A direct impedance tomography algorithm for locating small inhomogeneities

Transcription

1 A irect ipeance toography algorith or locating sall inhoogeneities Martin Brühl, Martin Hanke, Michael S. Vogelius Fachbereich Matheatik, Johannes Gutenberg-Universität Mainz, Mainz, Gerany; e-ail: Departent o Matheatics, Rutgers University, New Brunswick, NJ 0903, USA; e- ail: vogelius@ath.rutgers.eu Date: May 2, 200 Suary Ipeance toography seeks to recover the electrical conuctivity istribution insie a boy ro easureents o current lows an voltages on its surace. In its ost general or ipeance toography is quite ill-pose, but when aitional a-priori inoration is aitte the situation changes raatically. In this paper we consier the case where the goal is to in a nuber o sall objects (inhoogeneities) insie an otherwise known conuctor. Taking avantage o the sallness o the inhoogeneities, we can use asyptotic analysis to esign a irect (i.e., noniterative) reconstruction algorith or the eterination o their locations. The viability o this irect approach is ocuente by nuerical exaples. Matheatics Subject Classiication (2000): 65N2, 35R30, 35C20 Introuction Techniques or recovering the conuctivity istribution insie a boy ro easureents o current lows an voltages on the boy s surace go uner the heaing o electrical ipeance toography (EIT). The vast an growing literature relects the any possible applications o this etho, e.g. or eical iagnosis or nonestructive evaluation o aterials. For urther etails we reer to the recent survey paper []. Since the unerlying inverse proble is nonlinear an severely ill-pose it is generally avisable to incorporate all available a-priori knowlege Supporte by the Deutsche Forschungsgeeinschat (DFG) uner grant HA 22/2-3 Supporte by the National Science Founation uner grant DMS

2 2 Martin Brühl, Martin Hanke, Michael S. Vogelius about the unknown conuctivity. One such type o knowlege coul be that the boy consists o a sooth backgroun (o known conuctivity) containing a nuber o unknown, sall inclusions with a signiicantly higher or lower conuctivity. This situation arises or exaple in ine etection, where one tries to locate the position o burie anti-personnel ines ro electroagnetic ata. The ines have a higher (etal) or lower (plastic) conuctivity than the surrouning soil an they are sall relative to the area being iage. General purpose EIT reconstruction ethos are likely to ail: ue to the sallness o the ines the associate voltage potentials are very close to the potentials corresponing to the unperturbe eiu, so unless one knows exactly what patterns to look or, noise will largely oinate the inoration containe in the easure ata. Furtherore, in this application it is oten not necessary to reconstruct the precise values o the conuctivity o the ines or their shapes. The only inoration o real interest is their positions. In this work we propose a irect algorith or eterining the positions o sall conuctivity inhoogeneities. The algorith akes use o an asyptotic expansion o the voltage potentials, which has been erive by Ceio-Fengya et al. [6]; see also the prior work o Friean an Vogelius [] or the case o perectly conucting or insulating inhoogeneities. Aari et al. [] have also utilize this asyptotic expansion to esign a variationally base irect reconstruction etho, a etho that is quite ierent ro the one presente here. Our algorith is soewhat in the spirit o a etho evelope in [4,5], an it also has soe siilarities to a recent MUSIC-type etho evelope by Devaney [9]. A etaile iscussion o the relation between the latter (MUSIC) algorith an the linear sapling etho in inverse scattering (c. Kirsch [4]) is oun in Cheney [7]. This paper is organize as ollows. In the next section we review the asyptotic expansion o the voltage potentials in the presence o sall inhoogeneities, in particular we show how this leas to a siilar expansion or the associate Neuann-Dirichlet operator. The principal operator arising in the latter expansion will be exaine in etail; this operator is at the root o our algorith or the location o the inhoogeneities as explaine in Section 3 an Section 4. In Section 5 we show how certain inoration about the shape o the inhoogeneities ay be recovere, once their positions have been eterine. 2 The orwar proble with sall inhoogeneities Let,, enote a boune oain with a sooth bounary, an a sooth, positive backgroun conuctivity. We consier

