Ph.D. DissertationInternational Do torate S hool in Information andCommuni ation Te hnologiesDIT - University of TrentoMulti-Resolution Te hniques based onShape-Optimization for the Solution of InverseS attering ProblemsManuel BenedettiThesis in o-tutelle between University of Trento (Italy)and Université Paris Sud XI (Fran e).Thesis with European Label.Advisor:Andrea Massa, ProfessorUniversità degli Studi di TrentoCo-Advisor:Dominique Lesselier, Dire teur de Re her heLaboratoire des Signaux et Systèmes (Supéle -CNRS-UPS)De ember 2008

Thèse de Do toratSPECIALITE: PHYSIQUEE ole Do torale S ien es et Te hnologiesde l'Information des Télé ommuni ationset des SystèmesPrésentée par Manuel BenedettiSujet:Te hniques Multi-Résolution basées sur l'Optimisationde Forme pour la Résolution de Problèmes Inversesde Dira tionThèse en o-tutelle entre Université de Paris Sud 11 (Fran e)et University of Trento (Italie). Thèse ave le label européen.Soutenue le 1er De embre 2008 devant les membres du jury:M. LESSELIER DominiqueM. MASSA AndreaM. DORN OliverM. PICHOT ChristianM. PICHON LionelM. AZARO Renzo

A knowledgementsArriving at the end of my Ph.D., I would like to express mygratitude to all those who gave me the possibility of arryingout su h a wonderful professional experien e. First of all, Iwant to thank my thesis advisors Andrea Massa, Professor ofEle tromagneti Fields at the University of Trento and Dire -tor of the Ele tromagneti Diagnosti Laboratory (ELEDIALab.), and Dominique Lesselier, Dire teur de Re her he at theLaboratoire des Signaux et Systèmes (L2S) of Supéle , who had onden e in my abilities as well as in the o-tutelle programin whi h I was involved.My spe ial thanks also go to my tutors Mar Labert, re-sear her at L2S, and Massimo Donelli, resear her at the Uni-versity of Trento and member of the ELEDIA group. Certainly,I will never forget their helpfulness in providing me with somany theoreti al suggestions as well as with useful tips at amore pra ti al level.Moreover, I want to thank all my dear olleagues and friendsof the ELEDIA Resear h Group, whose skills and kindness on-tribute to make the Ph.D. a great professional experien e anda wonderful life experien e at the same time. I am deeply in-debted to all of them, stri tly in alphabeti al order by name(I hope not to forget anybody), Andrea Rosani, Anna Mar-tini, Aronne Casagranda, Davide Fran es hini, Edoardo Zeni,Federi o Caramani a, Federi o Viani, Gabriele Fran es hini,Gia omo Oliveri, Leonardo Lizzi, Lu a Ioriatti, Lu a Mani a,

Mauro Martinelli, Paolo Ro a, Renzo Azaro, and Ri ardoAramini.I also have to thank for the kindness the olleagues andfriends that I met during my stays at L2S, to be more pre- ise Andrea Cozza, Anthony Bourges, Arnaud Breard, BernardDu hene, Chin Yuan Chong, Christophe Conessa, Cyril Dahon,Frédéri Brigui, Frédéri Nouguier, Houmam Moussa, Jean-Philippe Groby, Juan Felipe Abas al, Karim Louertani, Ni o-las Ribiere-Tharaud, Tommy Gunnarsson, and Rami Kassab.A spe ial thank goes to the administrative sta of the lab aswell, namely Daniel Rouet and Maryvonne Giron.Finally, I am innitely grateful to my family, Stefano, Pie-rina, Susanna, and Mi òl, whose patient love and e ourage-ments enabled me to nish this work.

Abstra tIn the framework of inverse ele tromagneti s attering te h-niques, the thesis fo uses on the development and the analysisof the integration between a multi-resolution imaging pro edureand a shape-optimization-based te hnique. The arising method-ology allows, on one hand, to fully exploit the limited amount ofinformation olle table from s attering measurements by meansof the iterative multi-s aling approa h (IMSA) whi h enablesa detailed re onstru tion only where needed without in reas-ing the number of unknowns. On the other hand, the use ofshape-optimization, su h as the level-set-based minimization,provide an ee tive des ription of the lass of targets to be re-trieved by using a-priori information about the homogeneityof the s atterers. In order to assess strong points and draw-ba ks of su h an hybrid approa h when dealing with one ormultiple s atterers, a numeri al validation of the proposed im-plementations is arried out by pro essing both syntheti andlaboratory- ontrolled s attering data.KeywordsMi rowave Imaging, Inverse S attering, Level Sets, IterativeMulti-S aling Approa h, Homogeneous Diele tri S atterers.

RésuméLa re onstru tion non invasive de la position et de la formed'objets in onnus onstitue un thème de grand intérêt dansnombre d'appli ations, et on itera en parti ulier l'évaluation etle ontrle non destru tif (généralement référés par les abrévi-ations END et CND) pour la surveillan e et le ontrle indus-triel et le diagnosti de sous-surfa e [1. Dans un tel adre in-téressant, beau oup de méthodologies ont été proposées, prin- ipalement basées sur les rayons X [2, les ultrasons [3, et les ourants de Fou ault [4. Cependant, des appro hes dans le do-maine mi roonde (par lequel on entend de 300 MHz à 300 GHz )ont été ré emment re onnues omme orant des méthodologiesd'imagerie e a es, grâ e aux points lés suivants [1[5[8 :• les ondes éle tromagnétiques aux fréquen es mi roondespeuvent pénétrer matériaux naturels et arti iels sousréserve qu'ils ne soient pas des ondu teurs idéaux ;• les hamps dira tés par le ou les objets ibles sont repré-sentatifs non seulement des frontières de elui- i ou eux- i, mais aussi de la ou des stru tures intérieures ;i

• les mi roondes montrent une grande sensibilité au on-tenu en eau de la stru ture que l'on entreprend d'imager ;• les sondes de e domaine peuvent être employées sansau un onta t mé anique ave le spé imen testé .De plus, en omparaison aux rayons X voire aux appro hesbasées sur la résonan e magnétique, les méthodes mi roondesminimisent (ou évitent) des eets ollatéraux dans le spé imentesté. Don , elles peuvent par exemple être mises en ÷uvrede manière plus sûre en imagerie biomédi ale, en limitant en-tre autre le stress du patient dans la mesure où le onta tphysique ave le système d'imagerie peut être évité (e.g., ledépistage de an ers du sein [9), ou dans d'autres appli ations ritiques, telles que l'imagerie à travers les murs (dite Through-Wall Imaging ou TWI) [10.Une avan ée supplémentaire de l'inspe tion non invasivemi roonde est représentée par des appro hes de dira tion in-verse qui sont destinées à onstruire une image de la région soustest qui ontiennent de l'information quantitative bien dénie[11. La formulation mathématique du problème de dira tioninverse est présentée au hapitre 2 en se on entrant sur le asoù des informations a priori sur la géométrie sont ee tive-ment disponibles. Puisque les problèmes de dira tion inversen'ont généralement pas une solution sous une forme analytique,une résolution numérique basée sur la méthode des moments(MoM) est adoptée et les in onvénients prin ipaux du modèlequi apparaissent, tels que la non linéarité, le mauvais ondi-tionnement et la mauvaise lo alisation, sont dis utés [12[13.ii

En parti ulier, la mauvaise lo alisation est ausée par la perted'information entre le problème inverse et le problème dire t,alors que la non linéarité est due au fait que la solution nepeut pas être exprimée omme une somme linéaire d'élémentsindépendants. Par suite de la mauvaise lo alisation, le prob-lème inverse soure également de mauvais onditionnement,puisque la solution ne dépend pas en ontinuité des données.An de dis uter les stratégies de résolution qui ont étédéveloppées pour surmonter de tels in onvénients, le hapitre 3se on entre sur la situation a tuelle dans le adre de la dira -tion inverse. En parti ulier, les solutions régularisées [38 on-sistent à exprimer quelques propriétés physiques prévues desdira teurs au moyen de paramètres de régularisation, de efait onstruisant une famille de solutions appro hées. Mal-heureusement, le hoix du ou des paramètres de régularisationdevient alors le thème prin ipal, parti ulièrement dans le asdes problèmes non linéaires pour lesquels la littérature ne four-nit au un ritère. De façon analogue, l'utilisation des approx-imations, telles que Born, Rayleigh et Rytov [13, est limitéeà une lasse spé ique des problèmes de dira tion inverse, 'est-à-dire traitant les dira teurs de faibles ontrastes.À la diéren e des te hniques et approximations de régular-isation, les te hniques de minimisation tentent de faire fa e à lanon-linéarité du problème de dira tion inverse. Ces méthodolo-gies reformulent le problème omme une pro édure d'optimisationvisant à la minimisation de l'é art entre les hamps mesurés etla solution d'essai numériquement al ulée. À et eet, unefon tion de oût appropriée est dénie et l'espa e de re her heest exploré au moyen de stratégies adaptées au problème ef-iii

fe tif. Par onséquent, l'exa titude de la solution dépend del'e a ité de la stratégie de solution, puisque le problème peutposséder des solutions erronées de par sa non-linéarité. Dans le adre de stratégies de minimisation, la thèse dé rit stratégiesdéterministes et heuristiques de minimisation. Les plus large-ment onnues pour la première atégorie sont les méthodologiesbasées sur des minimisations par des ente selon le gradient ainsique présentées par Kleinman et al. [54. Ces méthodologiessont basées sur la dénition d'une série de solutions d'essai as-so iées à des valeurs stri tement dé roissantes de la fon tion de oût et elles sont ara térisées par la mise à jour du hamp éle -trique in onnu ainsi que des valeurs des propriétés éle tromag-nétiques (i.e., permittivité diéle trique et ondu tivité) dansle domaine de re her he. Malheureusement, la onvergen e dela minimisation déterministe dépend du point d'initialisation,puisque la solution peut être bloquée à des minima lo aux parsuite de la non-linéarité du problème. Au ontraire, les te h-niques heuristiques peuvent limiter le probléme de minima lo- aux grâ e à leur apa ité d'explorer l'espa e de re her he entotalité ainsi que à la possibilité d'in lure de l'information a pri-ori dans la solution. Une telle lasse de stratégies de minimi-sation se ompose généralement d'algo-rithmes sto hastiquesinspirés du omportement d'inse tes pour la mise à jour desin onnues.Sans élaborer au delà du né essaire à e stade, le ara -tère mal posé est fortement lié à la quantité d'information quel'on peut olle ter lors d'une expérien e de dira tion à butd'imagerie, et habituellement le nombre de données indépen-dantes est plus faible que la dimension de l'espa e des solutions;iv

des systèmes à vue multiple (on olle te le hamp dira té dansplusieurs dire tions ou sur plusieurs surfa es de l'espa e en-vironnant la zone étudiée) ou/et à illumination multiple (oné laire ette zone d'étude de plusieurs dire tions ou à partir desour es réparties dans plusieurs domaines) sont don générale-ment adoptés. Cependant, il est bien onnu que l'informationqui est ee tivement a essible par e ou es moyens est unequantité qui onnaît une limite supérieure [14[15. En on-séquen e, il est né essaire d'exploiter de manière e a e toutel'information ontenue dans les é hantillons re ueillis du hampdira té an d'atteindre une re onstru tion (d'obtenir une im-age) qui soit satisfaisante. Comme dis uté dans le hapitre3, an d'exploiter ee tivement toute l'information olle téeà partir des mesures de dira tion ee tuées, des stratégiesdites de multi-résolution ont été ré emment proposées. L'idéeest de viser une résolution spatiale performante ( 'est-à-direaméliorée par rapport à elle ouramment hoisie ou assuréedans la zone d'étude) seulement dans les régions d'intérêt (Re-gions of Interest dites RoIs) de l'espa e oû les dira teurs in- onnus sont lo alisés (plus pré isément, oû ils sont estimés êtrepar le pro essus d'imagerie qui est mis en ÷uvre) [16 et/oûdes dis ontinuités entre matériaux apparaissent être présentes[17[18. Quant aux réalisations pratiques, des stratégies déter-ministes ou impliquant des analyses statistiques des donnéesont été proposées an de déterminer le niveau de résolutionoptimal, tandis que des appro hes impliquant des fon tionssplines de degrés variés ont été employées an d'améliorer leniveau de résolution. Par ailleurs, des appro hes multi-étapesont été implémentées dans le but d'a roître de manière itéra-v

tive la résolution spatiale au moyen d'une pro édure analogueà elle d'un zoom [19 en gardant le rapport entre le nombred'in onnues (e.g., les paramètres éle tromagnétiques de ellulesave lesquelles on onsidère que la zone d'étude est divisée) etle nombre de données (e.g., les é hantillons re ueillis du oudes hamps dira tés) susamment faible et onstant de tellefaçon que le risque de la survenue de minima lo aux d'une fon -tionnelle oût (qui traduit de manière usuelle l'é art entre lesdonnées et elles que l'on pourrait asso ier par la simulationnumérique aux objets modélisés par la pro édure d'inversion,et qui orrespond au moins indire tement à la diéren e entreles objets réels et eux qui nous apparaissent re onstruits) dansle problème d'optimisation tel que onsidéré [15.Par ailleurs, l'absen e d'information ae tant la bonne ré-solution du problème inverse a été onsidérée, parti ulièrementen END-CND, à travers l'exploitation de la onnaissan e a pri-ori que l'on peut avoir sur la s ène de test, et du s énariode l'intera tion éle tromagnétique mis en jeu, au moyen d'unereprésentation e a e des in onnues de elle- i. En eet, dansbeau oup d'appli ations, le ou les objets in onnus sont ara -térisés par des propriétés éle tromagnétiques onnues (i.e., per-mittivité diéle trique et ondu tivité) et ils sont lo alisés dansune région hte onnue au moins à un ertain degré (des in er-titudes peuvent l'ae ter, e i menant à une omplexité addi-tionnelle qui pourrait être signi ative). De plus, ela dépen-dant de la pré ision re her hée, des s énarii plus omplexespeuvent être appro hés via l'introdu tion d'un ensemble de ré-gions homogènes ara térisées par des paramètres géométriques(de forme) et éle tromagnétiques diérents [20. Sous de tellesvi

hypothèses, un problème d'imagerie se réduit à un problèmede re onstru tion de forme, plus pré isément un problème pourlequel e sont les supports de es régions homogènes qui doiventêtre re onstruits. An d'atteindre un tel but, des te hniquesparamétriques qui sont destinées à représenter l'objet in onnuen terme de paramètres des riptifs de formes de référen e [21[22, des appro hes plus sophistiquées telles que l'évolution on-trlée de ourbes de type splines [23[25, des gradients deforme [26[28, ou des méthodes d'évolution d'ensembles deniveaux [31[32, ont été proposées. Plus en détail, quant auxstratégies d'optimisation de forme les plus importantes dansla formation d'images en mi roonde (le hapitre 4), des ap-pro hes paramétriques sont basées sur la des ription des objetsau moyen de formes de base qui sont orre tement paramétrées.Pour e qui on erne les méthodes d'ensembles de niveaux, le ontour zéro d'un tel ensemble dénit la frontière du ou desobjets homogènes re her hés, e qui, en ontraste aux straté-gies qui impliquent une des ription en pixels ou paramétriques,permet de représenter des formes omplexes ou des régionsd'une manière relativement simple (on parlera de méthodesd'inversion libres de ontraintes topologie). De plus, à la dif-féren e des appro hes paramétriques, les ensembles de niveaupermettent de ontrler la fusion et la division des objets d'unemanière naturelle.Dans le adre brossé i-dessus, la thèse se fo alise sur ledéveloppement et l'analyse de l'intégration d'une stratégie multi-é helle itérative (dite Iterative Multi-S aling Approa h ou IMSA)[19 et de la représentation en ensembles de niveaux (Level-Sets ou LS) [33. L'implémentation qui en résulte a pour butvii

d'exploiter de manière protable tant la onnaissan e a pri-ori disponible sur le s énario joué (e.g., l'homogé-néité du oudes dira teurs en est le point lé) que le ontenu informatifdes mesures ee tuées. Par raison de simpli ité, et sans pré-tention à exhaustivité, la formulation du problème inverse estréduite au as bidimensionnel de polarisation transverse mag-nétique (TM) quand on traitera d'une ou de plusieurs régionsd'intérêt.En parti ulier, l'ar hite ture de la stratégie proposée estprésentée au hapitre 4. La formulation mathématique del'appro he itérative multi-résolution ave la minimisation baséesur l'ensemble de niveaux (notée IMSA-LS) est on entrée surl'ar hite ture multi-étape. L'algorithme est basé sur des étapesoù la résolution spatiale est itérativement augmentée en on- entrant la région d'intérêt sur le se teur où l'objet est lo alisé.À la première étape, le domaine de re her he est dis rétisé etune solution brute est re her hée. Puis, à partir de la premièreévaluation, la première région d'intérêt est estimée et le niveaude résolution est augmenté seulement à l'intérieur de la régiond'intérêt. À et eet, une nouvelle fon tion multi-résolutiond'ensemble des niveaux est dénie et son évolution est menéeen résolvant un problème adjoint qui orrespond à la dérivationde la fon tion de oût.L'évaluation des possibilités de re onstru tion d'IMSA-LSest ee tuée premièrement en onsidérant des géométries sim-ples, telles qu'un ylindre ir ulaire ave un rayon de la demi-longueur d'onde, et des données synthétiques. L'algorithme estinitialisé ave la solution vraie an de réaliser un essai de stabil-ité, puis des re onstru tions sans et ave bruit sont ee tuées.viii

Pendant de telles expérien es, le omportement de la fon -tion multi-résolution d'ensemble des niveaux est également dis- uté. En outre, l'exé ution proposée est omparée à l'appro hedite "bare", 'est-à-dire la méthode standard. Généralement,l'IMSA-LS semble être plus pré is parti ulièrement ave defaibles rapports signal à bruit. Les mêmes on lusions tien-nent également en onsidérant des formes plus omplexes, tellesque le ylindre re tangulaire ou le ylindre reux. Pour mieuxinvestiguer l'évaluation, des données a quises en situation on-trlée de laboratoire pour quelques géométries d'essai ont étéaussi onsidérées. Dans de telles expérien es, IMSA-LS etl'appro he "bare" fournissent des résultats similaires en termesd'exa titude, puisque es données sont probablement ae téespar un faible niveau du bruit.Le hapitre 5 se on entre sur un développement ultérieurde l'IMSA-LS, ara térisé par la possibilité de traiter des ré-gions d'intérêt multiples, parti ulièrement pour re onstruireplusieurs objets de façon plus e a e en termes d'attributiondes in onnues. Plus en détail, une telle stratégie, appeléeIMSMRA-LS, est ara térisée par une ar hite ture multi-étape,où à haque niveau de résolution diérentes régions d'intérêtsont prises en onsidération simultanément. À la premièreétape, un problème "bare" est résolu en hoisissant le nom-bre de domaines selon la quantité d'information indépendantedans les données dira tées [14. Puis, à partir des évaluationsbrutes des objets, les régions d'intérêt sont dénies au moyend'une stratégie adaptée à nos besoins, basée sur l'identi ationdes ontours des formes re onstruites. La résolution spatialeest augmentée dans es régions en les dis rétisant ave une fra -ix

tion des ellules qui ont été onsidérées à la premiére étape, ande maintenir le nombre d'in onnues limité pendant la pro é-dure d'inversion. La re onstru tion multi-résolution est réitéréejusqu'à e que les paramétres des régions d'intérêt, 'est-à-direleurs bary entres et leurs dimensions, deviennent onstants.Le hapitre 5 dis ute également la performan e de re on-stru tion de l'implémentation multi-région. An de omparerl'exa titude de la re onstru tion fournie par IMSMRA-LS auxrésultats de l'appro he ave une seule région, une validationpréliminaire prend en onsidération un dira teur simple, telqu'un ylindre ir ulaire ave un rayon de la demi-longueurd'onde, situé dans un domaine arré de té deux longueursd'onde. Dans une telle expérien e, l'IMSMRA-LS s'avère pluse a e que l'IMSA-LS, parti ulièrement en raison de l'utilisationd'une stratégie de mise à jour de l'ensemble de niveaux plusappropriée. Après la dis ussion du hoix des paramètres pourle ritère d'arrêt, e hapitre 5 propose quelques expérien esnumériques ara térisées par des dira teurs multiples, tels quedeux ylindres ir ulaires, deux re tangles, ou trois objets dediérentes formes. Dans toutes es expérien es, l'IMSMRA-LSfournit une re onstru tion légèrement plus pré ise que l'appro he"bare", alors que l'exa titude des résultats de l'IMSA-LS estraisonnablement inférieure en raison de la résolution spatialeplus élevée. Cependant, en onsidérant des formes plus om-plexes, telles que les ylindres reux et des roix, aussi bienque des données bruitées, l'IMSMRA-LS surpasse l'appro he"bare". La dernière illustration onsidérée a trait à la re on-stru tion réalisée en traitant des données expérimentales delaboratoire. La géométrie de référen e se ompose de deuxx

ylindres ir ulaires diéle triques. Comme dans la dernière ex-périen e du hapitre 4, l'exé ution proposée semble fournir desrésultats tout à fait similaires à eux de la méthode ave uneseule région, puisque les données expérimentales sont ara -térisées par des rapports signal à bruit élevés. Cependant, en e qui on erne le dernier as du hapitre 4, l'IMSMRA-LSsemble être plus pré is en estimant la forme des ibles.En on lusion, e travail propose l'intégration entre l'appro- he multi-é helle et une méthode d'ensemble de niveaux and'exploiter de manière protable la quantité d'information ob-tenue via les mesures de la dira tion aussi bien que l'informa-tion disponible a priori sur le problème onsidéré. Deux réali-sations sont présentées an de traiter ee tivement des ong-urations ara térisées par un ou plusieurs objets. Les élémentsprin ipaux de l'appro he peuvent être ré apitulés omme suit:• représentation innovatri e multi-niveau des in onnues duproblème dans la te hnique de re onstru tion basée surles ensembles de niveaux ;• limitation du risque de blo age en des solutions erronéesgrâ e au rapport réduit entre données et in onnues ;• exploitation utile d'informations a priori (i.e., homogénéitéd'objets) sur le s énario à l'essai ;• résolution spatiale augmentée seulement dans les régionsd'intérêt . xi

En outre, de la validation numérique et expérimentale pro-posée, les on lusions suivantes peuvent être tirées :• l'IMSA-LS s'est habituellement avéré plus e a e quel'appro he "bare", parti ulièrement en traitant des don-nées bruitées dira tées par un objet de géométrie simpleaussi bien que omplexe ;• l'IMSMRA-LS a semblé être aussi e a e que l'appro he"bare" en traitant des géométries simples, alors qu'unear hite ture multi-région appropriée a amélioré l'exa ti-tude de la re onstru tion ave les diuseurs multiples ;• les stratégies intégrées (i.e., IMSA-LS et IMSMRA-LS)ont semblé né essiter moins de al uls que l'appro hestandard, en atteignant une re onstru tion possédant lemême niveau de rêsolution spatiale dans la des riptionde l'objet .

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Published Journals Papers

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592 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 6, 2007

A Two-Step Inverse Scattering Procedure for theQualitative Imaging of Homogeneous Cracks in

Known Host Media—Preliminary ResultsManuel Benedetti, Massimo Donelli, Dominique Lesselier, and Andrea Massa, Member, IEEE

Abstract—In the framework of nondestructive evaluation andtesting, microwave inverse scattering approaches demonstratedtheir effectiveness and the feasibility of detecting unknown anom-alies in dielectric materials. In this letter, an innovative techniqueis proposed in order to enhance their reconstruction accuracy. Theapproach is aimed at first estimating the region-of-interest (RoI)where the defect is supposed to be located and then at improvingthe qualitative imaging of the crack through a level-set-basedshaping procedure. In order to assess the effectiveness of the pro-posed approach, representative numerical results concerned withdifferent scenarios and blurred data are presented and discussed.

Index Terms—Genetic algorithms, level set, microwave imaging,nondestructive testing and nondestructive evaluation (NDT/NDE).

I. INTRODUCTION

NONDESTRUCTIVE testing and nondestructive evaluation(NDT/NDE) techniques are aimed at detecting unknown

defects and other anomalies buried in known host objects bymeans of noninvasive methodologies [1]–[3]. In such a frame-work, electromagnetic inverse scattering approaches can play animportant role. As an example, some approaches that approxi-mate defective regions with rectangular shapes have been pro-posed [4], [5]. Despite the satisfactory results, such techniquesare adequate when facing NDE/NDT problems where the re-trieval of the positions and the rough estimation of the sizes ofthe defects are enough, but they cannot be reliably used when anaccurate knowledge of the shapes of the defects is needed as insome industrial processes and usually in biomedical diagnosis.Notwithstanding, they are useful for providing a “first-step” in-formation concerned with a rough localization of the defects tobe further improved by means of a successive refinement recon-struction carried out with suitable contour detection methods.

Towards this end, this letter presents a two-step procedureaimed at improving the reconstruction of [4], [5]. More in de-tail, starting from the knowledge of the scattered field with andwithout the defect, the approximate problem in which the defect

Manuscript received July 16, 2007; revised October 9, 2007.M. Benedetti, M. Donelli, and A. Massa are with the Department of Infor-

mation and Communication Technology, University of Trento, 38050 Trento,Italy (e-mail: manuel.benedetti@dit.unitn.it; massimo.donelli@dit.unitn.it; an-drea.massa@ing.unitin.it).

D. Lesselier is with the Département de Recherche en Électromagnétisme-Laboratoire des Signaux et Systèmes, CNRS-SUPELEC-UPS 11, 91192 Gif-sur-Yvette CEDEX, France (e-mail: dominique.lesselier@lss.supelec.fr).

Color versions of one or more of the figures in this letter are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LAWP.2007.910954

is assumed of simple shape (e.g., a rectangle) is reformulated interms of an inverse scattering one and successively solved bymeans of the minimization of a suitably defined cost function[6]. After such a step, the region-of-interest (RoI) where the de-fect is supposed to be located is determined and the second re-trieval phase takes place by applying a shape-based optimizationtechnique based on the numerical evolution of a level-set func-tion [7].

The outline of this letter is as follows. The mathematical for-mulation of the proposed approach is presented in Section IIby focusing on the second step of the reconstruction procedure.Then, the effectiveness of the approach is discussed with refer-ence to a set of representative numerical results in dealing withblurred measurement data (Section III). Finally, some conclu-sions follow (Section IV).

II. MATHEMATICAL FORMULATION

Let us consider a 2-D scenario where a homogeneous defect(or crack) characterized of unknown position andshape lies in a cylindrical host region characterized byknown relative permittivity and conductivity . The de-fective host medium is probed by electromagnetic transversemagnetic (TM) plane waves with an incident field

, and the induced electromagnetic field is givenby

(1)

where is the free-space Green’s function andis the object function ( being the working

frequency), or analogously, in a more “practical” expression [8]

(2)

by considering the inhomogeneous Green’s functionand the total electric field in the scenario without defects

defined as follows:

(3)

where is the differential object given by

if

if(4)

1536-1225/$25.00 © 2007 IEEE

BENEDETTI et al.: TWO-STEP INVERSE SCATTERING PROCEDURE FOR THE QUALITATIVE IMAGING OF HOMOGENEOUS CRACKS 593

With reference to the “differential formulation,” the first stepof the approach considers the partitioning of in and theonly one computation of the inhomogeneous Green’s matrix

of entries according to the procedure detailedin [8]. Then, the RoI is modeled with a rectangular homo-geneous shape described through the coordinates of the center

, its length , its side , and the relative ori-entation . Accordingly, turns out to be fully described bymeans of the following object function profile:

if

and

otherwise

(5)

where and. Under these assumptions, the

unknown array

(6)

is determined by solving the inverse scattering problem formu-lated in terms of an optimization one.

In detail, starting from the knowledge of the data samplescollected in the observation domain (i.e., the total fieldwith the defect and without the defect ,

) and in the investigation domain (i.e.,, ), is obtained by minimizing the

mismatching between estimated and measured scattering dataevaluated through the computation of , as shown in (7)at the bottom of the page. As far as the minimization processis concerned, trial solutionsare randomly initialized ( , being the iterationindex) and an iterative procedure takes place until a stop-ping criterion holds true ( or ,

). At eachiteration, the following operations are performed:

1) the iteration index is updated ;2) a set of genetic operators described in [5] is applied to

in order to generate the th ;3) the best trial solution achieved so far,

being, is stored and its fitness

evaluated in order to check the thresholdcondition for the stopping criterion.

At the end of the first step, the genetic algorithm (GA)-basedoptimization returns the array that defines the RoI

(8)

where the superscript denotes the estimated values.The second step of the approach is aimed at refining the esti-

mate of the defect starting from the knowledge coming from thefirst step (i.e., the homogeneous defect lies in ). Towards thispurpose, a level-set-based strategy is employed. The algorithmis initialized by defining an elliptic trial shape centered at

, with axes equal to and , respectively, and ro-tated by . Then, the level set is defined in accordingto the rule based on the oriented distance function [9]. In par-ticular, is equal to if ,and otherwise, being a point be-longing to the contour of [7], [9]. Concerning the numericalimplementation, is discretized in cells and the followingsequence is iteratively applied.

1) The accuracy of the current trial shape in retrieving theactual shape of the defect is evaluated by computing thevalue of the metric shown in (9) at the bottom of the page,where is the differential object function equal to

if and 0

(7)

(9)

594 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 6, 2007

otherwise. Furthermore, is the solution of thefollowing equation:

(10)

2) The level-set-based process ends if a fixed number of it-eration is performed or and

is assumed as the crack profile. Otherwise, the levelset function is updated by solving aHamilton–Jacobi equation

(11)

where stands for the numerical counter-part of the Hamiltonian operator [9], [10] and is thetime-step parameter chosen according to the Courant–Friedrich–Leroy condition [11]. Moreover, isthe velocity function determined by solving the adjointproblem as detailed in [7] and [9].

III. NUMERICAL ANALYSIS

This section is devoted to a numerical analysis of the pro-posed approach. A set of selected and representative numericalresults related to a couple of experiments are reported and dis-cussed for pointing out the improvement in the crack detectionand shaping.

The first experiment (indicated as the “experiment A”) con-siders an unknown void defect of elliptical cross section that liesin a square lossless host medium of side and charac-terized by a dielectric permittivity equal to . The defectis located at and rotated by withaxes equal to and , respectively. The scenario hasbeen probed by orthogonal and equally spaced angulardirections and the field has been measured at points.Moreover, the scattering data have been blurred with an additivenoise of Gaussian-type characterized by a fixed signal-to-noiseratio (SNR).

Concerning the numerical procedure, has been discretizedin and in subdomains.

As an example, Fig. 1(a) shows the reconstruction result fromthe two-step procedure in correspondence with SNR 10 dB.As it can be observed, the support of the defect (whose actualperimeter is evidenced by the dotted line) belongs to the RoI(dashed–dotted line) estimated at the end of the first step. How-ever, the crack dimension is largely overestimated. On the con-trary, the shape of the crack is more faithfully retrieved, despitethe nonfavorable SNR. Such an event is quantitatively quanti-fied by the value of the localization error 1.2% [12] thatimproves by 30% with respect to the single-step inversion. Forcomparison purposes, Fig. 1(b) shows the reconstruction ob-tained by the “bare” level-set method setting and dis-cretizing the domain such that the spatial resolution is equalto that of Fig. 1(a). As it can be noticed, the reconstructionworsens.

As far as the area error [12] is concerned, Fig. 2 showsthe behavior of the error figure versus the SNR. As it can benoticed, the two-step approach turns out to be more robust than

Fig. 1. Experiment A: (a) reconstruction after the first step (i.e., the RoI )and dielectric distribution estimated at the end of the two-step procedure;(b) dielectric distribution estimated by means of the “bare” level set approach(i.e., ).

