UNIVERSITY OF TRENTO NEW RESULTS ON ELECTROMAGNETIC IMAGING BASED ON THE INVERSION OF MEASURED SCATTERED-FIELD DATA

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1 UNIVRSITY O TRNTO DIPARTIMNTO DI INNRIA SCINZA DLL INORMAZION 383 Poo Teto Italy, Via Sommaie 4 tt://www.isi.uit.it NW RSULTS ON LCTROMANTIC IMAIN BASD ON TH INVRSION O MASURD SCATTRD-ILD DATA R. Azao,. Bozza, C. statico, A. Massa, M. Pastoio, D. Pegolato, a A. Raazzo Jauay Tecical Reot # DISI--79

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3 New esults o electomagetic imagig base o te iesio of sytetic a measue scattee-fiel ata R. Azao,. Bozza, C. statico, 3 A. Massa, 4 M. Pastoio, D. Pegolato, 4 a A. Raazzo MC S..l., ia eto i Coigliao 6 65 eoa, Italy. Poe: Deatmet of Bioysical a lectoic gieeig, Uiesity of eoa, Via Oea Pia A, 645, eoa, Italy. Poe: ax: Deatmet of Matematics, Uiesity of eoa, Via Doecaeso 35, 646 eoa, Italy 4 Deatmet of Ifomatio a Commuicatio Tecology, Uiesity of Teto Via Sommaie 4, I-385 Teto, Italy. Poe: ax: Abstact A ew aoac fo te iesio of sytetic a measue scattee ata is oose i tis ae. Te aoac is base o a iteatie tecique i wic te oliea equatios of te iese scatteig oblem ae sole witi te -t oe Bo aoximatio. A egulaizatio sceme base o a Iexact-Newto meto is alie. Seeal umeical simulatios a exeimetal esults ae eote. Multile seaate ielectic cylies ae localize a ecostucte i a oisy eiomet. Keywos Imagig Systems, Micowaes, lectomagetic Scatteig, Iese Poblems I. INTRODUCTION Te eseac actiity coceig te eelomet of ew a efficiet imagig systems is i ai ogess. New imagig tecologies ae cotiuously oose, imoe a combie wit te existig oes ]. Amog te aious ew aoaces, electomagetic iagostic teciques ae facig a easig umbe of otetial alicatios, luig te etectio of tuels, ies a ote buie iomogeeities, ciil a iustial oestuctie tests a meical iagostics ]-6]. I tese teciques, iet electomagetic waes ae use to iteogate a obect. Te, te etube fiel te scattee fiel is collecte by oe o moe sesos i a cetai umbe of locatios outsie te obect. If cyliical obects ae cosiee a te illumiatig/eceiig system otates aou te coss sectio of te obect, tomogaic ifomatio is collecte. Te goal is usually to etiee te ielectic ofile of te obect o to isect it i oe to etect cacks o efects isie te mateial. om a matematical oit of iew, electomagetic imagig aoaces ae base o te iese scatteig oblem, wose equatios costitute te elatiosis betwee te ukow aametes of te tagets a te "iut" ata i. e., samles of te scattee electic fiel, wic ae collecte by a suitable measuemet aaatus. Te eelomet of efficiet imagig systems base o te electomagetic scatteig is still cosiee a ifficult task. Te mai easo is elate to te oliea atue of te equatios of te iese scatteig oblem. I aitio, te ill-oseess of te oblem makes te solutio still moe comlicate, se secific "egulaizatios" must be alie. ially, te comutatioal loa is usually aote limitig facto tat is, of couse, ot ieeet of te olieaity a ill-oseess. Desite te aboe limitig factos, seeal micowae-imagig aoaces ae bee ecetly oose. I oe to lieaize te iese scatteig oblem, simlifie imagig metos ae bee eeloe ue te fist-oe Bo aoximatio 7]-]. Ue tis aoximatio, te al electic fiel, i te wole sace, is witte i tems of te kow iet fiel te uetube fiel. Se te fist-oe Bo aoximatio ca be alie oly fo weakly scattees, a cetai imoemet i te etectio caabilities ca be obtaie by cosieig te seco-oe Bo aoximatio ]]. I tis case te oblem becomes oliea, altoug te ielectic aametes emai te oly ukows to be etemie. It soul be metioe tat, i ecet yeas, seeal aoaces ae bee oose fo te solutio of te full oliea iese scatteig, wic ae able, i ile, to isect ey stog scattees. Tey ae maily iteatie metos, bot etemiistic a stocastic oes. Howee, esite te sigificat imoemet aciee, tese stategies ae still quite comlicate a ey time cosumig. I te eset ae, a ew etemiistic a iteatie meto is oose. Te aoac is a two-ste Iexact-Newto meto, wic is eeloe i te famewok of te -t oe Bo aoximatio. As it is well kow, te Bo seies coeges fo weakly scattees. Cosequetly, fo tose alues fo wic te seies