3 0 0 / A irect ipeance toography algorith 3 conuctivity istributions o the or (2.) where. Here, the points inicate the positions o the centers o the inhoogeneities, an the sooth sets (with ) escribe their relative shapes. The average inhoogeneity size is speciie by the paraeter, which is assue to be sall. We will also assue that "!, so that the constant conuctivity o any inhoogeneity is ierent ro that o the ajacent backgroun. I one inuces a noral current low, with $&%(')+*-,., on the bounary, then this gives rise to a voltage potential / that solves the Neuann bounary value proble 02 3 /4 in 3 65 on With the aitional noralization conition $(%(' / *-, 7 the solution / becoes unique. The relation between the applie bounary currents an the bounary voltages /9 %(' eines a linear apping : <; >=?@/9 %&', the so-calle Neuann-Dirichlet operator. Here we consier : as an operator ro A B into itsel, where A B 7CDEA ; $ %('F<*-, GIH. In this topology : is copact an selajoint. We are intereste in the behavior o : as tens to zero, i.e., as the inhoogeneities shrink to the points, J. First o all, one expects that / %(' converges to the bounary values o the potential /6K corresponing to the backgroun conuctivity K. This is inee one conclusion o the ollowing uch ore etaile result ue to Ceio-Fengya et al. [6, Theore 2]. Theore 2. Assue the points, 2, satisy LNMPORQ ST U WVX* or K YZ! [ an \V]* (2.2) K or soe constant * K 0a. Let ^ `_ enote the Neuann unction or the ierential operator in the oain one has the asyptotic expansion /4 S b/ K c e hg ]n porq S LNMPORQ ji 0lk ^. Then, or U & 0 / K (2.3)?s. The syetric positive einite ut -atrix as is the so-calle polarization tensor corresponing to the -th inhoogeneity.

4 n k t t k 4 Martin Brühl, Martin Hanke, Michael S. Vogelius The Neuann unction ^ `_ is the unique solution to 0lk 0lk ^ `_ ^ `_ i in 65 k 7i with the noralization $(%&' ^ `_ It is easy to see that u LNM ^ syetric, i.e., ^ `_ G^ _ or `_ \ syetry 0lk LNM yiels a sooth extension o ^ to 0 t satisying ^ S ^ U. The polarization tensor epens on the relative shape o the -th inhoogeneity on is t. This t is a syetric positive einite atrix, that uctivity contrast an the con-. For its exact einition we reer to [6] (see also Appenix A o the present paper). In [] an [6] it is assue that the sets are star-shape, but as evience by [7] an [2] one ay ispense with this conition in case the are strictly positive. For the case o constant backgroun conuctivity there is a slight variation o Theore 2., that or any practical applications is ore useul, since it relies on an explicit unaental solution. Let `_ enote the unction `_ )i_ )i_ - Theore 2.2 Assue the backgroun conuctivity the points has the asyptotic expansion as /4 ji/ K S,, satisy conition (2.2). Then, or hg? with as in Theore 2.. We note that or EA B %(' /4 i/ K 65 k U -*-, Fi 0lk & 0 ]n U / K S is constant an that one porq (2.4) the unctions / an /K are not necessarily sooth up to the bounary. However, ue to elliptic regularity results, the ierence / S il/ K S is sooth (near) an up to the bounary. It is prove 9o in [6] that the reainer ters in (2.3) an (2.4) are boune by q, uniorly or, however, nuerical experients suggest it is even saller, naely o the orer. The constant in the bouns or the reainer ters epens on the oains,, the backgroun conuctivity, the constant * K an the noral current. The epenence on aniests itsel through the nee or an energy estiate an the nee or pointwise bouns on the values an the erivatives o /6K near the positions all o which only requires a boun on "! $ %('&%. In other