Fig. 2. Experiment A: area error versus SNR.

the blurring on data and the resulting performances are better inan amount between 150% and 100%.

The “experiment B” deals with a more complex cross-sec-tion shape of defect indicated by the dotted line in Fig. 3(a).As an example, let us analyze the case of SNR 20 dB, whenthe profile reconstructed by the proposed method is shown inFig. 3(a), while Fig. 3(b) gives the dielectric distribution esti-mated by the “bare” level set. Starting from the estimation of the

BENEDETTI et al.: TWO-STEP INVERSE SCATTERING PROCEDURE FOR THE QUALITATIVE IMAGING OF HOMOGENEOUS CRACKS 595

Fig. 3. Experiment B: (a) reconstruction after the first step (i.e., the RoI )and dielectric distribution estimated at the end of the two-step procedure;(b) dielectric distribution estimated by means of the “bare” level-set approach(i.e., ).

RoI, the two-step approach provides a satisfactory reconstruc-tion improving both the localization error and the area error withrespect to the first step ( 1.5%, 0.5%; 3.7%,

1.5%). Similar considerations hold true when smallerSNRs are considered, as pointed out by the values of the areaerror pictorially reported in Fig. 4.

IV. CONCLUSION

In this letter, an innovative two-steps procedure for NDE/NDT applications has been proposed and preliminarily as-sessed. The method consists of a first step aimed at determiningthe region of interest where the defect is supposed to be locatedand a successive shaping process for enhancing the qualitativeimaging. The approach has been evaluated by considering

Fig. 4. Experiment B: area error versus SNR.

blurred synthetic data and different crack cross sections. Theachieved results have pointed out the effectiveness of the ap-proach, thus suggesting its future employment in biomedicalimaging.

REFERENCES

[1] P. J. Shull, Nondestructive Evaluation: Theory, Techniques and Appli-cations. Boca Raton, FL: CRC Press, 2002.

[2] R. Zoughi, Microwave Nondestructive Testing and Evaluation. Am-sterdam, The Netherlands: Kluwer, 2000.

[3] J. C. Bolomey, “Recent European developments in active microwaveimaging for industrial, scientific and medical applications,” IEEETrans. Microw. Theory Tech., vol. 37, no. 12, pp. 2109–2117, Dec.1989.

[4] K. Meyer, K. J. Langenberg, and R. Schneider, “Microwave imaging ofdefects in solids,” presented at the 21st Annu. Rev. Progr. QuantitativeNDE, Snowmass Village, CO, Jul. 31–Aug. 5 1994.

[5] S. Caorsi, A. Massa, M. Pastorino, and M. Donelli, “Improved mi-crowave imaging procedure for nondestructive evaluations of two-di-mensional structures,” IEEE Trans. Antennas Propag., vol. 52, no. 6,pp. 1386–1397, Jun. 2004.

[6] Y. R. Samii and E. Michielssen, Electromagnetic Optimization by Ge-netic Algorithms. New York: Wiley, 1999.

[7] O. Dorn and D. Lesselier, “Level set methods for inverse scattering,”Inverse Problems, vol. 22, pp. R67–R131, 2006.

[8] S. Caorsi, G. L. Gragnani, M. Pastorino, and M. Rebagliati, “Amodel-driven approach to microwave diagnostics in biomedical appli-cations,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 10, pt. 2, pp.1910–1920, Oct. 1996.

[9] A. Litman, D. Lesselier, and F. Santosa, “Reconstruction of a two-di-mensional binary obstacle by controlled evolution of a level-set,” In-verse Problem, vol. 14, pp. 685–706, 1998.

[10] S. Osher and J. A. Sethian, “Fronts propagating with curvature-de-pendent speed: Algorithms based on Hamilton-Jacobi formulations,”J. Comp. Phys., vol. 79, pp. 12–49, 1988.

[11] J. A. Sethian, Levelset Methods and Fast Marching Methods, 2nd ed.Cambridge, U.K.: Cambridge Univ. Press, 1999, Monographs on Ap-plied and Computational Mathematics.

[12] S. Caorsi, M. Donelli, and A. Massa, “Detection, location, and imagingof multiple scatterers by means of the iterative multi-scaling method,”IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1217–1228, Apr.2004.

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)2 ∫

DI

τ (r′)Ev (r′)G2D (r, r′) dr′, r ∈ DI 8H9B<#+# λ 4= %<# *5 EA+'2(& B5?#3#(A%<. Ev 4= %<# %'%53 #3# %+4 >#3&. 5(& G2D (r, r′) =

− j4H

(2)0

(2πλ‖r − r′‖

) 4= %<# 1+##D=$5 # %B'D&46#(=4'(53 I+##(J= 12( %4'(. H(2)0 *#4(A %<#=# '(&DE4(&. K#+'%<D'+&#+ L5(E#3 12( %4'(:@( '+&#+ %' +#%+4#?# %<# 2(E('B( $'=4%4'( 5(& =<5$# '1 %<# %5+A#% Υ *7 =%#$D*7D=%#$ #(<5( 4(A %<# =$5%453 +#='32%4'( '(37 4( %<5% +#A4'(. 533#& +#A4'(D'1D4(%#+#=% 8RoI9.

R ∈ DI . B<#+# %<# = 5%%#+#+ 4= 3' 5%#& M- . %<# 1'33'B4(A 4%#+5%4?# $+' #&2+# '1 Smax=%#$= 4= 5++4#& '2%:O4%< +#1#+#( # %' P4A: -8+9 5(& %' %<# *3' E &45A+56 &4=$357#& 4( P4A: ,. 5% %<#>+=% =%#$ 8s = 1. s *#4(A %<# =%#$ (26*#+9 5 %+453 =<5$# Υs = Υ1. *#3'(A4(A %' DI .4= <'=#( 5(& %<# +#A4'( '1 4(%#+#=% Rs M Rs=1 = DI 4= $5+%4%4'(#& 4(%' NIMSA #C253=C25+# =2*D&'654(=. B<#+# NIMSA &#$#(&= '( %<# &#A+##= '1 1+##&'6 '1 %<# $+'*3#6 5%<5(& 5(& 4% 4= '6$2%#& 5 '+&4(A %' %<# A24&#34(#= =2AA#=%#& 4( MQ:@( 5&&4%4'(. %<# 3#?#3 =#% 12( %4'( φs 4= 4(4%4534K#& *7 6#5(= '1 5 =4A(#& &4=%5( #12( %4'( &#>(#& 5= 1'33'B= M,HM,RS

φs (r) =

−64(b=1,...,Bs

‖r − rb‖ 41 τ (r) = τC64(b=1,...,Bs‖r − rb‖ 41 τ (r) = 0

8 9B<#+# rb = (xb, yb) 4= %<# bD%< *'+&#+D #33 8b = 1, . . . , Bs9 '1 Υs=1:;<#(. 5% #5 < =%#$ s '1 %<# $+' #== 8s = 1, ..., Smax9. %<# 1'33'B4(A '$%464K5%4'($+' #&2+# 4= +#$#5%#& 8P4A: ,9S

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% 5• !"#$%& '()("*( +%,!%-%(./.0"( 6 78" &'$*49$ 01' $3&' 39 *"#*"9"'$"% 3'$"*:9 &0 $8" 2";"2 9"$ 01' $3&' 49 0&22&<9

τks(r) =

s∑

i=1

NIMSA∑

ni=1

τkiB

(rni

)r ∈ DI =5><8"*" $8" 3'%"? ks 3'%3 4$"9 $8" k6$8 3$"*4$3&' 4$ $8" s6$8 9$"# @ks = 1, ..., kopt

s -B

(rni

) 39 4 *" $4'B124* )4939 01' $3&' <8&9" 91##&*$ 39 $8" n6$8 91)6%&:43' 4$ $8"

i6$8 *"9&21$3&' 2";"2 @ni = 1, ..., NIMSA- i = 1, ..., s- 4'% $8" &"C 3"'$ τki

39 B3;"')D

τki=

τC 30Ψki

(rni

)≤ 0

0 &$8"*<39" =E>2"$$3'BΨki

(rni

)=

φki

(rni

) 30 i = s

φkopti

(rni

) 30 (i < s) 4'% (rni∈ Ri

) =F><3$8 i = 1, ..., s 49 3' =5>G• 10%$2 30-.!0#4.0"( ',2/.0(5 + (' " τks

(r) 849 )""' "9$3:4$"%- $8" "2" $*3 H"2% Evks(r) 39 '1:"*3 422D &:#1$"% 4 &*%3'B $& 4 #&3'$6:4$ 83'B ;"*93&' &0 $8"I"$8&% &0 I&:"'$9 =I&I> @J, 49

Evki

(rni

)=

∑NIMSApi=1 ζv

(rpi

) [1− τki

(rpi

)G2D

(rni

, rpi

)]−1,

rni, rpi

∈ DI

ni = 1, ..., NIMSA .

=/>

• 6"-. 14( .0"( 89/$4/.0"( : K$4*$3'B 0*&: $8" $&$42 "2" $*3 H"2% %39$*3)1$3&'=/>- $8" *" &'9$*1 $"% 9 4$$"*"% H"2% ξvks(rm) 4$ $8" m6$8 :"491*":"'$ #&3'$-

m = 1, ..., M(v)- 39 1#%4$"% )D 9&2;3'B $8" 0&22&<3'B "L14$3&'

ξvks(rm) =

s∑

i=1

NIMSA∑

ni=1

τki

(rni

)Ev

ki

(rni

)G2D

(rm, rni

) =M>4'% $8" H$ )"$<""' :"491*"% 4'% *" &'9$*1 $"% %4$4 39 ";4214$"% )D $8" :12$36*"9&21$3&' &9$ 01' $3&' Θ %"H'"% 49

Θ φks =

∑Vv=1

∑M(v)m=1

∣∣∣ξvks(rm)− ξv

ks(rm)

∣∣∣2

∑Vv=1

∑M(v)m=1

∣∣∣ξvks(rm)

∣∣∣2 . =,.>

• ;0(0&0</.0"( =.",,0(5 6 78" 3$"*4$3;" #*& "99 9$&#9 @3G"G- kopts = ks 4'% τ opt

s = τks

<8"'N =,> 4 9"$ &0 &'%3$3&'9 &' $8" 9$4)323$D &0 $8" *" &'9$*1 $3&' 8&2%9 $*1" &* =(><8"' $8" :4?3:1: '1:)"* &0 3$"*4$3&'9 39 *"4 8"% @ks = Kmax &* = > <8"' $8"

! #$%#& '( ) %'*#+ ,-. ,//0 1'+ $2*34 5%4'( 4( !"#$%# &$'()#*%6532# '1 %7# '8% 12( %4'( 48 89533#+ %75( 5 :;#& %7+#87'3& γth< !8 15+ 58 %7# 8%5*434%='1 %7# +# '(8%+2 %4'( 48 '( #+(#& > '(&4%4'( ?+@. %7# :+8% '++#8$'(&4(B 8%'$$4(B +4%#+4'( 48 85%48:#& C7#(. 1'+ 5 :;#& (29*#+ '1 4%#+5%4'(8. Kτ . %7# 95;4929(29*#+ '1 $4;#38 C74 7 65+= %7#4+ 6532# 48 89533#+ %75( 5 28#+ &#:(#& %7+#87'3& γτ5 '+&4(B %' %7# +#35%4'(874$95;j=1,...,Kτ

NIMSA∑

ns=1

|τks(rns

)− τks−j (rns)|

τC

< γτ ·NIMSA. ?--@D7# 8# '(& +4%#+4'(. 5*'2% %7# 8%5*434%= '1 %7# +# '(8%+2 %4'(. 48 85%48:#& C7#( %7# '8% 12( %4'( *# '9#8 8%5%4'(5+= C4%74( 5 C4(&'C '1 KΘ 4%#+5%4'(8 58 1'33'C8E

1

KΘ

KΘ∑

j=1

Θ φks −Θ φks−j

Θ φks

< γΘ. ?-,@

KΘ *#4(B 5 :;#& (29*#+ '1 4%#+5%4'(8 5(& γΘ *#4(B 28#+F&#:(#& %7+#87'3&8G< H7#(%7# 4%#+5%46# $+' #88 8%'$8. %7# 8'32%4'( τ opts 5% %7# sF%7 8%#$ 48 8#3# %#& 58 %7# '(#+#$+#8#(%#& *= %7# I*#8%J 3#6#3 8#% 12( %4'( φopt

s &#:(#& 58

φopts = 5+B [94(h=1,...,k

opts(Θ φh)

]. ?-K@

• !"#$!%&' ()*$!" F D7# 4%#+5%4'( 4(&#; 48 2$&5%#& >ks → ks + 1G

• +","- ."! ()*$!" F D7# 3#6#3 8#% 48 2$&5%#& 5 '+&4(B %' %7# 1'33'C4(B L5943%'(FM5 '*4 +#35%4'(874$φks

(rns) = φks−1 (rns

)−∆tsVks−1 (rns)Hφks−1 (rns

) ?-N@C7#+# H· 48 %7# L5943%'(45( '$#+5%'+ >K,>KK B46#( 58

H2 φks(rns

) =

95;2Dx−

ks; 0

+94(2

Dx+

ks; 0

+

+95;2Dy−

ks; 0

+94(2

Dy+

ks; 0

41 Vk(s)

(rn(s)

)≥ 094(2

Dx−

ks; 0

+95;2

Dx+

ks; 0

+

+94(2Dy−

ks; 0

+95;2

Dy+

ks; 0

'%7#+C48#?-O@

5(& Dx±ks

=±φks(xns±1,yns)∓φks(xns ,yns)

ls

. Dy±ks

=±φks(xns ,yns±1)∓φks (xns ,yns)

ls

< ∆ts 48 %7#%49#F8%#$ 7'8#( 58 ∆ts = ∆t1lsl1

C4%7 ∆t1 %' *# 8#% 7#2+48%4 533= 5 '+&4(B %' %7#34%#+5%2+# >,K. ls *#4(B %7# #33F84&# 5% %7# sF%7 +#8'32%4'( 3#6#3< Vks

48 %7# 6#3' 4%=

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% 501' $3&' &6#1$"% 0&22&73'8 $9" 813%"23'": :188":$"% 3' ;+< )> :&2?3'8 $9" 4%@&3'$#*&)2"6 &0 A/B 3' &*%"* $& %"$"*63'" $9" 4%@&3'$ C"2% F vks

D &*%3'82>-Vks

(rns) = −ℜ

∑V

v=1τCEv

ks(rns)F

vks(rns)∑V

v=1

∑M(v)

m=1 |ξvks(rm)|

2

,

ns = 1, ..., NIMSA

A,EB79"*" ℜ :$4'%: 0&* $9" *"42 #4*$DF9"' $9" sG$9 63'363H4$3&' #*& ":: $"*63'4$":- $9" &'$*4:$ 01' $3&' 3: 1#%4$"%;τ opts (r)= τks−1 (r)- r ∈ DI AIB 4: 7"22 4: $9" RoI ;Rs → Rs−1D J& %& :&- $9" 0&22&73'8&#"*4$3&': 4*" 4**3"% &1$K• !"#$%&%'!( !) %*+ ,&-. +(%+- !) %*+ 0!1 G $9" "'$"* &0 Rs &0 &&*%3'4$":

(xcs, y

cs) 3: %"$"*63'"% )> &6#1$3'8 $9" "'$"* &0 64:: &0 $9" *" &':$*1 $"% :94#":4: 0&22&7: ;,L ;M38D ,A(B

xcs =

∫DI

xτ opts (r)B (r) dx dy

∫DI

τ opts (r)B (r) dx dy

A,5B

ycs =

∫DI

yτ opts (r)B (r) dx dy

∫DI

τ opts (r)B (r) dx dy

; A,/B

• 23%'"&%'!( !) %*+ 4'5+ !) %*+ 0!1 G $9" :3%" Ls &0 Rs 3: &6#1$"% )> "?4214$3'8$9" 64N3616 &0 $9" %3:$4' " δc (r) =√(x− xc

s)2 + (y − yc

s)2 3' &*%"* $& "' 2&:"$9" : 4$$"*"*- '46"2>

Ls = 64Nr

2×

τ opts (r)

τC

δc (r)

. A,OB(' " $9" RoI 94: )""' 3%"'$3C"%- $9" 2"?"2 &0 *":&21$3&' 3: "'94' "% ;ks → ks−1 &'2>3' $93: *"83&' )> %3: *"$3H3'8 Rs 3'$& NIMSA :1)G%&643': ;M38D ,A B 4'% )> *"#"4$3'8$9" 63'363H4$3&' #*& ":: 1'$32 $9" :>'$9"$3 H&&6 )" &6": :$4$3&'4*> As = soptB- 3D"D-

|Qs−1 −Qs|

|Qs−1|× 100

< γQ, Q = xc, yc, L A+.B

γQ )"3'8 4 $9*":9&2% :"$ 4: 3' ;,L- &* 1'$32 4 64N3616 '16)"* &0 :$"#: Asopt = SmaxB3: *"4 9"%D $ $9" "'% &0 $9" 612$3G:$"# #*& ":: As = soptB- $9" #*&)2"6 :&21$3&' 3: &)$43'"% 4:

τ opt(rni

)= τ opt

s

(rni

)- ni = 1, ..., NIMSA- i = 1, ..., soptD

! #$%#& '( ) %'*#+ ,-. ,// 0'+ $1*23 4%3'( 3( !"#$%# &$'()#*% ! "#$%&' )* +)*',)-'./5( '+&#+ %' 466#66 %7# #8# %39#(#66 '0 %7# +,-./, 4$$+'4 7. 4 6#2# %#& 6#% '0+#$+#6#(%4%39# +#612%6 '( #+(#& :3%7 *'%7 6;(%7#%3 4(& #<$#+3=#(%42 &4%4 36 $+#6#(%#&7#+#3(> ?7# $#+0'+=4( #6 4 73#9#& 4+# #94214%#& *; =#4(6 '0 %7# 0'22':3(@ #++'+ A@1+#6B• /' 1)23142'! 5$$'$ δ

δ|p =

√(xc

s|p − xc|p)2−

(yc

s|p − yc|p)2

λ× 100 C,-D:7#+# rc|p =

(xc|p , yc|p

) 36 %7# #(%#+ '0 %7# pE%7 %+1# 6 4%%#+#+. p = 1, ..., P . P*#3(@ %7# (1=*#+ '0 '*F# %6> ?7# 49#+4@# 2' 423G4%3'( #++'+ < δ > 36 &#A(#& 46

< δ >=1

P

P∑

p=1

δ|p . C,,D

• -$#1 5%42*142'! 5$$'$ ∆∆ =

I∑

i=1

1

NIMSA

NIMSA∑

ni=1

Nni

× 100 C,HD:7#+# Nni

36 #I142 %' 1 30 τ opt(rni

)= τ

(rni

) 4(& 0 '%7#+:36#>!6 04+ 46 %7# (1=#+3 42 #<$#+3=#(%6 4+# '( #+(#&. %7# +# '(6%+1 %3'(6 749# *##($#+0'+=#& *; *21++3(@ %7# 6 4%%#+3(@ &4%4 :3%7 4( 4&&3%39# J416634( ('36# 74+4 %#+3G#&*; 4 63@(42E%'E('36#E+4%3' CSNRD

SNR = 102'@∑Vv=1

∑M(v)m=1 |ξ

v (rm)|2

∑Vv=1

∑M(v)m=1 |µv,m|2

C,KD

µv,m *#3(@ 4 '=$2#< J416634( +4(&'= 94+34*2# :3%7 G#+' =#4( 9421#>6787 ,9!4:#42 ;141 . <2$ =)1$ <9)2!>#$678787 &$#)2*2!1$9 ?1)2>142'! 5( %7# A+6% #<$#+3=#(%. 4 2'662#66 3+ 124+ '8E #(%#+#&6 4%%#+#+ '0 L(':( $#+=3%%393%; ǫC = 1.8 4(& +4&316 ρ = λ/4 36 2' 4%#& 3( 4 6I14+#3(9#6%3@4%3'( &'=43( '0 63&# LD = λ M,H> V = 10 TM $24(# :49#6 4+# 3=$3(@3(@ 0+'=%7# &3+# %3'(6 θv = 2π (v − 1)/V . v = 1, ..., V . 4(& %7# 6 4%%#+3(@ =#461+#=#(%6 4+# '22# %#& 4% M = 10 +# #39#+6 1(30'+=2; &36%+3*1%#& '( 4 3+ 2# '0 +4&316 ρO = λ>!6 04+ 46 %7# 3(3%3423G4%3'( '0 %7# +,-./, 42@'+3%7= 36 '( #+(#&. %7# 3(3%342 %+342'*F# % Υ1 36 4 &36L :3%7 +4&316 λ/4 4(& #(%#+#& 3( DI > ?7# 3(3%342 9421# '0 %7# %3=#6%#$ 36 6#% %' ∆t1 = 10−2 46 3( M,H> ?7# RoI 36 &36 +#%3G#& 3( NIMSA = 15 × 1561*E&'=43(6 4% #4 7 6%#$ '0 %7# 3%#+4%39# =12%3E+#6'21%3'( $+' #66> O'( #+(3(@ %7#

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% 56$&##3'7 *3$"*34- $8" 0&22&93'7 &':71*4$3&' &0 #4*4;"$"*6 846 )""' 6"2" $"% 4 &*%3'7$& 4 #*"23;3'4*< 423)*4$3&' %"423'7 93$8 63;#2" ='&9' 6 4$$"*"*6 4'% '&36"2"66 %4$4>Smax = 4 ?;4@3;1; '1;)"* &0 6$"#6A- γxc

= γ yc

= 0.01 4'% γL = 0.05 ?;12$3B6$"# #*& "66 $8*"68&2%6A- Kmax = 500 ?;4@3;1; '1;)"* &0 &#$3;3C4$3&' 3$"*4$3&'6A-γΘ = 0.2 4'% γτ = 0.02 ?&#$3;3C4$3&' $8*"68&2%6A- KΘ = Kτ = 0.15Kmax ?6$4)323$< &1'$"*6A- 4'% γth = 10−5 ?$8*"68&2% &' $8" &6$ 01' $3&'ADE371*" F 68&96 64;#2"6 &0 *" &'6$*1 $3&'6 93$8 $8" +,-./, D $ $8" :*6$ 6$"#GE37D F?0A B s = 1- $8" 6 4$$"*"* 36 &**" $2< 2& 4$"%- )1$ 3$6 684#" 36 &'2< *&1782<"6$3;4$"%D I84'=6 $& $8" ;12$3B*"6&21$3&' *"#*"6"'$4$3&'- $8" J1423$4$3K" 3;473'7 &0 $8"6 4$$"*"* 36 3;#*&K"% 3' $8" '"@$ 6$"# GE37D F?(A B s = sopt = 2 46 &':*;"% )< $8" "**&*3'%"@"6 3' I4)D ,D E&* &;#4*36&' #1*#&6"6- $8" #*&:2" *"$*3"K"% )< $8" 63'72"B*"6&21$3&';"$8&% G+F ?3'%3 4$"% 3' $8" 0&22&93'7 46 10$#./, A- 98"' DI 846 )""' %36 *"$3C"% 3'

NBare = 31× 31 "J142 61)B%&;43'6- 36 68&9' GE37D F? AD L' 7"'"*42- $8" %36 *"$3C4$3&'&0 $8" 10$#./, 846 )""' 8&6"' 3' &*%"* $& 4 83"K" 3' $8" 98&2" 3'K"6$374$3&' %&;43'4 *" &'6$*1 $3&' 93$8 $8" 64;" 2"K"2 &0 6#4$342 *"6&21$3&' &)$43'"% )< $8" +,-./, 3'$8" RoI 4$ s = soptD 2$8&178 $8" :'42 *" &'6$*1 $3&'6 GE376D F?(A? A 4 83"K"% )< $8" $9& 4##*&4 8"64*" 63;324* 4'% J13$" 2&6" $& $8" $*1" 6 4$$"*"* 64;#2"% 4$ $8" 6#4$342 *"6&21$3&' &0 10$#./, GE37D F?3A 4'% +,-./, GE37D F?(A- $8" +,-./, ;&*" 043$80122< *"$*3"K"6 $8"6<;;"$*< &0 $8" 4 $142 &)M" $- "K"' $8&178 $8" *" &'6$*1 $3&' "**&* 4##"4*6 $& )" 24*7"*$84' $8" &'" &0 $8" 10$#./, ?E37D NAD O1*3'7 $8" 3$"*4$3K" #*& "%1*"- $8" &6$ 01' $3&'

Θopt = Θ φopts 36 3'3$3422< 84*4 $"*3C"% )< 4 ;&'&$&'3 422< %" *"463'7 )"84K3&*D I8"'-

Θopt⌋IMSA

)" &;"6 6$4$3&'4*< 1'$32 $8" 6$&##3'7 *3$"*3&' %":'"% )< *"24$3&'683#6 ?,,A4'% ?,+A 36 64$36:"% ?E37D N B s = 1AD I8"'- 40$"* $8" 1#%4$" &0 $8" :"2% %36$*3)1$3&'3'%1 3'7 $8" "**&* 6#3=" 98"' s = sopt = 2 4'% ks = 1- Θopt⌋IMSA

6"$$2"6 $& 4 K421" &0

6.28×10−4 983 8 36 &0 $8" &*%"* &0 $8" 10$#B/, "**&* ?Θopt⌋Bare= 1.42×10−4AD P1 8 462378$ %3Q"*"' " )"$9""' Θopt⌋IMSA

4'% Θopt⌋Bare

%"#"'%6 &' $8" %3Q"*"'$ %36 *"$3C4$3&'G3D"D- $8" )4636 01' $3&'6 B (rn(i=2)

)- n(i) = 1, ..., NIMSA 4*" '&$ $8" 64;" 46 $8&6" &010$#./, - )1$ 3$ %&"6 '&$ 4Q" $ $8" *" &'6$*1 $3&' 3' $"*;6 &0 )&$8 2& 423C4$3&' 4'%4*"4 "6$3;4$3&'- 63' " δ⌋IMSA−LS < δ⌋Bare−LS 4'% ∆⌋IMSA−LS < ∆⌋Bare−LS ?I4)D ,ADE37D N 426& 68&96 $84$ $8" ;12$3B6$"# ;12$3B*"6&21$3&' 6$*4$"7< 36 84*4 $"*3C"% )<4 2&9"* &;#1$4$3&'42 )1*%"' )" 416" &0 $8" 6;422"* '1;)"* &0 3$"*4$3&'6 0&* *"4 83'7$8" &'K"*7"' " ?E37D N B ktot⌋IMSA = 125 K6D ktot⌋Bare = 177- )"3'7 ktot $8" $&$42'1;)"* &0 3$"*4$3&'6 %":'"% 46 ktot =∑sopt

s=1 kopts 0&* $8" +,-./, A- 4'% "6#" 3422< $& $8"

! " $%&$' () * &(+$, - . -!!/ 0(, %1+23 4&3() 3) !"#$%# &$'()#*%,$'1 $' )15+$, (0 6(4&3)78%(3)& (%$,4&3()9: "9 4 54&&$, (0 04 &. 93) $ &;$ (5%2$<3&=(0 &;$ +, 8+49$' 427(,3&;59 39 (0 &;$ (,'$, (0 O (2× η3). η = NIMSA, NBare >3:$:. &;$9(21&3() (0 &?( '3,$ & %,(+2$59 39 )$ $994,= 0(, (5%1&3)7 4) $9&354&$ (0 &;$ 9 4&&$,$'@$2' 4)' 0(, 1%'4&3)7 &;$ A$2( 3&= A$ &(,B. &;$ (5%1&4&3()42 (9& (0 &;$ -,./+, 4&$4 ; 3&$,4&3() 39 &?( (,'$,9 3) 547)3&1'$ 95422$, &;4) &;4& (0 &;$ 01$#/+, :234353 6'7%8 91:1 "9 0(, &;$ 9&4+323&= (0 &;$ %,(%(9$' 4%%,(4 ;. C371,$ D 9;(?9&;$ ,$ ()9&,1 &3()9 ?3&; &;$ -,./+, EC379: D>1B> B>#B (5%4,$' &( &;(9$ (0 &;$01$#/+, EC379: D>(B><B>= B ?3&; '3G$,$)& 2$A$29 (0 4''3&3A$ )(39$ () &;$ 9 4&&$,$''4&4 ESNR = 20 dB >:'>BH SNR = 10 dB >*7<<)#BH SNR = 5 dB >('::'*B: "9$<%$ &$'. ?;$) &;$ SNR '$ ,$49$9. &;$ %$,0(,54) $9 ?(,9$): I(?$A$,. 49 (1&23)$'+= &;$ +$;4A3(, (0 &;$ $,,(, @71,$9 3) J4+: -. +21,,$' '4&4 4)'K(, )(39= ()'3&3()94G$ & 5(,$ $A3'$)&2= &;$ 01$# 35%2$5$)&4&3() &;4) &;$ 512&38,$9(21&3() 4%%,(4 ;: C(, (5%2$&$)$99. &;$ +$;4A3(, (0 Θopt⌋IMSA

A$,919 &;$ 3&$,4&3() 3)'$< 39 ,$%(,&$' 3) C37:L 0(, '3G$,$)& 2$A$29 (0 SNR: "9 3& 4) +$ )(&3 $'. &;$ A421$ (0 &;$ $,,(, 4& &;$ $)' (0&;$ 3&$,4&3A$ %,( $'1,$ '$ ,$49$9 49 &;$ SNR 3) ,$49$9:M) &;$ 9$ ()' $<%$,35$)&. &;$ 945$ 3, 124, 9 4&&$,$,. +1& $)&$,$' 4& 4 '3G$,$)&%(93&3() ?3&;3) 4 24,7$, 3)A$9&374&3() 9N14,$ (0 93'$ LD = 2λ >ρO = 2λB. ;49 +$$),$ ()9&,1 &$': " (,'3)7 &( EO. M = 20H v = 1, ..., V ,$ $3A$,9 4)' V = 20 A3$?9 4,$ ()93'$,$' 4)' DI 39 '39 ,$&3P$' 3) NIMSA = 13× 13 %3<$29:C371,$ Q>1B 9;(?9 &;$ ,$ ()9&,1 &3() (+&43)$' 4& &;$ ()A$,7$) $ >sopt = 3B += -,./+, ?;$) SNR = 5 dB: J;$ ,$912& ,$4 ;$' += &;$ 01$#/+, >NBARE = 47× 47B39 ,$%(,&$' 3) C37: Q>(B 49 ?$22: "9 3& 4) +$ )(&3 $'. &;$ 512&38,$9(21&3() 3)A$,93() 39 ;4,4 &$,3P$' += 4 +$&&$, $9&354&3() (0 &;$ (+R$ & $)&$, 4)' 9;4%$ 49 ()@,5$' += &;$A421$9 (0 δ 4)' ∆ >δ⌋IMSA−LS = 0.59 A9: δ⌋Bare−LS = 2.72 4)' ∆⌋IMSA−LS = 0.48 A9:

∆⌋Bare−LS = 0.64B: "9 0(, &;$ (5%1&4&3()42 2(4'. &;$ 945$ () 2193()9 0,(5 %,$A3(19$<%$,35$)&9 ;(2' &,1$:2353 ,8!:?#:7 91:1 / @# :1!AB)1$ , 1::#$#$J;$ 9$ ()' &$9& 49$ '$429 ?3&; 4 5(,$ (5%2$< 9 4&&$,3)7 ()@71,4&3(): " ,$ &4)7124,(G8 $)&$,$' 9 4&&$,$, >L = 0.27λ 4)' W = 0.13λB ;4,4 &$,3P$' += 4 '3$2$ &,3 %$,53&&3A3&= ǫC = 1.8 39 2( 4&$' ?3&;3) 4) 3)A$9&374&3() '(543) (0 LD = 3λ 49 3)'3 4&$'+= &;$ ,$' '49;$' 23)$ 3) C37: /: M) 91 ; 4 49$. &;$ 35473)7 9$&1% 39 54'$ 1% (0 V = 309(1, $9 4)' M = 30 5$491,$5$)& %(3)&9 0(, $4 ; A3$? v EO: DI 39 %4,&3&3()$' 3)&(

