Modeling considerations for imaging with a standard metal detector * Pascal Druyts, Lionel Merlat, Marc Acheroy

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1 Modeling onsideaions fo iaging wih a sandad eal deeo * Pasal Duys, Lionel Mela, Ma Aheoy Signal Iage Cene, Royal Miliay Aadey Avenue de la Renaissane, Bussels, Belgiu Pasal.Duys@ele.a.a.be, lela@ele.a.a.be, Ma.Aheoy@ele.a.a.be ABSTRACT A sandad pulsed induion eal deeo is used o iage buied ealli objes by sanning an aea of inees. I is shown ha, unde speifi hypoheses, he oupu iage is he esul of he onvoluion of a age funion wih a kenel depending on he iniden agnei field. Seveal hypoheses ae onsideed, leading o diffeen kenel shapes and diffeen inepeaions of he age funion. As he deeo iaging funion is a low-pass file, shape s deails spead ou and he esuling aw iages ae blued. Sine a high-pass esoaion file us be used o deonvolve he aw iages, ae us be aken o avoid a song aplifiaion of noise. The iaging file is opued using a nueial siulaion of he iniden agnei field. Finally, he esoaion file is opued using he Wiene appoah. Resuls ae shown fo a ouple of ealli piees. Keywods: eleoagnei induion iaging, iage esoaion, buied landine deeion. NOTATIONS s Laplae vaiable (deivaion opeao). J uen densiy j oplex uniy: j = 1 elei sala poenial ω pulsaion o Fouie ansfo vaiable A agnei veo poenial a Sala * agnei sala poenial H espeively he agnei field geneaed by he 3D olun veo o equivalenly a fis a H oil, he agnei field geneaed by he oil fo ode 3D olun enso 0 H a uni uen and he agnei field saeed by he age. The sae subsip onvenion holds fo he ohe fields and poenials. a i oponen of he veo a (s) ie vaying volage indued in he deeo oil T a 3D line veo o equivalenly, fis ode 3D oupu of he deeo; i esuls fo he line enso esuling fo he ansposiion of a evaluaion of (s). by eans of ie inegaion a seond ode 3D enso M (s) agnei polaizabiliy a ij oponens of he enso a evaluaed (by he eal deeo) agnei Γ polaizabiliy 1 ideniy enso whose oponens ae δ ij (s) dipola oen 1a o 1 a uni veo in he dieion of a 1 a = a a. Nabla opeao = (,, ). x y z a b. sala podu of a and b a b veo podu of a and b ab dyadi o exenal podu ab = a b ij i j = whee ϕ ϕ ϕ Laplaien opeao ϕ = + + x y z gaviy ene of he eal deeo oil gaviy ene of a age * Pa of his wok has been funded by he EC unde he Espi 9870 HOPE poje. LM is on seonden fo he Fenh Gean Reseah Insiue on Sain-Louis, Fane (ela@isl..f). Copyigh 000 Soiey of Phoo-Opial Insuenaion Enginees. This pape was published in Poeedings of SPIE Deeion and Reediaion Tehnologies fo Mines and Minelike Tages, volue 4038, pages , Olando, FL, USA, Augus 000 and is ade available as an eleoni epin wih peission of SPIE. One pin o eleoni opy ay be ade fo pesonal use only. Syseai o uliple epoduion, disibuion o uliple loaions via eleoni o ohe eans, dupliaion of any aeial in his pape fo a fee o fo oeial puposes, o odifiaion of he onen of he pape

2 ab ab b T a enso podus whose oponens ae a ij b jk, a ij b j and a ij b i espeively R an inegaion vaiable. I appeas always in inegals ogehe wih one of he hee following line, sufae o volue eleen H agnei field dl line eleen B agnei induion field ds sufae eleen E elei field d volue eleen D X elei displaeen field Whee X is R, dl, ds o d indiaes whehe X he quaniy efes o he MD oil o he age a ( ) * b( ) Is he onvoluion of a and b 1 INTRODUCTION The eal deeo (MD) is widely used fo he deeion of buied ealli ies. Is appliaion field speads fo lassial easue huning o aheology and ivil engineeing. Sine Wold Wa II, iliaies and NGOs ae oonly using i fo deining opeaions [1, hap. 1]. Coon MD s ely on single oil, o a se of oils, illuinaing he gound wih a ie vaying agnei field. Eleooive foes ae indued in he onduive aeials and in un podue eddy uens. These uens indue a seonday (vey weak) agnei field, whih is piked up by a eeiving oil. The seonday field is onveed ino a signal, whih is up o now only lile poessed, o finally igge an audio waning. Alhough MDs pefoanes have been slighly ineased sine WWII, suh as in he deeion deph o he gound ejeion, hey sill have poo disiinaion apabiliies. Espeially in he ine deeion onex, i should be kep in ind ha a MD is only sensiive o onduive and/o agnei aeials, and no an explosive podus. Beause os of he ines onain a leas a iniu of eal, hese deeos ae able o loalize he. Theefoe, i uns ou ha he deeo doesn only igge on ines bu also on eal saps, bulle asings and ohe balefield debis, babed wies, e.. This yields a high false ala ae (up o 999/1000). Fuheoe, eah ala given by he MD us be anually heked ou by pobing he gound wih a bayone and he soue of he ala us be dug ou. This is he ain eason why deining is daaially slow. Soe effos have been iniiaed fo a ouple of yeas o ipove he MD in ode o edue is false ala ae. Seveal aks ae followed, whih ae based eihe on new ehnologies (uli-oils, uli-fequeny induion ode [], e.) o on signal poessing (fingepin analysis [4], iaging, e.). The wok epoed in his pape belongs o he seond aegoy. Ou ai is o povide he opeao wih valuable iages in whih ypial ealli objes ay be eognized and possibly, assoiae o soe buying deph infoaion. Mealli objes of inees ae boh pas of ines suh as a fiing pin o a eal sap, and (pas of) non-ine objes suh as shells, aidges, and babed wie. In soe ases, he iage will allow he use o onlude wih eainy if he obje is a ine o no. In favoable iusanes (vey haaeisi objes, lose o he sufae, e.), i ould even be possible o idenify he ype of ine o non-ine obje. In ohe ases, if he iage is no shap enough (obje oo sall, oo deeply buied, oo oplex, e.) o if he obseved shapes ae no disiinan enough, i will no be possible o onlude wih eainy whehe he obje is a ine o no. In his las ase, i would be neessay o exavae he obje and ea i like a ine. The iage ould howeve sill be useful fo adaping he exavaion ask. Iages ould hus be useful o speed up he deining by eduing he false ala ae and also o inease he safey of he opeao by poviding opleenay infoaion abou he obje suh as is buying deph, volue, oienaion, a lis of possible objes e. Even if his opleenay infoaion is no 100% eliable, i ould be quie valuable when opaed o a blind podding and exavaion. Iages ae he esul of a D san. The pahway does no need o follow a egula gid bu his eases he pos poessing. Thee ae seveal soues of bluing. On he eie side, he deeo geneaes a agnei field whih speads like a bunh. This agnei field indues eddy uens whih ay exend in egions of he age whee he field geneaed by he eiing oil was null. The field geneaed by hese eddy uens on is un speads like a bunh. Finally, on he eeive side, he deeo inegaes he flux hough is oil, whih pesens a lage seion. If he ages of inees wee lage han he deeo foopin, aw iages would epodue he shap deails. Unfounaely ages of inees ae uh salle (ypially a few o less) han he deeo foopin (ypially, a ean diaee of 5 ). A pos-poessing is hus equied o enhane he iage shapness in ode o ake he inepeable by he opeao. This de-bluing iplies a song aplifiaion of he high fequenies of he signal ha wee aenuaed by he iaging poess. Cae us be aken o avoid song noise aplifiaion. We will show ha unde speifi hypohesis he iaging poess ay be odeled by eans of a onvoluion file. A Wiene file [5] ay hen be used o esoe he iage aking ino aoun he haaeisis of he noise and of he iaging file.

