UNIVERSITY OF TRENTO DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL INFORMAZIONE 3823 Poo Trento (Italy), Via Sommarie 4 http://www.disi.unitn.it ITERATIVE MULTI SCALING-ENHANCED INEXACT NEWTON- METHOD FOR MICROWAVE IMAGING G. Olieri, G. Bozza, A. Massa, and M. Pastorino January 2 Technical Report # DISI--67
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Iteratie Multi Scaling Enhanced Inexact Newton Method for Microwae Imaging Giacomo Olieri (), Gioanni Bozza (2), Andrea Massa (), and Matteo Pastorino (2) () ELEDIA Group, DISI, Uniersity of Trento, I 385 Trento, Italy E mail: andrea.massa@ing.unitn.it (2) DIBE, Uniersity of Genoa, I 6 Genoa, Italy E mail: matteo.pastorino@unige.it Introduction In the last years, seeral approaches hae been proposed for soling inerse problems arising in microwae imaging [] and related applications including non inasie diagnostics, biomedical imaging, remote sensing, and subsurface prospecting [] [4]. In a microwae imaging problem, the targets are illuminated by incident waes and scattered field samples are measured outside the inestigation area [] [4]. In order to retriee the unknwon obects from the measurements, different stochastic [][3] and deterministic [2][4] approaches hae been proposed. As for these latter, they are usually based on iteratie procedures such as gradient or Newton type methods. In this framework, an approach based on an Inexact Newton method (IN) has been recently proposed for soling inerse scattering problems formulated through electric field integral equations (EFIEs) [4]. Such a method has been alidated on synthetic and experimental results as well as extended to contrast source formulations [5] showing seeral adantages in terms of stability, accuracy, and conergence rate with respect to state of the art techniques [4]. Howeer, it can suffer from local minima because of its deterministic nature. In order to oercome/mitigate such a drawback, the iteratie multiscaling approach () introduced in [2] for conugate gradient methods is considered in this paper. The is a synthetic zoom procedure that, thanks to an efficient exploitation of the aailable information from scattering data, guarantees higher resolution and enhanced reconstruction with respect to the corresponding bare approaches whateer the inersion technique [2][3]. Thanks to these features, it represents a candidate solution for improing the performances of IN and aoiding some intrinsic drawbacks caused by the limited amount of indipendent data and the deterministic nature of the same approach. In the following, the integration of the with the IN method ( IN technique) will be described and its performances will be compared to those of the standard IN implementation (Bare IN). Mathematical Formulation The Inexact Newton method (IN) [4] is an iteratie regularization technique aimed at soling nonlinear and ill posed problems. Under the assumption of cylindrical scatterers and Transerse Magnetic (TM) polarization of the incident fields with respect to the axes of the scatterers, the retrieal of the dielectric
properties, ε (r) and σ (r), of an inestigation region can be recast as the r solution of the following integral equations 2 Es () r = k τ ( r' ) E ( r' ) G( r; r' ) dr', r Dmeas () where τ () r ε () r E Din D in 2 () r = Ei ( r) k ( r' ) E ( r' ) G( r; r' ) dr', r Din Din σ () r τ (2) = r is the contrast function, is the th illumination, ωε G denotes the free space Green function. Moreoer, total electric field in i D meas D in, and the incident field, respectiely. E, E s, and E i are the, the scattered electric field in the obseration domain V By introducing the unknown array, x = [ τ, E,, E ] K V V y = [ E, K, E, E, K, E ] s s i i, the inerse problem can be written as ( x) y, and the known array, T = (3) where T is the nonlinear operator defined by () and (2). The Bare IN method discretizes D in N subdomains, and iteratiely linearizes the nonlinear problem (3) around the current solution the Fréchet deriatie T of T and updates x as follows ( outer IN loop [4]) where x h in x + x by means of + = x h (4) is found by using the truncated Landweber method [6] as a regularized solution of the linear problem h = y T x (5) T x ( ) ( inner IN loop [4]). The Bare IN outer and inner loops stop when satisfactory solution according to the user defined conergence criterion or when a maximum number of iterations ( I and I, respectiely) is reached. To better address the drawbacks ineherent with the deterministic nature of IN when dealing with nonlinear problems, the strategy is profitably exploited and integrated with the Bare IN. Towards this end, the Bare IN is iteratiely applied to reconstruct the dielectric distribution of the region of interest (RoI) belonging to the inestigation domani (equal to the