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% ,,NIMSA = 19× 19 51)6%&743'5 89:32" NBare = 33× 33; 4'% ∆t1 35 5"$ $& 0.06<+,-,., /0)12031'! '4 35# 63'771!8 9$13#$10 ="0&*" %35 1553'> $:" *" &'5$*1 $3&' 4#4)323$3"5- 2"$ 15 5:&9 4 *"512$ &' "*'"% 93$: $:" )":4?3&* &0 $:" #*&#&5"% 4##*&4 :9:"' ?4*@3'> $:" 15"*6%"A'"% $:*"5:&2%5 8γΘ- γτ - γxc - γyc - γ

L; &0 $:" 5$&##3'> *3$"*34<B3>1*" / %35#24@5 $:" *" &'5$*1 $3&'5 4 :3"?"% )@ 153'> $:" 5"$5 &0 #4*47"$"*5 >3?"' 3'C4)< D EΓ1 6 B3>< /80;F Γ2 6 B3>< /8(;F Γ3 6 B3>< /8 ;F Γ4 6 B3>< /82; 9:32" $:" )":4?3&*5&0 $:" &5$ 01' $3&' 4*" %"#3 $"% 3' B3>< H< 5 3$ 4' )" '&$3 "%- $:" $&$42 '17)"* &03$"*4$3&'5 ktot 3' *"45"5 45 $:" ?421"5 &0 $:" $:*"5:&2%5 γΘ 4'% γτ %" *"45"< I&9"?"*-3' 5#3$" &0 4 24*>"* ktot- 153'> 2&9"* $:*"5:&2% ?421"5 %&"5 '&$ #*&?3%" )"$$"* *"512$5- 455:&9' )@ $:" &7#4*35&' )"$9""' 5"$$3'>5 Γ2 4'% Γ4 EB3>5< /8(;682;- 4'% B3>< H< C:"5"$5 &0 #4*47"$"*5 :4*4 $"*3J"% )@ γΘ = 0.2 4'% γτ = 0.02 #*&?3%" 4 >&&% $*4%"6&K)"$9""' $:" 4*353'> &7#1$4$3&'42 )1*%"' 4'% $:" L1423$@ &0 $:" *" &'5$*1 $3&'5< 504* 45 $:" 5$&##3'> *3$"*3&' &0 $:" 712$36*"5&21$3&' #*& "%1*" 35 &' "*'"%- B3>1*" H425& 5:&95 $9& %3K"*"'$ )":4?3&*5 &0 $:" &5$ 01' $3&' 9:"' 153'> Γ2 4'% Γ3 82"$$3'>

γΘ = 0.2 4'% γτ = 0.02;< M' #4*$3 124*- $:" #*&#&5"% 4##*&4 : 5$&#5 4$ sopt = 3-3'5$"4% &0 sopt = 4- 9:"' 3' *"453'> )@ 4 %">*"" &0 74>'3$1%" $:" ?421"5 &0 γxc - γyc - 4'%

γL

< 2$:&1>: 93$: 4 :"4?3"* &7#1$4$3&'42 )1*%"'- $:" :&3 " γxc = γyc = 0.01 4'%

γL= 0.05 *"512$5 7&*" "K" $3?" E5"" B3>< /8(; ?5< B3>< /8 ;<+,-,-, ;'1%< =030 B3>1*"5 ,.6,+ 4'% C4)2" N 5:&9 $:" *"512$5 0*&7 $:" &7#4*4$3?"5$1%@ 4**3"% &1$ 3' &**"5#&'%"' " 93$: %3K"*"'$ ?421"5 &0 53>'426$&6'&35" *4$3& ESNR =

20 dB 6 B3>< ,.80; ?5< B3>< ,.8(;F SNR = 10 dB 6 B3>< ,.8 ; ?5< B3>< ,.82;F SNR =

5 dB 6 B3>< ,.8#; ?5< B3>< ,.84 ;< C:"@ 01*$:"* &'A*7 $:" *"234)323$@ 4'% "O 3"' @&0 $:" 712$36*"5&21$3&' 5$*4$">@ 3' $"*75 &0 L1423$4$3?" *" &'5$*1 $3&' "**&*5 8B3>< ,,;-"5#" 3422@ 9:"' $:" '&35" 2"?"2 >*&95< M' #4*$3 124*- $:" >0$# 37#2"7"'$4$3&' %&"5 '&$@3"2% "3$:"* $:" #&53$3&' &* $:" 5:4#" &0 $:" *" $4'>124* 5 4$$"*"* 9:"' SNR = 5 dB-9:"*"45 $:" ?6@AB6 #*&#"*2@ *"$*3"?"5 )&$: $:" )4*@ "'$"* 4'% $:" &'$&1* &0 $:"$4*>"$< 5 0&* $:" &7#1$4$3&'42 &5$- 3$ 5:&12% )" '&$3 "% $:4$ 42$:&1>: $:" ?6@AB6*"L13*"5 4 >*"4$"* '17)"* &0 3$"*4$3&'5 0&* *"4 :3'> $:" &'?"*>"' " 8B3>< ,+- C4)< N;-$:" $&$42 47&1'$ &0 &7#2"P Q&4$3'>6#&3'$ &#"*4$3&'5- fpos = O (2× η3)× ktot- 151422@*"512$5 57422"* 8C4)< N;<

! " $%&$' () * &(+$, ! - !../ 0(, %1+23 4&3() 3) !"#$%# &$'()#*%+,+, -.*#$/ 1) 2131 4 5'))'6 78)/!9#$56$ &63,' &$7& 47$ 37 () $,)$' 83&6 &6$ 3)9$,73() (0 &6$ '4&4 7 4&&$,$' +: 4 63;6$,%$,<3&&393&: =ǫC = 2.5> (?@ $)&$,$' :23)',3 42 ,3);- 2$&&3); LD = 3λA 56$ $B&$,)42,4'317 (0 &6$ ,3); 37 ρext =23λ- 4)' &6$ 3)&$,)42 ()$ 37 ρint =

λ3

A C: 4771<3); &6$74<$ 4,,4);$<$)& (0 $<3&&$,7 4)' ,$ $39$,7 47 3) D$ &3() EA!- &6$ 3)9$7&3;4&3() '(<43)37 '37 ,$&3F$' 83&6 NIMSA = 19× 19 4)' NBare = 35× 35 7G14,$ $227 0(, &6$ :;<4=;4)' &6$ >1$#4=; - ,$7%$ &39$2:A H(,$(9$,- ∆t1 37 3)3&3423F$' &( 0.003A"7 3& 4) +$ (+7$,9$' 0,(< I3;A E- 86$,$ &6$ %,(J2$7 86$) SNR = 20 dB KI3;7A E=1>=(> 4)' SNR = 10 dB KI3;7A E= >=9> ,$ ()7&,1 &$' +: <$4)7 (0 &6$ :;<4=;KI3;7A E=1>= > 4)' &6$ >1$#4=; KI3;7A E=(>=9> 4,$ 76(8)- &6$ 3)&$;,4&$' 7&,4&$;:171422: (9$, (<$7 &6$ 7&4)'4,' ()$ +(&6 3) 2( 4&3); &6$ (+M$ & 4)' 3) $7&3<4&3); &6$764%$A N) %4,&3 124,- 86$) SNR = 20 dB- &6$ '37&,3+1&3() 3) I3;A E=1> 37 4 043&6012$7&3<4&$ (0 &6$ 7 4&&$,$, 1)'$, &$7& =δ⌋IMSA−LS = 1.25 4)' ∆⌋IMSA−LS = 3.13>A *)&6$ ()&,4,:- &6$ ,$ ()7&,1 &3() 83&6 &6$ >1$#4=; 37 9$,: %((, =δ⌋Bare−LS = 65.2 4)'

∆⌋Bare−LS = 34.39>A O$,&43)2:- 4 7<422$, SNR 9421$ 3<%43,7 &6$ 3)9$,73() 47 76(8)3) I3;A E= > K (<%4,$' &( I3;A E=1>A P(8$9$,- 3) &637 47$- &6$ :;<4=; 37 4+2$ &(%,(%$,2: 2( 4&$ &6$ (+M$ & =δ⌋IMSA−LS = 1.7 97A δ⌋Bare−LS = 65.9> ;393); ,(1;6 +1&17$012 3)'3 4&3()7 4+(1& 3&7 764%$ =∆⌋IMSA−LS = 7.6 97A ∆⌋Bare−LS = 34.55>A+,?, ;8!3@#3/ 2131 4 :.)3/A)# ; 133#$#$%56$ 247& 7:)&6$&3 &$7& 47$ 37 43<$' 4& 32217&,4&3); &6$ +$6493(, (0 &6$ :;<4=; 86$)'$423); 83&6 P = 3 7 4&&$,$,7 =ǫC = 2.0> '37&4) $' 0,(< ()$ 4)(&6$,A 56$ &$7& ;$(<$&,:37 64,4 &$,3F$' +: 4) $223%&3 (?@ $)&$,$' :23)'$,- 4 3, 124, (?@ $)&$,$' 7 4&&$,$,- 4)'4 7G14,$ (?@ $)&$,$' (+M$ & 2( 4&$' 3) 4 7G14,$ 3)9$7&3;4&3() '(<43) 64,4 &$,3F$' +:

LD = 3λA C: 4'(%&3); &6$ 74<$ 4,,4);$<$)& (0 $<3&&$,7 4)' ,$ $39$,7 47 3) D$ &3()EAE- &6$ 3)9$7&3;4&3() '(<43) 37 '37 ,$&3F$' 83&6 NIMSA = 23× 23 4)' NBare = 31× 317G14,$ $227 0(, &6$ :;<4=; 4)' &6$ >1$#4=; - ,$7%$ &39$2:A H(,$(9$,- ∆t1 37 7$& &(

0.03AI3;1,$7 Q 4)' R 76(8 &6$ ,$712&7 0,(< &6$ (<%4,4&39$ 7&1': 4,,3$' (1&3) (,,$7%()'$) $ 83&6 '3?$,$)& 9421$7 (0 73;)42@&(@)(37$ ,4&3(A "7 76(8) +: &6$,$ ()7&,1 &3()7 =I3;A Q> 4)' 47 $B%$ &$'- &6$ <12&3@,$7(21&3() 4%%,(4 6 %,(93'$7 <(,$4 1,4&$ ,$712&7 4)' 4%%$4,7 &( +$ <(,$ ,$234+2$ &64) &6$ >1$#4=; $7%$ 3422: 83&6 2(8

SNRA 5637 () 2173() 37 01,&6$, ()J,<$' +: &6$ +$6493(, (0 &6$ ,$ ()7&,1 &3() $,,(,7

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% ,567389 ,:;- 0&* <=3 = $=" +,-./, 4 =3">"? 4 2&<"* 2& 423@4$3&' "**&* 4? <"22 4? 4 2&<"*4*"4 "**&* $=4' $=&?" &0 01$#./,2 "?#" 3422A 0&* SNR = 5 dB9 (' $=" &$="* =4'%-)&$= 428&*3$=B? #*&>3%" 8&&% "?$3B4$"? &0 $=" ? 4$$"*"* 1'%"* $"?$ <="' 3'>"*$3'8 %4$44C" $"% )A 2&< '&3?" DSNR = 20 d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ρ = 15BB 3' %34B"$"*- =4*4 $"*3@"% )A 4 '&B3'42>421" &0 $=" &)R" $ 01' $3&' "K142 $& τ(r) = 2.0 ± 0.3- 4'% 2& 4$"% 4$ xc = 0.0-

yc = −30BB <3$=3' 4' 3'>"?$384$3&' %&B43' 4??1B"% 3' $=" 0&22&<3'8 &0 ?K14*"8"&B"$*A 4'% "J$"'?3&' 20× 20 B29SA ?"$$3'8 ǫC = 3.0- $=" *" &'?$*1 $3&'? 4 =3">"% 4*" ?=&<' 3' 7389 ,T 6)#B6 ')<*!; &B#4*"% $& $=&?" 0*&B $=" ?$4'%4*% /, 6$;EF6 ')<*!; 4$ F = 4 %3C"*"'$ &#"*4$3&'0*"K1"' 3"?9 N=4$">"* $=" 0*"K1"' A- $=" 1'L'&<' ? 4$$"*"* 3? 4 1*4$"2A 2& 423@"% 4'%)&$= 428&*3$=B? A3"2%- 4$ &'>"*8"' "- ?$*1 $1*"? $=4$ & 1#A 4 24*8" ?1)?"$ &0 $=" $*1"&)R" $9 U1 = 4 ?3B324*3$A &0 #"*0&*B4' "?- 1?1422A >"*3Q"% 3' ?A'$="$3 "J#"*3B"'$? <="'$=" >421" &0 SNR 3? 8*"4$"* $=4' 20 d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

! " $%&$' () * &(+$, - . -//0 1(, %2+34 5&4() 4) !"#$%# &$'()#*%• 4))(65&46$ 723&483$6$3 ,$%,$9$)&5&4() (1 &:$ %,(+3$7 2);)(<)9 4) &:$ 9:5%$8'$1(,75&4()8+59$' ,$ ()9&,2 &4() &$ :)4=2$>• $?$ &46$ $@%3(4&5&4() (1 &:$ 9 5&&$,4)A '5&5 &:,(2A: &:$ 4&$,5&46$ 723&489&$%9&,5&$AB>

• 3474&5&4() (1 &:$ ,49; (1 +$4)A &,5%%$' 4) 1539$ 9(32&4()9 &:5);9 &( &:$ ,$'2 $' ,5&4(+$&<$$) '5&5 5)' 2);)(<)9>• 29$123 $@%3(4&5&4() (1 &:$ +,-$.'$. 4)1(,75&4() C4D$D. (+E$ & :(7(A$)$4&BF 5+(2& &:$9 $)5,4( 2)'$, &$9&>• $):5) $' 9%5&453 ,$9(32&4() 3474&$' &( &:$ ,$A4() (1 4)&$,$9&DG,(7 &:$ 6534'5&4() () $,)$' <4&: '4?$,$)& 9 $)5,4(9 5)' +(&: 9B)&:$&4 5)'$@%$,47$)&53 '5&5. &:$ 1(33(<4)A () 3294()9 5) +$ ',5<)H• &:$ /01,20 292533B %,(6$' 7(,$ $?$ &46$ &:5) &:$ 94)A3$8,$9(32&4() 47%3$7$)&58&4(). $9%$ 4533B <:$) '$534)A <4&: (,,2%&$' '5&5 9 5&&$,$' 1,(7 947%3$ 59 <$33 59 (7%3$@ A$(7$&,4$9 :5,5 &$,4I$' +B ()$ (, 9$6$,53 (+E$ &9>

• &:$ 4)&$A,5&$' 9&,5&$AB 5%%$5,$' 3$99 (7%2&5&4()533B8$@%$)946$ &:5) &:$ 9&5)'5,'5%%,(5 : 4) ,$5 :4)A 5 ,$ ()9&,2 &4() <4&: &:$ 957$ 3$6$3 (1 9%5&453 ,$9(32&4()<4&:4) &:$ 92%%(,& (1 &:$ (+E$ &D

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"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% ,5a( )

b( )

c( )

Actual Scatterer

Υ1

(xcs=1, y

cs=1)

rns=1

rns=1

Rs=2

φkopts=1

φks=1

s = 1, k = 1

s = 1, k = kopt

s = 2, k = 1

DI

DI

DI

Ls=1

rns=2

φks=2

!"#$% &' 6*4#73 42 *"#*"8"'$4$3&' &0 $7" +,-./, 9&&:3'; #*& "%1*"< =0> ?3*8$ 8$"#=k = 1>@ $7" 3'A"8$3;4$3&' %&:43' 38 %38 *"$39"% 3' N 81)B%&:43'8 4'% 4 &4*8" 8&21$3&'38 2&&C"% 0&*< =(> ?3*8$ 8$"# =k = kopt>@ $7" *";3&' &0 3'$"*"8$ $74$ &'$43'8 $7" D*8$"8$3:4$" &0 $7" &)E" $ 38 %"D'"%< = > F" &'% 8$"# =k = 1>@ 4' "'74' "% *"8&21$3&' 2"A"238 18"% &'2G 3'83%" $7" *";3&' &0 3'$"*"8$<

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Field Distribution Updating

Determine

Cost Function Evaluation

Level Set Update

Problem Unknown Representation

FALSE

TRUE

TRUE

Stop

Initialization

FALSEIMSA

Stopping Criterion

Stopping CriteriaLS

Compute Evki

(rni

)

rni∈ DI

xcs,y

cs,Ls

Determine ξvks

(rmv

)and compute Θ φks

rmv∈ DO

Compute φksfrom φks−1

τks(r) =

∑si=1

∑NIMSA

ni=1 τkiB(rni

)

rni∈ DI

s-th resolution level

ks = ks + 1

Υsopt

Υs = Υ1

ks = 0

s = s+ 1

γxc,γyc,γL

γτ , γΘ, γth, Kmax

!"#$% &' 52( 6 '347,48 '$9 ,3%&3() (0 &:$ +,-./, ;((83)7 %,( $'1,$<

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% ,5x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(2)

1 τ (x, y) 0 1 τ (x, y) 06+7 6(7

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 0 1 τ (x, y) 06 7 6-7

!"#$% & 819"*3 42 :4$4; <3* 124* =23'%"* 6ǫC = 1.8- LD = λ- .'/%#)#%% 0+%#7;>" &'?$*1 $3&'? @3$A 123452 4$ 6+7 s = 1 4'% 6(7 s = sopt = 2- 6 7 6+$#452 ;(#$3942 3'B"*?3&' 6-7;

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10-4

10-3

10-2

10-1

100

0 50 100 150 200

Θ

Iteration, k

s=1 s=2

IMSA (Θopt)

IMSA (Θ)

BARE

!"#$% &' 516$,3 42 74&48 93, 124, :23)'$, ;ǫC = 1.8. LD = λ. +',%#)#%% -.%#<8=$>4?3(, (0 &>$ (@& 01) &3()8

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% +,R

(2)

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 0 1 τ (x, y) 05+6 5(6(3)

R

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

(2)R

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 0 1 τ (x, y) 05 6 5-6

(3)R

R(2)

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 0 1 τ (x, y) 05#6 5. 6

!"#$% &' 73* 124* 823'%"* 5ǫC = 1.8- LD = λ- /'0%1 2+%#69 :" &';$*1 $3&';<3$= 345674 52"0$ &21>'6 4'% 8+$#674 5*3?=$ &21>'6 0&* %3@"*"'$ A421"; &0 SNRBSNR = 20 dB 5$&#6- SNR = 10 dB 5>3%%2"6- SNR = 5 dB 5)&$$&>69

! #$%#& '( ) %'*#+ ,- ../ 0'+ $1*23 4%3'( 3( !"#$%# &$'()#*%

10-3

10-2

10-1

100

101

0 20 40 60 80 100

Θo

pt

Iteration, k

SNR = 20 dBSNR = 10 dB

SNR = 5 dB

s = 1 s = 2

s = 1 s = 2

s = 2s = 1

!"#$% &' 516#+3 42 74%48 93+ 124+ :23(&#+ ;ǫC = 1.8- LD = λ<8 =#>4?3'+ '0 %># '@%01( %3'( ?#+@1@ %># ('3@# 2#?#28

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% +5

(2)R

(3)R

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 06+7

x

y λ

λ−0.5 0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

1 τ (x, y) 06(7

!"#$% &' 819"*3 42 :4$4; <3* 124* =23'%"* 6ǫC = 1.8- LD = 2λ- SNR = 5 dB7;>" &'?$*1 $3&'? @3$A 6+7 ,-./0- 4'% 6(7 1+$#/0- ;

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y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 06+7 6(7

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 06 7 6-7

!"#$% &' 829$,4 53 :5&5; <$ &5)=235, >34)'$, 6ǫC = 1.8. LD = 3λ. .'/%#)#%% 0+%#7;<$ ()?&,2 &4()? @4&A 123452 1(, &A$ '4B$,$)& ?$&&4)=? (1 C5+; D E6+7 Γ1. 6(7 Γ2. 6 7

Γ3. 6-7 Γ4;

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% +5

10-4

10-3

10-2

10-1

100

101

0 200 400 600 800 1000 1200 1400

Θopt

Iteration, k

Γ3Γ2 Γ4

Γ1

Γ1Γ2, Γ3

Γ4

!"#$% &' 617"*3 42 84$49 :" $4';124* <23'%"* =ǫC = 1.8- LD = 3λ- +',%#)#%% -.%#>9?"@4A3&* &0 $@" &B$ 01' $3&' &0 /01230 0&* $@" %3C"*"'$ B"$$3';B &0 D4)9 E9

! " $%&$' () * &(+$, -. //0 1(, %2+34 5&4() 4) !"#$%# &$'()#*%x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 06+7 6(7

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 06 7 6-7

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 06#7 6. 7 !"#$% &'( 829$,4 53 :5&5; <$ &5)=235, >34)'$, 6ǫC = 1.8. LD = 3λ. /'0%1 2+%#7;<$ ()?&,2 &4()? @4&A 345674 63$1& (329)7 5)' 8+$#674 6,4=A& (329)7 1(, '4B$,$)&C532$? (1 SNR DSNR = 20 dB 6&(%7. SNR = 10 dB 694''3$7. SNR = 5 dB 6+(&&(97;

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% +5

100

101

102

103

5 10 15 20

δ

SNR [dB]

IMSABARE

6+72.5

2.0

1.5

1.0

0.5

0.0 5 10 15 20

∆

SNR [dB]

IMSABARE

6(7

!"#$% &&' 819"*3 42 :4$4; <" $4'=124* >23'%"* 6ǫC = 1.8- LD = 3λ- ,'-%. /+%#7;?421"@ &0 $A" "**&* B=1*"@ C"*@1@ SNR;

! " $%&$' () * &(+$, -. //! 0(, %1+23 4&3() 3) !"#$%# &$'()#*%10

-3

10-2

10-1

100

101

0 200 400 600 800 1000 1200 1400

Θo

pt

Iteration, k

IMSABARE

5+610

-2

10-1

100

101

0 100 200 300 400 500 600

Θo

pt

Iteration, k

IMSABARE

5(6

10-1

100

101

0 100 200 300 400 500 600

Θo

pt

Iteration, k

IMSABARE

5 6 !"#$% &'( 718$,3 42 94&4: ;$ &4)<124, =23)'$, 5ǫC = 1.8. LD = 3λ. -'.%/ 0+%#6:>$?4@3(, (0 &?$ (A& 01) &3() @$,A1A &?$ 3&$,4&3() 3)'$B C?$) 5+6 SNR = 20 dB. 5(6

SNR = 10 dB. 4)' 5 6 SNR = 5 dB:

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% +5x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

2 τ (x, y) 0 2 τ (x, y) 06+7 6(7

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

2 τ (x, y) 0 2 τ (x, y) 06 7 6-7

!"#$% &'( 819"*3 42 :4$4; <&22&= >23'%"* 6ǫC = 2.5- LD = 3λ- .'/%0 1+%#7;?" &'@$*1 $3&'@ =3$A 234563 62"0$ &219'7 4'% 7+$#563 6*3BA$ &219'7 0&* %3C"*"'$D421"@ &0 SNR ESNR = 20 dB 6$&#7- SNR = 10 dB 6)&$$&97;

! " $%&$' () * &(+$, -./ -!!0 1(, %2+34 5&4() 4) !"#$%# &$'()#*%x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(3)

R(3)

R(3)

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 06+7 6(7

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(3)

R(3)

R(3)

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 06 7 6-7

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(3)

R(3)

R(3)

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 06#7 6. 7 !"#$% &'( 829$,4 53 :5&5; <23&4%3$ = 5&&$,$,= 6ǫC = 2.0/ LD = 3λ/ /'0%1 2+%#7;>$ ()=&,2 &4()= ?4&@ 345674 63$1& (329)7 5)' 8+$#674 6,4A@& (329)7 1(, '4B$,$)&C532$= (1 SNR DSNR = 20 dB 6&(%7/ SNR = 10 dB 694''3$7/ SNR = 5 dB 6+(&&(97;

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% 5,

2.0

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4.0

5 10 15 20

δ

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6+7

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2.0

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4.0

5.0

6.0

5 10 15 20

∆

SNR [dB]

IMSABARE

6(7

!"#$% &'( 812$3#2" 9 4$$"*"*9 6ǫC = 2.0- LD = 3λ- ,'-%. /+%#7: ;421"9 &0 $<" "**&*=>1*"9 ?"*919 SNR:

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y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

3 τ (x, y) 0 3 τ (x, y) 06+7 6(7

x

y λ

λ−0.5 0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

x

y λ

λ−0.5 0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

3 τ (x, y) 0 3 τ (x, y) 06 7 6-7

!"#$% &'( )* 89%$,4:$)&53 ;5&5 6;5&5<$& =>5,<$433$? @ A7C D4, 235, E34)'$,6=-.#)/01-# 234#56?7C F$ ()<&,2 &4()< G4&H 0789:7 63$1& (32:)7 5)' ;+$#9:7 6,4IH& (32:)7 5& '4J$,$)& 1,$K2$) 4$< f @f = 1GHz 6+76(7L f = 2GHz 6 76-7C

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% 55x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

3 τ (x, y) 0 3 τ (x, y) 06#7 6+ 7

x

y λ

λ−1.0 1.0

1.0

−1.0

2.0

−2.0

−2.0

2.0

x

y λ

λ−1.0 1.0

1.0

−1.0

2.0

−2.0

−2.0

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!" $%&$'()* &(+$,-./-0012(,%3+45 6&5()5) !"#$%#&$'()#*%

10-2

10-1

100

101

102

0 200 400 600 800 1000

Θo

pt

Iteration, k

IMSABARE

10-3

10-2

10-1

100

101

102

0 200 400 600 800 1000

Θo

pt

Iteration, k

IMSABARE

7+8 7(8

10-3

10-2

10-1

100

101

102

0 200 400 600 800 1000

Θo

pt

Iteration, k

IMSABARE

10-3

10-2

10-1

100

101

102

0 200 400 600 800 1000

Θo

pt

Iteration, k

IMSABARE

7 8 7-8

!"#$%&'(9:%$,5;$)&64<6&67<6&6=$&>?6,=$544$@A !8CD5, 346, E45)'$,

7>-.#)/01-# 234#56@8CF$G6H5(,(2&G$ (=&23) &5()H$,=3=&G$)3;+$,(25&$,6&5()=

IG$)7+8

f=1G

Hz/7(8

f=2G

Hz/7 8

f=3G

HZ/6)'7-8

f=4G

HzC

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% 56IMSA− LS Bare− LS

s = 1 s = 2

δ 6.58× 10−6 2.19× 10−6 5.21× 10−1

∆ 2.36 0.48 0.64

!"#$ %& 718"*3 42 94$4: ;3* 124* <23'%"* =ǫC = 1.8- +',%#)#%% -.%#>: ?**&* @A1*"B:

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SN

R=20

dB

SN

R=10

dB

SN

R=5dB

IM

SA−

LS

Bare−

LS

IM

SA−

LS

Bare−

LS

IM

SA−

LS

Bare−

LS

δ5.91×10−

12.72

2.28

2.45

6.78×10−

11.63

∆0.98

1.28

1.07

1.80

1.50

2.07

!"#$ %& 738$,5 64 96&6: ;5, 346, <45)'$, =ǫC = 1.8/ +',%- ./%#>: ?643$@ (2 &A$$,,(, 5)'$B$@ 2(, '5C$,$)& D643$@ (2 SNR:

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% 56

!" #$ %&'&(!"!') γΘ γτ γxc, γyc γL

Γ1 0.5 0.05 0.01 0.05

Γ2 0.2 0.02 0.01 0.05

Γ3 0.2 0.02 0.1 0.5

Γ4 0.02 0.002 0.01 0.05

!"#$ %& 718"*3 42 94$4: ;" $4'<124* =23'%"* >ǫC = 1.8- LD = 3λ- +',%#)#%% -.%#?:93@"*"'$ A"$$3'<A 0&* $B" #4*48"$"*A &0 $B" A$&##3'< *3$"*34:

! " $%&$' () * &(+$, -./ -00! 1(, %2+34 5&4() 4) !"#$%# &$'()#*%

SN

R=20

dB

SN

R=10

dB

SN

R=5dB

IM

SA−

LS

Bare−

LS

IM

SA−

LS

Bare−

LS

IM

SA−

LS

Bare−

LS

kto

t1089

41393

53410

28

N361

1089

361

1089

361

1089

f pos

1.02×10

11

1.02×10

11

3.70×10

10

1.37×10

11

3.86×10

10

7.23×10

10

!"#$ %& 627$,4 53 85&59 :$ &5);235, <34)'$, =ǫC = 1.8/ LD = 3λ/ +',%- ./%#>9?(7%2&5&4()53 4)'$@$A 1(, '4B$,$)& C532$A (1 SNR9

"#$"% &' ( $&)"* +,- +../ 0&* #1)23 4$3&' 3' !"#$%# &$'()#*% 56f = 1GHz f = 2GHz

IMSA− LS Bare− LS IMSA− LS Bare− LS

ktot 506 69 532 200

fpos 4.88× 109 1.22× 1011 5.14× 109 3.55× 1011

f = 3GHz f = 4GHz

IMSA− LS Bare− LS IMSA− LS Bare− LS

ktot 678 198 621 200

fpos 6.55× 109 3.51× 1011 5.99× 109 3.55× 1011

!"#$ %& 78#"*39"'$42 :4$4 ;:4$4<"$ =>4*<"322"? @5ACD E3* 124* F23'%"*;=+,#)-./+# 123#45?CD E&9#1$4$3&'42 3'%"8"<D

Contents1 Introdu tion 12 The Inverse S attering Problem 72.1 Mathemati al Formulation . . . . . . . . . . . . 82.1.1 Field S attered byInhomogeneous Obje ts . . . . . . . . . 82.1.2 Exploitation of the a-prioriInformation about the S atterers . . . . 112.2 Numeri al Solution ofthe Inverse S attering Problem . . . . . . . . . 122.3 Drawba ks of the InverseS attering Problem . . . . . . . . . . . . . . . . 143 An Overview on Inverse S attering Te hniques 173.1 Regularized Solutions andApproximations . . . . . . . . . . . . . . . . . . 183.2 An Overview onMinimization Te hniques . . . . . . . . . . . . . 193.2.1 Deterministi Approa hes . . . . . . . . 20xv

CONTENTS3.2.2 Heuristi Minimization Methodologies . 223.3 State-of-the-Art onMulti-Resolution Approa hes . . . . . . . . . . 243.3.1 Wavelet-basedmulti-resolution approa hes . . . . . . . 273.3.2 Adaptive Multis ale Te hniques . . . . . 283.3.3 The Iterative Multi-S aling Approa h . . 283.4 Shape-Optimization Algorithms . . . . . . . . . 313.4.1 Parametri Approa hes . . . . . . . . . . 323.4.2 Level-Set-based Shape Optimization . . 344 The Multi-ResolutionLevel Set Approa h 374.1 Mathemati al Formulation . . . . . . . . . . . . 384.2 Numeri al Validation:Syntheti Data . . . . . . . . . . . . . . . . . . 474.2.1 Initializing with the True Solution . . . . 484.2.2 Initializing with Exa t Data . . . . . . . 534.2.3 Inversion of Data s attered by a Cir ularCylinder . . . . . . . . . . . . . . . . . . 544.2.4 Re tangular S atterer . . . . . . . . . . 674.2.5 Hollow Cylinder . . . . . . . . . . . . . . 794.3 Numeri al Validation by means of Laboratory-Controlled Data . . . . . . . . . . . . . . . . . . 815 The Multi-Region Approa h 875.1 Mathemati al Formulation . . . . . . . . . . . . 885.2 Preliminary Validation . . . . . . . . . . . . . . 101xvi