3 This pape is oganized as follows. In seion, he eal deeo opeaion piniple is pesened. The ineaions beween he senso and he ages ae highlighed fo a physial poin of view. In seion 3 a nube of age odels ae developed. The eal ages ha ay be epesened by eah developed odel is also disussed. I will be shown ha unde soe hypohesis, he iaging poess ay be odeled by he onvoluion of a age funion wih an iaging kenel. The inepeaion of he age funion and he shape of he iaging kenel will be disussed fo eah odel developed. Seion 4 deals wih he iage esoaion poedue. I pesens he design of he Wiene esoaion file. Seion 5 pesens he expeienal seup and shows esuls wih ages of vaious sizes and shapes. Finally, seion 6 onludes he disussion. METAL DETECTOR Geneally speaking, a MD opeaes wih eihe oninuous wave (CW) o pulsed induion (PI) agnei fields a low fequenies (audio band). The low fequeny exiaion allows fo a deep peneaion of he field ino he gound. The wavelengh is lage when opaed o he volue of inees and he ages ae hus in he nea field of he illuinao [1]. The popagaion effes ay heefoe be negleed. The ie vaying agnei field geneaed by he deeo will indue eddy uens in onduive bodies. The eddy uen evoluion will be govened by he esisive and induive haaeisis of he pah hough whih hey flow. We ay hus onlude ha he only elevan paaees ae he onduiviy and agnei peeabiliy ogehe wih he age geoey. If a age is in he field of he deeo, he deeo answe will be influened by he age shape, elaive posiion and oienaion, onduiviy and agnei peeabiliy. The eleial peeabiliy of he age and is suounding is ielevan. The MD we use is a sandad odel and opeaes in PI ode [] wih a single oil. I geneaes a agnei field of vey sho duaion. Afe he piay field has vanished, he oil senses only he seonday field, whih is geneaed by he eddy uens indued in he ages (Figue 1.a). In ou ase, he exiaion uen wavefo is a saw ooh ain wih alenae sign (Figue 1.b). Is speu uppe bound is ainly defined by he fequeny oponens of he fas deeasing phase, whih lass fo abou 5 µs. Hene, he bandwidh aouns appoxiaely 00 khz. This oplies wih he above assupion of negligible popagaion effes and he whole syse an be odeled in he quasi-sai agnei egie. I should be noed ha suh a deeo is no able o dee non-onduive ages even if hey pesen a agnei peeabiliy onas wih hei suounding. Piay agnei field i ( ) s in B Seonday field T T d (a) (b) Figue 1 (a) A sall age laid in he piay vaying agnei field bak-saes a seonday field. (b) Coil uen wavefo of he pulsed induion eal deeo used. The lowe view is a zoo on a single saw ooh of he ain shown a he op. T is he haging peiod, wheeas T d is he fas dishaging one. The iniden agnei field indued a he poin by a hoizonal oil loaed a he oigin when diven by a uen ( s) I, whee s is he Laplae vaiable, is (Bio Sava law) [7]: I dl 1 µ 0 R B ( ) = 4π R oil This expession ay be used o opue he agnei field geneaed by a speifi eal deeo if he geoey of is oil is known. In paie, he field is opued on a 3D laie by eans of a nueial inegaion poedue. Wih he adoped noaions, B 0 is defined as he agnei iniden field pe uni uen ( B 0 = B I ). The volage indued ino he oil esuls fo he Lenz law ( EMF = dφ d ). As hee is no uen flowing ino he oil duing he easueen phase, he oal volage easued aoss he oil is due o he agnei field baksaeed by he age. s = s B R, s ds () ( ) ( ) S (1)

4 This ay also be wien using he agnei veo poenial as a funion of he uen densiy in he age J : µ J ( R, s) A (, s) = d 4π R µ s J ( R, s) d 4π R R ( s) = s A ( R, s) dl = dl. Equaion 4 ay be ewien by peuing he ode of inegaion: 0 s = s J ( R, s). A ( R ) d whee oil oil ( ) ag e 0 µ 1 A ( ) = 4π R oil dl (3) (4) (5) (6) is he agnei veo poenial oesponding o a uni uen in he oil. Equaion (5) is an expession of he eipoiy heoe [8] and ay be used o opue he volage indued in he oil by inegaing ove he age volue. The advanage of his foulaion is ha he volage indued in he oil by any age ay be opued one he uen densiy in he age is known wihou opuing he agnei field geneaed by he uens and wihou inegaing he flux on he (possibly oplex) shape of he oil. The volage expessed by (5) is a ie dependen funion. An evaluaion funion ha opeaes on ha ie vaying volage o podue a sala easueen is ipleened inside he deeo. This evaluaion funion is a linea opeao whih alulaes soe kind of aveaging on a given ie window: { ( s) } = O (7) The signal is hen fuhe poessed o podue he audio ala. This las poessing is no onsideed in his pape beause we use as inpu of he signal poessing algoihs he signal, whih we all he eal deeo oupu. The deeo we use povides his signal hough a nueial oupu inefae. Obviously, when going fo ( s) o, valuable infoaion abou he age signaue is los bu we don know of any oeial eal deeo ha povides aess o ( s). 1 Inoduion 3 TARGET MODELS We have shown in he pevious seion ha he only paaees influening he eal deeo oupu ae he agnei peeabiliy and eleial onduiviy. Theefoe, he geneal invese poble uses as inpu he answe of he eal deeo and podues a 3D iage of he onduiviy and peeabiliy disibuions of he undegound. This would be a vey useful infoaion, as onduive o agnei ages would appea shaply and well onased. The invese poble has howeve an infinie nube of soluions (ill-posed poble). This is due o he fa ha a veo of haaeisis (onduiviy and peeabili us be eonsued on a 3D volue using as inpu a D sanning. The invese poble is fuheoe vey sensiive o easueen noise. Indeed, he die poble aenuaes signifianly he high fequenies of he onduiviy and he peeabiliy. The poble us hus be egulaized. This is done by adding soe onsains on he onsideed ages. Those onsains epesen a-pioi knowledge o hypohesis on he age. One ould fo exaple ake he hypohesis ha he age is a single hoogeneous sphee. The invese poble would hen edue o he esiaion of he adius, peeabiliy and onduiviy of ha sphee. A lage vaiey of onsains ay be onsideed. The hoie should be ade in a way ha akes hose onsains, a leas appoxiaely, saisfied by he age of inees; in ou ase, ypial ealli pas enouneed on a inefield. As ou goal is o develop a syse ha is able o un on an ebedded opue in eal ie, we also onsideed oplexiy of he possible odels. I should be noed ha he oplexiy of he poble is highly ineased by he fa ha a sandad eal deeo is used, and heefoe he eiing oil is oved ogehe wih he eeiving oil (fo he deeo we use, a single oil is used). This iplies indeed ha he pah and inensiy of he eddy uens indued in he age vay fo one easueen o he ohe.