4 coeges, it is iteestig to ealuate imoemets tat ca be obtaie by usig ige-oe aoximatios a te coesoig comutatioal loa. Te aoac escibe i Sectios II a III combies a oute iteatio loo Newto iteatio wit a ie loo, i wic te tucate Lawebe meto is alie. Te meto as ey goo egulaizatio oeties, as oe by te simulatios tat ae eote i Sectio IV. ially, i Sectio V, elimiay exeimetal aliatios of te aoac ae oie. Te iut ata fo tis exeimetal aliatio ae measue i te aecoic cambe of te Joit Laboatoy DIB-MC S..l., eoa, Italy. utemoe, ata oie by te ésel Istitute, Maseilles, ace, ae use to test te meto 3], too. II. MATHMATICAL ORMULATION As it is well kow, te iese scatteig oblem cosists i etieig te electomagetic oeties of a egio illumiate by oe o moe kow souces, fom te fiel measue outsie te egio itself. Te geometical two-imesioal cofiguatio of te cosiee imagig system is eote i igue. Te aea ue test, i, is illumiate fom V iffeet iectios multiiew aagemet a, fo eac iew, te electic fiel is collecte at M oits locate i te omai, obs i obs. igue eometical cofiguatio of te imagig system. I tis ae, tasese-magetic illumiatio is aote, tat is, eey souce ouces a kow iet wae olaize otogoally to te lae i wic lies. Tis yotesis allows a scala fomulatio of te oblem i tems of te i oectio alog te z-axis of te iole fiels. o tis easo, we call a te electic fiel aiate by te -t souce i fee sace a i esece of te scattee, esectiely, actually meaig tei oectios. utemoe, witout losig i geeality, i is suose to be suoue by acuum, wose ielectic emittiity a magetic emeability ae eote by ε a μ. Ue tese yoteses a assumig tat o magetic meia ae iole, a ae elate as follows, i wee ω is te ulsatio at wic te fiels oscillate, Helmoltz equatio 4] a is te scatteig otetial efie by is te kow ee s fuctio fo te two-imesioal