5.2.1 The IMSMRA-LS when dealingwith Simple Geometries . . . . . . . . . 1015.2.2 Calibration of the Stopping Criterion . . 1075.3 Numeri al Validation: Syntheti Data . . . . . . 1095.3.1 Inversion of Data s attered by Two Di-ele tri Cylinders . . . . . . . . . . . . . 1095.3.2 Experiment with Three Obje ts . . . . . 1235.3.3 Experiments with Complex Shapes . . . 1255.4 Validation by usingLaboratory-Controlled Data . . . . . . . . . . . 1336 Con lusions and Open Problems 137A The Adjoint Problem 159B Algorithms 161B.1 The Edge Dete tion Operator ∂(·) . . . . . . . . 161B.2 How to Countthe Number of Obje ts . . . . . . . . . . . . . . 162

xvii

CONTENTS

xviii

List of Tables4.1 Numeri al Data. Cir ular ylinder (ǫC = 1.8,Noiseless Case). Error gures. . . . . . . . . . 574.2 Numeri al Data. Cir ular ylinder (ǫC = 1.8,Noisy Case). Values of the error indexes fordierent values of SNR. . . . . . . . . . . . . . 674.3 Numeri al Data. Re tangular ylinder (ǫC =

1.8, LD = 3λ, Noiseless Case). Dierent set-tings for the parameters of the stopping riteria. 714.4 Numeri al Data. Re tangular ylinder (ǫC =1.8, LD = 3λ, Noisy Case). Computational in-dexes for dierent values of SNR. . . . . . . . . 734.5 Experimental Data (Dataset Marseille [100).Cir ular ylinder (dielTM_de 8f.exp). Com-putational indexes. . . . . . . . . . . . . . . . . 825.1 Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 5 dB). Computational indexes. . . . . . 119xix

5.2 Numeri al Data. Two re tangular s atterers (ǫC =1.8, LD = 3λ, SNR = 20 dB).Values of the er-ror indexes. . . . . . . . . . . . . . . . . . . . . 121

xx

List of Figures2.1 Geometry of an inverse s attering problem. . . . 92.2 Geometry of an inverse s attering problem whena-priori information about the s atterer is avail-able. . . . . . . . . . . . . . . . . . . . . . . . . 114.1 Graphi al representation of the IMSA-LS zoom-ing pro edure. (a) First step (k = 1): the inves-tigation domain is dis retized in N sub-domainsand a oarse solution is looked for. (b) Firststep (k = kopt): the region of interest that on-tains the rst estimate of the obje t is dened.( ) Se ond step (k = 1): an enhan ed resolutionlevel is used only inside the region of interest. . 394.2 Blo k diagram des ription of the IMSA-LS zoom-ing pro edure. . . . . . . . . . . . . . . . . . . . 414.3 Numeri al Data. Cir ular ylinder (ǫC = 1.8,

LD = λ, Noiseless Case). Re onstru tions wheninitializing IMSA-LS with the true solution [(a)s = 1, (b) s = 2, and ( ) s = sopt = 3 . . . . . . 49xxi

LIST OF FIGURES4.4 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the ostfun tion when initializing IMSA-LS with thetrue solution. . . . . . . . . . . . . . . . . . . . 504.5 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noisy Case). Re onstru tions wheninitializing IMSA-LS with the true solution [(a)SNR = 20 dB, (b) SNR = 10 dB, and ( )SNR = 5 dB . . . . . . . . . . . . . . . . . . . 524.6 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noisy Case). Behavior of the ost fun -tion when initializing IMSA-LS with the truesolution. . . . . . . . . . . . . . . . . . . . . . 544.7 Cir ular ylinder (ǫC = 1.8, LD = λ, NoisyCase). Cir ular ylinder (ǫC = 1.8, LD = λ,Noiseless Case). Re onstru tions when initial-izing IMSA-LS with the exa t data. . . . . . . . 554.8 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Re onstru tions withIMSA-LS at (a) s = 1, (b) s = sopt = 2, ( )Bare-LS . Optimal inversion (d). . . . . . . . . . 564.9 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the ostfun tion. . . . . . . . . . . . . . . . . . . . . . . 584.10 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the levelset φks

at s = 1 [(a) k = 1, (b) k = 20. . . . . . 60xxii

LIST OF FIGURES4.11 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the levelset φks

at s = sopt = 2 [(a) k = 1, (b) k = kopt. . 614.12 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the ve-lo ity Vks

at s = 1 [(a) k = 1, (b) k = 20.. . . . . . . . . . . . . . . . . . . . . . . . . . . 624.13 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the velo -ity Vks

at s = sopt = 2 [(a) k = 1, (b) k = kopt.. . . . . . . . . . . . . . . . . . . . . . . . . . . 634.14 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noisy Case). Re onstru tions withIMSA-LS (a) and Bare-LS (b) for SNR = 20 dB. 644.15 Cir ular ylinder (ǫC = 1.8, LD = λ, NoisyCase). Re onstru tions with IMSA-LS (left ol-umn) and Bare-LS (right olumn) for SNR =10 dB. . . . . . . . . . . . . . . . . . . . . . . . 654.16 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noisy Case). Re onstru tions withIMSA-LS (left olumn) and Bare-LS (right ol-umn) for SNR = 5 dB. . . . . . . . . . . . . . . 664.17 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = 2λ, Noisy Case). Re onstru tions withIMSA-LS (a) and Bare-LS (b) when SNR =5 dB. . . . . . . . . . . . . . . . . . . . . . . . . 68xxiii

LIST OF FIGURES4.18 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = 2λ, SNR = 5 dB). Behavior of the levelset φks

and the velo ity Vks. (a) φks

at k = 2,(b) Vksat k = 2, and ( ) φks

at k = 3. . . . . . 694.19 Numeri al Data. Re tangular ylinder (ǫC =1.8, LD = 3λ, Noiseless Case). Re onstru tionswith IMSA-LS for the dierent settings of Tab.4.3 [(a) Γ1, (b) Γ2, ( ) Γ3, (d) Γ4. . . . . . . . . 704.20 Numeri al Data. Re tangular ylinder (ǫC =1.8, LD = 3λ, Noiseless Case). Behavior of the ost fun tion of IMSA-LS for the dierent set-tings of Tab. 4.3. . . . . . . . . . . . . . . . . . 724.21 Numeri al Data. Re tangular ylinder (ǫC =1.8, LD = 3λ, Noisy Case). Re onstru tionswith IMSA-LS (a) and Bare-LS (b) for SNR =20 dB. ( ) Behavior of the ost fun tion versusthe iteration index. . . . . . . . . . . . . . . . . 754.22 Numeri al Data. Re tangular ylinder (ǫC =1.8, LD = 3λ, Noisy Case). Re onstru tionswith IMSA-LS (a) and Bare-LS (b) for SNR =10 dB. ( ) Behavior of the ost fun tion versusthe iteration index. . . . . . . . . . . . . . . . . 764.23 Numeri al Data. Re tangular ylinder (ǫC =1.8, LD = 3λ, Noisy Case). Re onstru tionswith IMSA-LS (a) and Bare-LS (b) for SNR =5 dB. ( ) Behavior of the ost fun tion versusthe iteration index. . . . . . . . . . . . . . . . . 77xxiv

LIST OF FIGURES4.24 Numeri al Data. Re tangular ylinder (ǫC =1.8, LD = 3λ, Noisy Case). Values of the errorgures versus SNR. . . . . . . . . . . . . . . . . 784.25 Numeri al Data. Hollow ylinder (ǫC = 2.5,LD = 3λ, Noisy Case). Re onstru tions withIMSA-LS (left olumn) and Bare-LS (right ol-umn) for SNR = 20 dB . . . . . . . . . . . . . . 794.26 Numeri al Data. Hollow ylinder (ǫC = 2.5,LD = 3λ, Noisy Case). Re onstru tions withIMSA-LS (left olumn) and Bare-LS (right ol-umn) for SNR = 10 dB . . . . . . . . . . . . . . 804.27 Experimental Data (Dataset Marseille [100).Cir ular ylinder (dielTM_de 8f.exp). Re on-stru tions with IMSA-LS (left olumn) and Bare-LS (right olumn) at dierent frequen ies f [f =1 GHz (a)(b); f = 2 GHz ( )(d). . . . . . . . . 834.28 Experimental Data (Dataset Marseille [100).Cir ular ylinder (dielTM_de 8f.exp). Re on-stru tions with IMSA-LS (left olumn) and Bare-LS (right olumn) at dierent frequen ies f [f =3 GHz (a)(b); f = 4 GHz ( )(d). . . . . . . . . 844.29 Experimental Data (Dataset Marseille [100).Cir ular ylinder (dielTM_de 8f.exp). Behav-ior of the ost fun tion versus the number of it-erations when (a) f = 3 GHz, and (b) f = 4 GHz. 854.30 Experimental Data (Dataset Marseille [100).Cir ular ylinder (dielTM_de 8f.exp). Behav-ior of the ost fun tion versus the number of it-erations when (a) f = 1 GHz, and (b) f = 2 GHz. 86xxv

LIST OF FIGURES5.1 Blo k diagram des ription of the IMSMRA-LSzooming pro edure. . . . . . . . . . . . . . . . . 905.2 Graphi al representation of the morphologi alpro essing: (a) prole re onstru ted at step s(Qs = 1), (b) dete tion of the seeds by means ofthe erosion pro edure, ( ) walking around theedges, and (d) dete tion of obje t's boundariesand of the regions of interest for the resolutionlevel s + 1. . . . . . . . . . . . . . . . . . . . . . 985.3 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = 2λ, Noisy Case). Re onstru tions withIMSMRA-LS at (a) s = 1, (b) s = 2, and ( )s = sopt = 3. . . . . . . . . . . . . . . . . . . . 1025.4 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = 2λ, Noisy Case). Behavior of the level setφks

at the end of the step (a) s = 1, (b) s = 2,and ( ) s = sopt = 3. . . . . . . . . . . . . . . . 1045.5 Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = 2λ, SNR = 5 dB). Behavior of the ostfun tion when using IMSMRA-LS . . . . . . . . 1055.6 Numeri al Data. Re tangular ylinder (ǫC =1.8, LD = 3λ, Noiseless Case). Behavior of the ost fun tion of IMSMRA-LS when varying γL. 1065.7 Numeri al Data. Re tangular ylinder (ǫC =1.8, LD = 3λ, Noiseless Case). Re onstru tionsobtained at the end of the iterative pro edurewith IMSMRA-LS when varying γL [(a) γL =5.0, (b) γL = 1.5, ( ) γL = 0.7. . . . . . . . . . 108xxvi

LIST OF FIGURES5.8 Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,Noiseless Case). Re onstru tions with IMSMRA-LS at (a) s = 1 and (b) s = 2, ( ) s = sopt = 3.Optimal inversion (d). . . . . . . . . . . . . . . 1115.9 Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,Noiseless Case). Behavior of the ost fun tionwhen using IMSMRA-LS . . . . . . . . . . . . . 1125.10 Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 20 dB). Re onstru tions with IMSMRA-LS (a), IMSA-LS (b), and Bare-LS ( ). . . . . 1145.11 Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 10 dB). Re onstru tions with IMSMRA-LS (a), IMSA-LS (b), and Bare-LS ( ). . . . . 1155.12 Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 5 dB). Re onstru tions with IMSMRA-LS (a), IMSA-LS (b), and Bare-LS ( ). . . . . 1165.13 Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 5 dB). Values of the error gures versusSNR. . . . . . . . . . . . . . . . . . . . . . . . 1175.14 Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 5 dB). Behavior of the ost fun tion. . . 1185.15 Numeri al Data. Two re tangular s atterers (ǫC =1.8, LD = 3λ, SNR = 20 dB). Re onstru tionswith IMSMRA-LS (a), IMSA-LS (b), and Bare-LS ( ). . . . . . . . . . . . . . . . . . . . . . . . 1205.16 Numeri al Data. Two re tangular s atterers (ǫC =1.8, LD = 3λ, SNR = 20 dB). Behavior of the ost fun tion. . . . . . . . . . . . . . . . . . . . 121xxvii

LIST OF FIGURES5.17 Numeri al Data. Three s atterers (ǫC = 1.8,LD = 3λ, SNR = 20 dB). Re onstru tions withIMSMRA-LS (a) and Bare-LS (b). . . . . . . . 1235.18 Numeri al Data. Three s atterers (ǫC = 1.8,LD = 3λ, SNR = 10 dB). Re onstru tions withIMSMRA-LS (a) and Bare-LS (b). . . . . . . . 1245.19 Numeri al Data. Three s atterers (ǫC = 1.8,LD = 3λ, SNR = 5 dB). Re onstru tions withIMSMRA-LS (a) and Bare-LS (b). . . . . . . . 1255.20 Numeri al Data. Three s atterers (ǫC = 1.8,LD = 3λ, Noisy Case). Values of the error g-ures versus SNR. . . . . . . . . . . . . . . . . . 1275.21 Numeri al Data. Two hollow ylinders (ǫC =2.5, LD = 5λ, SNR = 20 dB). Re onstru tionswith (a) IMSMRA-LS and (b) Bare-LS . Behav-ior of the ost fun tion ( ). . . . . . . . . . . . 1285.22 Numeri al Data. Two hollow ylinders (ǫC =2.5, LD = 5λ, SNR = 10 dB). Re onstru tionswith (a) IMSMRA-LS and (b) Bare-LS . Behav-ior of the ost fun tion ( ). . . . . . . . . . . . . 1305.23 Numeri al Data. Three obje ts hara terized by omplex shapes (ǫC = 1.8, LD = 5λ, SNR =20 dB). Re onstru tions with (a) IMSMRA-LSand (b) Bare-LS . Behavior of the ost fun tion( ). . . . . . . . . . . . . . . . . . . . . . . . . 1315.24 Numeri al Data. Three obje ts hara terizedby omplex shapes (ǫC = 1.8, LD = 5λ, NoisyCase). Values of the error gures versus SNR. . 132xxviii

LIST OF FIGURES5.25 Experimental Data (Dataset Marseille [100).Two ir ular ylinders (twodielTM_4f.exp). Re- onstru tions with IMSMRA-LS (left olumn)and Bare-LS (right olumn) at dierent frequen- ies f [f = 2 GHz (a)(b); f = 4 GHz ( )(d). . 1345.26 Experimental Data (Dataset Marseille [100).Two ir ular ylinders (twodielTM_4f.exp). Be-havior of the ost fun tion versus the number ofiterations when (a) f = 2 GHz, and (b) f =4 GHz. . . . . . . . . . . . . . . . . . . . . . . . 135

xxix

LIST OF FIGURES

xxx

Stru ture of the ThesisThe thesis is stru tured in hapters a ording to the organiza-tion detailed in the following.The rst hapter deals with an introdu tion to the thesis,fo using on the main motivations and on the subje t of thiswork.Then, Chapter 2 presents the mathemati al formulation ofthe inverse s attering problem, pointing out the main draw-ba ks su h as non-linearity, ill- ondition, and ill-posedness.Chapter 3 is on erned with the state-of-the-art. The ex-ploitation of regularized solutions and approximations to opewith the ill-posedness of the inverse problem is dis ussed. More-over, both deterministi and heuristi minimization te hniquesare presented. Finally, a brief overage of the literature onmulti-resolution te hniques and shape-optimization is given.The iterative multi-s aling approa h with level-set-basedxxxi

LIST OF FIGURESoptimization is dis ussed in Chapter 4. The mathemati alformulation is fo used on the multi-step ar hite ture and theproposed numeri al validation, arried out when onsideringboth numeri ally-synthesized and laboratory- ontrolled data,assesses the re onstru tion apabilities of the proposed method-ology by onsidering targets hara terized by simple and om-plex shape.Chapter 5 deals with the iterative multi-s alingmulti-regionapproa h with level-set-based minimization, ustomized for ge-ometries hara terized by multiple obje ts. After presentingthe mathemati al formulation by fo using on the main dier-en es with respe t to the single-region version, the ee tivenessof the approa h is evaluated by means of the dis ussion of asele ted set of results, when dealing with both numeri al andlaboratory- ontrolled data.Con lusions, further developments, and open problems arepresented in Chapter 6. Finally, two appendi es gives moredetail on the adjoint problem (whose solution allows to om-pute the gradient of the ost fun tion) and on two imagingalgorithms (the edge dete tion operator and a te hnique for ounting the number of obsta les in an image).xxxii

Chapter 1Introdu tionIn the introdu tion, the motivation of the thesis is pointed outstarting from a brief overview about the framework of te h-niques for non-destru tive evaluation and testing.

1

The non-invasive re onstru tion of position and shape ofunknown targets is a topi of great interest in many appli a-tions, su h as non-destru tive evaluation and testing (usuallyreferred to with the a ronyms NDE and NDT) for industrialmonitoring and subsurfa e sensing [1. In su h an interestingframework, many methodologies have been proposed, mainlybased on x-rays [2, ultrasoni s [3, and eddy urrents [4. How-ever, mi rowave approa hes have been re ently re ognized asee tive imaging methodologies be ause of the following keypoints [1 [5-[8:(a) ele tromagneti elds at mi rowave frequen ies anpenetrate non-ideal ondu tor materials;(b) the eld s attered by the target is representativenot only of its boundary, but also of its inner stru -ture;( ) mi rowaves show a high sensivity to the water on-tent of the stru ture under test;(d) mi rowave sensors an be employed without me- hani al onta ts with the spe imen.In addition, ompared to x-ray and magneti resonan e, mi ro-wave-based approa hes minimize (or avoid) ollateral ee ts inthe spe imen under test. Therefore, they an be safely em-ployed in biomedi al imaging, limiting the stress for the pa-tient sin e the physi al onta t with the imaging system anbe avoided (e.g., the breast s reening [9), or in other riti alappli ations, su h as through-wall imaging (TWI) [10.2

CHAPTER 1. INTRODUCTIONA further advan e in mi rowave non-invasive inspe tion isrepresented by inverse s attering approa hes aimed at re on-stru ting the image of the region under test in a quantita-tive fashion [11. Unfortunately, the underlying mathemati- al model is hara terized by several drawba ks that urrentlylimit their massive employment, espe ially in the NDE/NDT'sframework. More in detail, the inverse s attering problems areintrinsi ally ill-posed [12 as well as non-linear [13.Sin e the ill-posedness is strongly related to the amountof olle table information and usually the number of indepen-dent data is lower than the dimension of the solution spa e,multi-view/multi-illumination systems are generally adopted.However, it is well known that the olle table information isan upper-bounded quantity [14[15. Consequently, it is ne es-sary to ee tively exploit the overall information ontained inthe s attered eld samples for a hieving a satisfa tory re on-stru tion.In order to exploit the whole amount of information ol-le ted from s attering measurements, multi-resolution strate-gies have been re ently proposed. The idea is that of usingan enhan ed spatial resolution only in those regions of interest(RoI s) where the unknown s atterers are found to be lo ated[16 and/or where dis ontinuities o ur [17[18. As for theproposed implementations, deterministi or statisti ally-baseddata pro essing strategies have been proposed to determinethe optimal resolution level, and spline-based approa hes havebeen employed to improve the resolution level. Furthermore,multi-step approa hes have been implemented to iteratively in- rease the spatial resolution by means of a so- alled zooming3

pro edure [19 by keeping the ratio between unknowns and datasuitably low and onstant, thus redu ing the risk of o urren eof lo al minima [15 in the arising optimization problem.On the other hand, the la k of information ae ting theinverse problem has been addressed, espe ially in NDE/NDT,through the exploitation of the a-priori knowledge on the s e-nario under test by means of an ee tive representation of theunknowns. In many appli ations, the unknown defe t is har-a terized by known ele tromagneti properties (i.e., diele tri permittivity and ondu tivity) and it lies within a known hostregion. Moreover, depending on the desired degree of a u-ra y, more omplex s enarios an be approximated by a setof homogeneous regions hara terized by dierent shape andele tromagneti parameters [20. Under these assumptions, animaging problem redu es to a shape re onstru tion problem,namely to a problem where the support of the homogeneousregions needs to be retrieved. Towards this end, parametri te hniques aimed at representing the unknown obje t in termsof des riptive parameters of referen e shapes [21-[22 and moresophisti ated approa hes su h as evolutionary- ontrolled spline urves [23-[25, shape gradients [26-[28 or level-sets [31-[32have been proposed. As far as level-set-based methods are on- erned, the homogeneous obje t is dened as the zero level ofa ontinuous fun tion and, unlike pixel-based or parametri -based strategies, su h a des ription enables one to represent omplex shapes or regions in a simple way.Within su h a framework, the thesis fo uses on the develop-ment and the analysis of the integration of the iterative multi-s aling strategy (IMSA) [19 and the level-set (LS ) representa-4

CHAPTER 1. INTRODUCTIONtion [33. The implementation is aimed at protably exploitingboth the available a-priori knowledge on the s enario undertest (e.g., the homogeneity of the s atterer) and the informa-tion ontent from the s attering measurements. For the sake ofsimpli ity and with no laim of exhaustivity, the inverse prob-lem formulation is restri ted to the two-dimensional transverse-magneti (TM ) ase when dealing with one and with multipleRoI s. As for the assessment of the proposed strategy, the nu-meri al validation deals with diele tri lossless s atterers, by onsidering both syntheti and experimental (i.e., laboratory- ontrolled) data.

5

6

Chapter 2The Inverse S atteringProblemThe ele tromagneti inverse s attering problem is presented inthis hapter, fo using on its mathemati al formulation as wellas on the main drawba ks, su h as non-linearity, ill- onditioning,and ill-posedness.

7

2.1. MATHEMATICAL FORMULATION2.1 Mathemati al Formulation2.1.1 Field S attered byInhomogeneous Obje tsLet us onsider a region, alled investigation domain DI , har-a terized by a relative permittivity ǫ(r) and ondu tivity σ(r).As shown in Fig. 2.1, su h a region is probed by a set of Vtransverse-magneti (TM) plane waves sour es, with ele tri eld ζv(r) = ζv(r)z (v = 1, . . . , V ), r = (x, y), and the s at-tered eld, ξv(r) = ξv(r)z, is olle ted at M(v), v = 1, ..., V ,measurement points rm(v) distributed in the observation do-main DO.In order to ele tromagneti ally des ribe the investigationdomain DI , let us dene the ontrast fun tionτ(r) = [ǫ(r)− 1]− j σ(r)

2πfε0

r ∈ DI ,

(2.1)where f is the frequen y of operation (the time dependen eej2πft being implied). Under the hypothesis of a linear, isotropi and non-magneti propagation medium, the s attered eld dis-tribution ξv(r) is the solution of the following Helmholtz equa-tion (see [13[35[36 for a more detailed explanation)

∇2ξv(r)−(

2π

λ

)2

ξv(r) = −j2πfµ0Jv(r) . (2.2)where λ is the ba kground wavelength. Moreover, Jv(r) is the8

CHAPTER 2. THE INVERSE SCATTERING PROBLEMy

xO

(ǫ0, µ0, σ = 0)

M (v)

(ǫr, µr, σ)

DO

DI

rm(v)

(ǫr, µr, σ)

1

m (v)

r

EM Source

v = 1, ..., VFigure 2.1: Geometry of an inverse s attering problem.equivalent urrent density radiating in free-spa e, dened inDI as follows

Jv(r) = j2πfε0τ (r) Ev (r) (2.3)Ev being the total ele tri eld. By imposing that ξv(r) satis-es the Sommerfeld radiation ondition, namelylim|r|→+∞

√|r|

(∂ξv(r)

∂ |r| − jκ(r)ξv(r)

)= 0 , (2.4)the solution of (2.2) is given by the following pair of Lippmann-9

2.1. MATHEMATICAL FORMULATIONS hwinger integral equationsξv

(rm(v)

)=

(2πλ

)2 ∫DI

τ (r′)Ev (r′)G2D

(rm(v)/r

′)dr′

rm(v) ∈ DO ,

(2.5)ζv (r) = Ev (r)−

(2πλ

)2 ∫DI

τ (r′)Ev (r′)G2D (r/r′) dr′

r ∈ DI .

(2.6)Moreover, G2D (r/r′) is the free-spa e two-dimensional Green'sfun tion dened as followsG2D (r/r′) = −j

4H

(2)0

(2π

λ‖r − r′‖

), (2.7)

H(2)0 being the se ond-kind zeroth-order Hankel fun tion.The aim of an inverse s attering te hnique is the re onstru -tion of both the ele tromagneti properties τ(r) and the totaleld distribution Ev (r), with r belonging to DI [i.e., the equiv-alent urrent density Jv(r), starting from the knowledge of themeasurements ξv

(rm(v)

), rm(v) ∈ DO, and of the in ident eldζv (r) radiated by the known sour e. Unfortunately, a losedform solution of integral equations (2.5) and (2.6) does notgenerally exist. Consequently, the inverse s attering problemhas to be reformulated and ee tive inversion methodologieshave to be employed in order to retrieve the solution. Chap-ter 3 will fo us on the state-of-the-art of the inverse s atteringapproa hes. 10

CHAPTER 2. THE INVERSE SCATTERING PROBLEMy

xO

M (v)

DO

DI

rm(v)

(ǫC, µC, σ = 0)

1

m (v)

(ǫC, µC, σ = 0)

(ǫ0, µ0, σ = 0)

τ0 = 0

r

τC

Figure 2.2: Geometry of an inverse s attering problem whena-priori information about the s atterer is available.2.1.2 Exploitation of the a-prioriInformation about the S atterersLet us now assume that a ylindri al homogeneous non-magneti obje t with known relative permittivity ǫC and ondu tivity σCo upies a non-homogeneous region Υ belonging to an inves-tigation domain DI . How ould su h an amount of a-prioriinformation be protably exploited? A possible solution on-sists in re-dening the ontrast fun tion τ(r) as followsτ(r) =

τC

0r ∈ Υotherwise (2.8)11

2.2. NUMERICAL SOLUTION OFTHE INVERSE SCATTERING PROBLEMlettingτC = (ǫC − 1)− j

σC

2πfε0. (2.9)Sin e the diele tri properties of the obsta les are a-priori knownand homogeneous, the shape of Υ, or its ontour, be omes asu ient parameter for the hara terization of the domain un-der test. Consequently, suitable inversion te hniques an beapplied in order to retrieve the boundaries of Υ by solvingequations (2.5) and (2.6). Su h te hniques, usually known asshape-optimization methods [37, are dealt with in Se t. 3.4.Although the hypothesis of homogeneous obje ts and a-priori known permittivity and ondu tivity appears to be strongand restri tive, many inverse problems an be redu ed to thesear h of homogeneous obsta les inside homogeneous or a-prioriknown ba kgrounds [32[33.2.2 Numeri al Solution ofthe Inverse S attering ProblemIn order to allow a numeri al solution of the inverse s atteringproblem, equations (2.5) and (2.6) an be dis retized a ordingto a point-mat hing version of the Method of Moments (MoM)[39. More in detail, the investigation domain DI is partitionedin N square sub-domains Dn with bary entres rn, n = 1, ..., N .In ea h sub-domain, a pulse basis fun tion is dened as

B (rn) =

1 if rn ∈ Dn

0 if rn /∈ Dn, (2.10)12

CHAPTER 2. THE INVERSE SCATTERING PROBLEMwhile the dis rete ontrast fun tion is given by the followingrelationshipτ (r) =

∑N

n=1 τnB (rn)r, rn ∈ DI .

(2.11)Consequently, by assuming that the in ident eld ζv andthe total eld Ev are onstant inside ea h sub-domain Dn, thedis rete form of the Lippmann-S hwinger integral equations isgiven by the following relationshipsξvm(v)

(rm(v)

)=

∑Nn=1 τnEv

n (rn) G2D

(rm(v)/rn

)

rm(v) ∈ DO ,

(2.12)ζvn (rn) = Ev

n (rn)−∑N

p=1 τpEvp

(rp

)G2D

(rn/rp

)

rn ∈ DI ,

(2.13)where G2D (rm/rn) is the two-dimensional dis rete Green op-erator given byG2D (rm/rn) =

=

− j

2

[πκoAH

(2)1 (κoA)− 2j

] ifm = n

− jπκoA

2J1 (κoA) H

(2)0 (κo ‖rm − rn‖) ifm 6= n(2.14)where A is the area of the square sub-domain, κo = 2π

λisthe free-spa e wavenumber, H

(2)1 is the se ond kind rst orderHankel fun tion, and J1 is the rst kind Bessel fun tion.13

2.3. DRAWBACKS OF THE INVERSESCATTERING PROBLEM2.3 Drawba ks of the InverseS attering ProblemUnlike the forward s attering problem, whi h is aimed at de-termining the elds ξv and Ev (i.e., the ee ts) starting fromζv and the τ (i.e., the ause), the inverse s attering prob-lem should re onstru t the ause starting from the observationof the measurable ee ts. It is also well known (see [13 and[38 for an exhaustive and detailed overview) that the inverseproblems are intrinsi ally hara terized by several drawba ks,whi h are detailed in the following:• Ill-posedness - The forward problem is hara terizedby a loss of information, sin e its solution represents atransformation from a physi al quantity (i.e., the om-plete des ription of the s atterer τ and the knowledge ofthe ele tromagneti sour e ζv) with a ertain information ontent to the s attered eld, whi h is hara terized bya lower information ontent. In addition, the s atteredeld provided by a band-limited system is smoother thanthe one provided by the a tual obje t. As a onsequen e,the orresponding inverse s attering problem requires again of information in order to retrieve a solution as loseas possible to the ause. Su h a loss of information isknown as ill-posedness of the inverse problem. To opewith the ill-posedness, the golden rule onsists in addingsome additional information to ompensate the loss of in-formation of the imaging pro ess. Su h a information isdened as additional sin e it annot be derived neither14

CHAPTER 2. THE INVERSE SCATTERING PROBLEMfrom the s attered eld nor from the properties of themapping between the data and the spa e of unknowns,whi h des ribes the imaging pro ess. It omes from otherinformative sour es or from previous information gainedon the obje t [40 and it is usually referred to as a-prioriinformation.• Non-linearity - With referen e to equations (2.12) and(2.13), the inverse problem is non-linear be ause the vari-able τ to be solved an not be written as a linear sumof independent omponents. As a matter of fa t, the to-tal eld Ev depends on the diele tri properties of thedomain under test. On the other hand, the inverse s at-tering problem is linear with respe t to the equivalent urrent density Jv(r) = τ(r)Ev(r), a ording to a on-trast sour e formulation [41[42.• Ill- onditioning - As a onsequen e of the ill-posednessand due to the band-limited nature of the system as well,the numeri al ounterpart of the inverse s attering prob-lem appears to be ill- onditioned, sin e the solution doesnot depend ontinuously on the data. As a matter offa t, the numeri al solution may suer from numeri alinstability and a small error in the initial data an resultin mu h larger errors in the answers.These drawba ks have to be taken into a ount when solvingan inverse s attering problem. In Ch. 3, a brief overview onthe state-of-the-art of the inversion methodologies will be pre-15

2.3. DRAWBACKS OF THE INVERSESCATTERING PROBLEMsented, fo using on the multi-resolution approa hes and theshape-optimization methods.