5 In his seion six siplified age odels will be developed. Fo hose odels, he invesion poedue has a odeaed oplexiy and no poble is expeed fo eal ie poessing. Alhough siple, hose odels ae expeed o be appopiae fo a lage vaiey of ages of inees. Soe heoeial jusifiaions ae povided in his seion and hese ae also suppoed by he expeienal esuls pesened in seion 5. Fo hose siple age odels we will no be able o eove he eleoagnei haaeisis of he ages bu only a deived haaeisi ha we all he age funion. Fo eah ase, he eaning of he age funion will be disussed. We will show ha even if i is no uniquely linked o he loal eleoagnei popeies of he age, i onains valuable infoaion on he shape and eleoagnei popeies of he age. Single sall age If he age is sall enough, he iniden agnei field beoes hoogeneous on he age volue. The oesponding veo poenial is hen v v v ( ) 0 ( v B0 A ) = (8) whee v is he gaviy ene of he age. The oesponding volage indued in he eal deeo is: whee s = ag e 0 0 (9) ( s) J ( R, s).( B ( ) R ) d = sb ( ). ( s) v is he agnei dipole oen of he age [7]. 1 = ( R, s)) d ( s) ( R J ag e Fo linea aeials in an hoogeneous agnei field, he agnei oen us be a linea funion of he agnei field: ( s) = M ( s). H (, s) (11) whee (s) is he agnei polaizabiliy of he age; by eipoiy [1], his is a 3 by 3 syei enso: M M ( s) M = M M xx xy xz ( s) M xy ( s) M xz ( s) ( s) M yy ( s) M xz ( s) ( s) M ( s) M ( s) The agnei polaizabiliy enso enopasses he speifi eleial and geoeial feaues of he age. I is fequeny dependen. Fingepin analysis fo age lassifiaion an benefi fo is popeies. Soe ypial agnei polaizabiliy enso ae epoed in [1 hape 8]. Finally, he volage indued aoss oil by he bak-saeed field is: T s = sµ H M H I s (13) ( ) ( ) ( ) ( ) 0 This las equaion was obained fo a oil loaed a he oigin in a hoizonal plane. If he oil is anslaed o, he above equaion beoes: T, s sµ H M H I s (14) xz 0 ( ) ( ) ( ) ( ) = 0 0 Sily speaking equaion (14) is only valid fo a punual age suh as a dipole. Fo all ohe ages, he field is no hoogeneous on he age volue and a odel noise is inodued by assuing ha he field is hoogeneous. To analyze he effe of ha odeling noise, he ohe soue of noise us also be aken ino aoun. We ewie 14 as T, s = sµ H M H I s + (15) ( ) ( ) ( ) ( ) η whee η is he oal noise. The vaious soues of noise ae disussed in seion 1. 0 The oal noise will lii he eahable esoluion (see seion 4). As long as he age odeling noise is sall when opaed o he ohe soues of noise, he hoogeneiy hypohesis is valid and does no deease he eahable esoluion. This will be he ase if ax < f ( d) whee ax is he axial adius of he age and d is he bes esoluion ha ay eahed in absene of age odeling noise (fo a dipole age). Clealy, if a age is lage han he esoluion, he hypohesis ay no hold and one us eihe aep a lowe esoluion o use oe adequae odel. Soe alenaive odels ae poposed in he nex seions. The peise uppe bound fo he age size is age 0 zz (10) (1)

6 dependen. Fuhe eseah should allow he definiion of his uppe bound oe peisely fo a nube of ypial ages. This ould be done by eans of easueen o siulaion. Inoduing he evaluaion funion of he deeo (7) in (14), he oupu signal beoes: T ( ) B0 ( ) Γ B ( ) (16) The enso si ( s) Γ = O M µ = 0 ( s) Γ = Γ Γ is alled he evaluaed (by he deeo) agnei polaizabiliy enso. I efles he age s haaeisis easued by he deeo. Is eigenveos indiae he oienaion of he age. The eigenvalues ae invaian fo a given age; hey ould hus be used as inpu of a lassifie. Cae should howeve be aken beause diffeen ages ould have he sae eigenvalues. Fuheoe, he eigenvalues ould be diffeen fo diffeen exeplas of a given age and hey ould also hange wih he age of he age (ypially, he oxidaion of a eal sap ould signifianly hange is esisane a he lii, he sap ould beak, leading o an open iui insead of a losed one), o wih is envionen (epeaue, e.). Equaion (16) is a non linea equaion in he 9 unknowns ( x, y, z, Γ, Γ, Γ, Γ, Γ ) Theefoe, if he eal deeo is used o sense he age answe in 9 well hosen posiions, he 3D posiion of he age as well as is haaeisi aix A ould be eoveed by solving a non linea syse of equaions. To deease he noise sensiiviy, oe poins should be used and an eo ieia should be iniized. The eigenvalues of he aix ae invaian fo a given age and he eigenveos povide infoaion on he oienaion of he age. This appoah is uenly unde invesigaion. The ain open quesions ae: how any easueen poins should be used, wha is he opial sanning pah and how o hoose he iniial values o ensue onvegene o a valid soluion? The possibiliy o eove he ie dependen agnei polaizabiliy enso ( M (s) ) is also onsideed. This ould allow fo a bee age disiinaion hanks o fingepin analysis bu, as explained above, his is no possible wih a sandad eal deeo ha does no povide aess o he ie deay of he oil volage. Siple adapaions of he eal deeo ha ould solve he poble ae unde invesigaion. 3 Two sall ages The ase of wo sall ages in he field of he deeo is now invesigaed. The objeive is o poin ou he effe of he ineaion of he ages on he easued EMF. The esul developed hee will be helpful fo he analysis of ohe age aangeens (see nex seion). I should be noed ha even if he ages ae sall enough o be onsideed as dipoles fo he deeo s poin of view, hey ay inea and i is in geneal neessay o ake ino aoun he exa shapes of he wo ages o opue he oupling. Only if he wo ages ae suffiienly fa fo eah ohe, he dipole appoxiaion ay be used o opue he oupling. Two ages, loaed in 1 and, ae exied by boh he oil s iniden field and by he field indued by he ohe age. If he dipole appoxiaion is valid: in s 1 = M1( H1 + H1 ) (18) in s1 = M ( H + H ) in whee H sj i is he iniden field of he MD on age i and H i is he saeed field fo age j on age i. The field saeed by eah dipole is: As C 1 C1 H H s1 s 1 1 = 4π 1 1 = 4π xx xy xz Γ Γ Γ ( 1 1 I ) 1 1 xy yy xz xx Γ Γ Γ xz xz zz xy 1 ( 1 1 I ) = C 1 1 = C =, le s noe C his enso. Cobining (18) and (19) leads o: in in [ M CM C] M ( H + CM H ) 1 = The expession of is obained by swapping indexes 1 and. 1 xz yz zz (17) (19) (0)