inestigation domani at the first step of the process). At each step, a fixed discretization of the RoI is used by considering N subdomains ( N << N, N being the number of degrees of freedom of the inerse problem and the geometry at hand) and the IN reconstruction is performed. From two successie steps, the RoI is updated exploiting the information on the location and extension of the scatterers acquired by processing the reconstructed profile [3]. The synthetic zooming process is iterated until the stationarines of the RoI is reached [2][3]. The result is that a high resolution IN reconstruction problem (as required to achiee a suitable image of the inestigation domain) is recast as a set of low resolution out in x is a
ones [2][3] allowing improed conergence speed and accuracy of the oerall inersion as well as an enhanced robustness to local minima problem. Numerical Results In the first numerical example, a homogeneous lossless square cylinder of.8λ side is considered [2]. The obect is located in an inestigation domain of L = 2.4λ side (free space background) and it is characterized by ε r =. 5. A set of V=8 line sources equally spaced on a circle of ρ S = 2. 4λ radius is employed. For each source, the total field is measured at M = 2 equally spaced detectors located oer a circle of ρ M =. 8λ radius (the noiseless case is considered). The inersion data hae been synthetically computed by means of the MoM method and different discretization grids hae been adopted for the direct and inerse procedure in order to aoid the inerse crime problem [2]. Actual obect -IN reconstruction -IN reconstruction.6.6.6.5.4.3.2. -. - -.5.5 - -.5.5.5.4.3.2. -. - -.5.5 - -.5.5.5.4.3.2. -. - -.5.5 - -.5.5 (a) (b) (c) Figure Square cylinder: (a) actual obect, (b) Bare IN reconstruction, and (c) IN reconstruction. The plots in Fig. show the effectieness of the IN method ( N = 4, I = 5, and I = 3 ) in localizing the obect and proiding a out in good approximation of the actual distribution. Howeer, the shape of the obect is distorted and some artifacts appear [Fig. (b)]. Otherwise, the reconstruction obtained by the IN approach after S = 4 steps ( N = 36, = 3, and I in = 3 at each step) confirms the effectieness of the technique in reducing the reconstruction error [Fig. (c)]. Such an obseration is confirmed by the alues of the error indexes (total tot, internal int, and external ext [2]) in Tab. I. Table I Reconstruction indexes. Bare IN IN Obect int ext tot int ext tot Square.36 3.93 6.37 7.2 3 8.25.69 Hollow.39 6.86 9.85 9. 3.37 3 4.77 The second example deals with the reconstruction of a hollow square cylinder with L =.2λ and L =.4λ. The same parameters of the preious example out in I out
hae been employed and the has been stopped after S = 3 steps. As it can be obsered (Fig. 2), although the Bare IN approach proides quite good performances, the shape of the scatterer does not exactly match with the actual one. On the contrary, the integration allows significant improements in terms of accuracy of the retrieed profile [Fig. 2(c) Tab. ] (e.g., 3 = 9.85 s. = 4.77 ). tot tot Actual obect -IN reconstruction -IN reconstruction.6.6.6.5.4.3.2. -. - -.5.5 - -.5.5.5.4.3.2. -. - -.5.5 - -.5.5.5.4.3.2. -. - -.5.5 - -.5.5 (a) (b) (c) Figure 2 Hollow square cylinder: (a) actual obect, (b) Bare IN reconstruction, and (c) IN reconstruction. Also from the computational point of iew, the scheme enables a nonnegligible reduction of the computational burden. As a matter of fact, the Bare IN inersion required about minutes on an Intel Core Duo PC, while less than 7.5 seconds were required by the IN inersion. References [] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, Eolutionary optimization as applied to inerse problems, Inerse Problems, ol. 25, no. 2, pp. 233 ( 4), Dec. 29. [2] S. Caorsi, M. Donelli, D. Franceschini, and A. Massa, A new methodology based on an iteratie multiscaling for microwae imaging, IEEE Trans. Microwae Theory Tech., ol. 5, no. 4, pp. 62 73, Apr. 23. [3] M. Donelli, G. Franceschini, A. Martini, and A. Massa, An integrated multiscaling strategy based on a particle swarm algorithm for inerse scattering problems, IEEE Trans. Geoscience Remote Sensing, ol. 44, no. 2, pp. 298 32, Feb. 26. [4] G. Bozza, C. Estatico, M. Pastorino, and A. Randazzo, An inexact Newton method for microwae reconstruction of strong scatterers, IEEE Antennas and Wireless Propagation Letters, ol. 5, no., pp. 6 64, December 26. [5] G. Bozza and M. Pastorino, An inexact Newton based approach to microwae imaging within the contrast source formulation, IEEE Trans. Antennas Propagat., ol. 57, no. 4, pp. 22 32, Apr. 29. [6] L. Landweber, An iteration formula for Fredholm integral equations of the first kind, American Journal of Mathematics, ol. 73, no. 3, pp. 65 624, July 95.