16

Chapter 3An Overview on InverseS attering Te hniquesThis hapter deals with an overview on the inverse s atteringte hniques. The exploitation of regularized solutions and ap-proximations to ope with the ill-posedness of the inverse prob-lem will be dis ussed. Then, both deterministi and heuristi minimization te hniques will be presented. Finally, the hapterfo uses on the state-of-the-art on multi-resolution te hniquesand shape-optimization.17

3.1. REGULARIZED SOLUTIONS ANDAPPROXIMATIONS3.1 Regularized Solutions andApproximationsAs pointed out in Se t. 2.3, the inverse problem is hara -terized by ill-posedness, whi h is a onsequen e of the loss ofinformation between ause and ee t in the forward prob-lem. A possible solution onsists in mathemati ally expressingsome expe ted physi al properties of the s atterer and to ex-pli itly use su h a knowledge to build families of approximatesolutions. For instan e, A. N. Tikhonov and V. Y. Arsenin [38introdu ed a family of approximate solutions depending on aregularization parameter. For noise-free data, the approximatesolutions onverge to the true solution when the regularizationparameter tends to zero. Otherwise, an optimal approxima-tion of the exa t solution exists for a non-zero value of theregularization parameter. However, the hoi e of the value ofthe regularization parameter be omes the main issue, sin e it ould depend on the geometry under test. Furthermore, whilemathemati al methods and e ient numeri al algorithms arealready available for linear inverse s attering problems, the s i-enti literature does not provide any simple rule for the op-timal hoi e of the regularization oe ient when nonlinearproblems are dealt with [43. Looking for riteria, in [44 and[45 the authors onsidered this parameter an additional un-known whi h has to be ontrolled by the optimization pro ess.On the other hand, the improvement of the robustness withrespe t to false solutions and the onvergen e rate as well as thenon-uniqueness and ill- onditioning inherent to inverse s atter-18

CHAPTER 3. AN OVERVIEW ON INVERSESCATTERING TECHNIQUESing problems has been dealt with as well [46-[50. As far asnon-linearity is on erned, some attempts have been devotedto the problem linearization by using suitable approximations.However, the use of Born, Rayleigh and Rytov approximations[13 is suitable only for dealing with weakly s attering obje ts[51. The iterative Born method [52 and its modied form [53 ould extend their availability, but they still remain te hniquesthat an be applied only to diele tri s atterers.3.2 An Overview onMinimization Te hniquesIn order to ope with the non-linearity, the inverse s atteringproblem an be re ast as an optimization pro edure, wherea suitable ost fun tion depending on the mismat h betweenthe measured elds and the numeri ally evaluated one is min-imized. Su h a ost fun tion is omputed on the basis of thetrial solution τ = τn; n = 1, ..., N and it is usually expressedin matrix form as follows [13Φ τ = α

‖ξ−GEXT

eτ eE‖2‖ξ‖2

+β‖ζ− eE+G

INTeτ eE‖2

‖ζ‖2(3.1)where G

EXTand G

INTare the M×N external Green's matrixand the N ×N internal Green's matrix, respe tively, and α, β19

3.2. AN OVERVIEW ONMINIMIZATION TECHNIQUESare two user-dened regularization parameters. Furthermore,ζ is the N × 1 in ident eld array, ξ is the M × 1 measuredeld, and E is the N × 1 estimated total ele tri eld.The optimal solution τ opt is found as the N × 1 array thatminimizes the relationship (3.1), namely

τ opt = arg mink=1,...,K [Φ τk] (3.2)where τ k is the trial solution a hieved at the step k by the iter-ative updating pro edure. As a onsequen e, the quality of thenal solution depends mostly on the sear h strategy, sin e theproblem may have false solutions due to its non-linearity. Thenext two subse tions dis uss both deterministi and heuristi approa hes.3.2.1 Deterministi Approa hesAs far as deterministi approa hes are on erned, the mostwidely used are the gradient-based te hniques introdu ed byKleinman et al. in [54. Su h methodologies are based on asimultaneous update of the unknown eld Ek and s atterers ontrast τ k in order to avoid the full solution of the forwardproblem at ea h iteration. Towards this end, a sequen e of trialsolutions χk

=[τk, Ek

], k = 1, ..., K, is dened with stri tlyde reasing ost fun tion values. At ea h iteration k, the trialsolution is updated as followsχ

k+1= χ

k+ νk · Uk (3.3)20

CHAPTER 3. AN OVERVIEW ON INVERSESCATTERING TECHNIQUESwhere Uk is the value of the updating fun tion at the iterationk, k = 1, ..., K, and

νk = argminp

[Φ

χ

k+ νp · Uk

]. (3.4)The iterative pro edure is iterated until the value of (3.1) issmaller than a xed threshold or a stationary ondition isrea hed.The updating of the unknowns Uk onsists in the evalu-ation of the steepest des ent Γk = −∇τk

(Φ

χ

k

). Severalupdating strategies have been developed, but one of the mostee tive ar hite ture in terms of onvergen e rate is the Polak-Ribiere pro edure [55. Su h an algorithm denes the followingupdate termUk = Γk −

ΓTk

(Γk − Γk−1

)

ΓTk−1Γk−1

Uk−1 (3.5)where the supers ript T indi ates the transpose operator.As for some examples of other updating strategies, the bi- onjugate gradient method (see [56 and [57 for more details),or its development alled the bi- onjugate gradient stabilizedmethod (see [58 for more details), have been implemented tosolve nonsymmetri systems. Moreover, the generalized min-imum residual (GM-RES ) method (see [59 for more details)and the quasi minimal residual method (see [60 for more de-tails) have a dierent approa h based on the reation of properbasis fun tions to represent the solution spa e. However, su hstrategies are often more demanding, even though a faster on-vergen e rate is reported. Moreover, at the best of the author's21

3.2. AN OVERVIEW ONMINIMIZATION TECHNIQUESknowledge, these updating approa hes are generally used tosolve linear problems.3.2.2 Heuristi Minimization MethodologiesUnfortunately, the onvergen e of the deterministi minimiza-tion strategies is stri tly dependent on the initialization point.As a matter of fa t, sin e the problem at hand is non-linear,the solution an be easily trapped in lo al minima representingfalse re onstru tions if the initialization is not appropriate.Sin e false solutions are physi ally una eptable solutions,the minimization methods should meet the following require-ments to avoid lo al minima:(a) the possibility of easily in luding the whole amountof available a-priori information on the unknownsolution;(b) on-line ontrol of the solution quality in order to as-sure that trial solutions, estimated during the sam-pling of the sear h spa e, are physi ally admissiblesolutions;( ) suitable operators able to fully exploit the informa-tion on the solution gained during the minimizationand/or arising from the a-priori information;(d) operators able to easily repla e non-feasible solu-tions by newly feasible ones, without introdu inguser-dened penalty fun tions.22

CHAPTER 3. AN OVERVIEW ON INVERSESCATTERING TECHNIQUESSu h properties are usually owned by the heuristi optimizationte hniques based on evolutionary algorithms. These strategiesare hara terized by the denition of a set of W trial solutionχ

k=

χw

k; w = 1, ..., W

, ommonly alled population, that isiteratively updated a ording to the following general relation-shipχ

k+1= χ

k+ U

k

χ

ℓ; ℓ = 1, ..., k

(3.6)where the updating operator Uk· depends on the urrentsolution and on the previous k−1 trial solution. Consequently,the optimal solution is found as the one minimizing the ostfun tion (3.1), namely

χopt

= argmink=1,...,K

[minw=1,...,W

(Φ

χw

k

)]. (3.7)where τ k is the trial solution a hieved at the step k by theiterative updating pro edure.As far as the updating strategies are on erned, many ap-proa hes have been developed. Simulated Annealing [61 isone of the rst and most ommon single agent algorithm (i.e.,

W = 1). The basis idea is to dene Uka ording to a temper-ature parameter that in rementally de reases at ea h iteration onverging to a stable ondition. However, the smaller the up-date be omes, the lesser the hill- limbing1 apability is.In order to in rease the hill- limbing apabilities, multipleagent te hniques (i.e., W > 1) have been also adopted. Some1The hill- limbing apability refers to the possibility of dealing withmulti-minima ost fun tion, without the solution being trapped in a lo alminima. 23

3.3. STATE-OF-THE-ART ONMULTI-RESOLUTION APPROACHESwell known examples are the Geneti -Algorithms (GA) [62-[64 [21 and the Parti le Swarm Optimizer (PSO) [65. InGA, a population of trial solution is dened and it is evolvedby mimi ing Darwin's evolution theory by means of a set ofupdated operators, known as sele tion, rossover, andmutation[66. The pro edure is based on a ompetitive pro ess aimed atextra ting the best individual among the population. On the ontrary, the PSO is based on a ooperative logi inspired bythe behavior of o ks of birds looking for food [67[68. Ea hparti le (i.e., trial solution) olle ts information and makes itavailable to the whole swarm in order to ooperate in sear hingthe global minimum. As a matter of fa t, the update term Ukis hara terized by a personal term and a so ial term.Evolutionary te hniques are mu h more demanding thanthe deterministi pro edures in terms of omputational resour- es, but they provide a suitable exploitation of the a-prioriinformation together with an e ient representation of the so-lution spa e.3.3 State-of-the-Art onMulti-Resolution Approa hesIn order to deal with the ill-posedness of the inverse s atter-ing problems, many approa hes to olle t a greater amountof information have been onsidered. More in detail, multi-illumination [69 and/or multi-view [70 and/or multi-sour e[71 and/or multi-frequen y systems [72 are generally used,24

CHAPTER 3. AN OVERVIEW ON INVERSESCATTERING TECHNIQUESbut the information olle table from the s attering experimentsstill remains limited [13 to an upper bound that depends onthe geometri al hara teristi s of the imaging system and onthe working frequen y [12. In order to redu e the dimensionof the spa e of unknowns, some a-priori information (whenavailable) on the s enario under test [73[40 an be added byimposing a set of onstraints [15 on the retrievable diele tri prole.In order to restore the well-position of the inverse prob-lem, a dierent approa h onsists in fully exploiting all avail-able information on the s enario under test. In more detail, therepresentation of the unknown s atterer belongs to a nite di-mensional spa e, whose dimension is smaller than the essentialdimension of data (i.e., the number of retrievable unknownsshould be lower than the amount of information in data). Be- ause of the analyti al nature of the s attering operator, su han optimal dimension is a known quantity sin e it depends onthe extension of the investigation domain with respe t to thewavelength [74 and on the hara teristi s of the a quisitionsystem [12. Starting from these onsiderations and sin e thes attered eld is a spatial band-limited fun tion, the optimalnumbers of views V and of measurement points M depend onthe size of the investigation domain. As a matter of fa t, thedegrees of freedom of the eld s attered by an obsta le lo atedin free spa e is [15Rf = 2κ0α (3.8)where α is the radius of the minimum ir le en losing the s at-terer. Moreover, the number I of independent measurements25

3.3. STATE-OF-THE-ART ONMULTI-RESOLUTION APPROACHESthat an be olle ted, starting from the number of total mea-surement (i.e., M × V ), is I = R2f/2. Consequently, when

M × V measurements are available, the optimal number Noptof unknowns (i.e., independent equations) to be allo ated isgiven asNopt = min I, M × V/2 . (3.9)Unfortunately, the optimal number Nopt of retrievable unknowns(equal to the essential dimension of the s attered data) doesnot usually meet the riterion given in [39 for a suitable repre-sentation (in terms of spatial resolution) for both the diele tri prole of the s atterer and the indu ed ele tri eld.In order to suitably represent the unknowns and keep theratio between unknown and data, namely Nopt/(M×V ), smalland onstant during the inversion pro edure, a more ee tiverepresentation of the problem unknown should be adopted. Apossible solution onsists in the use of multi-resolution meshes,by de reasing the size of the sub-domains where the targetsare lo ated and using a oarse resolution level otherwise [75.A ording to su h a basi idea, various strategies based ona multi-resolution expansion of the unknowns have been pro-posed. These methods dene dis retization s hemes and orre-sponding tailored basis fun tions to better represent the geom-etry under test (i.e., higher resolution level near the dis ontinu-ities and oarse grid in the external homogeneous ba kground).Taking advantage of su h a kind of expansion, it is possible todistribute in a non-uniform fashion the unknowns inside thes attering domain (for more details see [76[77).26

CHAPTER 3. AN OVERVIEW ON INVERSESCATTERING TECHNIQUES3.3.1 Wavelet-basedmulti-resolution approa hesIn order to better allo ate the number of unknowns of the in-verse problem, a statisti al model is employed in [17 to gatherinformation on the suitable resolution levels to be adopted ina linear inverse problem. In pra ti e, the relative error ovari-an e matrix (RECM ) provides a rational basis for dealing withresolution/a ura y trade-os and to identify the optimal spa-tial resolution. As a matter of fa t, su h an approa h is ableto identify those regions in spa e where the information pro-vided by data is higher, thus in reasing the spatial resolutionby means of wavelet de omposition of the signals. The arisinginverse s attering problem is then re ast into the minimizationof a two-term ost fun tion enfor ing delity to s attering dataand mat hing with the statisti al prior model for the ontrast,respe tively. As a onsequen e, the inverse problem is regular-ized through the multis ale approa h that allows the olle tionof a-priori information.A similar approa h has been onsidered in [16. Startingfrom a-priori onsiderations on the mathemati al nature ofthe problem and from the intrinsi features exhibited by the lass of retrievable fun tions, su h a multi-resolution strategyasso iates part of the optimal number of unknowns Nopt to a oarse representation of the whole domain. Then, it on en-trates the remaining ones in those parts of the region undertest where a better resolution an be a hieved. Mathemati- ally, the arising problem is solved through the minimization(only on e) of a ost fun tion related to the s attering equa-27

3.3. STATE-OF-THE-ART ONMULTI-RESOLUTION APPROACHEStions. Furthermore, the unknowns are represented with thea-priori multi-resolution expansion by onsidering a suitablewavelet transformation.3.3.2 Adaptive Multis ale Te hniquesA dierent approa h based on a step-wise renement pro e-dure is developed [78 for a nonlinear s attering model. A se-quen e of dierent tests based on a-priori hypotheses (i.e., a olle tion of anomaly ongurations) are employed rst to lo- alize anomalous behaviors in large areas and then to renethese initial estimates in order to better hara terize the a -tual stru tures. The proposed stable oarse-to-ne lo alizationmethod denes a de omposition pro edure (able to zoom onstrong s atterers before rening other stru tures) and a prun-ing step to remove unreliable andidate anomalies. In furtherworks (for details see [18 and referred works), a dierent ap-proa h based on an adaptive iterative strategy is proposed toimprove spatial resolution by means of spline pyramids. Fi-nally, the multis ale distribution of the ontrast is determinedby adding detail lose to the surfa e and dening a oarse s aledeeper into the the material, pruning away unne essary degreesof freedom.3.3.3 The Iterative Multi-S aling Approa hIn the framework of adaptive multi-resolution approa hes, Caorsiet al. developed an iterative te hnique, alled iterative multi-s aling approa h (IMSA), where the distribution of the un-28

CHAPTER 3. AN OVERVIEW ON INVERSESCATTERING TECHNIQUESknowns is a-posteriori determined by exploiting the informa-tion gathered during the multi-step pro edure [79. Taking ad-vantage of a onstant multi-s aling pie e-wise pulse representa-tion able to deal with all possible multi-resolution ombinations(unlike wavelet expansion), su h an approa h is aimed at de-termining rstly the regions-of-interest (RoIs) where the s at-terers are lo ated, thus in reasing the spatial resolution onlywhere needed. The amount of information olle ted throughthe eld measurements is suitably exploited by allowing anenhan ed spatial resolution only in the RoIs and keeping theratio Nopt/(M × V ) low a ording to the riterion previouslydis ussed. As a onsequen e, the risk for the solution of beingtrapped in a lo al minima is sensibly redu ed [15, even thoughit annot be fully removed.At ea h step of the iterative pro edure, the re onstru tionis performed through the minimization of a suitable multi-resolution ost fun tion. The pro ess is iterated until the un-knowns' distribution rea hes a stationary ondition. The op-erations hara terizing ea h step s, s = 1, ..., S, of the IMSA an be summarized as follows (for more details, see [79):• Multi-Resolution Expansion - The representation ofthe unknowns is updated a ording to the new resolutionlevel s, s = 1, ..., S, by means of the following multi-resolution expansion

τ (r) =s∑

i=1

N∑

ni=1

τniB

(rni

), r ∈ DI (3.10)29

3.3. STATE-OF-THE-ART ONMULTI-RESOLUTION APPROACHESwhere B (rni

) is a re tangular basis fun tion whose sup-port is the ni-th sub-domain of the region of interest Riat the i-th resolution level, i = 1, ..., S.• Multi-Resolution Prole Retrieval - Minimize themulti-resolution ost fun tion at the resolution level s,given as

Φχ

s

= α‖ξ−Ps

i=1 Gi

EXTeτ i

eEi‖2‖ξ‖2

+β‖Ps

i=1[ζi− eEi+Gi

INTeτ i

eEi]‖2‖Ps

i=1 ζi‖2

(3.11)by onsidering the multi-resolution representation for theunknowns, namely χs

=[

τ(rns

), Ev

(rns

)]; ns =

1, ..., N ; i = 1, ..., s. At ea h resolution level, the Green'sfun tions Gi

EXTand Gi

INT, i = 1, ..., s, need to be re om-puted (for more detail see [79).

• Update of the Resolution Level - The resolution levelis updated (s← s + 1).• RoIs Estimation - The lo ation (xc

s, ycs) and the exten-sion (Lc

s) of the RoI is dened a ording to the lusteringpro edure des ribed in [19, by using the previous stepre onstru tion image and performing a noise ltering toeliminate some artifa ts in the re onstru ted image [79.• Termination Pro edure - Go to Multi-Resolution Ex-pansion until a stationary ondition on the qualitative30

CHAPTER 3. AN OVERVIEW ON INVERSESCATTERING TECHNIQUESre onstru tion parameters |Ωs−1 − Ωs||Ωs−1|

× 100

< γΩ, Ω = xc, yc, L (3.12)is rea hed (s = sopt), where γΩ are xed thresholds ex-perimentally determined.The multi-s aling pro edure has shown to be ee tive alsowhen dealing with multiple s atterers [19[80[81. The pro-posed lustering pro edure is able to manage dierent non- onne ted RoI s with a satisfa tory a ura y.3.4 Shape-Optimization AlgorithmsIn order to properly address the ill-posedness of the inverses attering problem, the full exploitation of whole amount ofinformation ontent appears as a key issue. That an be pro-vided though multi-resolution methodologies whi h allows tokeep the ratio between unknowns and data low during the in-version pro ess. In addition, several approa hes, su h as IMSA(Se t. 3.3.3), in rease the information ontent step-by-step,providing an iteratively-in reasing amount of a-priori infor-mation about the proper spatial resolution to be employed.However, multi-resolution te hniques are not generally aimedat taking into a ount the a-priori information about the ge-ometry at hand. As a matter of fa t, many pra ti al imagingproblems an be redu ed to the sear h of homogeneous obje tsinside known host media (Se t. 2.1.2), by assuming that the31

3.4. SHAPE-OPTIMIZATION ALGORITHMSele tromagneti properties of the targets are a-priori known[5-[8. In su h a ase, the inverse s attering problem is limitedto a qualitative imaging problem, where only the sear h of lo- ation and shape of the targets is arried out, unlike standardquantitative imaging problems [11, where the estimation of theele tri al ondu tivity and permittivity values is required.To protably exploit the a-priori information about thes atterers, shape-optimization algorithms an be onsidered asee tive strategies to solve qualitative imaging problems. Inthe following, a brief state-of-the-art about these methodolo-gies is provided, fo using basi ally on two lasses: the paramet-ri approa hes [21-[25, where the target is represented by a setof parameters, and the level-set-based strategies [31-[32 [82,where the shape of the s atterer is identied by the zero-levelof a ontinuous fun tion.3.4.1 Parametri Approa hesStarting from the knowledge of the unperturbed geometry andof the ele tromagneti properties of the media, the targets tobe retrieved an be protably dened as in lusions in a knownstru ture and approximated with a limited set of essential pa-rameters [21. Su h a parametrization and the use of a suitableGreen's fun tion [73 allow a redu tion of the number of un-knowns and onsequently a non-negligible omputational sav-ing during the re onstru tion pro ess arried out in terms ofthe optimization of a suitable ost fun tion.In general, these shape-optimization strategies des ribe thetargets as basi shapes, su h as re tangular domains, to be32

CHAPTER 3. AN OVERVIEW ON INVERSESCATTERING TECHNIQUESproperly parametrized. More in detail, by onsidering some a-priori assumptions, the s atterer is dened by the oordinatesof its enter (xc, yc), its length L, its side W , and the orientationθ. Therefore, the value of the ontrast fun tion in the n-thsub-domain, n = 1, ..., N , an be re-dened a ording to thefollowing relationship

τn =

τC ifP ∈ [

−L2, L

2

] andQ ∈ [−W

2, W

2

]

0 otherwise (3.13)whereP = (xn − xc) osθ + (yn − yc) sinθ (3.14)andQ = (xn − xc) sinθ + (yn − yc) osθ . (3.15)A ordingly, the set of parameters to be retrieved during there onstru tion pro ess is

χ = (xc, yc) , L, W, θ; Ev (rn) , n = 1, ..., N (3.16)where the total ele tri eld Ev (rn) an be updated by meansof ee tive forward solvers [83 or estimated during the opti-mization pro ess.In order to determine the optimal solution χopt

of the re- onstru tion problem, the problem at hand is re ast as an op-timization one via the denition of a suitable ost fun tionΦ

χ

= α‖ξ−G

EXTℵ(χ)‖2

‖ξ‖2

+β‖ζ− eE+G

INTℵ(χ)‖2

‖ζ‖2(3.17)33

3.4. SHAPE-OPTIMIZATION ALGORITHMSwhere ℵ (χ)

= τ(χ)E

(χ) is the equivalent urrent density omputed starting from χ. Due to the dis rete nature of the ar-ray of unknowns, the resulting minimization pro edure is usu-ally solved by means of evolutionary optimization approa hes,su h as GA. As for the use of more omplex parametriza-tion s hemes, su h as spline fun tions, it usually requires us-tomized strategies [23[24.3.4.2 Level-Set-based Shape OptimizationSin e 1988 when the paper of Osher and Sethian [29 appearedin the literature, level set methods are onsidered ee tive te h-niques for dealing with propagating fronts and interfa es [32.Su h methodologies have been su essfully exploited in manyframeworks, su h as modelling and simulation [84 and inverseproblems [85-[87, or more spe i ally in imaging [88-[90,medi al appli ations [91-[94, and geology [95.More in detail, on e the ee tiveness of these optimiza-tion strategies has been proved for the retrieval of position andshape of unknowns targets [31, they have been also used inele tromagneti s, for inverse s attering problems [33[34 [97-[98. In su h a framework, the rst works were mainly fo usedon binary geometries, where the permittivity and ondu -tivity of targets and ba kground were assumed to be a-prioriknown, thus turning the inverse s attering problem into a qual-itative imaging problem, where only the shape of the s attererhas to be re onstru ted. However, re ent advan es deal withmore omplex problems, su h as the retrieval of the values ofthe binary ontrast fun tion, the re overy of the support of34

CHAPTER 3. AN OVERVIEW ON INVERSESCATTERING TECHNIQUESseveral targets, or more omplex geometries [20[99.As for the main aspe t of level-set-based optimization, the ontrast fun tion is dened with respe t to the shape Υ of thes atterer as followsτ (r) =

τC φ (r) ≤ 0 and r ∈ Υ0 φ (r) > 0 and r /∈ Υ

(3.18)where φ is a ontinuous fun tion, alled level set, and the on-tour of Υ is identied by φ = 0. In order to retrieve theoptimal shape Υopt of the re onstru tion problem, the level setalgorithm starts from an initial guess Υk=1 and iteratively per-forms the following sequen e of operations (k = 1, ..., K beingthe iteration index):• Update of the Field Distribution - The values of thes attered eld ξ

k(rm), m = 1, ..., M with rm ∈ DO, andof the total eld Ek (r), r ∈ DI , are omputed start-ing from φk (r) and τk (r) by solving a forward s atteringproblem.

• Gradient Computation - The shape derivative∇ΥΘ φkof the ost fun tionΘ φk =

∥∥∥ξ −GEXT

τ k (φk) Ek (φk)∥∥∥

2 (3.19)is omputed in order to get a velo ity fun tion Vk (r)for the update of the level set fun tion. Vk (r) an beobtained by al ulating the Eulerian derivative of (3.19)[32 or similarly by solving an adjoint problem [33. Moredetails about the adjoint problem for the two-dimensionalTM ase are given in Appendix A.35

3.4. SHAPE-OPTIMIZATION ALGORITHMS• Iteration Update - The iteration index is updated (k ←

k + 1).• Level Set Update - The level set fun tion φk is updateda ording to the following relationship

φk (r) = φk−1 (r) + νk−1Uk−1 (r) (3.20)where Uk−1 is the update term and νk−1 is the step sizewhi h an be determined by a line-sear h strategy. WhenUk−1 (r) is hosen as

Uk−1 = |∇φk−1| Vk−1 · n , (3.21)with n = ∇φk−1/ |∇φk−1|, relationship (3.20) is a Hamilton-Ja obi-type equation.Sin e the updates Uk ould provide rough boundaries of thetrial shape Υk, k = 1, ..., K, regularization strategies for shapeinversion are often onsidered [32[87. Su h te hniques onsistin employing smoothing operators (e.g., by assuming that thesolution belongs to a parti ular fun tion spa e onsisting ofsmooth fun tions) or penalization terms based on the desiredgeometri properties of the trial solutions. These onstraints an be determined through the analysis of the data at handand of the a hievable resolution.36

Chapter 4The Multi-ResolutionLevel Set Approa hIn this hapter, the iterative multi-s aling approa h with level-set-based optimization is presented. The mathemati al for-mulation is fo used on the multi-step ar hite ture and theproposed numeri al validation, arried out both with numeri- al synthesized data and laboratory- ontrolled data, dis ussesthe re onstru tion apabilities of the proposed multi-resolutionmethodology by onsidering simple and omplex targets.37

4.1. MATHEMATICAL FORMULATION4.1 Mathemati al FormulationWith referen e to Ch. 2 where the inverse s attering problemhas been mathemati ally formulated, this se tion is aimed atpresenting the integration of the iterative multi-s aling strategy(IMSA) and the level-set (LS ) representation. Let us onsidera ylindri al homogeneous non-magneti obje t with relativepermittivity ǫC and ondu tivity σC that o upies a region Υbelonging to an investigation domain DI . Su h a s atterer isprobed by a set of V transverse-magneti (TM) plane waves,with ele tri eld ζv(r) = ζv(r)z (v = 1, . . . , V ), r = (x, y),and the s attered eld, ξv(r) = ξv(r)z, is olle ted at M(v),v = 1, ..., V , measurement points rm(v) distributed in the ob-servation domain DO.In order to retrieve the unknown position and shape of thetarget Υ by step-by-step enhan ing the spatial resolution onlyin that region, alled region-of-interest (RoI), R ∈ DI , wherethe s atterer is lo ated [19, the following iterative pro edureof Smax steps is arried out.With referen e to Fig. 4.1(a) and to the blo k diagramdisplayed in Fig. 4.2, at the rst step (s = 1, s being the stepnumber) a trial shape Υs = Υ1, belonging to DI , is hosenand the region of interest Rs [ Rs=1 = DI is partitioned intoNIMSA equal square sub-domains, NIMSA being the number ofdegrees of freedom of the problem at hand [14. In addition,the level set fun tion φs is initialized as follows [33[97:

φs

(rns

)=

−minb=1,...,Bs

dnsb if τ (rns

)= τCminb=1,...,Bs

dnsb if τ (rns

)= 0

(4.1)38

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHa( )

b( )

c( )

Actual Scatterer

Υ1

(xcs=1, y

cs=1)

rns=1

rns=1

Rs=2

φkopts=1

φks=1

s = 1 - k = 1

s = 1 - k = kopt

s = 2 - k = 1

DI

DI

DI

Ls=1

rns=2

φks=2

Figure 4.1: Graphi al representation of the IMSA-LS zoomingpro edure. (a) First step (k = 1): the investigation domain isdis retized in N sub-domains and a oarse solution is lookedfor. (b) First step (k = kopt): the region of interest that on-tains the rst estimate of the obje t is dened. ( ) Se ond step(k = 1): an enhan ed resolution level is used only inside theregion of interest. 39

4.1. MATHEMATICAL FORMULATIONwhere dns,b =√

(xns− xb)

2 + (yns− yb)

2, rns= (xns

, yns) and

rb = (xb, yb) are the enter of the ns-th pixel and the b-thborder- ell (b = 1, . . . , Bs) of Υ1, respe tively.Then, at ea h step s of the pro ess (s = 1, ..., Smax), thefollowing optimization pro edure is repeated (Fig. 4.2):• Problem Unknown Representation - The ontrastfun tion is represented in terms of the level set fun tionas follows

τks(r) =

s∑

i=1

NIMSA∑

ni=1

τkiB

(rni

)r ∈ DI (4.2)where the index ks indi ates the k-th iteration at the s-th step [ks = 1, ..., kopt

s , B (rni

) is a re tangular basisfun tion whose support is the n-th sub-domain at the i-th resolution level [ni = 1, ..., NIMSA, i = 1, ..., s, andthe oe ient τkiis given by

τki=

τC ifΨki

(rni

)≤ 0

0 otherwise (4.3)lettingΨki

(rni

)=

φki

(rni

) if i = sφk

opti

(rni

) if (i < s) and (rni∈ Ri

)(4.4)with i = 1, ..., s as in (4.2).40

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHField Distribution Updating

Determine

Cost Function Evaluation

Level Set Update

Problem Unknown Representation

FALSE

TRUE

TRUE

Stop

Initialization

FALSEIMSA

Stopping Criterion

Stopping CriteriaLS

Compute Evki

(rni

)

rni∈ DI

xcs,y

cs,Ls

Determine ξvks

(rmv

)and compute Θ φks

rmv∈ DO

Compute φksfrom φks−1

τks(r) =

∑si=1

∑NIMSA

ni=1 τkiB

(rni

)

rni∈ DI

s-th resolution level

ks = ks + 1

Υsopt

Υs = Υ1

ks = 0

s = s + 1

γxc,γyc,γL

γτ , γΘ, γth, Kmax

Figure 4.2: Blo k diagram des ription of the IMSA-LS zoom-ing pro edure. 41

4.1. MATHEMATICAL FORMULATION• Field Distribution Updating - On e τks

(r) has beenestimated, the ele tri eld Evks

(r) is numeri ally om-puted a ording to a point-mat hing version of the Methodof Moments (MoM) [39 asEv

ki

(rni

)=

∑NIMSA

pi=1 ζv(rpi

) [1− τki

(rpi

)G2D

(rni

/rpi

)]−1,

rni, rpi∈ DI

ni = 1, ..., NIMSA .(4.5)• Cost Fun tion Evaluation - Starting from the totalele tri eld distribution (4.5), the re onstru ted s at-tered eld ξv

ks

(rm(v)

) at the m(v)-th measurement point,m(v) = 1, ..., M(v), is updated by solving the followingequationξvks

(rm(v)

)=

s∑

i=1

NIMSA∑

ni=1

τki

(rni

)Ev

ki

(rni

)G2D

(rm(v)/rni

)(4.6)and the t between measured and re onstru ted data isevaluated by the multi-resolution ost fun tion Θ denedasΘ φks

=

∑Vv=1

∑M(v)m(v)=1

∣∣∣ξvks

(rm(v)

)− ξv

ks

(rm(v)

)∣∣∣2

∑Vv=1

∑M(v)m(v)=1

∣∣ξvks

(rm(v)

)∣∣2 .(4.7)42

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACH• Minimization Stopping - The iterative pro ess stops[i.e., kopt

s = ks and τ opts = τks

when: (a) a set of ondi-tions on the stability of the re onstru tion holds true or(b) when the maximum number of iterations is rea hed[ks = Kmax, or ( ) when the value of the ost fun tion issmaller than a xed threshold γth. As far as the stabil-ity of the re onstru tion is on erned [ ondition (a), therst orresponding stopping riterion is satised when,for a xed number of iterations, Kτ , the maximum num-ber of pixels, the value of whi h has hanged, be omessmaller than a user-dened threshold γτ a ording to therelationshipmaxj=1,...,Kτ

NIMSA∑

ns=1

∣∣τks

(rns

)− τks−j

(rns

)∣∣τC

< γτ ·NIMSA.(4.8)The se ond riterion, about the stability of the re on-stru tion, is satised when the ost fun tion be omesstationary within a window of KΘ iterations as follows:

1

KΘ

KΘ∑

j=1

Θ φks −Θ φks−jΘ φks

< γΘ. (4.9)KΘ being a xed number of iterations and γΘ being user-dened thresholds;. When the iterative pro ess stops,the solution τ opt

s at the s-th step is sele ted as the onerepresented by the best level set fun tion φopts denedas

φopts = arg [minh=1,...,k

opts

(Θ φh)]. (4.10)43

4.1. MATHEMATICAL FORMULATION• Iteration Update - The iteration index is updated [ks →

ks + 1;• Level Set Update - The level set is updated a ordingto the following Hamilton-Ja obi relationship

φks

(rns

)= φks−1

(rns

)−∆tsVks−1

(rns

)H

φks−1

(rns

)(4.11)where H· is the Hamiltonian operator [29[30 given asH2

φks

(rns

)=

max2Dx−

ks; 0

+ min2

Dx+

ks; 0

+

+max2Dy−

ks; 0

+ min2

Dy+

ks; 0

if Vks

(rns

)≥ 0min2

Dx−

ks; 0

+ max2

Dx+

ks; 0

+

+min2Dy−

ks; 0

+ max2

Dy+

ks; 0

otherwise (4.12)with• Dx±

ks=

±φks(xns±1,yns)∓φks(xns ,yns)

ls,

• Dy±ks

=±φks(xns ,yns±1)∓φks (xns ,yns)

ls.