7 in I appeas lealy fo (0) ha he oupling indues wo addiional es ( M 1CM C and CM H ). Fo fis ode spheial ages, he polaizabiliy aix is diagonal M i = Ki s [ 1 + sti ]1. Fuheoe, if he wo ages ae idenial, he fis addiional e is he squae of he seond one. Boh es inease wih he fequeny and deeases wih he disane. Fo he fis-ode ages onsideed, he oupling ineases wih he fequeny and beoes onsan fo fequenies highe han he u-off fequeny of he ages ( ν > T 1 = T ) whee i eahes a plaeau. If highe fequeny poles ae exied, he oupling will deease fo fequenies highe han he pole haaeisi fequeny. C deeases asypoially as he seond one as Fo fequenies on he plaeau, his gives espeively: 1K ( T1T ). Theefoe, a a given fequeny, he fis addiional e deeases as K and K To suaize, if he disane is ineased, o he fequeny deeased, he oupling will deease: T 3 6 and Fo song oupling, equaion (0) should be used The fis e ( M 1 CM C ) will fis vanish. Equaion 0 hen beoes: in in = M H + CM H (1) This is he ase of weak oupling. ( ) Finally, he oupling opleely vanishes, Equaion (0) beoes: = in M H () and he volage sensed by he oil is: T T ( ) = B 0 ( 1 ) Γ 1B ( 1 ) + B 0 ( ) Γ B ( ) (3) 0 As fo a single age, if a nube of easueens ade in well-hosen posiions ae available, i should be possible o eove he ages posiion and he evaluaed agnei polaizabiliy ensos by eans of a non linea opiizaion. The nube of unknowns is howeve signifianly ineased and oe easueens ae needed. If he oupling ay be negleed, he nube of unknowns is uliplied by wo. If he oupling us be onsideed, 36 ij kl ij kl addiional unknowns Ο ( M 1 ( s) M ( s)) = Cijkl A1 A appea. We ae uenly invesigaing his appoah. The sae quesions as fo he single age ase us be answeed. Moe pobles ae howeve expeed beause he hoie of iniial values beoes oe oplex and he esoluion ie ineases. 4 Disee nube of sall ages In ase of song oupling, he esuls deived fo wo ages ay no be genealized. Fo weak oupling, he genealizaion of (1) gives: = in i Mi Hi + CM jh j i If he oupling an opleely be negleed, he above equaion beoes: in = M H (5) An he deeo oupu beoes: i i i in j T ( ) = B ( ) Γ B ( ) i 0 (4) 0 i i 0 i (6) The posiions of he dipoles, hei haaeisi aies and he ouplings ould heoeially be eoveed by eans of a non-linea opiizaion as aleady disussed in he ase of a single o wo ages. An addiional poble is o deeine he suiable nube of ages. Thee is fuheoe a fas inease of he nube of unknowns wih he nube of ages. The inease is less daai in absene of oupling; bu sill he nube of ages should eain liied o keep a aable poble. The axiu nube of ages fo whih he syse ay be solved is uenly unde invesigaion. 5 Disibuion of independen dipoles Fo a plana disibuion of dipoles a a given deph, equaion (6) beoes:

8 i ( ) = B ( R ) Γ ( R ) B0 ( R ) S ds 0 (7) whih ay be expended as follows: = B Γ + B Γ + B Γ + B B Γ + B B Γ + B ( ) x xx y yy z zz x y xy x z xz y Bz Γyz Equaion (8) is a su of D onvoluions ha anno be inveed. This poble an only be solved fo speifi ages fo whih he agnei polaizabiliy enso ay be siplified. Aong ohes, hee ae wo suh ases: Fo a spheial hoogeneous age, he agnei polaizabiliy enso is diagonal and is hee eleens ae equal ( Γ = Γ = Γ T ). Then, he easued volage is: xx yy zz = ( ) = B T Fo a hoizonal loop he only non-null e of he agnei polaizabiliy enso is Γ zz = T. Then, he easued volage is: 0 z ( ) = B T Fo hese wo ases, he age funion ay be eoveed by eans of a deonvoluion poess using a suiable kenel. This kenel depends on he disibuion of he iniden induion field B 0 ( ) and of he deph of he age disibuion. One igh hink ha a 3D deonvoluion poess ould be used o eove a 3D disibuion of ages fo a 3D sanning. Obviously suh a 3D sanning would be diffiul o pefo in paie beause a vey long ie would be needed o pefo he sanning. Fuheoe, hee is a oe fundaenal eason ha pevens suh a 3D eonsuion. Indeed, he inpu of he eonsuion should be he MD oupu on a 3D laie spanning he oplee spae whee he easueen is signifian. This inludes easueen ove and below he buied objes whih is obviously ipossible. One ay even show ha one he easueen of he agnei field is known on one plane above he soil, i ay be opued on any ohe plane above he soil. This is due o he fa ha in absene of uens, he agnei field ay be expessed as he gadien of a sala poenial: * B = gad( ) (31) The soluion of suh an equaion is uniquely deeined by he knowledge of he bounday ondiions. I is suffiien o know he poenial on he bounday (Diihle poble) o is noal deivaive (Neuann poble)[7]. Theefoe, one he veial oponen of he agnei field (noal deivaive of he poenial) is known on a plane above he soil, he agnei field ay be opued in all he half-spae above ha plane. The above easoning is based on he assupion ha he agnei field is sensed siulaneously on he whole sanning plane o equivalenly ha he uen densiy in he age does no hange fo one easueen o he ohe. This would be he ase if he eiing oil eains fixed and only he eeiving oil is oved o pefo he sanning. By eipoiy, a fixed eeiving and a oving eiing oil would be equivalen. Wih a sandad eal deeo, he eiing and eeiving oils (whih ae in soe ase he sae physial oil) ae igidly fixed and ove ogehe. Theefoe, he onsan exiaion hypohesis is no valid. Howeve, fo he age odels ha lead o a onvoluion (sphee and loop), equaions (9) and (30) show ha he onfiguaion is equivalen o a fiive fixed eiing and oving eeiving oil. As an exaple, fo spheial age, he equivalen fixed eiing oil would geneae a file of onsan odulus on he egion of inees and he eeiving oil would geneae a field whose odule is he squae of he odule of he field geneaed by he eal oil (see equaion (30)). By analogy o axial seeovision, one igh hink ha i ould be possible o eove he deph of he ages if he sanning is pefoed on wo planes a diffeen heigh. The above disussion shows howeve ha his is ipossible. Fo any hypohesized deph, a possible dipole disibuion will be found. Wihou a pioi knowledge on he disibuion, i is ipossible o find ou he eal dipole disibuion and hus he eal deph. Indeed, eah disibuion found will auaely pedi any easueen ha ould be ade in he aessible half spae. We ae howeve invesigaing ehniques siila o he fousing of a aea o find he deph of he age. The iage will look blued o pesen aifas if he deph used is no he igh one. The deph will hen be hosen o ge he bes esoed iage. Fo an auoai deph esiaion, a ieia aing he qualiy (no bluing, no aifas) of an iage us be defined. An alenaive is o pesen he esul a diffeen deph o he use ha will hen use his own ieia o sele he bes iage. The syse will hen eun he oesponding deph. As disussed above, he sanning a diffeen heighs is useless. Howeve, sanning wih diffeen oienaions of he iniden field ay bing oe infoaion on he age. This ould aongs ohe allow fo assessing all eleens of 0 (8) (9) (30)

