UNIVERSITY OF TRENTO. G. Franceschini, M. Donelli, D. Franceschini, M. Benedetti, P. Rocca, and A. Massa. January Technical Report # DISI

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1 UNIRSITY OF TRNTO DIPARTINTO DI INGGNRIA SCINZA DLL INFORAZION 3813 Poo Trento (Italy), ia Sommarie 14 RCONSTRUCTION OF DILCTRIC OBJCTS FRO APLITUD-ONLY DATA ADANTAGS AND OPN PROBLS OF A TWO-STP ULTI-RSOLUTION STRATGY G. Francechini,. Donelli, D. Francechini,. Benedetti, P. Rocca, and A. aa January 11 Technical Report # DISI-11-4

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3 Recontruction of Dielectric Object from Amplitude Only Data Adantage and Open Problem of a Two Step ulti reolution Strategy G. Francechini,. Donelli, D. Francechini,. Benedetti, P. Rocca and A. aa Department of Information and Communication Technology, Unierity of Trento ia Sommarie 14, 385 Trento, Italy, mail: gabriele.francechini@dit.unitn.it, maimo.donelli@dit.unitn.it, daide.francechini@dit.unitn.it, manuel.benedetti@dit.unitn.it, paolo.rocca@dit.unitn.it, andrea.maa@ing.unitn.it Abtract In the following contribution an innoatie trategy for the inerion of amplitude only data in microwae imaging application i preented. The method conit of two tep. At the firt tep the ource i yntheized in order to compute the incident field in the inetigation domain. In the econd tep the profile of the object i recontructed thank to the iteratie multi caling approach combined to the Particle Swarm Optimier, an innoatie and effectie eolutionary minimization technique. The effectiene of the algorithm i preliminary aeed through the inerion of experimental data concerning an inhomogeneou dielectric catterer. I. INTRODUCTION The recontruction of the geometrical and phyical characteritic of an unknown object i a topic of great interet in many different field, uch a biomedical and indutrial diagnotic. The microwae imaging technique are potentially uitable for thee problem, but they hae ome drawback related to the nature of the mathematical model and to the hardware etup required to collect the neceary field urement. A a matter of fact, the inere cattering problem are ill poed, highly non linear and the amount of collectable information i limited alo if multi illumination, multi iew and multi frequency ytem are conidered. In [1] and [] a criterion related to the geometrical and phyical characteritic of the ytem i proided in order to ealuate the upper bound to the achieable information and to choe the optimal number of unknown. oreoer, the data acquiition require complex and expenie hardware etup. In particular, the urement of the phae ditribution turn out to be critical when high frequencie are conidered. A a matter of fact, holographic and interferometric technique (generally ued in optical application [3][4]) allow to retriee the phae information tarting from amplitude only data, but they require undeired additional pot proceing. In order to realize a reliable and cot effectie imaging apparatu, ome different trategie baed on phaele data hae been deeloped in the pat. Two main path of reearch eem to be uually taken into account: (a) the direct application of a recontruction algorithm for the proceing of phaele field data (Single Step Strategy) (ee for example [5] [6]); (b) the plitting of the phaele data recontruction into a two tep proce (Two Step Strategy) where the firt tep deal with a phae retrieal problem for completing the amplitude only inerion data and the latter i concerned with a tandard recontruction from complete field data (ee for example [7][8]). In thi contribution an innoatie two tep trategy belonging to the econd cla i propoed and preented in Sect. II. In Sub Sect. II.A an inere ource problem i preented and oled through the modeling of the electric field according to the Ditributed Cylindrical Wae odel (DCW odel). In Sub Sect. II.B, a multi reolution cot functional [9] i defined and minimized uing the Particle Swarm Optimizer [1], one of the mot effectie eolutionary iteratie procedure. In ection III, ome experimental reult are preented in order to draw ome preliminary concluion (Sect. I) on the effectiene of the propoed methodology. II. TWO STP ALGORITH Let u conider the claical tomographic imaging configuration in which an unknown cylindrical dielectric object i located in an inacceible inetigation domain and i illuminated by a et of T polarized incident electromagnetic wae. The aim of the propoed algorithm i the recontruction of the contrat function dielectric permittiity. τ () r = ε ( r) r 1 DI τ ( r) defined in (1), where ε i the relatie r r (1)