∆ts is the time-step hosen as ∆ts = ∆t1lsl1with ∆t1 to beset heuristi ally a ording to the literature [33, ls beingthe ell-side at the s-th resolution level. Vksis the velo ityfun tion omputed following the guidelines suggested in[33 by solving the adjoint problem of (4.5) in order to44

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHdetermine the adjoint eld F vks

(for more details, pleaserefer to the Appendix A). A ordingly,Vks

(rns

)= −ℜ

PVv=1 τCEv

ks(rns)F

vks

(rns)PVv=1

PM(v)m(v)=1|ξv

ks(rm(v))|2

,

ns = 1, ..., NIMSA

(4.13)where ℜ stands for the real part.When the s-th minimization pro ess terminates, the ontrastfun tion is updated [τ opts (r)= τks−1 (r), r ∈ DI (4.2) as well asthe RoI [Rs → Rs−1. To do so, the enter of Rs of oordinates

(xcs, y

cs) is determined as follows [19 [Fig. 4.1(b):

xcs =

∑si=1

∑NIMSA

ni=1 xniτ opts

(rni

)B

(rni

)∑s

i=1

∑NIMSA

ni=1 τ opts

(rni

)B

(rni

) (4.14)yc

s =

∑s

i=1

∑NIMSA

ni=1 yniτ opts

(rni

)B

(rni

)∑s

i=1

∑NIMSA

ni=1 τ opts

(rni

)B

(rni

) . (4.15)The estimated side Ls of Rs is omputed asLs = 2

∑s

i=1

∑NIMSA

ni=1

[dni,cs

τ opts

(rni

)B

(rni

)]∑s

i=1

∑NIMSA

ni=1

[τ opts

(rni

)B

(rns

)] (4.16)where dni,cs=

√(xni− xc

s)2 + (yni

− ycs)

2.On e the RoI has been identied, the level of resolution isenhan ed [ks → ks−1 only in this region by dis retizing Rs into45

4.1. MATHEMATICAL FORMULATIONNIMSA sub-domains [Fig. 4.1( ) and by repeating the mini-mization pro ess until the syntheti zoom be omes stationary(s = sopt), i.e.,

|Ωs−1 − Ωs||Ωs−1|

× 100

< γΩ, Ω = xc, yc, L (4.17)

γQ being a threshold set as in [19, or until a maximum num-ber of steps (sopt = Smax) is rea hed. As far as the regionsoutside Rs are on erned, the spatial resolution is left un- hanged as dened in the previous steps (rni, ni = 1, ..., NIMSA,

i = 1, ..., s − 1) and the values of the problem unknowns are omputed a ording to equation (4.4).At the end of the multi-step pro ess (s = sopt), the problemsolution is obtained as τ opt(rni

)= τ opt

s

(rni

), ni = 1, ..., NIMSA,i = 1, ..., sopt.

46

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACH4.2 Numeri al Validation:Syntheti DataIn order to assess the ee tiveness of the IMSA-LS approa h, asele ted set of representative results on erned with syntheti data is presented herein. The performan es a hieved are eval-uated by means of the following error gures:• Lo alization Error δ

δ|p =

√(xc

s|p − xc|p)2

−(

ycs|p − yc|p

)2

λ× 100 (4.18)where rc|p =

(xc|p , yc|p

) is the enter of the p-th trues atterer, p = 1, ..., P , P being the number of obje ts.The average lo alization error < δ > is dened as< δ >=

1

P

P∑

p=1

δ|p . (4.19)• Area Estimation Error ∆

∆ =

[I∑

i=1

1

NIMSA

NIMSA∑

ni=1

Nni

]× 100 (4.20)where Nni

is equal to 1 if τ opt(rni

)= τ

(rni

) and 0 oth-erwise. 47

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAAs far as the numeri al experiments are on erned, the re on-stru tions have been performed by blurring the s attering datawith an additive Gaussian noise hara terized by a signal-to-noise-ratio (SNR)SNR = 10log∑V

v=1

∑M(v)m(v)=1

∣∣ξv(rm(v)

)∣∣2∑V

v=1

∑M(v)m(v)=1 |µv,m(v)|2

(4.21)µv,m(v) being a omplex Gaussian random variable with zeromean value.4.2.1 Initializing with the True SolutionIn the rst experiment, a lossless ir ular s atterer of knownpermittivity ǫC = 1.8 and radius ρ = λ/4 is entered at xc =yc = λ/6 [33 in a square investigation domain of side LD = λ.V = 10 TM plane waves are impinging from the dire tions θv =2π (v − 1)/V , v = 1, ..., V , and the s attering measurementsare olle ted at M(v) = 10 re eivers uniformly distributed ona ir le of radius ρO = λ.As far as the initialization of the IMSA-LS algorithm is on erned, the initial trial obje t Υ1 is the true solution sam-pled at the resolution level s = 1. The initial value of the timestep is set to ∆t1 = 10−2 as in [33. The RoI is dis retizedin NIMSA = 15 × 15 sub-domains at ea h step of the itera-tive multi-resolution pro ess. Con erning the stopping riteria,the following onguration of parameters has been sele ted a - ording to a preliminary alibration dealing with simple knowns atterers and noiseless data: Smax = 4 (maximum number of48

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHx

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

(2)R

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50R =

(2)R

(3)

1 τ (x, y) 0( )Figure 4.3: Numeri al Data. Cir ular ylinder (ǫC = 1.8, LD =λ, Noiseless Case). Re onstru tions when initializing IMSA-LS with the true solution [(a) s = 1, (b) s = 2, and ( )s = sopt = 3 . 49

4.2. NUMERICAL VALIDATION:SYNTHETIC DATA

10-5

10-4

10-3

10-2

10-1

100

101

0 50 100 150 200

Θopt

Iteration, k

s=1 s=2 s=3

IMSA

Figure 4.4: Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the ost fun tion wheninitializing IMSA-LS with the true solution.

50

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHsteps), γexc

= γeyc

= 0.01 and γeL = 0.05 (multi-step pro essthresholds), Kmax = 500 (maximum number of optimizationiterations), γΘ = 0.2 and γτ = 0.02 (optimization thresholds),

KΘ = Kτ = 0.15 Kmax (stability ounters), and γth = 10−5(threshold on the ost fun tion).Figure 4.3 shows the re onstru tion a hieved at the end ofthe iterative steps of the multi-resolution pro edure [(a) s = 1- k = kopt, (b) s = 2 - k = kopt, and ( ) s = sopt = 3 - k = koptwhile the behavior of the ost fun tion is reported in Fig. 4.4.The regions of interest determined at steps s = 2 and s = 3are reported on the re onstru ted prole by the dashed/dotted ontour [Figs. 4.3(b) and ( ).As for the quality of the re onstru tions, it is worth notingthat Fig. 4.3(a) is the true obje t sampled at the level s = 1(i.e., it is equal to the the initial guess). Then, as the resolu-tion in reases, the re onstru tion improves as expe ted. As amatter of fa t, the shape of the obje t re onstru ted at the endof the pro ess [Fig. 4.3( ) appears to be better retrieved thanin Figs. 4.3(a) and (b). Su h an improvement is obvious alsoin the behavior of the ost fun tion, sin e the error jumps toa lower value as the spatial resolution improves. For the sakeof ompleteness, let us emphasize that the spikes of the ostfun tion between two adja ent steps are due to the updating ofthe eld distribution in orresponden e with the new resolutionlevel.When dealing with noisy data, the method still onverges tothe exa t solution, even though the quality of the re onstru -tions gets worse with respe t to the noiseless ase. Figure 4.5shows the solutions a hieved with dierent levels of additive51

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAx

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(3)

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(3)

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(2)

1 τ (x, y) 0( )Figure 4.5: Numeri al Data. Cir ular ylinder (ǫC = 1.8, LD =λ, Noisy Case). Re onstru tions when initializing IMSA-LSwith the true solution [(a) SNR = 20 dB, (b) SNR = 10 dB,and ( ) SNR = 5 dB . 52

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHnoise on s attered data [(a) SNR = 20 dB; (b) SNR = 10 dB;( ) SNR = 5 dB. The re onstru tions are similar to the a -tual obje t sampled at the highest resolution level. More indetail, the symmetry of the a tual obje t appears to be betterretrieved as the noise on the data de reases. As for the be-havior of the ost fun tion (Fig. 4.6), the nal error in reasesas the SNR in reases, as expe ted. Moreover, the jump ofthe ost fun tion due to the in rease of the spatial resolutionis less visible than in the noiseless ase, and it disappears withthe lowest SNR.4.2.2 Initializing with Exa t DataFor the sake of ompleteness, this subse tion deals with thebehavior of IMSA−LS when the initial guess is the true obje tsampled at the same resolution level used to solve the forwardproblem that generates the data (i.e., the inverse rime o urs).In order to perform the inversion, the number of sub-domainshas been for ed to NIMSA = Ndirect = 51 × 51, while theremaining parameters are the same of the previous se tion. Asexpe ted, the inversion algorithm stops immediately (sopt = 1- kopt = 0), sin e the stopping riterion on the value of the ost fun tion holds true. As a onsequen e, the re onstru tiona hieved is equal to the initial trial shape (Fig. 4.7).53

4.2. NUMERICAL VALIDATION:SYNTHETIC DATA

10-3

10-2

10-1

100

101

0 20 40 60 80 100

Θo

pt

Iteration, k

SNR = 20 dBSNR = 10 dBSNR = 5 dB

s = 1 s = 2 s = 2 s = 3

s = 1 s = 2

s = 2s = 1s = 3s = 2

Figure 4.6: Numeri al Data. Cir ular ylinder (ǫC = 1.8, LD =λ, Noisy Case). Behavior of the ost fun tion when initializingIMSA-LS with the true solution.4.2.3 Inversion of Data s attered by a Cir u-lar CylinderThe rst test ase with an initial guess dierent from the truesolution deals with the inversion of data s attered by the ir u-lar diele tri ylinder des ribed in Se t. 4.2.1. In su h a ase,the initial trial obje t Υ1 is a disk with radius λ/4 and enteredin DI while the other parameters are those sele ted previously.54

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHx

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 0(a)Figure 4.7: Cir ular ylinder (ǫC = 1.8, LD = λ, Noisy Case).Cir ular ylinder (ǫC = 1.8, LD = λ, Noiseless Case). Re on-stru tions when initializing IMSA-LS with the exa t data.Figure 4.8 shows samples of re onstru tions with the IMSA-LS. At the rst step [Fig. 4.8(a) - s = 1, the s atterer is or-re tly lo ated, but its shape is only roughly estimated. Thanksto the multi-resolution representation, the qualitative imagingof the s atterer is improved in the next step [Fig. 4.8(b) -s = sopt = 2 as onrmed by the error indexes in Tab. 4.1.For omparison purposes, the prole retrieved by the single-resolution method [33 (indi ated in the following as Bare-LS ),when DI has been dis retized in NBare = 31 × 31 equal sub-domains, is shown [Fig. 4.8( ). In general, the dis retiza-tion of the Bare-LS has been hosen in order to a hieve in55

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAx

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(2)

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 0 1 τ (x, y) 0( ) (d)Figure 4.8: Numeri al Data. Cir ular ylinder (ǫC = 1.8, LD =λ, Noiseless Case). Re onstru tions with IMSA-LS at (a) s =1, (b) s = sopt = 2, ( ) Bare-LS . Optimal inversion (d).56

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHIMSA − LS Bare − LS

s = 1 s = 2

δ 6.58 × 10−6 2.19 × 10−6 5.21 × 10−1

∆ 2.36 0.48 0.64Table 4.1: Numeri al Data. Cir ular ylinder (ǫC = 1.8, Noise-less Case). Error gures.the whole investigation domain a re onstru tion with the samelevel of spatial resolution obtained by the IMSA-LS in the RoIat s = sopt.Although the nal re onstru tions [Figs. 4.8(b)( ) a hievedby the two approa hes are similar and quite lose to the trues atterer sampled at the spatial resolution of Bare-LS [Fig.4.8(d) and IMSA-LS [Fig. 4.8(b), the IMSA-LS more faith-fully retrieves the symmetry of the a tual obje t, even thoughthe re onstru tion error appears to be larger than the oneof the Bare-LS (Fig. 4.9). During the iterative pro edure,the ost fun tion Θopt = Θ φopts is initially hara terized bya monotoni ally de reasing behavior. Then, Θopt⌋IMSA

be- omes stationary until the stopping riterion dened by re-lationships (4.8) and (4.9) is satised (Fig. 4.9 - s = 1). Then,after the update of the eld distribution indu ing the errorspike when s = sopt = 2 and ks = 1, Θopt⌋IMSAsettles to avalue of 6.28 × 10−4 whi h is of the order of the Bare-LS er-ror (Θopt⌋Bare

= 1.42 × 10−4). The slight dieren e betweenΘopt⌋IMSA

and Θopt⌋Baredepends on the dierent dis retiza-57

4.2. NUMERICAL VALIDATION:SYNTHETIC DATA

10-4

10-3

10-2

10-1

100

0 50 100 150 200

Θ

Iteration, k

s=1 s=2

IMSA (Θopt)

IMSA (Θ)

BARE

Figure 4.9: Numeri al Data. Cir ular ylinder (ǫC = 1.8, LD =λ, Noiseless Case). Behavior of the ost fun tion.tion [i.e., the basis fun tions B (

rn(i=2)

), n(i) = 1, ..., NIMSA arenot the same as those of Bare-LS , but it does not ae t the re- onstru tion in terms of both lo alization and area estimation,sin e δ⌋IMSA−LS < δ⌋Bare−LS and ∆⌋IMSA−LS < ∆⌋Bare−LS(Tab. 4.1).Fig. 4.9 also shows that the multi-step multi-resolutionstrategy is hara terized by a lower omputational burden be- ause of the smaller number of iterations for rea hing the on-vergen e (Fig. 4.9 - ktot⌋IMSA = 125 vs. ktot⌋Bare = 177,letting ktot the total number of iterations dened as ktot =58

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACH∑sopt

s=1 kopts for the IMSA-LS ), and espe ially of the redu ednumber of oating-point operations. As a matter of fa t, sin ethe omplexity of the LS -based algorithms is of the order of

O (2× η3), η = NIMSA, NBare (i.e., the solution of two dire tproblems is ne essary for omputing an estimate of the s at-tered eld and for updating the velo ity ve tor), the omputa-tional ost of the IMSA-LS at ea h iteration is two orders inmagnitude smaller than the one of the Bare-LS.The behavior of the multi-resolution level set fun tion Ψksduring the inversion pro edure is shown in Figs. 4.10 and 4.11.At iteration k = 1 of the step s = 1, the level set is initializeda ording to the oriented distan e fun tion (4.1) by onsideringthe initial guess Υ1. As it an be noti ed from Fig. 4.10(a), thelevel set fun tion is dened on the resolution level s = 1 andits value is lower than zero in the enter of the investigationdomain, namely inside the obje t dened by the trial solution.Then, the update of Ψksis arried out by means of the velo ityfun tion Vks

, whose behavior in the investigation domain atiteration k = 1 and at step s = 1 is reported in Fig. 4.12(a).After k = 20 iterations, the trial solution appears to be similarto the a tual obje t, sin e the region where φks=1 < 0 is en-tered on the bary entre of the a tual obje t [Fig. 4.10(b), andthe values of the velo ity fun tion are lower than at iterationk = 1 [Fig. 4.12(b), espe ially in the region where the trues atterer is lo ated. At the next step (i.e., s = sopt = 2), Ψks

atresolution level s = 2 is dened in the new RoI, while the levelset at the previous resolution level, φkopts=1

, is onsidered outsideRs=2. As for the update of the multi-resolution level set, thevelo ity fun tion is omputed only in Rs=2 [Fig. 4.13(a) and,59

4.2. NUMERICAL VALIDATION:SYNTHETIC DATA-0.005

0

0.005

0.01

-1/2-1/4

01/4

1/2-1/2

-1/4

0

1/4

0.01

0

-0.01

φ(x,y)

x/λ

y/λ(a)-0.005

0

0.005

0.01

-1/2-1/4

01/4

1/2-1/2

-1/4

0

1/4

0.01

0

-0.01

φ(x,y)

x/λ

y/λ(b)Figure 4.10: Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the level set φks

at s = 1[(a) k = 1, (b) k = 20. 60

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACH-0.005

0

0.005

0.01

-1/2-1/4

01/4

1/2-1/2

-1/4

0

1/4

0.01

0

-0.01

φ(x,y)

x/λ

y/λ(a)-0.005

0

0.005

0.01

-1/2-1/4

01/4

1/2-1/2

-1/4

0

1/4

0.01

0

-0.01

φ(x,y)

x/λ

y/λ(b)Figure 4.11: Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the level set φks

ats = sopt = 2 [(a) k = 1, (b) k = kopt.61

4.2. NUMERICAL VALIDATION:SYNTHETIC DATA-0.2

-0.1

0

0.1

0.2

-1/2-1/4

01/4

1/2-1/2

-1/4

0

1/4

0.20.1

0-0.1-0.2

V(x,y)

x/λ

y/λ(a)-0.2

-0.1

0

0.1

0.2

-1/2-1/4

01/4

1/2-1/2

-1/4

0

1/4

0.20.1

0-0.1-0.2

V(x,y)

x/λ

y/λ(b)Figure 4.12: Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the velo ity Vks

at s = 1[(a) k = 1, (b) k = 20. 62

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACH-0.006

-0.003

0

0.003

0.006

-1/2-1/4

01/4

1/2-1/2

-1/4

0

1/4

-0.006-0.003

0 0.003 0.006

V(x,y)

x/λ

y/λ(a)-0.006

-0.003

0

0.003

0.006

-1/2-1/4

01/4

1/2-1/2

-1/4

0

1/4

-0.006-0.003

0 0.003 0.006

V(x,y)

x/λ

y/λ(b)Figure 4.13: Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noiseless Case). Behavior of the velo ity Vks

at s =sopt = 2 [(a) k = 1, (b) k = kopt.63

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAx

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(2)

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)Figure 4.14: Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noisy Case). Re onstru tions with IMSA-LS (a) andBare-LS (b) for SNR = 20 dB.as expe ted, its ontribution be omes more and more negligibleas the iteration index ks=2 in reases [Fig. 4.13(b).As for the stability of the proposed approa h, Figures 4.14-4.16 show the re onstru tions with the IMSA-LS [Figs. 4.14-4.16(a) ompared to those of the Bare-LS [Figs. 4.14-4.16(b)with dierent levels of additive noise on the s attered data[Fig. 4.14 - SNR = 20 dB; Fig. 4.15 - SNR = 10 dB; Fig.4.16 - SNR = 5 dB (bottom). As expe ted, when the SNRde reases, the performan es worsen. However, as outlined bythe behavior of the error gures in Tab. 2, blurred data and/ornoisy onditions ae t more evidently the Bare implementationthan the multi-resolution approa h.64

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHx

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(3)

R(2)

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)Figure 4.15: Cir ular ylinder (ǫC = 1.8, LD = λ, Noisy Case).Re onstru tions with IMSA-LS (left olumn) and Bare-LS(right olumn) for SNR = 10 dB.In the se ond experiment, the same ir ular s atterer, but entered at xc = −yc = 7λ/15 within a larger investigationsquare of side LD = 2λ (ρO = 2λ), has been re onstru ted. A - ording to [15, M(v) = 20; v = 1, ..., V re eivers and V = 20views are onsidered and DI is dis retized in NIMSA = 13× 13pixels. Figure 4.17(a) shows the re onstru tion obtained atthe onvergen e (sopt = 3) by IMSA-LS when SNR = 5 dB.The result rea hed by the Bare-LS (NBARE = 47 × 47) isreported in Fig. 4.17(b) as well. As it an be noti ed, themulti-resolution inversion is hara terized by a better estima-tion of the obje t enter and shape as onrmed by the valuesof δ and ∆ (δ⌋IMSA−LS = 0.59 vs. δ⌋Bare−LS = 2.72 and65

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAx

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(3)

R(2)

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)Figure 4.16: Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = λ, Noisy Case). Re onstru tions with IMSA-LS (left olumn) and Bare-LS (right olumn) for SNR = 5 dB.∆⌋IMSA−LS = 0.48 vs. ∆⌋Bare−LS = 0.64). As for the ompu-tational load, the same on lusions from previous experimentshold true.As far as the behavior of the level set in this se ond exper-iment is on erned, Figure 4.18 shows that the fun tion φks=1at the iteration k = 2 presents some irregularities. The levelset is hara terized in the entre of the investigation domainby several spikes due to the update pro edure at the previousiteration [Fig. 4.18(a). Unfortunately, these peaks ompro-mise the omputation of the update term at k = 3, althoughthe behavior of the velo ity fun tion is regular [Fig. 4.18(b).As a matter of fa t, the level set at k = 3 appears to be even66

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHIMSA − LS Bare − LS

SNR = 20 dB

δ 5.91 × 10−1 2.72

∆ 0.98 1.28

SNR = 10 dB

δ 2.28 2.45

∆ 1.07 1.80

SNR = 5 dB

δ 6.78 × 10−1 1.63

∆ 1.50 2.07Table 4.2: Numeri al Data. Cir ular ylinder (ǫC = 1.8, NoisyCase). Values of the error indexes for dierent values of SNR.more irregular than at iteration k = 2.4.2.4 Re tangular S attererThe se ond test ase deals with a more omplex s attering onguration. A re tangular s atterer (L = 0.27λ and W =0.13λ) hara terized by a diele tri permittivity ǫC = 1.8 is entered at xc = −2

3λ, yc = λ within an investigation domainof LD = 3λ as indi ated by the red dashed line in Fig. 4.19. Insu h a ase, the imaging setup is made up of V = 30 sour esand M = 30 measurement points for ea h view v [15. DI ispartitioned into NIMSA = 19×19 sub-domains (while NBare =

33× 33) and ∆t1 is set to 0.06.67

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAx

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(2)

R(3)

x

y λ

λ−0.5 0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)Figure 4.17: Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = 2λ, Noisy Case). Re onstru tions with IMSA-LS (a)and Bare-LS (b) when SNR = 5 dB.Before dis ussing the re onstru tion apabilities, let us showa result on erned with the behavior of the proposed approa hwhen varying the user-dened thresholds (γΘ, γτ , γexc, γeyc , γeL)of the stopping riteria. Figure 4.19 displays the re onstru -tions a hieved by using the sets of parameters given in Tab. 4.3[Γ1 - Fig. 4.19(a); Γ2 - Fig. 4.19(b); Γ3 - Fig. 4.19( ); Γ4 - Fig.4.19(d) while the behaviors of the ost fun tion are depi tedin Fig. 4.20. As it an be noti ed, the total number of iter-ations ktot in reases as the values of the thresholds γΘ and γτde rease. However, in spite of a larger ktot, using lower thresh-old values does not provide better results, as shown by the omparison between settings Γ2 and Γ4 [Figs. 4.19(b)-(d), and68

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACH-0.08

0

0.08

0.16

-1 -1/2 0 1/2 1 -1-1/2

01/2

-0.08

0

0.08

0.16

φ(x,y)

x/λ

y/λ

-3-2-1 0 1 2

-1 -1/2 0 1/2 1 -1-1/2

01/2

-3-2-1 0 1 2

V(x,y)

x/λ

y/λ(a) (b)-0.08

0

0.08

0.16

-1 -1/2 0 1/2 1 -1-1/2

01/2

-0.08

0

0.08

0.16

φ(x,y)

x/λ

y/λ( )Figure 4.18: Numeri al Data. Cir ular ylinder (ǫC = 1.8,LD = 2λ, SNR = 5 dB). Behavior of the level set φks

and thevelo ity Vks. (a) φks

at k = 2, (b) Vksat k = 2, and ( ) φks

atk = 3.

69

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0( ) (d)Figure 4.19: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noiseless Case). Re onstru tions with IMSA-LS forthe dierent settings of Tab. 4.3 [(a) Γ1, (b) Γ2, ( ) Γ3, (d)Γ4. 70

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHSet of Parameters γΘ γτ γexc , γeyc γeL

Γ1 0.5 0.05 0.01 0.05

Γ2 0.2 0.02 0.01 0.05

Γ3 0.2 0.02 0.1 0.5

Γ4 0.02 0.002 0.01 0.05Table 4.3: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noiseless Case). Dierent settings for the parame-ters of the stopping riteria.Fig. 4.20. The sets of parameters hara terized by γΘ = 0.2and γτ = 0.02 provide a good trade-o between the arising omputational burden and the quality of the re onstru tions.As far as the stopping riterion of the multi-resolution pro e-dure is on erned, gure 9 also shows two dierent behaviorsof the ost fun tion when using Γ2 and Γ3 (letting γΘ = 0.2and γτ = 0.02). In parti ular, the proposed approa h stopsat sopt = 3, instead of sopt = 4, when in reasing by a degreeof magnitude the values of γexc , γeyc , and γeL. Although with aheavier omputational burden, the hoi e γexc = γeyc = 0.01 andγeL = 0.05 appears to be more ee tive [see Fig. 4.19(b) vs.Fig. 4.19( ).Figures 4.21-4.24 and Table 4.4 show the results from the omparative study arried out in orresponden e with dierentvalues of signal-to-noise ratio [SNR = 20 dB - Fig. 4.21(a) vs.Fig. 4.21(b); SNR = 10 dB - Fig. 4.22(a) vs. Fig. 4.22(b);SNR = 5 dB - Fig. 4.23(a) vs. Fig. 4.23(b). They fur-71

4.2. NUMERICAL VALIDATION:SYNTHETIC DATA

10-4

10-3

10-2

10-1

100

101

0 200 400 600 800 1000 1200 1400

Θopt

Iteration, k

Γ3Γ2 Γ4

Γ1

Γ1Γ2, Γ3

Γ4

Figure 4.20: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noiseless Case). Behavior of the ost fun tion ofIMSA-LS for the dierent settings of Tab. 4.3.

72

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHIMSA − LS Bare − LS

SNR = 20 dB

ktot 1089 41

N 361 1089

fpos 1.02 × 1011 1.02 × 1011

SNR = 10 dB

ktot 393 53

N 361 1089

fpos 3.70 × 1010 1.37 × 1011

SNR = 5 dB

ktot 410 28

N 361 1089

fpos 3.86 × 1010 7.23 × 1010Table 4.4: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noisy Case). Computational indexes for dierentvalues of SNR.

73

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAther onrm the reliability and e ien y of the multi-resolutionstrategy in terms of qualitative re onstru tion errors (Fig. 4.24),espe ially when the noise level grows. In parti ular, the Bareimplementation does not yield either the position or the shapeof the re tangular s atterer when SNR = 5 dB, whereas theIMSA-LS properly retrieves both the bary enter and the on-tour of the target. As for the omputational ost, it should benoti ed that although the IMSA-LS requires a greater numberof iterations for rea hing the onvergen e (Figs. 4.21-4.23( ),and Tab. 4.4), the total amount of omplex oating-point op-erations, fpos = O (2× η3)× ktot, usually results smaller (Tab.4.4).

74

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)

10-3

10-2

10-1

100

101

0 200 400 600 800 1000 1200 1400

Θopt

Iteration, k

IMSABARE approach

( )Figure 4.21: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noisy Case). Re onstru tions with IMSA-LS (a)and Bare-LS (b) for SNR = 20 dB. ( ) Behavior of the ostfun tion versus the iteration index.75

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)

10-2

10-1

100

101

0 100 200 300 400 500 600

Θopt

Iteration, k

IMSABARE approach

( )Figure 4.22: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noisy Case). Re onstru tions with IMSA-LS (a)and Bare-LS (b) for SNR = 10 dB. ( ) Behavior of the ostfun tion versus the iteration index.76

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)

10-1

100

101

0 100 200 300 400 500 600

Θopt

Iteration, k

IMSABARE approach

( )Figure 4.23: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noisy Case). Re onstru tions with IMSA-LS (a)and Bare-LS (b) for SNR = 5 dB. ( ) Behavior of the ostfun tion versus the iteration index.77

4.2. NUMERICAL VALIDATION:SYNTHETIC DATA

100

101

102

103

5 10 15 20

δ

SNR [dB]

IMSABARE

(a)2.5

2.0

1.5

1.0

0.5

0.0 5 10 15 20

∆

SNR [dB]

IMSABARE

(b)Figure 4.24: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noisy Case). Values of the error gures versus SNR.