9 he agnei polaizabiliy enso. This ould be done using a synhonized newok of oils. An appopiae oils geoey and exiaion, will allow fo geneaing an iniden agnei field fo whih only one oponen ( B, B o B z ) is non-null in a defined egion of he spae. If siilaly, an appopiae obinaion of he volage sensed by he eeiving oils is used, i will be possible o eove independenly eah oponen of he evaluaed agnei polaizabiliy (17). In seion 4, a geneal odel suied fo a disee nube of sall ages has been developed. Wih ha odel, he invesion is pefoed by a non-linea opiizaion. If he ages ae a onsan deph, if he oupling an be negleed and if he agnei polaizabiliy is spheial o loop like, he invesion is siplified and beoes a deonvoluion. Indeed, he poble ay hen be seen as a paiula ase of he dipole disibuion and equaions (9) o (30) yields. The age funion beoes a su of Dia funions. A Dia funion appeas a eah age loaion and is apliude haaeizes he oesponding age. 6 Elongae ages and disks The field eied by a plana disk o a plana elongaed age ay be odeled by a disibuion of dipoles (see Figue ). To use (9) o (30), we have o show ha he oesponding dipoles ae independen and ha hey ae equivalen o sall sphees o loops. This is in geneal no he ase. Typially, fo a losed loop, he equivalen dipoles us be oupled by any ean o podue he sae uen in eah dipole as he one flowing hough he oiginal iui. This song oupling (even if i is no a uual oupling by agnei induion) violaes he hypohesis of dipole independene. Theefoe, (30) ay no be used o odel an exended plana loop and he odel developed in he nex seion should be used insead (equaion (36)). I I x y Figue - The equivalene of a uen loop aying a uen I and a uen esh. A uen flowing in inne esh is anelled by uens flowing in suounding eshes. The oveall eaining uen esh is he oiginal bodeing loop. To be odeled auaely by a disibuion of independen dipoles, he uen densiy indued in he age by a punual exiaion (a foalized agnei field fo insane) should eain loalized inside a sphee of adius d (see seion ). The uen ouside ha sphee ay be negleed if he agnei field geneaed by hose uens ay be negleed. In fis appoxiaion, his is equivalen o equie a negligible dipola oen. Typially, fo iula uens, d J θ ( ). d us be negligible. Theefoe, he uen should deease fase han. To use equaion (30) he spheial syey us fuheoe be valid. To quanify he odel liis, nueial siulaions o expeienal esuls should be pefoed. The expeienal esuls pesened in seion 5 shows ha he odel is valid fo he es ages onsideed. Nueial siulaions ae uenly unde invesigaion. The following disussion pesens a qualiaive disussion on he odel liis fo a fla disk. A low fequenies, he disk anno be odeled by independen dipoles beause he field geneaed by he age is 1 negligible in opaison o he exiaion field. The uen hen deeases as (sae EMF fo eah ile and esisane linea in ) and he agnei oen inease as. In ohe wods, he exenal pa of he disk eis he ajo pa of he agnei field. Fo highe fequenies, he field geneaed by he disk ineases. Lage uen loops enopasses he exiaion field as well as he field geneaed by he inne uen loops whih paially opensaes he exiaion agnei field. Theefoe, he uen densiy deeases fase and he independen dipole odel beoes valid. This phenoenon is linked o he skin effe. A sandad eal deeo, geneaes low as well as high fequenies. The lowe fequenies will be esponsible fo a odel eo. This effe is howeve liied by he fa ha he volage indued in he deeo oil is popoional o he deivaive of he uen in he age. Theefoe, highe fequenies have oe influene on he deeo oupu.

10 7 Single exended losed winding Fo a single lage losed loop, lying on a hoizonal plane a a given deph, In his expession, I( s) (, s) = s K ( ) (3) Z ( s) Z ( s) = + sl is he ipedane of he loop and K ) is he uual oupling beween he deeo (loaed a ) and he age whih ay be expessed as follows: z K = B 0 ( R ). ds = B0 ( R ). N ( R ). ds ( ) S ag e plane whee S is he Region inside he age and N ( R ) is defined as follows: 1: inside he age aea S, 0: ouside he age aea. Equaion (33) shows ha he oupling ay be expessed as a D onvoluion: deeo is: ( z (33) K = T * B. The oupu of he eal whee s I ( s) Z ( s) = O{ ( s) } = O K = αk (34) s I d ( s) α = O (35) Z ( s) Hene: N Y = ± = * Bz = T * B α z (36) To opue Y knowing, he sign abiguiy us be esolved. This ay be done by using a heuisi algoih ha uses he fa ha a null uual is seldo a loaions whee no sign hange ous. The age funion T is popoional o N whih has been defined above and ay be eoveed by eans of a deonvoluion poess. The definiion of he age funion shows ha hee is a lea link beween ha funion and he loop shape. The above deivaion ay be genealized o a lage plana losed winding. The value of N funion a a given poin ay hen be inepeed as he nube of uns of he winding aound ha poin. Equaion (36) ay no be genealized o oe han one age. Indeed, fo wo ages he equaion beoes: Y = ± T '* B (37) 1 ± = T1 '* Bz + T '* Bz = One sees ha Y ay no be opued using he only available easueen poble oes fo he fa ha 1 + z z. Apa fo he sign abiguiy, he exep if in any poin only one of he ages indues a signifian volage in he deeo oil. This will be he ase if he disane beween he ages is lage enough o obseve well sepaaed peaks in he aw iage. Equaion (36) ay howeve be used fo seveal lose ages if i is possible o fis sepaae hei onibuion. We ae uenly invesigaing he possibiliy o sepaae he onibuion of eah age by eans of pole esiaion of he ie-doain ipulse esponse. This is howeve no possible wih a sandad eal deeo ha uses an evaluaion funion (5) o opue a single value fo he age ipulse esponse. The ipulse esponse is hus no available fo daa poessing. The feasibiliy o enhane he eal deeo o be able o ge he ipulse esponse in invesigaed. 8 Conlusion In he pevious seions, six age odels have been developed. They ay be used o epesen a lage vaiey of eal ages as ine fiing pins o eal saps, babed wies, e. We have enioned he possibiliy o eove he posiion and he evaluaed agnei polaizabiliy enso of a disee nube of ages hough a non-linea opiizaion poess. We have also shown ha fo a nube of age odels (seion 5, 6 and 7), he iaging poess ay be odeled by a onvoluion file. The esoaion ehnique pesened in seion 4 will hen be used o iage he age.