5 ωε ε, beig ε te elatie ielectic emittiity of i at oit. By efiig te scattee fiel, equatio ca be witte i oeatoial fom as wee te oliea itegal oeato is efie as follows scatt, ], K V, 3 scatt,, ]. 4 ν i I te iese oblem, te al electic fiel i te iestigatio omai i is as ukow as. utemoe, because of te measuemet ocess, te electic fiel i obs is coute by oise. Hece, te solutio of te iese oblem equies te iesio of te oliea itegal equatio,,, 5 ] scatt* obs wee scatt * eotes te iffeece betwee te measue al electic fiel a te iet oe. Te oliea itegal equatio 5 is ey citical a comutatioally exesie to sole. Moeoe it is ot sufficiet to etemiate bot a,, but oly te so-calle equialet cuet esity J. I oe to oecome tis oblem, i tis wok te tem, i ] is elace by its -t oe Bo aoximatio. Te -t oe Bo aoximatio allows te scattee fiel to be witte i tems of te scatteig otetial a of te kow iet fiel. I tis way, te kowlege of te al electic fiel i te iestigatio omai is ot ecessay. I fact, te oeato ] exessig te -t oe Bo aoximatio is ecusiely efie by te followig elatios ] 6 i ] i By usig tis aoximatio, equatio 5 becomes ] ]. 7 ] scatt*,, 8 obs wee te oly ukow is te scatteig otetial. Se is a comositio of eolm itegal oeatos o a ifiite imesioal sace, equatio 8 is ill-ose 5]. utemoe, fo > te oblem tus out to be oliea. Moeoe, it is fuametal to emak tat te sequece efie by elatios 6 a 7 coeges to te exact scattee fiel fo oly if te scattee is ot too stog 7]. Amog iffeet matematical tools, equatio 8 is sole by usig a iteatie Newto meto, i wic at eac ste te oliea ill-ose equatio 8 is lieaize by meas of te écet eiatie of. I fact, it is ossible to wite σ ] ] σ ] σ O, 9, wee, is te écet eiatie of te oeato ecusie elatios see Aeix fo te oof at oit, tat is, te liea oeato gie by te followig

6 σ ] σ, i i ] ], ] i ], III. TH RULARIZIN INVRSION PROCDUR As aleay metioe, te -t oe Bo equatio 8 is sole by meas of a egulaizig Iexact-Newto meto. Te meto is base o te followig iteatie sceme: At ay ste, equatio 8 is fist lieaize, a te geealize solutio of suc a liea equatio is aoximate a egulaize by usig a iteatie meto, amely te tucate Lawebe meto fo liea systems. A egulaizig sceme is equie to sole te lieaize equatio because it esults as ill-ose as te oliea oe 5]. Te algoitm ca be summaize by usig a oute-ie iteatie sceme, wee ay oute iteatio is te Newto ste fo te lieaizatio, a ay ie iteatio is te Lawebe ste fo te egulaize esolutio of te liea equatio aisig fom te oute ste. Te oose meto is able to imoe te efomaces wit esect to te wiely use Lawebe meto fo oliea iese oblems, i tems of umeical cost, a ece it ca be cosiee as a ali tool fo eal alicatios. Iee, te oose meto equies te comutatio of a limite umbe of écet eiaties, wic is geeally te exesie at of ay solig oceue by lieaizatio. If L i is a suitable iitial guess wic aoximates te tue obect fuctio, te Newto meto ca be scematize as follows. Let a sole, fo σ L i, te liea system σ ],,.., V, scatt* te uate σ a eeat wit :, util a stoig ule is satisfie o we max. At ay Newto ste, se te liea equatio is ill-ose, we seac fo a egulaize solutio by comutig a fixe umbe of stes of te followig tucate Lawebe meto. Set σ, a let kmax be a maximum umbe of iteatios. Iitialize k a cose β suc tat < β < V * s, comute σ, k β σ V, k * σ, k * scatt* ] 3 te uate k : k a eeat util k k max. Hee * eotes te aoit oeato of te écet eiatie, a s is te sectal om, tat is, te lagest moulus of te sigula alues of te agumet. Te Lawebe meto 3 is a iteatie egulaizatio meto, tat is, a iteatie meto wee te umbe of iteatios lays te ole of a egulaizatio aamete. Iee, i te case of oisy ata, it ca be easily oe tat te fist iteatios oie oise filteig, wile te subsequet oes estoe fom comoets wit ige oise, so tat te accuacy of te iteatie solutio stats to become wose. Te meto equies te coice of two aametes: te coegece aamete β a te umbe of iteatios k max to be comute.