78

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACH4.2.5 Hollow CylinderThe third test ase is on erned with the inversion of the datas attered by a higher-permittivity (ǫC = 2.5) ylindri al ring entered at xc = yc = 34λ, letting LD = 3λ. The outer radiusof the ring is ρext = 2

3λ, and the inner one is ρint = λ

3. Byassuming the same arrangement of emitters and re eivers asin Se tion 4.2.4, the investigation domain is dis retized with

NIMSA = 19×19 and NBare = 35×35 square ells for the IMSA-LS and the Bare-LS, respe tively. Moreover, ∆t1 is initializedto 0.003.x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

2 τ (x, y) 0 2 τ (x, y) 0(a) (b)Figure 4.25: Numeri al Data. Hollow ylinder (ǫC = 2.5, LD =3λ, Noisy Case). Re onstru tions with IMSA-LS (left olumn)and Bare-LS (right olumn) for SNR = 20 dB .79

4.2. NUMERICAL VALIDATION:SYNTHETIC DATAx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

2 τ (x, y) 0 2 τ (x, y) 0(a) (b)Figure 4.26: Numeri al Data. Hollow ylinder (ǫC = 2.5, LD =3λ, Noisy Case). Re onstru tions with IMSA-LS (left olumn)and Bare-LS (right olumn) for SNR = 10 dB .As it an be observed from Figs. 4.25-4.26, where the pro-les when SNR = 20 dB [Figs. 4.25(a)(b) and SNR = 10 dB[Figs. 4.26(a)(b) re onstru ted by means of the IMSA-LS[Figs. 4.25-4.26(a) and the Bare-LS [Figs. 4.25-4.26(b) areshown, the integrated strategy over omes the standard oneboth in lo ating the obje t and in estimating the shape. In par-ti ular, when SNR = 20 dB, the distribution in Fig. 4.25(a)is a faithful estimate of the s atterer under test (δ⌋IMSA−LS =1.25 and ∆⌋IMSA−LS = 3.13). On the ontrary, the re on-stru tion with the Bare-LS is very poor (δ⌋Bare−LS = 65.2and ∆⌋Bare−LS = 34.39). Certainly, a smaller SNR valueimpairs the inversion as shown in Fig. 4.26(a) [ ompared80

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHto Fig. 4.25(a). However, in this ase, the IMSA-LS isstill able to properly lo ate the obje t (δ⌋IMSA−LS = 1.7 vs.δ⌋Bare−LS = 65.9) giving rough but useful indi ations aboutits shape (∆⌋IMSA−LS = 7.6 vs. ∆⌋Bare−LS = 34.55).4.3 Numeri al Validation by means ofLaboratory-Controlled DataIn order to further assess the ee tiveness of the IMSA-LSalso in dealing with experimental data, the multiple-frequen yangular-diversity bi-stati ben hmark provided by Institut Fres-nel in Marseille (Fran e) has been onsidered. With refer-en e to the experimental setup des ribed in [100, the datasetdielTM_de 8f.exp has been pro essed. The eld samples[M(v) = 49, V = 36 are related to an o- entered homo-geneous ir ular ylinder ρ = 15mm in diameter, hara ter-ized by a nominal value of the obje t fun tion equal to τ(r) =2.0 ± 0.3, and lo ated at xc = 0.0, yc = −30mm within aninvestigation domain assumed in the following of square geom-etry and extension 30× 30 m2.By setting ǫC = 3.0, the re onstru tions a hieved are shownin Figs. 4.27-4.28 (left olumn) ompared to those from thestandard LS (right olumn) at F = 4 dierent operation fre-quen ies. Whatever the frequen y, the unknown s atterer isa urately lo alized and both algorithms yield, at onvergen e,stru tures that o upy a large subset of the true obje t. Su ha similarity of performan es, usually veried in syntheti ex-81

4.3. NUMERICAL VALIDATION BY MEANS OFLABORATORY-CONTROLLED DATAf = 1 GHz f = 2 GHz

IMSA − LS Bare − LS IMSA − LS Bare − LS

ktot 506 69 532 200

fpos 4.88 × 109 1.22 × 1011 5.14 × 109 3.55 × 1011

f = 3 GHz f = 4 GHz

IMSA − LS Bare − LS IMSA − LS Bare − LS

ktot 678 198 621 200

fpos 6.55 × 109 3.51 × 1011 5.99 × 109 3.55 × 1011Table 4.5: Experimental Data (Dataset Marseille [100). Cir- ular ylinder (dielTM_de 8f.exp). Computational indexes.periments when the value of SNR is greater than 20 dB, seemsto onrm the hypothesis of a low-noise environment as it wasalready eviden ed in [101.Finally, also in dealing with experimental datasets, the IMSA-LS proves its e ien y sin e the overall amount of omplexoating point operations still remains two orders in magnitudelower than the one of the Bare-LS (Tab. 5 - Figs. 4.30-4.29).82

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACHx

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

3 τ (x, y) 0 3 τ (x, y) 0(a) (b)x

y λ

λ−0.5 0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

x

y λ

λ−0.5 0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

3 τ (x, y) 0 3 τ (x, y) 0( ) (d)Figure 4.27: Experimental Data (Dataset Marseille [100).Cir ular ylinder (dielTM_de 8f.exp). Re onstru tions withIMSA-LS (left olumn) and Bare-LS (right olumn) at dier-ent frequen ies f [f = 1 GHz (a)(b); f = 2 GHz ( )(d).83

4.3. NUMERICAL VALIDATION BY MEANS OFLABORATORY-CONTROLLED DATAx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

3 τ (x, y) 0 3 τ (x, y) 0(a) (b)x

y λ

λ−1.0 1.0

1.0

−1.0

2.0

−2.0

−2.0

2.0

x

y λ

λ−1.0 1.0

1.0

−1.0

2.0

−2.0

−2.0

2.0

3 τ (x, y) 0 3 τ (x, y) 0( ) (d)Figure 4.28: Experimental Data (Dataset Marseille [100).Cir ular ylinder (dielTM_de 8f.exp). Re onstru tions withIMSA-LS (left olumn) and Bare-LS (right olumn) at dier-ent frequen ies f [f = 3 GHz (a)(b); f = 4 GHz ( )(d).84

CHAPTER 4. THE MULTI-RESOLUTIONLEVEL SET APPROACH10

-3

10-2

10-1

100

101

102

0 200 400 600 800 1000

Θo

pt

Iteration, k

IMSABARE approach

(a)10

-3

10-2

10-1

100

101

102

0 200 400 600 800 1000

Θo

pt

Iteration, k

IMSABARE approach

(b)Figure 4.29: Experimental Data (Dataset Marseille [100).Cir ular ylinder (dielTM_de 8f.exp). Behavior of the ostfun tion versus the number of iterations when (a) f = 3 GHz,and (b) f = 4 GHz. 85

4.3. NUMERICAL VALIDATION BY MEANS OFLABORATORY-CONTROLLED DATA10

-2

10-1

100

101

102

0 200 400 600 800 1000

Θo

pt

Iteration, k

IMSABARE approach

(a)10

-3

10-2

10-1

100

101

102

0 200 400 600 800 1000

Θo

pt

Iteration, k

IMSABARE approach

(b)Figure 4.30: Experimental Data (Dataset Marseille [100).Cir ular ylinder (dielTM_de 8f.exp). Behavior of the ostfun tion versus the number of iterations when (a) f = 1 GHz,and (b) f = 2 GHz. 86

Chapter 5The Multi-RegionApproa hThis hapter deals with the iterative multi-region multi-s alingapproa h with level-set-based minimization, ustomized for ge-ometries hara terized by multiple obje ts. After presentingthe mathemati al formulation by fo using onto the main dif-feren es with respe t to the single-region version, the ee tive-ness of the approa h is evaluated by means of the dis ussionof a sele ted set of results, when dealing both with numeri aldata and with laboratory- ontrolled data.87

5.1. MATHEMATICAL FORMULATION5.1 Mathemati al FormulationThis se tion is aimed at presenting the multi-s aling multi-region approa h (IMSMRA) integrated with level-set-based op-timization (LS ). Let us onsider a set of P homogeneous ob-sta les with relative permittivity ǫC and ondu tivity σC thato upy the regions Υp, p = 1, ..., P , belonging to an investiga-tion domain DI . As in the previous hapter, su h a s enario isprobed by a set of V transverse-magneti (TM) plane waves,with ele tri eld ζv(r) = ζv(r)z (v = 1, . . . , V ), r = (x, y).The s attered eld, ξv(r) = ξv(r)z, is olle ted at M(v), v =1, ..., V , measurement points distributed in a region, alled ob-servation domain DO, external to the investigation domain.In order to retrieve the unknown position and shape of thetarget Υp, p = 1, ..., P , a multi-step pro edure aimed at suit-ably in reasing the spatial resolution only in a set of regions ofinterest (RoIs) ontaining the s atterers is onsidered [80[81.With respe t to the IMSA-LS dis ussed in Se t. 4.1, the strat-egy presented herein is able to deal with a set of q = 1, ..., Qmaxregions to better allo ate the unknowns when the s enario is hara terized by several obje ts distan ed from one another.With referen e to Fig. 5.1, at the rst step (s = 1, s beingthe step index) an initial guess shape Υs=1 belonging to DIis hosen. Sin e no a-priori information on the s enario undertest is assumed, the region of interest R

(qs)s , qs = 1, ..., Qs, at

s = 1 (i.e., R(qs)s=1 = DI , Qs=1 = 1) is partitioned into NMRequal square sub-domains, NMR being the number of degreesof freedom of the problem at hand [14. The initialization ofthe level set fun tion φ

(qs)s , with s = 1 and qs = 1, is arried88

CHAPTER 5. THE MULTI-REGION APPROACHout by means of the oriented distan e fun tion (4.1) as in Se t.4.1. Then, at ea h step s of the pro ess (s = 1, ..., Smax), thefollowing optimization pro edure is repeated:• Problem Unknown Representation - The problemunknown is represented at the ks-th iteration, ks = 1, ..., kopt

s ,as followsτks

(r)|qs=

s∑

i=1

Qi∑

qi=1

N(qi)⌋MR∑

n(qi)=1

τki|qiB

(rn(qi)

)r ∈ DI(5.1)where N(qi)⌋MR is the number of sub-domains used todis retize R

(qi)i and B (

rn(qi)

) is a re tangular basis fun -tion whose support is the n-th sub-domain of the qi-thregion of interest at the i-th resolution level. Moreover,the oe ient τ(qi)ki

is given byτki|qi

=

τC ifΨki

(rn(qi)

)≤ 0

0 otherwise (5.2)lettingΨki

(rn(qi)

)=

φki

(rn(qi)

) if i = s

φkopti

(rn(qi)

) if i < s and rn(qi)∈ R

(qi)i

.(5.3)• Field Distribution Updating - After updating the prob-lem unknown τ

(qi)ki

, the value of the ele tri eld Evks

(r) inthe n(qi)-th sub-domains of the qi-th region of interest is89

5.1. MATHEMATICAL FORMULATIONProblem Unknown Representation

Initialization

Field Distribution Updating

Cost Function Evaluation

FALSE

TRUE

Stopping CriteriaLS

Determine

FALSE

TRUE

Stop

IMSAStopping Criterion

Level Set Update

qi = 1, ..., Qi

qs = 1, ..., Qs

qs = 1, ..., Qs

τks(r) |qs

=∑s

i=1

∑Qi

qi=1

∑N(qi)⌋MR

n(qi)=1 τki|qiB

(rn(qi)

)

rn(qi)∈ DI

s-th resolution level

Υs|qs= Υ1 Qs=1 = 1

Compute Evki

(rn(qi)

)rn(qi)

∈ DI

Determine ξvks

(rmv

)and Θ φks

rmv∈ DO

ks ← ks + 1γH, γΘ, γth, Kmax

xcs|qs

,ycs|qs

,Ls|qs,Ws|qs

ks ← 0

s← s + 1

Υsopt|qs

γxcs|qs

,γycs|qs

,γLs|qs

,γWs|qs

Compute φks|qs

Figure 5.1: Blo k diagram des ription of the IMSMRA-LSzooming pro edure. 90

CHAPTER 5. THE MULTI-REGION APPROACH omputed by means of a ustomized numeri al te hniquebased on the point-mat hing version of the Method ofMoments (MoM) [39 a ording to the following relation-shipEv

ki

(rn(qi)

)=

=∑Qi

qi=1

∑N(qi)⌋MR

n(qi)=1 ζv(rpi

)×

×[1− τki

(rp(qi)

)G2D

(rn(qi)

/rp(qi)

)]−1,

rn(qi), rp(qi)

∈ DI

n(qi) = 1, ..., N(qi)⌋MR . (5.4)• Cost Fun tion Evaluation - The re onstru ted s at-tered eld ξv

ks

(rm(v)

) at the m(v)-th measurement point,m(v) = 1, ..., M(v), is updated by the following relation-ship

ξvks

(rm(v)

)=

∑si=1

∑Qi

qi=1

∑N(qi)⌋MR

n(qi)=1 τki

(rn(qi)

)×

×Evki

(rn(qi)

)G2D

(rm(v)/rn(qi)

)(5.5)where the total ele tri eld distribution Evki

(rn(qi)

) isgiven by (5.4). Then, the ost fun tion is evaluated atthe iteration ks by onsidering the relationship (4.7).• Minimization Stopping - The iterative pro ess stops(i.e., kopt

s = ks and τ opts = τks

) when a set of onditions on erned with the stability of the re onstru tion be- omes true or when the maximum number of iterations is91

5.1. MATHEMATICAL FORMULATIONrea hed (ks = Kmax) or when the value of the ost fun -tion is smaller than a xed threshold γth. As far as thestability of the re onstru tions is on erned, the followingdouble riterion is onsidered: The rst ondition is aimed at assessing if the shapeof the trial solution does not hange during the iter-ative pro ess. In order to over ome the limitations hara terizing the standard riterion (4.8) when theiterative solution is hara terized by a blinking be-havior1, a new strategy based on the Hausdor dis-tan e L [102-[105 is adopted. More in detail, su ha riterion is satised when in the region of interestR

(qs)s , qs = 1, ..., Qs, for a xed number of iterations,

KL, the value of the Hausdor distan e omputedbetween the ontour of the (ks − j)-th trial solution,j = 1, ..., KL, and the ontour of the urrent trial so-lution is smaller than a user dened threshold γL.A ordingly, the following relationship is evaluatedmaxj=1,...,KL

L∂

(τks|qs

), ∂

(τks−j|qs

)

ls|qs

< γL(5.6)where the operator ∂ (·) (Se t. B.1) performs theedge dete tion, providing the Bks|qs

ontour sub-1 The iterative solution is hara terized by a blinking behavior whena small amount of pixels of the re onstru tion turns up intermittently (i.e.,the level set emerges intermittently), without signi antly modifying theestimated shapes. 92

CHAPTER 5. THE MULTI-REGION APPROACHdomains of the re onstru ted shapes, and ls|qsisthe average value of the ell-sides in the region qs atthe step s, omputed as follows

ls|qs=

∆xs|qs+ ∆ys|qs

2(5.7)

∆xs and ∆ys being the ell-side along x and y, re-spe tively. For the sake of ompleteness, the Haus-dor distan e L is dened asL

∂

(τks|qs

), ∂

(τks−j |qs

)=

= maxmaxrb(qs)minrp(qs)

dks,qs,j,b,p ,maxrp(qs)minrb(qs)

dks,qs,j,b,p

b(qs) = 1, ..., Bks|qs

p(qs) = 1, ..., Bks−j|qs(5.8)wheredks,qs,j,b,p =

[(xB

ks

∣∣b(qs)− xB

ks−j

∣∣p(qs)

)2

+

+(

yBks

∣∣b(qs)− yB

ks−j

∣∣p(qs)

)2] 1

2

,(5.9)(xB

ks

∣∣b(qs)

, yBks

∣∣b(qs)

) and (xB

ks−j

∣∣p(qs)

, yBks−j

∣∣p(qs)

) be-ing the oordinates of the Bks|qs

ontour sub-domainsdete ted by ∂ (·). 93

5.1. MATHEMATICAL FORMULATION The se ond riterion, about the stability of the re- onstru tion, is satised when the ost fun tion be- omes stationary within a window of KΘ iterationsas in (4.9).• Iteration Update - The iteration index is updated (ks →

ks + 1).• Level Set Update - As for IMSA-LS, the level set in theregion qs at the step s is updated by solving the followingHamilton-Ja obi equation

φks

(rn(qs)

)= φks−1

(rn(qs)

)−

− ∆ts|qsVks−1

(rn(qs)

)H

φks−1

(rn(qs)

)(5.10)where H· is the Hamiltonian operator given asH2

φks

(rn(qs)

)=

=

max2Dx−

ks

∣∣qs

; 0

+ min2Dx+

ks

∣∣qs

; 0

+

+max2Dy−

ks

∣∣qs

; 0

+ min2Dy+

ks

∣∣qs

; 0if Vks

(rn(qs)

)≥ 0min2

Dx−

ks

∣∣qs

; 0

+ max2Dx+

ks

∣∣qs

; 0

+

+min2Dy−

ks

∣∣qs

; 0

+ max2Dy+

ks

∣∣qs

; 0otherwise (5.11)94

CHAPTER 5. THE MULTI-REGION APPROACHwith• Dx±

ks

∣∣qs

=±φks(xn(qs)±1,yn(qs))∓φks(xn(qs),yn(qs))

∆xs|qs

,• Dy±

ks

∣∣qs

=±φks(xn(qs),yn(qs)±1)∓φks(xn(qs),yn(qs))

∆ys|qs

.Furthermore, Vks

(rn(qs)

) is the value of the velo ity fun -tion in the n(qs)-th sub-domains of the qs-th region of in-terest at the step s and∆tks|qs

is the time-step hosen bymeans of the Courant-Friedri h-Leroy onstraint [29[30as follows∆tks|qs

=min

∆xs|qs, ∆ys|qs

maxn(qs)Vks

(rn(qs)

) . (5.12)In order to get the velo ity fun tion in the qs-th region ofinterest, the adjoint problem of (5.4) is solved and start-ing from the adjoint eld F vks

∣∣qsthe following relationshipis evaluated

Vks

(rn(qs)

)= −ℜ

PVv=1 τC

eEvks

(rn(qs))Fvks

(rn(qs))PV

v=1

PM(v)m(v)=1|ξv

ks(rm(v))|2

,

n(qi) = 1, ..., N(qi)⌋MR .(5.13)• Level Set Re-Initialization - Unfortunately, the re-sult of the update pro edure on the level set is not adistan e fun tion, and, in general, the level set repre-sentation is not unique [32[106. To restore the ori-95

5.1. MATHEMATICAL FORMULATIONented distan e fun tion thus redu ing numeri al prob-lems on erned with the omputations of the nite dif-feren es Dks, a fun tional approa h is onsidered herein.In parti ular, if H

φks

(rn(qs)

), whi h is an approxima-tion of ∣∣∇φks

(rn(qs)

)∣∣, is greater than a ertain thresholdγφ, n(qs) = 1, ..., N(qi), then the level-set fun tion is re-initialized as

φks

(rn(qs)

)=

−minb=1,...,Bqs

dn(qs),b

if τ (rn(qs)

)= τCminb=1,...,Bqs

dn(qs),b

if τ (rn(qs)

)= 0 ,(5.14)where dn(qs),b is the distan e between rn(qs) and rb, rbbeing b-th border- ell (b = 1, . . . , Bqs

) of the trial shapere onstru ted in the region qs, qs = 1, ..., Qs.At the end of the s-th minimization pro ess, the ontrast fun -tion is updated [ τ opts |qs

= τks|qs, r ∈ DI and qs = 1, ..., Qsby means of (5.1) and the new regions of interest R

(qs)s , qs =

1, ..., Qs, are dened. To do so, the following set of operationsis repeated for all the regions determined at the step s (Fig.5.2):• Find the Number of Seeds - Sin e the re onstru -tions provided by the level-set-based strategies are bi-nary, the rst step of the morphologi al pro essing2 is2 The term morphologi al refers to the study of the shapes of the re- onstru ted s atterers in order to get the most suitable regions of interest.96

CHAPTER 5. THE MULTI-REGION APPROACHaimed at ounting the number of targets whi h have beenre onstru ted. Su h a pro edure, alled erosion [80, on-sists in reating a new image τEs

∣∣qsof the shapes re on-stru ted in the qs-th region, qs = 1, ..., Qs, as follows

τEs

(xn(qs), yn(qs)

)=

=

τC if τ opts

(xn(qs), yn(qs)

)= τC and∑1

p=−1

∑1j=−1 τ opt

s

(xn(qs)−p, yn(qs)−j

)< 9 · τC

0 otherwise .(5.15)The qs-th arising image ontains at least one pixel forea h obje t [Fig. 5.2(b). Unfortunately, these pixels, alled seeds, bring no information about the a tual num-ber of targets (e.g., a hollow dis has two seeds, the rston the inner ontour and the se ond on the outer on-tour).• Count the Number of Obje ts - Sin e the seeds arene essarily lo ated on the ontours of the re onstru tedshapes, the number of obje ts is ounted by exploitingboth the information provided by the previous step andthe edge dete tion operator ∂ (·). That is, the ontour

∂(

τ opts |qs

) of the shapes re onstru ted in the qs-th re-gion of interest at the step s is found [Fig. 5.2( ) andthe boundaries of the re onstru ted s atterers3 are de-te ted by walking along the arising edges starting from3 The term boundary refers to a ontour of one obje t (e.g., thehollow ylinder has two boundaries).97

5.1. MATHEMATICAL FORMULATIONrn

τ (rn) = 0

τ (rn) = τC

Seeds(a) (b)Contour of the shapes

RoI 1

RoI 2

Boundary of object 2

Boundary of object 1

A

A

( ) (d)Figure 5.2: Graphi al representation of the morphologi alpro essing: (a) prole re onstru ted at step s (Qs = 1), (b)dete tion of the seeds by means of the erosion pro edure,( ) walking around the edges, and (d) dete tion of obje t'sboundaries and of the regions of interest for the resolution levels + 1. 98

CHAPTER 5. THE MULTI-REGION APPROACHthe seeds (more details about the algorithm are givenin Se t. B.2). As a onsequen e, the a tual number ofobje ts is determined by ounting the number of bound-aries, negle ting those that are in luded within other on-tours [Fig. 5.2(d).• Dene the Parameters of the Region of Interest -

Qs+1 regions of interest are dened at the step s + 1 byevaluating the maximum size of the boundaries dete tedin the previous step. In order to prevent the RoIs to bein luded in the obje ts or overlapping between adja entregions, the oordinates of the enter xcs+1

∣∣qs+1

, ycs+1

∣∣qs+1and the sides Ls+1

∣∣∣qs+1

, Ws+1

∣∣∣qs+1

are omputed by in- reasing the a tual size of the regions by an allowan evalue As+1, s = 1, ..., Smax − 1. Moreover, Ls+1

∣∣∣qs+1

,Ws+1

∣∣∣qs+1

have to be hosen su h thatmin(Ls+1

∣∣∣qs+1

, Ws+1

∣∣∣qs+1

)max(Ls+1

∣∣∣qs+1

, Ws+1

∣∣∣qs+1

) >3

4(5.16)in order to allow an a urate solution of the forward prob-lem a ording to the point-mat hing version of the MoM[39, sin e the sides of the RoIs are dis retized using thesame number of sub-domains.After the denition of the parameter of the RoIs, the step index99

5.1. MATHEMATICAL FORMULATIONis in reased (s← s + 1) and a onvergen e he k of the multi-s aling pro edure is performed. The re onstru tion algorithmis stopped if both lo ation and sides of the region of interestbe ome stationary (sopt = s) [19, i.e., |Ωs−1 − Ωs||Ωs−1|

× 100

< γΩ, Ω = xc

s|qs, yc

s|qs, Lc

s

∣∣∣qs

, W cs

∣∣∣qs(5.17)

γΩ being a user dened threshold, or when a maximum numberof steps (sopt = Smax) is rea hed.If the onvergen e he k does not hold true, the total amountNMR of available basis fun tions is split in the Qs regions tokeep the ratio between data and unknown onstant, a ordingto the following relationship

N(qs)⌋MR = L

NMR

eLs|qs× fWs|

qsPQsps=1

eLs|ps

×fWs|ps

qs = 1, ..., Qs

(5.18)where the operator L returns the smaller integer part of itsargument.Finally, at the end of the multi-step pro ess (s = sopt),the problem solution is obtained as τ opt(rn(qi)

)= τ opt

s

(rn(qi)

),n(qi) = 1, ..., N(qi)⌋MR, qi = 1, ..., Qi, i = 1, ..., sopt.

100

CHAPTER 5. THE MULTI-REGION APPROACH5.2 Preliminary Validation5.2.1 The IMSMRA-LS when dealingwith Simple GeometriesIn order to preliminary test the multi-region approa h, this test ase deals with the geometry onsidered in the se ond experi-ment of se tion 4.2.3, namely with the ylinder having radiusλ/4 and entered at xc = −yc = 7λ/15 in an investigationsquare of side LD = 2λ. The purpose of this se tion is twofold:on one hand, to assess the re onstru tion apabilities of themulti-region approa h when the geometry is hara terized bya single obje t. On the other hand, to dis uss the behaviorof the multi-resolution level set during the iterative pro ess,pointing out benets and drawba ks of the update strategyadopted in the multi-region approa h.As for the initialization of the IMSMRA-LS te hnique, theinitial guess is a ir ular s atterer of radius λ/4 entered in theinvestigation domain and sampled at the resolution level s = 1.The RoI is dis retized in NMR = 13× 13 sub-domains at ea hstep of the iterative multi-region multi-resolution pro edure,while the maximum number of RoI is set to Qmax = 10. Thestopping riterion has been ongured as follows: γφ =

√2(maximum value for the numeri al Hamiltonian), Smax = 5(maximum number of steps), γexc

= γeyc

= 0.01 and γeL = 0.05(multi-step pro ess thresholds), Kmax = 500 (maximum num-101

5.2. PRELIMINARY VALIDATIONx

y λ

λ−0.5 0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

(2)R

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.25 0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R(2)

R(3)

1 τ (x, y) 0( )Figure 5.3: Numeri al Data. Cir ular ylinder (ǫC = 1.8, LD =2λ, Noisy Case). Re onstru tions with IMSMRA-LS at (a)s = 1, (b) s = 2, and ( ) s = sopt = 3.102

CHAPTER 5. THE MULTI-REGION APPROACHber of optimization iterations), γΘ = 0.2 and γL = 1.5 (opti-mization thresholds), KΘ = KL = 0.10 Kmax (stability oun-ters), and γth = 10−5 (threshold on the ost fun tion). In ad-dition, at the s-th step and in the q-th region the value ∆tks|qsof the time-step, with s = 1, ..., sopt and qs = 1, ..., Qmax, is hosen a ording to the CFL-based relationship, while thevalues of the allowan e As for estimating the size of the re-gions of interest are set to As=2 = 40%, As=3 = 30%, and

As=4 = As=5 = 20%.Figure 5.3 shows the re onstru tion a hieved when onsid-ering syntheti data blurred by noise hara terized by SNR =5 dB. Be ause of the low SNR and of the oarse resolutionlevel, the result obtained at the end of step s = 1 [Fig. 5.3(a)appears to be ina urate in terms of shape estimation, whilethe bary entre of the s atterer is quite a urately estimated.However, in spite of the ina urate re onstru tion at the rststep, the behavior of the level set φks

|qs, qs = 1, appears tobe quite regular [Fig. 5.4(a), espe ially if ompared with thebehavior of the level set at the rst iterations of the se ond ex-periment of Se t. 4.2.3. Su h a regular shape, even at the lastiteration of the step, is mainly due to the re-initialization pro- edure des ribed in Se t. 5.1 and to the CFL-based pro edurefor hoosing ∆tks

|qs.Thanks to the in rease of the spatial resolution a hieved inthe RoI at s = 2, the shape of the a tual s atterer is betterestimated [Fig. 5.3(b). Then, at the nal step, s = sopt =

3, the RoI is further fo used on the area that ontains thetrue obje t and the a ura y of the re onstru tion is furtherimproved [Fig. 5.3( ). Furthermore, Figs. 5.4(b) and ( ) show103

5.2. PRELIMINARY VALIDATION-0.02

0

0.02

0.04

0.06

-1-1/2

01/2 -1

-1/2

0

1/2-0.02 0

0.02 0.04 0.06

φ(x,y)

x/λ

y/λ

-0.02

0

0.02

0.04

0.06

-1-1/2

01/2 -1

-1/2

0

1/2-0.02 0

0.02 0.04 0.06

φ(x,y)

x/λ

y/λ(a) (b)-0.02

0

0.02

0.04

0.06

-1-1/2

01/2 -1

-1/2

0

1/2-0.02 0

0.02 0.04 0.06

φ(x,y)

x/λ

y/λ( )Figure 5.4: Numeri al Data. Cir ular ylinder (ǫC = 1.8, LD =2λ, Noisy Case). Behavior of the level set φks

at the end of thestep (a) s = 1, (b) s = 2, and ( ) s = sopt = 3.104

CHAPTER 5. THE MULTI-REGION APPROACH

10-1

100

0 50 100 150 200

Θopt

Iteration, k

s = 1 s = 2 s = 3

IMSA

Figure 5.5: Numeri al Data. Cir ular ylinder (ǫC = 1.8, LD =2λ, SNR = 5 dB). Behavior of the ost fun tion when usingIMSMRA-LS .that the solving shape of φopt

s,qs, when s = 2 and s = sopt = 3respe tively, is quite symmetri al with respe t to the enter ofthe s atterer. As for the quality of the nal re onstru tion [Fig.5.3( ), the lo alization error is slightly higher than the onea hieved in the re onstru tion of Fig. 4.17(a) (δ⌋IMSMRA−LS =

0.73 vs. δ⌋IMSA−LS = 0.59), while the shape of the obje t isbetter retrieved (∆⌋IMSMRA−LS = 0.45 vs. ∆⌋IMSA−LS = 0.48). Finally, Figure 5.5 shows the behavior of the multi-resolution105

5.2. PRELIMINARY VALIDATION

10-4

10-3

10-2

10-1

100

101

0 100 200 300 400 500 600 700 800

Θopt

Iteration, k

γL=5.0 γL=1.3γL=0.7

s=1 s=2 s=3

Figure 5.6: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noiseless Case). Behavior of the ost fun tion ofIMSMRA-LS when varying γL. ost fun tion Θopt when Q = 1. It is worth noting that thespikes between two adja ent steps hara terizing IMSA-LS 'serror urves have disappeared for IMSMRA-LS, sin e the elddistribution is updated before the evaluation of Θ

∆tks|qs

,with ks = 1. Furthermore, as the spatial resolution improves,the ost fun tion is hara terized by a jump whose amplitudeis higher between s = 1 and s = 2 than between s = 2 ands = sopt = 3. 106

CHAPTER 5. THE MULTI-REGION APPROACH5.2.2 Calibration of the Stopping CriterionWith referen e to the mathemati al formulation presented inSe t. 5.1, this se tion is aimed at dis ussing the alibration ofthe stopping riterion of IMSMRA-LS. More in detail, the fol-lowing onsiderations will fo us on the hoi e of the thresholdγH, sin e the values of the other parameters have already beendis ussed in Se t. 4.2.4.In order to hoose the proper value of γH, the s attering onguration is the same as in Se t. 4.2.4, namely a re tan-gular s atterer (L = 0.27λ and W = 0.13λ) hara terized bya diele tri permittivity ǫC = 1.8 and entered at xc = −2

3λ,

yc = λ within an investigation domain of side LD = 3λ. Figure5.6 shows the behavior of the ost fun tion, while the re on-stru tions a hieved by using dierent numeri al values of γLare depi ted in Fig. 5.7 [γL = 5.0 - Fig. 5.7(a), γL = 1.5 - Fig.5.7(b), γL = 0.7 - Fig. 5.7( ). As previously dis ussed (Se t.4.2.4), the number of total iterations ktot de reases as the valueof the threshold de reases. As a matter of fa t, γL representsthe maximum Hausdor distan e (expressed in terms of ls|qs)between the trial solution at iterations ks and the one at ks +1.Therefore, the smaller the value of γL is, the fewer the dier-en es between two trial solutions should be in order to meetthe requirements of the stopping riterion. Su h more stri t ondition usually o urs when the iteration index in reases.However, a good trade-o between the arising omputationalburden and the a ura y of the re onstru tion an be a hievedwhen setting γL = 1.5 (i.e., the Hausdor distan e betweentwo trial solutions an be 1.5× ls|qs

at most).107

5.2. PRELIMINARY VALIDATIONx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0( )Figure 5.7: Numeri al Data. Re tangular ylinder (ǫC = 1.8,LD = 3λ, Noiseless Case). Re onstru tions obtained at theend of the iterative pro edure with IMSMRA-LS when varyingγL [(a) γL = 5.0, (b) γL = 1.5, ( ) γL = 0.7.108