11 Depending on he odel, diffeen iaging kenels, diffeen aw iage inpu and diffeen age funion inepeaion us be used. The esuls ae suaized as follows: Disibuion of independen spheial dipoles Disibuion of independen hoizonal loop like dipoles Tage ype Raw iage Kenel Tage funion ( x, 0 ( x, z B 0 Single plana exended losed winding ± ( x, B T ( x, = Γxx ( x, = Γyy ( x, = Γzz ( x, z B 0 T ( x, = Γ ( x, Table 1 : iaging onvoluion haaeisi fo vaious age odels zz N( x, T ( x, = whee N(x, is he α nube of uns of he winding aound he poin (x, and s I d ( s) α = O efles he Z ( s) eleial haaeisi of he winding. The vaious kenels ay be opued one he deph of he age and he field geneaed by he eal deeo fo a uni uen ( B 0 ) is known. We have shown ha sanning a vaious heighs is useless o esiae he deph of he age. The deph ould howeve be povided by anohe senso o seleed o povide he bes iage (kind of fousing). Fo he plana winding, no quesionable hypohesis wee ade; fo a disee nube of sall spheial o loop like age, he only open quesion is he axial allowable size fo he age. Fo he elongaed o disk ages howeve, a nube of hypohesis ha ould fail fo soe eal ages whee ade. Nueial siulaion should be pefoed o assess oe peisely he liis of his odel. Expeienal esuls pesened in seion 5 ae howeve poising. 1 Poble saeen 4 IMAGE RESTORATION We have shown in seion 3 ha he iaging poess ay be odeled fo a nube of ages of inees by a onvoluion poess: g( x, = T ( x, * h( x, + η( x, (38) whee g( x, is he aw iage buil fo he eal deeo oupu by an appopiae sanning, T ( x, is he age funion, h( x, is he iaging kenel and η( x, is he noise. Table 1 suaizes he aw iage, he age funion and he iaging kenel o use fo he vaious age odels fo whih equaion (38) ay be used. aious soues of noise have o be onsideed suh as: use of an eoneous age odel use of an esiaion of he agnei field geneaed by he oil H 0. Esiaion eos ae due o an eoneous odeling of he geoey of he oil bu also o an eo on he oil posiion and oienaion eleoni and quanifiaion noise. Effe of he soil. The pesene of he soil has no been aken ino aoun in he odeling. This is jusified if he soil is no onduive and no agnei. A agnei soil will disub he agnei field geneaed by he oil and bak saeed by he age; he shape of he iaging kenel will hange. Fuheoe, non-lineaiies ould appea. The effe of a onduive soil is wofold. Fis, i bak-saes enegy and will heefoe be a he oigin of lue. Seond, he field illuinaing he age is he su of he field geneaed by he oil and by he soil. Hene, he iaging kenel will be disoed. This disoion will howeve eain liied beause he age deph is in geneal uh salle han he skin deph of soils a he fequenies used. As a ypial exaple, Figue 3 shows he kenel fo he spheial dipole disibuion a vaious deph. Figue 4 pesens he oesponding Fouie ansfo. These figues show ha he iaging poess is a low pass file and ha he esoaion poess will have o aplify signifianly he high fequenies. Theefoe, ae should be aken o avoid a song aplifiaion of he noise.

12 The eahable esoluion is funion of he signal o noise aio and he shape of he iaging kenel. The eahable esoluion will deease when he signal o noise aio o he u-off fequeny of he iaging kenel is edued. The noise is funion of he age deph and deeo heigh above he gound. Indeed, he eleoni noise is independen of he age deph o deeo heigh, bu he ohe soue of noises suh as he oil odeling eo ay vay. The signal deeases vey fas wih he deph. Fuheoe, he iaging kenel beoes a songe low pass file when he deph is ineased. Theefoe, he eahable esoluion deeases fas when he disane beween he age and he deeo is ineased. Resoaion file I has been shown [11] ha iage deonvoluion is an ill-posed, eaning ha he soluion of he poble does no always exis, and in ase i exiss i is no unique and highly sensible on he inpu daa. The poble an howeve be egulaized and any ehods have been developed (see[10] fo a eview) o his end. Figue 3 Saples of kenels aken along a plane paallel o he deeo oil s plane and a heighs of 5, 15, 40 and 95 (fo lef o igh and op o boo). These fou views do no shae he sae oloap. Maxiu values a 5 ae obviously uh highe han he one a 95. Figue 4 Coesponding PSF of pevious kenels. The null fequenies ae eneed. The log of he PSF is displayed (fo a bee visualizaion) and he iages do no shae he sae oloap. A widely used appoah is he Wiene-esoaion [9]. This iniizes he ean squae eo beween he esiae and he ue iage using a linea opeao unde he assupion of signal-independen noise, a linea degadaion, he

13 saionnaiy of he iages and degadaion file and a iulan appoxiaion. expession is hen obained fo he esoaion file: whee S nn ( u, v) and ( u v) M ( u, v) ( u, v) Snn ( u, v) + S ( u, v) The following Fouie-doain * H = (39) H ( u, v) ff S ff, ae he speal densiies of he noise and of he age funion. In he absene of noise ( S nn = 0 ), he Wiene file edues o he ideal invese file. The esoed iage ay hen be opued: ( u, v) M ( u, v) G( u v) T ˆ =, (40) This appoah is no opial fo ou poble beause any undelying hypoheses ae no valid. Typially, he odeling noise (ha ould well be he pinipal soue of noise) is oelaed wih he ideal iage. This noise is fuheoe diffiul o esiae. Ideally, S should be opued on a daabase of epesenaive age funions. I is ff diffiul and expensive o gahe suh a daabase. Bee esuls ould even be obained using a heuisi esiae of S based on he blued iage. One hen ipliily ges a esoed iage ha is a non-linea funion of he blued ff iage. This shows ha he fa ha he ehod esis he esoed iage o a linea funion of he blued one is a sevee dawbak. The ehod is howeve easy o ipleen and expeienal esuls show ha good esuls ae obained. We ae uenly invesigaing oe sophisiaed ehods beause hey ould povide shape iages bu he ain goal is o avoid he anual paaee uning. 3 Paial onsideaions The esoaion file was opued using equaion 39. Snn was hosen onsan ( S nn = σ ) and fo S ff an heuisi esiae of he speal densiy of he age funion was used. The age funion was supposed o be a gaussian of adjusable vaiane and apliude. A 4-pixels vaiane was hosen fo all he expeiens. The apliude of he gaussian (A) ay be opued using he DC oponen of he invese file. ( x, ( x, g A = T ( x, = (41) h This leads o he following S S aio: nn ff S S nn ff σ = (4) g S h ( x, ( x, whee S is a speal densiy of he gaussian wih he seleed vaiane and uni apliude. σ is anually adjused o ge good esuls. In paie, he age funion is no a gaussian and bee esuls wee obained using: S S nn ff σ = (43) ax( g( x, ) S h( x, The advanage of he ehod is ha σ does no vay in a wide ange. Hene i ay easily be uned wih a few ieaions. 5 EXPERIMENTAL RESULTS In his seion, he expeienal seup will fis be pesened. Resuls will hen be shown fo ages ha an be odeled by an independen dipole disibuion and by a losed winding. A disussion on he field deph and on he ipoane o sele he appopiae odel will hen be pesened. Finally, soe onlusion will be dawn.

14 1 Expeienal seup The expeienal seup is pesened in Figue 5. The MD is owed by a ehanized X-Y able onolled by a PC. A squae aea of 40 x 40 is sanned wih a 5 sep. The nueial oupu of he eal deeo is eoded in eah poin o build a aw iage of 81 by 81 pixels. Figue 5: Piue of he lab failiies. The MD is owed by a ehanized X-Y able onolled by a PC. As explained ealie, os soils do no influene signifianly he deeo oupu. This is he ase wih he soil we used and, o siplify he expeiens, he ages whee laid on he gound. To ake i sho, he disane beween he age plane and he deeo oil is alled he age deph. In paie, he deeo is sanned as lose as possible o he soil and hose wo quaniies ae nealy equal. Dipole disibuion To hek he validiy of he dipole disibuion odel (equaions 9 and 30), a sall sphee (o hek equaion 9 ) and a sall loop (o hek equaion 30 ) have been used. As he esuls fo he sphee and he loop ae siila, only he esuls wih he loop have been pesened. Moe oplex ages have hen been esed. Fo hose shapes, he spheial dipole disibuion odel (equaions 9) has been used. In ode o oply wih he odel, hese ages us no inlude lage uen loops. Theefoe, hese ages ae all ade up of hin oppe wie. Aong he, we epo hee he ase of a saigh wie 10 long, a X like shape ade up of wo ben oppe wies of oveall diension 8 x 1 and finally a flag like shape of oveall diension 8x16. The used sphee is a 11 diaee seel ball. Figue 6 shows he aw iage gaheed fo a ball buied 30 unde he deeo and he esoed iage ogehe wih he file and eo iages (in he spaial doain). The noise 3 fao is σ = 510. The pixel size is 0.5. The esoed iage shows a foused disk of abou 3 pixels diaee, whih is lose o he eal ball diaee. The eo iage is opued aking he esoed iage as inpu of he bluing poess. The eo is hen: e = g Tˆ h. The eo iage pesens a ghos of he oil. This esuls fo a isalignen of he deeo head. Indeed he head suppo was slighly wised aound he veial axis and he oil plane was likely a lile iled (planes of he oil and of he san wee no pefely paallel). Anyway, he esuled iage shows a good ageeen wih he oiginal age size and shape. Figue 7 shows he sae ball buied a 5.5 ( σ = 510 ) and 8 ( σ = 510 ). As explained in seion 1, he ahievable esoluion deeases wih he disane of he age o he oil. Theefoe, he esoed shape speads ou and exend lealy he size of he ball. In ode o assess he esoluion ahievable by of he used Wiene file in absene of 1 (a) (b) () (d) Figue 6 Reonsuion of a seel ball a a deph of 3. (a) oiginal aw daa. (b) esoed iage. () file. (d) eo.