4 ( m() For the propoed methodology, only the knowledge of the amplitude of the al field, ) the amplitude and phae of the incident electric field, ( inc r ), in m () point, D m( ) r i aumed, together with r, being the oberation domain external to. Auming the knowledge of the phae, we do not limit the phaele nature of the algorithm becaue the urement can be executed only once and off line for each hardware etup and they are not o expenie being limited to a reduced number of point in the oberation domain. The relation between unknown ( τ () r and (r) ) and data i expreed by the following equation D m( ) inc m( ) m( ) ( r ) = ( r ) + j τ ( r') ( r') G( r / r') dr' ωμ () G( r / r') inc ( r) = ( r) j τ ( r') ( r') G( r / r') dr' ωμ (3) where i the free pace green function. It can be notice that in (3) the urement of the amplitude of the incident field in the inetigation domain are neceary. From a practical point of iew, it i a critical iue becaue the urement hae to be performed in a large number of point if a atifactory reolution leel i deired. oreoer, the experimental ytem (and in particular the electromagnetic enor) i moed by mean of a mechanical apparatu with ome tolerance in the poitioning. Therefore, a reduced ampling ditance between adjacent poition in would reult in an inaccurate ure of the field and, conequently, each field ample would be corrupted by a non negligible error. For aoiding uch a drawback, a uitable model of the radiating ource will be defined in the following. A. SOURC SYNTHSIS Becaue of the complexity and of the difficultie in collecting reliable and independent ure in a dene grid of point, let u aume that the incident field, ( r, i only aailable at the urement point belonging to the oberation inc m( )) domain. Therefore, in order to apply the contraint tated through (3) and before facing with the data inerion, it i mandatory to deelop a uitable model able to predict the amplitude of the incident field radiated by the actual electromagnetic ource in the inetigation domain. In the DCW odel, the antenna i repreented by mean of a linear array of W equally paced line ource and therefore the electric field can be expreed a k 8π f ε ς r ) (4) W ( () = A ) ( w H kd w w= 1 where i the uclidean ditance between the poition of w th element of the array and r, k i the free pace d w H waenumber and A w i the th order econd kind Hankel function. The optimal configuration of the unknown coefficient,, i determined minimizing the difference between the ure of the incident field and the yntheized alue in the oberation domain D A opt = arg min A ( ) = 1 m( ) = 1, ( ) inc rm ς rm ( ) ( ) = 1 m( ) = 1, inc ( r ) m( ) Such a problem i oled uing the well known Singular alue Decompoition algorithm and once the parameter are tuned, the electric field can be ealuated in eery point of the inetigation domain according to (4). (5)

5 B. OBJCT FUNCTION RCONSTRUCTION The amount of information in phaele data i extremely limited. In fact, beyond the typical limitation of the inere cattering problem, when amplitude only data are conidered the collectable information i reduced further on. Therefore, the iteratie multi caling approach [9] ha been cutomized for amplitude only data in order to control the dimenion of the earch pace and to improe the quality of the recontructed profile. Such an iteratie procedure i initialied auming a uniform ditribution of the unknown which are choen according to [1][]. oreoer, the alue of the incident field in each ubdomain of i ealuated and the ytem () (3) numerically oled through the minimization of a uitable cot function. Then, at each tep the reolution i adaptiely improed in the Region of Interet (RoI) where the object i uppoed to be located [9]. Accordingly, a multi reolution grid i obtained and a multi reolution cot function i defined Φ ISA PD = N ( r ) = 1 r= 1 n( r ) = 1 ς = 1 r= 1 n( r ) = 1 ( r ) ζ r n r n( r ) N ( r ) ς ( r ) n( r ) () 1 N t, m ) = inc ( rm ) + ωq( t ) t= q() t = 1 + ( ) = 1 m( ) = 1, ( ) r ξ m rm ( ) ( ) = 1 m( ) = 1, ( r ) m( ) q() t q() t m( ) q() t { [ τ ( r ) ξ ( r ) G( r / r )]} ξ ( r (7) (6) () 1 N t ) = ξ ( r ) ω n r n r q() t t= q() t = 1 { [ τ ( r ) ξ ( r ) G( r / r )]} ζ ( r (8) q() t q() t n( r ) q() t where the weighting function ω q( t ) can aume or 1 alue [9]. In order to completely exploit all the achieed information, each intermediate recontruction i ued a initial olution of the ucceie minimization proce. Howeer, the cot function (6) i till highly non linear and uffer of local minima problem. Therefore it i minimized uing the Particle Swarm Optimier (for a detailed decription ee [1] [1]), one of the mot effectie recent eolutionary technique baed on the oberation of the moement of warm of inect looking for food. Finally, the multi reolution procedure i iterated until a tationary condition i reached [9]. III. XPRINTAL ALIDATION In order to ae the robutne and the effectiene of the algorithm the real dataet of urement kindly proided by. Saillard and K. Belkebir (for detail ee [13]) ha been ued. The conidered tet cae the o called FoamDielIntT cattering configuration characterized by the following quantitie: τ =. ±.3, = m, =.45 ±. 15, R obj = 4. 1 m. The object i located in a quare inetigation domain of ide = 3. 1 m and = 8 different iew 1 obj 1 R obj 1 L I and = 41 urement point hae been taken into account. Single frequency data ( f = GHz ) hae been inerted. According to the tudie in [1] [1], the following configuration of PSO parameter i elected: contant inertial weight 5 ω =.4, acceleration coefficient C1 = C =., warm dimenion I = U, being U the number of unknown. 1 τ obj