CHAPTER 5. THE MULTI-REGION APPROACH5.3 Numeri al Validation: Syntheti DataIn order to assess the ee tiveness of the IMSMRA-LS te h-nique, some representative results on erned with geometries hara terized by more than one obje t are presented in thisse tion. In order to assess the a ura y of the re onstru tions,the error gures des ribed in Se t. 4.2 will be onsidered.5.3.1 Inversion of Data s attered by Two Di-ele tri CylindersThe rst syntheti test ase is hara terized by two diele tri ylinders pla ed inside a square investigation domain of sideLD = 3λ. In the rst experiment, P = 2 ir ular s atterers hara terized by a radius ρ = 7λ/15 and a diele tri permit-tivity ǫC = 1.8 are entered at (xc|1 = 5λ/6, yc|1 = 5λ/6)and (xc|2 = −5λ/6, yc|2 = −5λ/6). A ording to the guide-lines pointed out in [15, the imaging setup is made up ofV = 30 sour es and M = 30 measurement points for ea h viewv. Consequently, in order to fully exploit the multi-resolutionapproa h, DI is partitioned into NMR = 19× 19 sub-domains.Figure 5.8 shows the re onstru tions a hieved at the end ofthe steps of the multi-resolution pro ess [Figs. 5.8(a)-( ) andthe optimal inversion (i.e., the true s atterer sampled at thespatial resolution used to generate data) [Fig. 5.8(d). Theregions of interest dened at the resolution levels s = 2 ands = sopt = 3 are tra ed by the green dashed line in Fig. 5.8(b)109

5.3. NUMERICAL VALIDATION: SYNTHETIC DATAand in Fig. 5.8( ), respe tively. At step s = 2, only one RoI isdete ted (Qs=2 = 1), although two s atterers are found at theprevious step. Su h a behavior is due to the riterion used forthe denition of the region of interest, whose size is omputedwith a variable allowan e A in order to in rease gradually thespatial resolution during the multi-step pro edure. At steps = 3, when the allowan e on the size of the RoIs de reases,the number of regions of interest in reases as expe ted up tothe number of s atterers (Qs=3 = 2). As for the quality ofthe re onstru tion, the in rement of the spatial resolution isobvious espe ially between the steps s = 2 and s = sopt =3. Both s atterers are better lo alized at sopt, as onrmedby the lo alization errors: δ|1, s=2 = 1.30 vs. δ|1, s=3 = 1.26,δ|2, s=2 = 1.72 vs. δ|2, s=3 = 0.26. Moreover, the area errorde reases when the spatial resolution in reases: ∆|s=2 = 1.65vs. ∆|s=3 = 1.09.The improvement of the quality of the re onstru tions dur-ing the multi-s aling pro edure is also learly visible in thebehavior of the ost fun tion, whi h is hara terized by twojumps at the beginning of both steps s = 2 and s = sopt = 3.For the sake of ompleteness, both regions of interest are dis- retized in N(qs)⌋MR = 13× 13 sub-domains at s = sopt = 3,a ording to the multi-region pro edure explained in Se t. 5.1.As a onsequen e, sin e the omplexity of the algorithm is ofthe order of O (2× η3

s), ηs =∑Qs

qs=1 N(qs)⌋MR, the omputa-tional ost at ea h iteration of the step sopt is lower than the ost at s2, sin e ηs=2 = 361 and ηs=3 = 338.110

CHAPTER 5. THE MULTI-REGION APPROACHx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(2)

q=1

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R

R

R

(3)

q=2

(3)

q=1(2)

q=1

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0( ) (d)Figure 5.8: Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,Noiseless Case). Re onstru tions with IMSMRA-LS at (a)s = 1 and (b) s = 2, ( ) s = sopt = 3. Optimal inversion (d).111

5.3. NUMERICAL VALIDATION: SYNTHETIC DATA

10-3

10-2

10-1

100

0 50 100 150 200 250

Θopt

Iteration, k

s = 1 s = 2 s = 3

IMSMRA

Figure 5.9: Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,Noiseless Case). Behavior of the ost fun tion when usingIMSMRA-LS .In order to further analyze the proposed approa h, Fig-ures 5.10-5.12 shows the re onstru tions when dealing withnoisy data [SNR = 20 dB - Figs. 5.10(a), SNR = 10 dB- 5.11(a), and SNR = 10 dB - 5.12(a). The multi-regionte hnique is ompared with IMSA-LS [SNR = 20 dB - Figs.5.10(b), SNR = 10 dB - 5.11(b), and SNR = 10 dB - 5.12(b)as well as the Bare-LS approa h [SNR = 20 dB - Figs. 5.10( ),SNR = 10 dB - 5.11( ), and SNR = 10 dB - 5.12( ). As pre-viously dis ussed in Ch. 4, the multi-s aling approa h usuallyappears to a hieve better re onstru tions than the Bare-LS112

CHAPTER 5. THE MULTI-REGION APPROACHstrategy when data are ae ted by a high level of noise. As ex-pe ted, the quality of the re onstru tions of IMSMRA-LS doesnot worsen when the SNR de reases, while the performan e ofthe IMSA-LS is limited by the distan e between the s atterers,sin e only a region of interest is exploited. Su h a behavior isfurther onrmed by the re onstru tion errors versus the SNRshown in Fig. 5.13. While the area error of IMSA-LS is aboutthree times higher than the one of IMSMRA-LS, the lo aliza-tion errors of the multi-s aling te hniques appears to be similarwhen SNR = 5 dB. As for the Bare approa h, bary entre andshape of the s atterers are usually a urately estimated andthe re onstru tion at SNR = 20 dB appears to be better thanthe one of IMSMRA-LS. As a matter of fa t, the multi-s alingpro edure of IMSMRA-LS stops at s = 2 when SNR = 20 dBand only one region of interest is dete ted (Qs=2 = 1). As aresult, the spatial resolution is not in reased, sin e the zoomingon the regions of interest is not arried out. Su h a situationis a onsequen e of the high value of the allowan e A at s = 2(As=2 = 40% vs. As=3 = 30%). As a matter of fa t, A hasto be large enough to in lude orre tly the estimated shapesin the regions of interest, but at the same time it should be assmall as possible to enhan e the spatial resolution.In order to dis uss the omputational ee tiveness of IMS-MRA-LS, let us onsider the behavior of the ost fun tion when

SNR = 5 dB (Fig. 5.14). 113

5.3. NUMERICAL VALIDATION: SYNTHETIC DATAx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(2)

q=1

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

= RR(2) (3)

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0( )Figure 5.10: Numeri al Data. Two dis s (ǫC = 1.8, LD =3λ, SNR = 20 dB). Re onstru tions with IMSMRA-LS (a),IMSA-LS (b), and Bare-LS ( ).114

CHAPTER 5. THE MULTI-REGION APPROACHx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R

R

R

(2)

q=1

q=1

(3)

q=2

(3)

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(2)

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0( )Figure 5.11: Numeri al Data. Two dis s (ǫC = 1.8, LD =3λ, SNR = 10 dB). Re onstru tions with IMSMRA-LS (a),IMSA-LS (b), and Bare-LS ( ).115

5.3. NUMERICAL VALIDATION: SYNTHETIC DATAx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R

R

R

(2)

q=1

q=1

(3)

q=2

(3)

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(2)

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0( )Figure 5.12: Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 5 dB). Re onstru tions with IMSMRA-LS (a), IMSA-LS (b), and Bare-LS ( ). 116

CHAPTER 5. THE MULTI-REGION APPROACH

1.0

1.5

2.0

2.5

3.0

5 10 15 20

δ

SNR [dB]

IMSMRAIMSABARE

(a)0.0

1.0

2.0

3.0

4.0

5.0

5 10 15 20

∆

SNR [dB]

IMSMRAIMSABARE

(b)Figure 5.13: Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 5 dB). Values of the error gures versus SNR.117

5.3. NUMERICAL VALIDATION: SYNTHETIC DATA

10-1

100

101

0 50 100 150 200 250

Θopt

Iteration, k

s=1 s=2 s=3

IMSMRAIMSABARE

Figure 5.14: Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 5 dB). Behavior of the ost fun tion.In agreement with the quality of the re onstru tions andthe error gures at the lowest SNR, the nal value of the ost fun tion of the multi-region approa h is hara terized bya lower value than those of the Bare-LS and IMSA-LS. More-over, IMSMRA-LS requires less iterations to a hieve the opti-mal solution (ktot⌋IMSMRA−LS = 198 vs. ktot⌋IMSA−LS = 212).On the other hand, IMSMRA-LS is not hara terized by afaster onvergen e with respe t to the Bare approa h. How-ever, the multi-region methodology appears to be, as expe ted,more omputational ee tive than the other approa hes. That118

CHAPTER 5. THE MULTI-REGION APPROACHIMSMRA − LS IMSA − LS Bare − LS

SNR = 5 dB

kopts=1 95 84 101

Ns=1 361 361 900

kopts=2 53 84 −

Ns=2 361 361 −

kopts=3 50 − −

Ns=3 338 − −

fpos 8.89 × 109 9.97 × 109 7.36 × 1010Table 5.1: Numeri al Data. Two dis s (ǫC = 1.8, LD = 3λ,SNR = 5 dB). Computational indexes.is, the total number of omplex oating point operations fpos =∑sopt

s=1O (2× η3s)×kopt

s , ηs =∑Qs

qs=1 N(qs)⌋MR, NIMSA, NBare−LS,of IMSMRA-LS is lower than the ones of the other methods(Tab. 5.1).In the se ond experiment, P = 2 re tangular s atterers(L|1 = λ/2, W |1 = 5λ/6, L|2 = 5λ/6, and W |2 = λ/2) arelo ated inside a square investigation domain (LD = 3λ) withbary entres (xc|1 = 2λ/3, yc|1 = 2λ/3) and (xc|2 = −2λ/3,yc|2 = −2λ/3). The re onstru tions a hieved by IMSMRA-LS,IMSA-LS, and Bare-LS when SNR = 20 dB are shown in Fig.5.15. The result obtained in this experiment further onrmsthe ee tiveness of the multi-region methodology, sin e bothpositions and shapes of the re tangular s atterers appear tobe a urately estimated [Fig. 5.15(a). As a onrmation, the119

5.3. NUMERICAL VALIDATION: SYNTHETIC DATAx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R

R

Rq=1

(2)

q=1

(3)

q=2

(3)

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(2)

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0( )Figure 5.15: Numeri al Data. Two re tangular s atterers(ǫC = 1.8, LD = 3λ, SNR = 20 dB). Re onstru tions withIMSMRA-LS (a), IMSA-LS (b), and Bare-LS ( ).120

CHAPTER 5. THE MULTI-REGION APPROACHIMSMRA − LS IMSA − LS Bare − LS

SNR = 20 dB

< δ > 2.98 3.40 2.43

∆ 0.34 0.61 0.57Table 5.2: Numeri al Data. Two re tangular s atterers (ǫC =1.8, LD = 3λ, SNR = 20 dB).Values of the error indexes.

10-3

10-2

10-1

100

101

0 50 100 150 200 250

Θopt

Iteration, k

s=1 s=2 s=3

IMSMRAIMSABARE

Figure 5.16: Numeri al Data. Two re tangular s atterers (ǫC =1.8, LD = 3λ, SNR = 20 dB). Behavior of the ost fun tion.121

5.3. NUMERICAL VALIDATION: SYNTHETIC DATAre onstru tion errors of IMSMRA-LS are lower than those ofthe other methods, ex ept for the average lo alization errorwhi h is of the same order of magnitude as the one of Bare-LS.The behavior of the ost fun tions of IMSMRA-LS, IMSA-LS, and Bare-LS is depi ted in Fig. 5.16. As just indi ated,the nal value of the error is lower than the ones of the othermethods. As for the omputational burden, the IMSMRA-LSis hara terized by a faster onvergen e with respe t to IMSA-LS. Furthermore, the arising total number of omplex oatingpoint operations appears to be the lowest (fpos⌋IMSMRA−LS=

6.24× 109 vs. fpos⌋Bare−LS= 4.52× 1010).

122

CHAPTER 5. THE MULTI-REGION APPROACH5.3.2 Experiment with Three Obje tsThis experiment is aimed at illustrating the behavior of theIMSA-LS when dealing with P = 3 s atterers (ǫC = 2.0) hara terized by simple shapes but distan ed from one an-other. The test geometry is hara terized by an ellipti ylin-der (xc|1 = −45λ, yc|1 = −4

5λ, and α|1 = 1

2λ, β|1 = 0.43λ asaxes), a ir ular s atterer (xc|2 = 0, yc|2 = 4

5λ, ρ|2 = 1

4λ),and a square obje t (xc|3 = 4

5λ, yc|3 = 4

5λ, L|3 = 0.83λ and

W |3 = 12λ) lo ated in a square investigation domain hara -terized by LD = 3λ. By adopting the same arrangement ofemitters and re eivers as in Se tion 5.2.2, the investigation do-

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(2)

q=1

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)Figure 5.17: Numeri al Data. Three s atterers (ǫC = 1.8,LD = 3λ, SNR = 20 dB). Re onstru tions with IMSMRA-LS (a) and Bare-LS (b). 123

5.3. NUMERICAL VALIDATION: SYNTHETIC DATAx

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R(2)

q=1 R(3)

q=1

(3)Rq=2

Rq=3

(3)

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)Figure 5.18: Numeri al Data. Three s atterers (ǫC = 1.8,LD = 3λ, SNR = 10 dB). Re onstru tions with IMSMRA-LS (a) and Bare-LS (b).main is dis retized with NMR = 23×23 and NBare−LS = 31×31square ells for the IMSMRA-LS and the Bare-LS, respe tively.Moreover, ∆t1 is set to 0.03.Figures 5.17-5.20 show the results from the omparativestudy arried out in orresponden e with dierent values ofsignal-to-noise ratio. As shown by the re onstru tions (Figs.5.17-5.19) and as expe ted, the multi-resolution approa h pro-vides more a urate results and appears to be more reliablethan the Bare-LS espe ially with low SNR. This on lusion isfurther onrmed by the behavior of the re onstru tion errors(Fig. 5.20), for whi h the IMSMRA-LS a hieves a lower lo al-ization error as well as a lower area error than those of Bare-LS,124

CHAPTER 5. THE MULTI-REGION APPROACHespe ially for SNR = 5 dB (N(qs)⌋MR = 16×16, qs = 1, ..., Qs,with s = sopt = 3 and Qs=3 = 3). On the other hand, bothalgorithms provide good estimates of the s atterer under testwhen inverting data ae ted by low noise [SNR = 20 dB - Fig.5.17(a) vs. Fig. 5.17(b); Fig. 5.20(a) and (b).5.3.3 Experiments with Complex ShapesThe third test ase with syntheti data is on erned with s at-terers hara terized by omplex shapes. In the rst experi-ment, two higher permittivity (ǫC = 2.5) ylindri al rings are entered at (xc|1 = λ/4, yc|1 = 3λ/4) and (xc|2 = λ, yc|2 = λ)x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R

Rq=1

(2)

q=1R

(3)

(3)

q=2

R(3)

q=3

x

y λ

λ−0.75 0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1 τ (x, y) 0 1 τ (x, y) 0(a) (b)Figure 5.19: Numeri al Data. Three s atterers (ǫC = 1.8,LD = 3λ, SNR = 5 dB). Re onstru tions with IMSMRA-LS(a) and Bare-LS (b). 125

5.3. NUMERICAL VALIDATION: SYNTHETIC DATAin a square investigation domain of side LD = 5λ. The exter-nal radius of the rings is ρext = λ/3 and the internal one isρint = λ/6, so that the s atterer are not joined. The imagingsetup is made up of V = 40 sour es and M = 40 measure-ment points for ea h view v, while the investigation domainis dis retized in NMR = 29 × 29 and NBare−LS = 50 × 50 forIMSMRA-LS and Bare-LS, respe tively.Figures 5.21 and 5.22 show the re onstru tion a hievedwhen onsidering SNR = 20 dB and SNR = 10 dB, respe -tively. The multi-region approa h obtains the nal result atthe end of the step s = sopt = 2, with Qs=2 = 1. The RoI is hara terized by a re tangular shape, in order to better repre-sent the area where the s atterer are lo ated. The re onstru -tion is hara terized by a orre t estimation of position andshape of the ir ular s atterer, even though the obje ts arenot separated [Fig. 5.21(a). On the other hand, the result ofBare-LS is noti eably less a urate, sin e the symmetry of thetrue shapes is not retrieved [Fig. 5.21(b). However, the ostfun tion of Bare ends up with an error lower than the valuea hieved by IMSMRA-LS.The same on lusions hold true when in reasing the level ofnoise on the data (SNR = 10 dB - Fig. 5.22). In su h a ase,the re onstru tion obtained by the multi-region [Fig. 5.22(a)appears to be as a urate as the result in Fig. 5.21(a). On theother hand, the Bare approa h is not able to re onstru t thes atterer, even though the artifa ts are mainly lo ated on thedis s. This result is further onrmed by the behavior of theerror fun tions, sin e the nal value of IMSMRA-LS is lowerthan that of Bare-LS. 126

CHAPTER 5. THE MULTI-REGION APPROACH

2.0

2.5

3.0

3.5

4.0

5 10 15 20

δ

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IMSABARE

(a)1.0

2.0

3.0

4.0

5.0

6.0

5 10 15 20

∆

SNR [dB]

IMSABARE

(b)Figure 5.20: Numeri al Data. Three s atterers (ǫC = 1.8,LD = 3λ, Noisy Case). Values of the error gures versus SNR.127

5.3. NUMERICAL VALIDATION: SYNTHETIC DATAx

y λ

λ−1.25 1.25

1.25

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x

y λ

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10-2

10-1

100

101

0 100 200 300 400 500 600

Θopt

Iteration, k

s=1 s=2

IMSMRABARE

( )Figure 5.21: Numeri al Data. Two hollow ylinders (ǫC = 2.5,LD = 5λ, SNR = 20 dB). Re onstru tions with (a) IMSMRA-LS and (b) Bare-LS . Behavior of the ost fun tion ( ).128

CHAPTER 5. THE MULTI-REGION APPROACHThe se ond experiment deals with three s atterers har-a terized by the omplex shapes depi ted by the red dashedline in Figs. 5.23(a) and 5.23(b). The obje ts are enteredat (xc|1 = −5λ/4, yc|1 = 6λ/5), (xc|2 = λ, yc|2 = λ), and(xc|2 = λ, yc|2 = λ), letting LD = 5λ. The measurementsetup is the same as the one in the rst experiment and theinvestigation domain is dis retized in NMR = 29 × 29 andNBare−LS = 50 × 50 for IMSMRA-LS and Bare-LS, respe -tively.The re onstru tions a hieved when SNR = 20 dB are har-a terized by an a urate estimation of lo ation and shape of thes atterer, both for IMSMRA-LS and Bare-LS. More in detail,the multi-region approa h retrieves the shape of ross-shapeds atterer better than Bare, while the ir ular obsta le is sligthlyoverestimated (N(qs = 1)⌋MR = 13 × 13, N(qs = 2)⌋MR =13 × 13, N(qs = 3)⌋MR = 21 × 21, with s = sopt = 3 andQs=3 = 3). As for the re tangular s atterer, the re onstru -tions a hieved by IMSMRA-LS and Bare-LS are similar. Forwhat on erns the behavior of the ost fun tion [Fig. 5.23( ),the nal error of IMSMRA-LS appears to be lower than that ofBare-LS. Furthermore, the total number of oating point op-erations required by the multi-region strategy is lower with re-spe t to the single-region's ase (fpos⌋IMSMRA−LS

= 5.78×1011vs. fpos⌋Bare−LS= 2.52×1012). Finally, the error gures [< δ >- Fig. 5.24(a), ∆ - Fig. 5.24(b) onrm that in general themulti-region approa h is more ee tive in the estimation of theshape, espe ially with noisy data.129

5.3. NUMERICAL VALIDATION: SYNTHETIC DATAx

y λ

λ−1.25 1.25

1.25

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q=1

x

y λ

λ−1.25 1.25

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10-1

100

101

0 100 200 300 400 500 600

Θopt

Iteration, k

s=1 s=2

IMSMRABARE

( )Figure 5.22: Numeri al Data. Two hollow ylinders (ǫC = 2.5,LD = 5λ, SNR = 10 dB). Re onstru tions with (a) IMSMRA-LS and (b) Bare-LS . Behavior of the ost fun tion ( ).130

CHAPTER 5. THE MULTI-REGION APPROACHx

y λ

λ−1.25 1.25

1.25

−1.25

2.50

−2.50

−2.50

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R(2)

q=1

R(3)

q=1 q=2R

(3)

q=3

(3)R

x

y λ

λ−1.25 1.25

1.25

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10-2

10-1

100

101

0 200 400 600 800 1000 1200

Θopt

Iteration, k

s=1 s=2 s=3

IMSMRABARE

( )Figure 5.23: Numeri al Data. Three obje ts hara terized by omplex shapes (ǫC = 1.8, LD = 5λ, SNR = 20 dB). Re on-stru tions with (a) IMSMRA-LS and (b) Bare-LS . Behaviorof the ost fun tion ( ). 131

5.3. NUMERICAL VALIDATION: SYNTHETIC DATA

2.0

2.5

3.0

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5.5

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SNR [dB]

IMSMRABARE

(a)2.0

2.5

3.0

3.5

4.0

5 10 15 20

∆

SNR [dB]

IMSMRABARE

(b)Figure 5.24: Numeri al Data. Three obje ts hara terized by omplex shapes (ǫC = 1.8, LD = 5λ, Noisy Case). Values ofthe error gures versus SNR.132

CHAPTER 5. THE MULTI-REGION APPROACH5.4 Validation by usingLaboratory-Controlled DataThe last experiment of this thesis deals with the data olle tedat the laboratories of the Institut Fresnel in Marseille (Fran e)in 1999 [100. The multiple-frequen y angular-diversity bi-stati dataset twodielTM_4f.exp onsists of the eld sampless attered by two o- entered homogeneous ir ular ylinders(ρ = 15mm) hara terized by a nominal value of the obje tfun tion equal to τ(r) = 2.0 ± 0.3 and lo ated in an inves-tigation domain assumed in the following of square geometryand extension 30 × 30 m2. The eld samples are olle tedin M(v) = 49 measurement points, a ording to the V = 36dierent angular dire tion of the sour e (v = 1, ..., V ).In Fig. 5.25, the re onstru tions a hieved by IMSMRA-LS[Figs. 5.25(a) and ( ) are ompared with those of Bare-LS[Figs. 5.25(b) and (d) at F = 2 dierent frequen ies. By set-ting ǫC = 3.0, the s atterers are always a urately lo alized.On the other hand, the estimation of the shape of the targetsappears to be less a urate, espe ially for the Bare-LS andat the highest frequen y (f = 4 GHz). However, the re on-stru ted stru tures over in all ases a large part of the trueobje t, depi ted by the red dashed line in Fig. 5.25. These on lusions are further onrmed by the behavior of ost fun -tions, reported in Fig. 5.26.Su h a good performan e of both the re onstru tion al-gorithms when dealing with laboratory- ontrolled data oulddepend on to the low level of noise hara terizing the measure-133

5.4. VALIDATION BY USINGLABORATORY-CONTROLLED DATA−0.05

0.05

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y [m]

0.05 0.10 −0.05

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y [m]

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2 τ (x, y) 0 2 τ (x, y) 0(a) (b)−0.05

0.05

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y [m]

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0.10

x [m]

y [m]

0.05 0.10

2 τ (x, y) 0 2 τ (x, y) 0( ) (d)Figure 5.25: Experimental Data (Dataset Marseille [100).Two ir ular ylinders (twodielTM_4f.exp). Re onstru tionswith IMSMRA-LS (left olumn) and Bare-LS (right olumn)at dierent frequen ies f [f = 2 GHz (a)(b); f = 4 GHz( )(d). 134

CHAPTER 5. THE MULTI-REGION APPROACH

10-3

10-2

10-1

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101

102

0 50 100 150 200 250 300

Θo

pt

Iteration, k

IMSMRABARE

(a)10

-3

10-2

10-1

100

101

102

0 100 200 300 400 500

Θo

pt

Iteration, k

IMSMRABARE

(b)Figure 5.26: Experimental Data (Dataset Marseille [100).Two ir ular ylinders (twodielTM_4f.exp). Behavior of the ost fun tion versus the number of iterations when (a) f =2 GHz, and (b) f = 4 GHz. 135

5.4. VALIDATION BY USINGLABORATORY-CONTROLLED DATAments, thus further onrming the hypothesis of a low-noiseenvironment as already eviden ed in [101.

136

Chapter 6Con lusions and OpenProblemsIn this hapter, some on lusions are drawn and further ad-van es are envisaged in order to address the open problems.

137

In this thesis, a multi-resolution approa h for qualitativeimaging purposes based on shape optimization has been pre-sented. The proposed strategy integrates the multi-s ale methodand the level set representation of the problem unknowns in or-der to protably exploit the amount of information olle tablefrom the s attering experiments as well as the available a-prioriinformation on the s atterer under test. Two implementationshave been presented in order to ee tively deal with ongu-rations hara terized by one or multiple obje ts.In general, the main key features of the multi-resolutionlevel set approa h an be summarized as follows:• innovative multi-level representation of the problem un-knowns in the shape-deformation-based re onstru tionte hnique;• ee tive exploitation of the s attering data through theiterative multi-step strategy;• limitation of the risk of being trapped in false solutionsthanks to the redu ed ratio between data and unknowns;• useful exploitation of the a-priori information (i.e., ob-je t homogeneity) about the s enario under test;• enhan ed spatial resolution limited to the region of inter-est.From the validation on erned with dierent s enarios andboth syntheti and experimental data, the following on lu-sions an be drawn: 138

CHAPTER 6. CONCLUSIONS AND OPEN PROBLEMS• the IMSA-LS usually proved more ee tive than the single-resolution implementation, espe ially when dealing with orrupted data s attered from simple as well as omplexgeometries hara terized by one target;• the IMSMRA-LS appeared to be as ee tive as the single-region implementation when dealing with simple geome-tries, while the ee tive multi-region ar hite tures im-proved the re onstru tion a ura y when onsidering mul-tiple s atterers;• the integrated strategy (i.e., IMSA-LS and IMSMRA-LS ) appeared less omputationally-expensive than thestandard approa h in rea hing a re onstru tion with thesame level of spatial resolution within the support of theobje t.However, the a tual implementation is still hara terized byopen problems. For instan e, the regularity of the level setfun tion needs to be preserved during the level set update a - ording to the distan e-fun tion-based initialization. Further-more, dierent velo ity fun tions should be onsidered in orderto nd the optimal ompromise between the re onstru tion a - ura y and the numeri al stability. These and other relatedaspe ts will have to be further investigated, espe ially to dealwith the more attra tive but also more numeri ally ompli atedthree-dimensional inverse s attering problem.

139

140

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157

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158

Appendix AThe Adjoint ProblemThe adjoint eld F v (r) is dened by the following relationship

∇2F v (r)− κ2C(r)F v (r) = −

[ξvks

(rm(v)

)− ξv

ks

(rm(v)

)]∗δm(v)

r ∈ Υ

∇2F v (r)− κ20(r)F v (r) = 0

r /∈ Υ (A.1)where κC(r) = 2πf√

µoǫo [τC + 1] is the wavenumber in theregion inside the s atterer hara terized by a ontrast fun tionτC , κ0(r) = 2πf

√µoǫo is the free-spa e wavenumber, and δm(v)is the Krone ker delta, m(v) = 1, ..., M(v), v = 1, ..., V . Inorder to determine F v (r), let us onsider the adjoint in ident159

eld Iv (r) whi h is the solution of∇2Iv (r)− κ2

0(r)Iv (r) = −[ξvks

(rm(v)

)− ξv

ks

(rm(v)

)]∗δm(v)(A.2)with rn ∈ DI . By assuming that Iv (r) satises the Sommerfeldradiation ondition, namelylim|r|→+∞

√|r|

(∂Iv (r)

∂ |r| − jκ0(r)Iv (r)

)= 0 , (A.3)it an be determined by solving the following equation

Iv (r) =∑M(v)

m(v)=1

[ξvks

(rm(v)

)− ξv

ks

(rm(v)

)]∗G2D

(rm(v)/r

)

rm(v) ∈ DO ,(A.4)that an be a hieved by repla ing ξv (r) with Iv (r) andjωµ0J

v (r) with the right member of (A.2) in equation (2.5).Consequently, F v (r) is obtained by solving the following rela-tionshipIv (r) = F v (r)−

(2πλ

)2 ∫DI

τ (r′)F v (r′) G2D (r/r′) dr′

r ∈ DI ,(A.5)for a given τ (r), r ∈ DI .160

Appendix BAlgorithmsB.1 The Edge Dete tion Operator ∂(·)Let us onsider a ontrast fun tion τ dened in the investiga-tion domain DI by the following relationship

τ (rn) =

(ǫC − 1)− j σC

2πfε0

0

rn ∈ Υotherwisen = 1, ..., N

(B.1)where Υ is the shape of a s atterer lo ated in DI and hara ter-ized by the permettivity ǫC and the ondu tivity σC . In orderto retrieve the ontour of the shape Υ, the fun tion τB (rn) is161

B.2. HOW TO COUNTTHE NUMBER OF OBJECTS omputed as followsτB (rn) = ∂ [τ (rn)] =

=

⌈1−

Pn+1p=n−1

Pn+1b=n−1 τ(xp, yb)

9τC

⌉τ (xn, yn) = τC

0 otherwise (B.2)where ⌈·⌉ is the eiling fun tion⌈υ⌉ = min ς ∈ Z | υ ≤ ς (B.3)andτC = (ǫC − 1)− j

σC

2πfε0(B.4)More in detail, τB (rn) = 1 only when rn is a border sub-domain of Υ. Su h pixels are then grouped in a set Π denedas

Π :=rB

n =(xB

n , yBn

)| τB

(rB

n

)= 1

. (B.5)

B.2 How to Countthe Number of Obje tsLet us onsider a ontrast fun tion τ , the set of ontour sub-domains, Π :=rB

n =(xB

n , yBn

)| τB

(rB

n

)= 1

, and the Nseedsseeds rEi , i = 1, ...,Nseeds. τB and τE an be a hieved bymeans of the operators ∂(·) and erosion, respe tively. The162

APPENDIX B. ALGORITHMSnumber of obje ts in the re onstru tion an be estimated a - ording to the pseudo- ode lines reported in Alg. 1 and in Alg.2. More in detail, the program des ribed in Alg. 1 returns thenumber of regions Q in the ontrast fun tion τ . If the obje tsare separated enough from ea h other, then Q is equal to thenumber of obje ts.

163

B.2. HOW TO COUNTTHE NUMBER OF OBJECTSAlgorithm 1 Pseudo- ode lines des ribing how the number ofobje ts in a re onstru tion τ is ounted.function Q = ount-the-number-of-obje ts (

τB, τE)

i = 1M = N seeds

while i <MΣi = walk-along-τB-starting-from(

rEi

)

if there are other seeds rEj

j = 1, ..., i− 1, i + 1, ...,Nseeds, in the set Σi thenremove those seeds from the set ΣiupdateMendifnd the region Ri that holds Σi

i = i + 1end

while i <Mif Ri is overlapping other regions thenredene RiupdateMendif

i = i + 1end

returnM

164

APPENDIX B. ALGORITHMSAlgorithm 2 Pseudo- ode lines des ribing the fun tion walk-along-τB-starting-from(

rEi

).function Σ = walk-along-τB-starting-from (

rE)nd the seed rE in the set Πdene the rst pixel rΣ

i=1 of the boundary Σ as rΣi=1 = rE

i = 2while T = TRUEnd the adja ent pixel rΣ

i su h that τB(rΣ

i

)= 1

if rΣi /∈ Σ thenadd rΣ

i to Σelse

T = FALSEendif

i = i + 1end

return Σ

165