5 B n 0 0 k A k A irect ipeance toography algorith 5 wors, the epenence o on is only a epenence on "! %('&%. The etails o this arguent are oun in [7] an [2] or the case o the Maxwell equations. Let us now introuce the operator hg S ; A ji 0 ^ B U? & 0 B / K S eine by (2.5) Since / K epens linearly on this operator is linear, an ro Theore 2. (an the reark about the constant o the reainer ter) it ollows that ]n 9o q (2.6) :Wi : K 9o q is boune by 9o q UA B 0 k ). Here we also use the act that ^ or E. The operator is selajoint on A B S: i:wk. where the reainer ter q in the operator nor o (an thus in the operator nor o 0 SA B UA B U ^ U, since it is the liit o the selajoint operators In the rest o this section we will point out soe aitional properties o the operator, which turn out to be crucial or our approach to eterine the positions. We begin by introucing another linear operator ; A B? / K Here the potential /K correspons to the input current an can thus be represente as / K _ $ %(' ^ `_ -*-, with the stanar Eucliean inner prouct, hg or we then obtain hg hg hg 0 %&' &0 / K 0 ^ / K S h or _E \ ^ U U. Enowing ` I*-, h %('&%

6 K A 6 Martin Brühl, Martin Hanke, Michael S. Vogelius or arbitrary is given by Z hg 0. Thereore ; ^ U? B h (2.7) Lea 2. is injective. Proo Suppose that or wors, hg &0 solves the Cauchy prob- Then the unction le 0 0 G in hg ^ U G hg 0 CD ^ U or ) H %(' G, or in other 65 %&' G an ro the uniqueness o this proble (see or exaple [5, Theore 9.II]) we euce that M. I enotes the -th unit vector in, then in particular we have S 0, an thus. Inee, otherwise the ipole singularity o M ^ U at woul iply K. This proves that G, an thus the assertion o this lea. Corollary 2. is surjective. Proo This ollows ro Lea 2. an the well-known relation between the ranges an null spaces o ajoint operators with inite rank. ;? i Using the above orulae or an an the einition (2.5) o, we see that these operators are relate by (2.) where operator is given by S ji S! or. Fro the positive einiteness o the atrices " we conclue that is positive (respectively negative) seieinite, i S (respectively ) or all 2.

7 g T g T H H A irect ipeance toography algorith 7 Let us take a closer look at the range o. First we observe that is inite iensional with iension at ost ; ore precisely we have O &0 ^ U %(' ; Next we show that this inclusion is actually an equality. Proposition 2. The range o has iension an is given by O &0 ^ U %(' ; Proo The surjectivity o an iplies. This proposition is then an ieiate consequence o the orula (2.7) or. Now we present the ain tool or the ientiicaton o the positions 0 Proposition 2.2 Let *E C IH, >, an 2* ^ Then, i an only i "CD Proo Assue that ay be represente as ^ U are solu- But then both tions to the Cauchy proble 0> 0 G in lr hg &0 hg D0 C ; H.. U %('.. As a consequence o Proposition 2., ^ U ^ U, an * or E D0 C IH %(' 65 %&' G an ro the -0 uniqueness o solutions to this proble we conclue that * ^ U or C C -H. This is only possible i E C r ; J H, an so we have establishe the necessity o this conition. The suiciency ollows irectly ro Proposition 2.. Since the operator : ig: K is selajoint an copact on A B it aits a spectral ecoposition :W i : K T T T T $ %('&% 2 with eigenvalues T ecaying to zero. Siilarly, the inite-iensional selajoint operator can be ecopose as T T T DT $ %('&% 2