6 Figure. atching between yntheized and ured amplitude of the incident field in D Firtly, let u conider the ynthei of the ource. It ha proided a good agreement between ured and yntheized alue of the phae of the electric incident field a hown in Fig.. On the other hand, the amplitude turn out to be more critical, but it i till an acceptable approximation to our aim. A a matter of fact the recontruction of Fig. 3 point out the preence of two different leel of contrat. The catterer i well located and dimenioned alo if the recontructed hape i not o accurate. From a quantitatie point of iew, the dielectric propertie are atifactorily etimated. Figure 3. Recontructed profile of the real part of the object function I. CONCLUSIONS In thi contribution an innoatie two tep trategy ha been preented and it performance analyed conidering experimental data. The reult point out the effectiene of the approach and the feaibility of the inerion of amplitudeonly data without the need of expenie pot proceing of the data or phae retrieal algorithm. The DCW odel allow u to aoid the critical urement in the inetigation domain and the iteratie multi caling approach integrated with the Particle Swarm Optimier ha hown a good effectiene alo in dealing with phaele real data. RFRNCS [1] O.. Bucci, and G. Francechetti, On the degree of freedom of cattered field, I Tran. Antenna Propagat., ol. 37, pp ,1989. [] O.. Bucci and T. Iernia, lectromagnetic inere cattering: retrieable information and urement trategie, Radio Science, pp , 1997 [3]. Wolf, Determination of the amplitude and the phae of the cattered field by holography, J. Opt. Soc. Am. A, ol. 6, pp. 18, 197. [4] G. W. Fari and H.. Hertz, Tunable differential interferometer for optical tomography, Appl. Opt., ol. 8, pp , 1989.

7 [5] S. Caori, A. aa,. Patorino, and A. Randazzo, "lectromagnetic detection of dielectric catterer uing phaele ynthetic and real data and the memetic algorithm," I Tran. Geoci. Remote Sening, ol. 41, pp , Dec. 3. [6] T. Takenaka, D. J. N. Wall, H. Harada, and. Tanaka, "Recontruction algorithm of the refractie index of a cylindrical object from the intenity urement of the al field," icrowae Optical Technol. Lett., ol. 14, pp , Feb [7]. H. aleki, A. J. Deaney, and A. Schatzberg, "Phae retrieal and intenity only recontruction algorithm from optical diffraction tomography," J. Opt. Soc. Am. A, ol. 1, pp , [8] L. Crocco,. D Uro, and T. Iernia, Inere cattering from phaele urement of the al field on a cloet cure, J. Opt. Soc. Am. A, ol. 1, Apr. 4. [9] S. Caori,. Donelli, D. Francechini, and A. aa, A new methodology baed on an iteratie multicaling for microwae imaging, I Tran. on icrowae Theory Tech., ol.51, pp , Apr. 3. [1] J. Robinon and Y. Rahmat Sami, Particle warm optimization in electromagnetic, I Tran. on Antenna and Propagation, ol.5, pp , ar. 4. [11] J. Kennedy, R. C. berhart, and Y. Shi, Swarm Intelligence, San Francico, organ Kaufmann Publiher, 1. [1]. Donelli and A. aa, Computational approach baed on a particle warm optimizer for microwae imaging of two dimenional dielectric catterer, I Tran. on icrowae Theory Tech., ol.53, pp , ay 5. [13] K. Belkebir and. Saillard, Special iue on Teting inerion algorithm againt experimental data: inhomogeneou target, Inere Problem, ol.1, pp. 1 3, Dec. 5.