15 (a) (b) () (d) (e) (f) (g) (h) Figue 7 Reonsuion of he sae seel ball a deph of 5.5 (a-d) and 80 (e-h). (a) (b) () Figue 8 Assessen of he ahievable esoluion a he pevious 3 dephs (a: 3, b: 5.5, :8 ). odeling eo on he die file, Figue 8 shows he podu of he die and he Wiene file: H ( u, v). M ( u, v) ansfoed bak in he spaial doain. The hee views deonsae ha, he esoluion an no be bee han a few en pixels as he deph goes beyond 5. Figue 9 o Figue 11 pesen espeively he esuls fo a single saigh wie an X and a flag like shape. All shape ae ade of oppe wie of 3 diaee. The saigh wie is 10 long and 1 deep. The X like shape i has an oveall diension 8 x 1 and is 4 deep. Finally, he flag like shape has an oveall diension 8x 16 and is 4 deep. The esuls ae enouaging as in all ases, he shapes ay be eognized. One ay also noie ha he diensions of he iaged shapes ae oe and ha he ones ae ahe well eoveed. As ould be expeed, he bes esuls ae obained wih he saigh wie ha is he loses o he oil (1). Fo he saigh wie a noise fao σ = 10 was used. The hoie of ha value is no iial beause he iaging file has a high bandwidh due o he sall deph and he invesion is ahe easy. Fo deepe ages, he low pass haae of he iaging file beoe oe ponouned and he seleion of he igh noise fao beoes oe iky. If he fao is oo low, he file aplifies oo uh he high fequenies and oo uh noise beoe appaen in he esoed iage.. On he opposie, if he fao is oo high; he aw iage is oo lile esoed and he esuling iage eains so blued ha he shape ay no be eognized. In paie, only few ials whee needed o find he igh fao. The esuls pesened whee obained wih σ = 1. (a) (b) Figue 9 Resul of a saigh oppe wie, 10 long, 1 deep. The wie is iled of 30º. (a) aw iage, (b) esoed iage. (a) (b) () Figue 10 Resul of a X like shape, ade up of ben oppe wie (8x1, 4 deep). (a) age piue (oaed of ¼ un), (b) aw iage and () esoed one.

16 3 Closed winding (a) (b) () Figue 11 Resul of a flag like shape ade of oppe wie (8x16, 4 deep). (a) age piue (b) aw iage and () esoed one. To hek he validiy of he losed winding odel, (equaions 36), a lage eangula loop of highly esisive wie has been buil. I was andaoy o use highly esisive wie o avoid sauaion of he eal deeo. In paie, he seleion of he wie esisane and he nube of uns o use was iial. The answe of os esed ages pesened eihe a sauaed o a oo weak answe. The poble is ha we don have any onol on deeo s gain. This is fine fo deeion of he age bu o use he deeo fo iaging pupose, sauaion us be avoided and he hadwae should be adaped o allow he gain o be onolled. The es eangle is pesened in Figue 1. I is build of wo (seial) eangula loops of esisive wie. Is size is 1x 8 and i is 8 deep. To undesand he pesened esuls, eebe ha he aw iage is fis be ansfoed using Y ( x, = ± ( x, and ha he age funion a a given poin is popoional o he nube of uns of he winding aound ha poin. Ideally, fo he es age, he esoed iage should pesen a onsan value inside he eangle and zeo ouside. The esul obained is in good onodane wih he odel. The shape ay be eognized and he size is oe. The noise fao used was σ = (a) (b) () (d) (e) (f) Figue 1 Resoaion of a age ade up of a eangula loop of low onduive wie buied a 8. (a) age piue, (b) aw iage, (b) oeed inpu () esoed iage, (d) file in he spaial doain, (e) eonsuion eo. 4 Kenel seleion H, ) should be used. As explained in seion 7, i is ipossible o find he igh odel o he igh deph using only he eal deeo oupu. Indeed, fo eah hypohesis, a age shape will be found and he eonsuion eo will no be bigge fo a wong odel han fo he good one. This is due o he fa ha an infinie nube of ages ay bak-sae he filed sensed by he deeo. To ge an inepeable iage of he buied age, he igh kenel ( ( u v) The only way o sele he igh age odel is o use a pioi knowledge on he expeed ages. If he igh age odel is used, a shap inepeable iage will be obained. In paie, he esoaion ay be pefoed wih seveal age odels and he use ay hen sele he bes iage. Typially, if a oplex known shape is eoveed, hee is lile hane ha his was aidenal and he igh odel was os pobably used. If a ieia fo a (good iage ay be defined, his seleion ould also be auoaed. This appoah is uenly unde invesigaion. To deeine he deph of he obje, a siila appoah ould be used bu he deph ould also be povided by anohe senso as a gound peneaing ada. The poble of deph seleion ay be opaed o he fousing of an opial syse. Thee also, a-pioi knowledge on a good iage is used o fous he syse. If he field deph is lage, he deph ay no be eoveed wih a high auay. Fo ou iaging poble, he sae holds, bu apa fo he ipossibiliy o eove he obje deph, a lage field deph has he advanage ha a genei kenel opued a ediu deph ay be used fo all deph of inees. The esuls pesened in Figue 13 show ha his is no he ase and ha he igh kenel should be seleed. The onsequene fo ha is ha he deph of he obje ay be deeined wih a good auay. Fo eah view, he noise fao used is σ = Iages a- esul fo he esoaion of a ball 3 deep using he kenel opued fo 1, 3 and 9. One sees ha wih a wong deph, he esuling iage is blued and a ghos ay even appea. Resuls ae even wose fo iages e-f whee a ball buied a 8 is esoed using he kenel opued a 1, 3 and 8.