7 V * Coceig β, it is simle to sow tat te costaits < β < guaatee te coegece of σ, k towas scatt σ, fo k. I ou tests, we always set V * β. Se k max is te egulaizatio aamete, i oe to eet te aoximate solutio s s σ,k max, a quite small umbe of iteatios is to be efome. I ou exeiece, a goo coice is k max. IV. NUMRICAL RSULTS to suffe fom te oise o Te oose meto as bee teste by meas of seeal umeical simulatios. A sigle-fequecy cofiguatio is assume. utemoe, fo all te tagets cosiee i tis sectio, te imagig cofiguatio igue is comose by V 8 iews, i wic te souces ae locate at ositios i ola cooiates π s R,,,..., V. 4 4 o eac iew, M measuemet oits ae use. Suc oits lie o a ac of cicumfeece of wit m-t measuemet oit at te -t iew as cooiates gie by π Δθ aias. Te π Δθ π Δθ m R, m. 5 4 M Moeoe, i te umeical comutatio, te test aea i is atitioe i to N squae subomais a bot te fiel a ielectic quatities ae assume to be costat i eac subomai. Te accuacy of te iesio algoitm is ealuate by usig te followig eo aametes ob ε ε eε # N ob N ob ε 6 m ε ε # ε 7 N ob N ob e bg ε # N bg N bg ε ε ε 8 N ε ε e 9 ε N ε V e meas wee N ob a N bg ae te sets of iexes coesoig to te subomais wic ae occuie by te obects a te backgou, esectiely, # N ob a # N ae te umbes of elemets of N ob a N bg, esectiely, ε bg a ε ae te actual a ecostucte alues of te elatie ielectic emittiity i te -t subomai, a a ae te aays meas cotaiig te actual a ecostucte alues of te al electic fiel at te M measuemet oits fo te -t iew. utemoe, i all te simulatios, te followig alues fo te aametes of te iesio algoitm ae use: k max 5 a max. Te fist efome simulatios ae bee itee to ealuate te ifluece of te oe of te Bo aoximatio o te taget ecostuctio. To tis e, cicula omogeeous cylies of seeal ielectic emittiities ae bee ecostucte at

8 te oeatig fequecy of 6 MHz. I aticula, cylies of aius a λ /5.m a of elatie ielectic emittiity i te age., 3] ae use. Te locatios of te souces a of te eceies ca be obtaie fom elatios 4 a 5 by settig R. 67 m, π M 4, a Δ θ. Te iut ata ae comute by a aalytical exessio of te scattee fiel 6] a tey ae 3 coute by a aussia oise wit zeo mea alue so tat te sigal-to-oise atio SNR is equal to 5 B. Te iestigatio omai is a squae lae egio wit sie L λ m, wic is iscetize i 5 5 squae subomais. Table I eots te obect ecostuctio eo, fo te cosiee oes of te Bo aoximatio. ob ob Table I os o te ielectic ecostuctio of cicula cylies of aius a λ /5 a of iffeet emittiity fo seeal alues of te -t Bo aoximatio. SNR 5 B. ε ob ob Te table sows tat, at least fo ot too stog scattees, te obect ecostuctio eo eceases if te oe of aoximatio eases. I aticula, oe ca obsee a sigificat imoemet by siftig fom te fist to te seco oe moel. m Table II oies te mea alues of te ecostucte ielectic emittiity,. ε Table II Mea alues of te ecostucte elatie ielectic emittiity of te obect fo iffeet oe of te Bo aoximatio. SNR 5 B. ε ob ε m Tese esults sow tat, we te aoximatio oe eases, te age of te ielectic emittiities tat ca be ecostucte is extee. Howee, te ige is te aoximatio oe, te loge is te executio time. I aticula, te time of executio eases liealy wit te aoximatio oe. o tis easo, a tae-off betwee accuacy a see is eee. A goo coice seems to aot te seco oe aoximatio to ecostuct ote moe comlex sceaios. I all cases, te electic fiel ata ae simulate by usig te meto of momets wit a mes of 4 4 subomais. π Te cofiguatio aametes ae R.5 m, M 55, Δ θ. Te iestigatio omai is a squae omai wit sie L 3λ.5 m, wic is iscetize ito 3 3 squae subomais. I te seco examle, two seaate ielectic cylies ae ecostucte. I aticula, te fist cylie is a omogeeous cicula oe, wose cete a aius ae.75λ,. a a.3λ, esectiely. Te seco cylie as a omogeeous ob squae coss sectio, wose cete a sie ae.75λ,. a l.6λ, esectiely. Bot cylies ae te same ob