8 n g T T n g T n Martin Brühl, Martin Hanke, Michael S. Vogelius Fig. 2.. Unit isk with three inhoogeneities. V V (say with V ). Using (2.6) an stanar arguents ro perturbation theory or linear operators [3], we get (by appropriate enueration o the eigenvalues o : i :WK ) the ollowing asyptotic orulae as?s, ]n porq T Y (2.9) Here we have set T or Y. Let ; A B O O s? C H an ; A B? C H enote the orthogonal projectors T T an DT T respectively. Suppose or siplicity the eigenvalues o are siple; the liiting relationship (2.6) together with stanar arguents ro perturbation theory or linear operators [3] then gives ]n q or The sae stateent hols even i the eigenvalues are not siple, provie one akes appropriate choices o eigenvectors I T an DT, Y We illustrate the asyptotic behaviour o the eigenvalues by eans o a nuerical exaple. In the unit isk we choose circular inhoogeneities, which are shown or ( in Figure 2.. The conuctivity within each inhoogeneity is I whereas is the the hoogeneous backgroun conuctivity. The eigenvalues T o : ï :WK or three ierent values o are shown in Figure 2.2. Accoring to (2.9) we expect to see eigenvalues o orer while all the the reaining eigenvalues have saller agnitue (no bigger than the reainer ter in Theore 2.). In line with what we entione earlier this exaple suggests that po the q reainer ter in Theore 2. is inee, an not just the asserte by our estiate.

9 T g Q i n A irect ipeance toography algorith 9 Srag replaceents PSrag replaceents Fig Eigenvalues o or right) or the exaple ro Figure 2.. PSrag replaceents,, an (ro let to 3 Deterining the locations o the inhoogeneities base on the expansion ro Theore 2. Beore we present our approach to eterine the positions we briely recall the explicit characterization o the inclusions which has been erive in [4,5]. There it has been shown (provie : i:wk is seieinite) that the paraeter point lies within one o the inclusions i an only i the series %('&% T T converges. This is teste nuerically by irst estiating an then coparing the ecay o the square Fourier coeicients T $ %&' % with the ecay o the eigenvalues T. In orer to apply such an algorith, it is necessary to copute at least a ew o these ters in a stable anner. While the orer o the Fourier coeicients is essentially inepenent o, the size o the largest eigenvalues ecreases at least like as tens to. Consequently, in the presence o ata errors, the estiation o the ecay rate o the eigenvalues will be ipossible when the size o the unknown objects gets sall. The reconstruction etho we propose here involves only the actual size o the Fourier coeicients T %('&% (an to soe extent the size o the eigenvalues) but not their ecay rate. The present etho is thus uch less sensitive to noise. The eigenvalues are only use to ientiy the subspace corresponing to the largest eigenvalues. This is possible i these eigenvalues excee the noise level. In Proposition 2.2 we have seen that a test point coincies with one -0 o the positions i an only i * ^ U %(', or equivalently, i i. In other wors, i we ecopose the test unction orthogonally as an eine the angle \ by i! S

10 Q Q Q Q Q " 0 Martin Brühl, Martin Hanke, Michael S. Vogelius PSrag replaceents Fig. 3.. Deinition o the angle. c. Figure 3., then we have that lc ; H G! Unortunately, we cannot copute, because epens on the unknown positions. However, or sall sizes the projecte test unction is well approxiate by, an the projections can be copute or each by eans o the eigenunctions o the easure operator : i : K. This ay serve as a otivation or the ollowing einition o the angle S by! D S i T T T T $ %('&% %('&% For bc ; H all ters in the enoinator are o orer R as? i is chosen equal to. The nuerical value o ay be estiate by looking or a gap in the set o eigenvalues o : i :WK. I we plot! as a unction o, we thus expect to see large values or points which are close to the actual positions. Since none o the eigenvectors o : i : K corresponing to eigenvalues T, Y -, are exactly o the or, CD ; H, we expect the sae to be true i we plot! or oerate size. The calculation o! 0 requires the calculation o * ^ U %&', soething that in ost cases will be quite expensive. One notable exception occurs when, the backgroun eiu has constant conuctivity, an is a isk (or exaple C ; q b H ). In this case the Neuann unction has an explicit expression, that we ay use to calculate the bounary values o its graient. This calculation yiels the siple orula 0 ^ U i or )i W"

