procedure has been employed for the imaging in the reduced inestigation domain. The reconstructed image is rotated in the clockwise direction about the centre of the imaging domain by 360 n M. The reconstructed image after the 4 th iteration of the Newton Kantoroich algorithm without localization in the inestigation domain is shown in figure 4a. The reconstructed image after the 4 th iteration when localization in the inestigation domain is done is shown in figure 4b. The result with the proposed technique is seen to be much better. There is also a considerable reduction in computation time compared to the case where the entire imaging domain is employed. Figure 4a: The reconstructed image after the 4 th iteration. The entire inestigation domain is considered. Figure 4b: the reconstructed image after the 4 th iteration with the proposed technique. Conclusion A noel preconditioning technique for localizing the scatterer in the imaging domain in microwae imaging of two-dimensional circularly symmetric scatterers has been presented. The proposed technique reduces the degrees of freedom in the inerse scattering problem of microwae imaging, and also results in considerable reduction in the computation time and improement in the conergence rate. Acknowledgement Author Vinu Thomas thanks the Institute of Human Resource Deelopment, Got. of Kerala for proiding sponsorship for the research. Authors Anil Lonappan and G. Bindu acknowledge CSIR, Human Resource Group, Ministry of HRD, New Delhi, India for awarding Senior Research Fellowship. Author Jaimon Yohannan acknowledge CSIR, Human Resource Group, Ministry of HRD, New Delhi, India for awarding Research Associateship. Author A. V. Praeen Kumar is thankful to KSCSTE, Got. of Kerala, India for proiding research fellowship. Author K. T. Mathew thanks the Department of Science and Technology, Goernment of India, for the project SP/SO/D-22/98. REFERENCES 1. Chew, W. C., Waes and Fields in Inhomogeneous Media, IEEE Press, New York, Hadamard, J., Lectures on Cauchys Problem in Linear Partial Differential Equations, New Haen, C.T, Yale Uniersity Press, Wang, Y. M., W. C. Chew, An Iteratie Solution of Two Dimensional Inerse Scattering Problem, Int. J Imag Syst Tech, Vol. 1, 100C108, Chew, W. C., Y. M. Wang, Reconstruction of Two Dimensional Permittiity Using Distorted Born Iteratie Method, IEEE Trans Med Imag MI-9, , 1990.

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