9 elatie ielectic emittiity, i. e., ε ε ε. ob ob ob Te oose aoac as bee teste ue iffeet oeatig coitios. I aticula, iffeet alues of te ielectic emittiity a of te SNR ae cosiee. Table III eots te eos o te ecostucte istibutio of te elatie ielectic emittiity fo iffeet alues of a by assumig a SNR equal to 5 B. Te same table also oies te eos o te ecostuctio of te electic fiel at te measuemet oits. As ca be see, te eos ease as te elatie ielectic emittiity of te obects eases. Moeoe, quite accuate ecostuctios ae obtaie i te wole age of emittiity alues cosiee. utemoe, Table IV sows te same quatities fo iffeet alues of SNR fo te case i wic ε ob.6. ially, igue eots a examle of te ecostuctio of te elatie ielectic emittiity obtaie by te oose aoac i te case i wic ε ob.6 a SNR 5 B. As ca be see, bot cylies ae coectly locate. utemoe, te two coss sectios ae ietifie wit quite goo accuacies. Table III os o te ecostuctio of two cylies wit iffeet coss sectios fo iffeet alues of ε ob. SNR 5 B. ε ob ob bg e Ε ε ob Table IV os o te ecostuctio of two cylies wit iffeet coss sectios fo iffeet alues of te sigal-to-oise atio. ε ob.6. SNR ob bg e Ε y/λ x/λ igue Recostucte istibutio of te elatie ielectic emittiity. Two cylies wit iffeet saes. ε ob.6. SNR 5 B. I te ti examle, te cosiee imagig system is use to ecostuct tee seaate omogeeous cicula cylies of aius a.3λ. 3 Te cylies ae locate at.75λ,.75λ,.75λ,.75λ, a.,.75λ, esecti-ely. ob 3 Moeoe, all te cylies ae te same elatie ielectic emittiity, i. e., ε ε ε ε. ob ob ob ob ob ob Te eos o te ecostucte istibutio of te elatie ielectic emittiity ae sow i Table V fo iffeet alues of a by assumig a SNR equal to 5 B. utemoe, te eos o te ecostuctio of te al electic fiel at te ε ob

10 measuemet oits ae also oie. Table V os o te ecostuctio of tee cicula cylies fo iffeet alues of ε ob. SNR 5 B. ε ob ob bg e Ε As ca be see, te oose aoac is able to yiel quite goo esults fo all te cosiee alues of ε ob. utemoe, Table VI oies te same quatities fo iffeet alues of SNR i te case i wic ε ob.4 a te obtaie image fo ε ob.4 a SNR 5 B is eote i igue 3. I tis image, all te tee cylies ae coectly ietifie, a te eicte alues of te elatie ielectic emittiity ae ey close to te exact oes. Table VI os o te ecostuctio of tee cicula cylies fo iffeet alues of te sigal-to-oise atio. ε ob.4. SNR ob bg e Ε y/λ x/λ igue 3 Recostucte istibutio of te elatie ielectic emittiity. Tee cicula cylies. ε ob.4. SNR 5 B. V. XPRIMNTAL RSULTS ially, te oose meto as bee also teste by usig exeimetal ata. Te fist iete ata ae bee collecte i te aecoic cambe of te Joit Lab. DIB - MC S..l. igue 4. Te obect ue test is a squae ielectic cylie sie: l λ /5; cete:.,.; ε ob.5, wic is eclose i a squae iestigatio omai sie: L λ /; umbe of subomais: N 4. Te obect is illumiate by a tasmittig atea wit a ase cete locate at., 8.4λ a wokig at fequecy f 6 MHz. Te scattee electic fiel is collecte by a λ /4 iole atea at M 4 oits locate at i ola cooiates m 3λ /5, 7π/6m-π/3, m,,4. 5 o tis case, a sigle iew as bee use. igue 5 sows te ecostucte istibutio of te elatie ielectic emittiity, obtaie by usig te followig alues fo te aametes of te imagig meto: k max 3 a max.