11 Q Q k A irect ipeance toography algorith Srag replaceents PSrag replaceents PSrag replaceents Fig Exaple with three inclusions: or,, an. When the backgroun eiu has constant conuctivity, but the oain is not a isk, one ay utilize the orula with an * 0 ^ * U %&' %(' ji : K &0 U enoting the projection operator ro A i %(' -*-, G 35 CDIH onto A B %(', (3.) The presence o the operator :\K reners the coputation o soewhat expensive, an since this coputation typically has to be carrie out or a very large nuber o test points, one ay, or non-circular, preer to use the approach we escribe in Section 4. For the two-iensional exaple ro Figure 2., using synthetic ata without noise, we show plots o! in Figure 3.2 or three values o. The centers o the three circular inhoogeneities are clearly eterine in each case. Note that a-priori knowlege o is not require or the coputation o ; only the spectral ecoposition o the easure operator : i :WK is involve. In Figure 3.3 we consier an exaple with circular inhoogeneities with raius & (still insie the two-iensional unit isk) an we plot! or ierent values o. Inee, or V the seven centers are well reconstructe, whereas or saller the plots give isleaing results. This easily coputable sequence o plots yiels rearkably goo inoration on both the nuber o inhoogeneities as well as on their location. When coparing the etho propose here to other ethos that have been esigne or iaging sall inhoogeneities, it is air to point out that it uses inoration about the entire Neuann-Dirichlet ap. In contrast, the iterative approach taken in [6] only relies on a single set o Cauchy ata (a single

12 2 Martin Brühl, Martin Hanke, Michael S. Vogelius Fig Actual positions (upper let picture) o seven inclusions an. or point on the graph o the Neuann-Dirichlet ap) an not surprisingly, it is ore liite in its ability to eectively locate a high nuber o inhoogeneities. At this point it also sees relevant to point out that the use o spectral ata ro the (Neuann-Dirichlet) bounary ap is not new in ipeance iaging. It was or instance very early notice, that the use o ipose bounary currents closely approxiating eigenvectors is very avantageous [2]. 4 Deterining the locations o the inhoogeneities base on the expansion ro Theore 2.2 As entione in the last section the require calculation o the graient o the Neuann unction is oten quite costly. I the backgroun eiu is truly inhoogeneous (non-constant) then there is not uch that can be one about this. However, i the backgroun eiu has constant conuctivity, then it is possible to slightly change the algorith we have evelope so ar, such as to lower this cost signiicantly. For that purpose we rely on the

13 A k k A 0 k A irect ipeance toography algorith 3 asyptotic orula o Theore 2.2. This orula asserts that ]n 9o q :W i: K (4.) where is the inite rank operator A B?sA eine by an is the copact operator A hg Fi 0 %&' U?sA 65 k The reainer is boune by 9o q A B UA. The ajoint ; A in the operator nor on? o the operator has the representation A siple calculation shows that $ %(' %(' 65 _ -*-, 0 / K -*-, eine by (4.2) -*-, i $ %(' -*-,, ro which it ollows that aps A B into itsel. Furtherore, is one-to-one an onto as an operator A B Z? B this ollows ro its irect relationship to the solution o the interior Neuann proble (see or exaple [0, Proposition 3.37]). By appropriate ultiplication o (4.) by an by the projector ro (3.) we obtain n porq :W ie: K (4.3) The operators S: i:wk an are selajoint on A, an just as in the previous section we ay now use the easure ata S: i :WK to approxiately calculate the range o. I is a isk (in two iensions) then on A B an, an the orula (4.3) is thus ientical with (2.6). In the general case we have a result siilar to Proposition 2.. Proposition 4. The range o is given by O &0 U %(' ; 2 Proo The act that aps A onto A B yiels that. As in Section 2, the operator Fi 0 >i ;? / K S / K