17 (a) (b) () (d) (e) (f) Figue 13 Illusaion of a bad fousing. (a-) a ball buied a 3 and is esoed a 1 (a), 3 (b) and 9 (). (d-f) a ball is buied a 8 is esoed a 1 (d), 5 (e) and 8 (f) Finally, Figue 14 deonsae ha he igh age odel us be used. On ha exaple, a bass loop of 30 diaee is iaged using he odel of a spheial age (whih was suessfully used fo he onduive ball o he open oppe wies). Using ha kenel, i is ipossible o eieve he eal shape of he age whaeve he noise fao is ( σ = 0. 5 fo he pesened esul). (a) (b) () Figue 14 Illusaion of he esoaion using a bad kenel. (a) aw iage of a bass loop of 30 diaee buied a 3 esoed using (b) he igh kenel, () he kenel fo a spheial age 5 Conlusion Seveal expeiens have been ondued o hek he validiy of he odels developed in seion 3. Good esuls wee obained. I was indeed possible o eove inepeable shape infoaion. The qualiy of he esoed iage deeases wih he age deph and good esuls wee obained up o abou 10. The exa deph lii is funion of he size of he obje of inees and bee esoaion ehniques ould inease he deph lii. The neessiy and he way o hose he igh kenel deived fo he appopiae odel and deph was disussed. In seion 4, a nube of opened quesions didn eeive a oplee answe fo a heoeial poin of view. Fo insane: wha is he axial size fo a age o be odeled by a single dipole? Is he independen spheial dipole disibuion a good odel fo elongaed ages? The expeiens showed ha a 1 diaee sphee ay auaely be odeled by a spheial dipole and ha fo all he elongaed ages esed (saigh wie, X, flag), he odel is appopiae and allows a good shape eovey. The esoaion poedue is based on he deonvoluion of he aw iage using a Wiene file. To opue he esoaion file, he die iaging kenel us be known and he noise fao us be seleed. The opuaion of he kenel was based on a nueial siulaion of he agnei field geneaed by he oil's head. The noise fao was seleed by a ial and eo poedue. Expeiens showed ha a good fao ay be found wih a liied nube of ials. Invesigaions ae ondued o adjus auoaially his noise fao. Expeiens showed ha he ain noise is due o an eo on he iaging kenel. This eo does no oe fo an eo on he siulaed agnei field bu is ahe due o an eo on he oil posiion and oienaion. To be oe peise, he X-Y able povides a peise posiioning of he senso bu he oil suffes fo isplaeen and isalignen. We ae uenly inoduing soe oeing paaees in ode o ake ino aoun boh he oaion of he oil aound is axis and is il wih espe o he hoizonal plane. 6 CONCLUSION This pape has paved he way fo designing signal poessing algoihs, and oe espeially iaging algoihs, elying on soe basi bu ye fundaenals physial piniples of he agnei induion ehnology. We have saed wih a glane a he opeaion of he eal deeo. I was shown ha, wih he fequeny band used by he deeo, he quasi-sai agnei egie ay be used fo he odeling. Then, a ouple of basi age odels have been passed in eview, fo he single sall one, whih edues o a agnei dipole o he oe oplex lage losed loop. I has been shown ha a pioi knowledge is equied o eove soe paaees of inees suh as he deph of he age o he odel o use. Eah odel has been deailed and is validiy bounds have been disussed.

18 Soe expeiens have been ondued o show he validiy of he odels developed. Boh poin-like and open wie objes have been iaged using an iaging algoih uned fo unoupled dipoles-like ages. The lage losed loops winding, oplana o he deeo oil, has also been expeienally heked ou. Boh show a good ageeen wih he heoeial developens sine well-foused iages have been obained wih only a lile need of hand-uned paaees (he noise fao). The exa liis of he developed odels have sill o be deeined. I ould well be ha soe ages of inees fo he deine lead o poo esuls beause he odels ae no appopiae. Typially, fo soe ages, signifian uens ould be indued in lage loops o anisoopi dipoles ould be exied. The axial deph o ge usable esuls wih he ages enouneed on a eal inefield us also be deeined. This wok has iggeed a wide aoun of quesions. To answe hese quesions, oe siulaion and ials should be pefoed. Soe ipoveens ae uenly unde invesigaion suh as he possibiliy o use oe oplex esoaion ehods. The ain objeive is o avoid he anual uning of he paaees. The file should auoaially adap iself o he (loal) onen of he iage. The possibiliy o auoaially find he age deph is also onsideed. The ain objeive of he eseah was o ge as uh infoaion as possible on he ages wih a lassial eal deeo. Poising esuls wee obained, bu i appeaed ha soe ino adapaion of he deeo ould lead o a uh bee syse. Typially, a gain onol should be povided o avoid sauaion. The possibiliy o saple he volage ie deay would bing addiional useful infoaion (fingepin analysis). Many hypoheses on he age ay be elaxed if a ue bisai senso is used. Theefoe, fewe odels would be needed and bee iages ould be expeed. This kind of senso should inegae wo sepaaed oils. Fo insane, a lassial eal deeo ould be used as oving eie. The eeive ould be lef a a speifi loaion when he eie is sanned ove he field. I should be noied ha suh a bisai appaaus is well known in non-desuive esing appliaions. Theefoe, a lile effo should be equied o adap i o he ase of deining. Soe advaned induion ehniques ae uenly invesigaed. The objeive is o eove all of he paaees of he agnei polaizabiliy enso and hei speal haaeisis (se of poles). A se of oils of sae diaee wih appopiae elaive loaions and exiaion uens will allow fo onolling he oienaion of he iniden agnei field along a given dieion in a speifi egion of he spae undegound. Las bu no leas, iaging ehniques equie he knowledge of a posiion o assoiae o eah easueen poin, whaeve he san pah is, a egula gid o a anual sanning. A eal ie posiioning syse ha povides he 3D posiion and oienaion of he senso is hus needed. A posiion auay of less han 1 is andaoy. We ae uenly developing a lighweigh poable syse based on a aea and an aeleoee [1]. The feasibiliy sudy is opleed and a pooype will soon be available. ACKNOWLEDGEMENT We wish o expess ou gaiude o D. Ada Lewis, fo JRC/ISPRA/SAI/TDP, fo ou fuiful disussions abou he age odeling. 7 REFERENCES [1] C. E. Bau, Eds, Deeion and Idenifiaion of isually Obsued Tages, Taylo & Fanis, [] Y. Das e al, Analysis of an Eleoagnei Induion Deeo fo Real-Tie Loaion of Buied Objes, IEEE Tans. Geosiene and Reoe Sensing., vol. 8, no.3, pp , [3] I. J. Won e al., GEM3: a Monosai Boadband Eleoagnei Induion Senso, J. of Envionenal and Engineeing Geophysis, vol., no.1, pp , Mah [4] P. Gao, L. Collins, Ipoved Signal Poessing Appoahes fo Landine Deeion, Po. SPIE on Deeion and Reediaion Tehnologies fo Mines and Minelike Tages III, vol. 339, pp , [5] A. Rosenfield, A. C. Kak, Digial Piue Poessing, Chap. 7, Aadei Pess, 198. [6] D. Daniels, Sufae-Peneaing Rada, IEE books, [7] J. A. Saon, Eleoagnei heoy, MGaw-Hill Book Copany, [8] G. Faneshei, Eleoagneis Theoy, Tehniques, and Engineeing Paadigs, Plenu Pess, [9] H.C. Andews and B.R. Hun, Digial Iage Resoaion, Penie Hall In., Englewood Cliffs, New Jesey, [10] Guy Deoen. Iage eonsuion and esoaion: Oveview of oon esiaion suues and poble s. IEEE Tansaions on Aousis, Speeh and Signal Poessing, vol. 37, no. 1, pp , Deebe [11] A. K. Kasaggelos. Digial Iage Resoaion, Spinge-elag, [1] C. Beuie e al., Moion esiaion of a hand-held ine deeo, Signal Poessing Syposiu, Hilvaenbeek, NL, 3-4h Mah 000.