11 igue 4 Measuemet set u of te aecoic cambe of te Joit Lab DIB - MC S..l., eoa, Italy. y/λ x/λ igue 5 Recostucte istibutio of te elatie ielectic emittiity. Squae cylie. Real ata. Te meto as also bee assesse by ietig ata measue i te aecoic cambe of te esel Istitue of Maseilles, ace, mae aailable by coutesy of of. K. Belkebi a of. M. Sailla 3]. Te obects ue test ae two taget cicula omogeeous cylies. Te cete of te fist oe is locate at -.55,. a its aius is.5m. Its elatie ielectic emittiity is equal to 3., wic coesos to stog scattee of ig cotast wit esect to te backgou. Te cete of te seco oe is locate at te oigi of te fame system a its aius is.4m. Te elatie emittiity of tis cylie is.45. Te iews ae 8 a ouble ige o ateas ae use as souce a eceie. Te -t iew is locate at π s.67,,..., V. 4 o eac iew, M 4 measuemet oits ae use, wose ola cooiates ae π π 4π m.67, m,..., V M Te iestigatio omai is a squae wit a sie of. m a te electic fiel is measue at 3Hz. Te scattee-fiel ata ae bee iete by usig bot te seco oe a te fout oe Bo aoximatio. igue 6 eots te obtaie esults fo te cases a 4. I aticula, te figue sows te oigial a ecostucte ielectic ofiles obtaie by cuttig te obects by a te lae y, b te lae x -.5, a c te lae x.75. I bot cases, te localizatio of te obects is ey accuate a te emittiity of te lage cylie is ecostucte ey well. Moeoe, te fout oe Bo aoximatio allows a bette ecostuctio of te emittiity of te smalle cylie wit esect to te case. Ayway, suc a emittiity is ueestimate because te smalle cylie is a stog scattee a te Bo aoximatio o ot efom well, as it is well kow, i tis case.

12 VI. CONCLUSIONS I tis ae, ew esults o micowae imagig ae bee eote. A effectie iesio meto base o te oliea equatios of te iese scatteig oblem as bee oose a teste. A egulaizatio sceme costitute by two stes as bee successfully alie. Accuate ecostuctios, ee i oisy eiomets, ae bee umeically obtaie. Bo aoximatios of seeal oes ae bee use a a imoemet as bee fou by usig ige oe aoximatios. Seaate ielectic cylies ae bee cosiee a ecostucte ue tasese magetic illumiatio coitios. A multi-illumiatio cofiguatio as bee alie a te eos o te ecostucte aametes ae bee caefully ealuate. Moeoe, elimiay esults coceig te iesio of eal scattee-fiel ata, measue isie aecoic cambes, ae also bee eote. 3 Recostucte alues Recostucte alues 4 Actual alues.5 ε x m] a Recostucte alues Recostucte alues 4 Actual alues.5 ε y m] b

13 3 Recostucte alues Recostucte alues 4 Actual alues.5 ε c igue 6 Oigial a ecostucte ielectic ofiles wit seco a fout oe Bo aoximatios. Two cotiguous cicula cylies. a y, b x -.5, a c x.75. y m] APPNDIX Let be : X Y a oeato betwee te Hilbet saces X a Y. As it is well kow, is sai to be écetiffeetiable at oit x X if tee exists a uique liea a cotious oeato : X Y suc tat, fo X x x x x lim A wee eotes a suitable om. x is calle te écet eiatie of at x. om te equatio A it follows tat x x x O. A quatio A allows to obtai te écet eiatie of te oeato : L i L obs. Let us stat wit te case. It esults ] i i i ] A3 ece we coclue o >, let us elimiaily obsee tat, ] A4 i

14 ] i i i i ] ] ] ] ] A5 Moeoe it esults ] ] i ] ] ] ] ], O i A6 Usig equatio A5, we obtai ] ] ] ] ] ] ] ],,, O O O i i i i A7 ] ] ] ] ] ] ] ],, O O O i i i i i A8 Relatios A7 a A8 imly tat

15 ece we coclue ] ] i ], i ] O ] A9, ] i ], i ] ] A quatio A alog wit equatio A4 allows us to comute ecusiely te écet eiatie of te oeato eey alue of. fo RRNCS ]. C. iakos, "megig Imagig Tecologies: om Aeosace to Meical Imagig, Bigig te a," Poc. I It. Wokso o Imagig Systems a Teciques, Stesa, Italy, May 4, 4. ] M. Pastoio, "Recet iesio oceues fo micowae imagig i biomeical, subsuface etectio a oestuctie ealuatio," Measuemet, lseie, secial issue o "Imagig Measuemet Systems," ol. 36, , 4. 3] C. Picot et al., "Recet oliea iesio metos a measuemet systems fo micowae imagig," Poc. I It. Wokso o Imagig Systems a Teciques, Stesa, Italy, May 4, 4. 4] R. Zougi, Micowae Noestuctie Testig a aluatio, Kluwe Acaemic Publises, Te Netelas,. 5] S. K. Mooe, "Bette beast cace etectio," I Sectum, May issue, 5,. 6] M. Pastoio, A. Massa, S. Caosi, "A global otimizatio tecique fo micowae oestuctie ealuatio," I Tas Istum. Meas., ol. 5, o. 4, , Aug.. 7] P. M. Mose a H. esbac, Metos of Teoetical Pysics. Mcaw-Hill, New Yok, ] M. Slaey, A. C. Kak, a L.. Lase, "Limitatio of imagig wit fist-oe iffactio tomogay," I Tas. Micowae Teoy Tec., ol. 3, , ] A. Abubaka, P. a e Beg, a S. Semeo, "A obust iteatie meto fo Bo iesio," I Tas. eosci. Remote Ses., ol. 4, , eb. 4. ] N. Zaiig a Y. Zag, "Hybi Bo iteatie meto i low-fequecy iese scatteig oblem," I Tas. eosci. Remote Ses., ol. 36, , May 998. ] S. Caosi, A. Costa, a M. Pastoio, "Micowae imagig witi te seco-oe Bo aoximatio: Stocastic otimizatio by a geetic algoitm," I Tas. Ateas Poagat., ol. 49, o.,. -3, Ja.. ] R. Piei a. Leoe, "Iese scatteig of ielectic cylies by a seco-oe Bo aoximatio," I Tas. eosci. Remote Ses., ol. 37, , Ja ] K. Belkebi a M. Sailla, Secial sectio: Testig iesio algoitms agaist exeimetal ata, Iese Poblems, ol. 7,. 6-6, No.. 4] J. H. Ricmo, Scatteig by a ielectic cylie of abitay coss-sectio sae, I Tas. Ateas a Poagat., ol. AP-3, , ] M. Beteo a P. Boccacci, Itouctio to Iese Poblems i Imagig, IOP, Bistol, UK, ] C. A. Balais, Aace gieeig lectomagetics, Jo Wiley & Sos, 989

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