14 g o o i * i % 4 Martin Brühl, Martin Hanke, Michael S. Vogelius aps A B onto. It is now obvious ro the einition o, c. (4.2), that O 0 %(' ; ro which the assertion ollows. U To obtain an equivalent o Proposition 2.2 we ust insist that Assuption I For any set o istinct points CD in the &0 unctions C U %&' H are linearly inepenent Proposition 4.2 Suppose Assuption I is satisie. Let *u B 0 enote the unction, an let A Then i an only i lcd hg H hg ; H. C IH, > U %('. Proo This is an ieiate consequence o the linear inepenency assuption (Assuption I) applie to the points CDIH CD H, an the characterization o obtaine in Proposition 4.. As we have seen earlier a two-iensional isk satisies the Assuption I (this is veriie in the proo o Proposition 2.2) but there are any other such oains, as witnesse by the ollowing result. Proposition 4.3 Assuption I is satisie or any boune, convex, twoiensional oain, the bounary o which contains a straight line segent. Proo Pick a coorinate syste, such that the bounary o shares a line segent S with the, -axis an such that lies in the hal-plane. Let, enote the coorinates o the points,. In the, coorinate syste the unction p0 k U k has the expression % % &0, i, U. There- where ore % % &0 U are the, %&', i, coorinates o the vector %, ic,, ic, % ji hg `

15 o Y o Y ", o " 5, " o, o A irect ipeance toography algorith 5 on the coon line segent (say ) that shares with the, -axis. In orer to veriy this lea we have to show that hg &0 U " or soe set o vectors, iplies that We o this by showing that % %, ic, ji ` G hg, i, %&' G G or all 2 ". (4.4) iplies % % or all. The expression that appears on the let han sie o (4.4) is a eroorphic unction (o, ) an i either % or % is not zero then it has proper poles at hg,,. Here we use that C, H are istinct coplex hg S hg nubers, as ollows ro the act that the points CD H C, UH are istinct an the act that each is positive. The presence o these proper poles, however, woul contraict the act that the" let han sie o (4.4) ientically vanishes on the real interval. We conclue that % %, as esire. G, The oiie reconstruction algorith procees just as in Section 3, using the spectral ecoposition o the operator S: i : K in place o that o : ic: K, an using the explicit unctions in place o. 5 Recovering other geoetric inoration In this section we presue or the oent that the positions,, o the inhoogeneities have alreay been reconstructe, an that we now seek to recover inoration about their shape. First, let us ake the sipliying assuption that is the unit isk in, an that the backgroun conuctivity is constant. In this case the approach base on the expansion ro Theore 2. is quite siple, since the unction 0 ^ given by 0 U %(' is known explicitly; as entione earlier it is i ^ U or W" )i 5 We shall consier the two input currents,, or which / K 0 are the resulting haronic potentials, an so / K.

16 ' 6 Martin Brühl, Martin Hanke, Michael S. Vogelius Fig. 5.. Selection o the points an. We assue or siplicity that only one inhoogeneity is present in having center %, an polarization tensor. We choose the two points % an on the bounary so that % ic % Figure 5.. Then we have % Fi & )i % % i Fro our ata we have available the voltage ierences % % S:W i : K ic, 2, c. (5.) i an we ay thus approxiately recover the scale polarization tensor Fi We now iscuss the kin o inoration about the unknown object that can be gaine ro this ata. [ Suppose the inhoogeneity has the or where is the ellipse given by C ; H (here is a syetric positive einite Ft -atrix). Let with! O Ò M ",,,, be the rotation atrix, or which the ocal interval o the rotate ellipse! lies on the -axis, i.e., or which C ; DH with " Here, V, enote the lengths o the two sei axes o the ellipse. In Appenix A we sketch how the associate polarization tensor o ay be calculate explicitly. The resulting orula is Fi &% $ i] R ( o*) o ( ),+

17 % % % % % 5 % % A irect ipeance toography algorith 7 with i] ` % &% an i] ` &% % (5.2) Here ( an C H enote the aspect ratio o the sei axes o the ellipse an the contrast o the conuctivities, respectively. Note that or all aissible values o an. The scale polarization tensor o the original ellipse is then given by. We note that the atrix escribing the geoetry o the ellipse an the scale polarization tensor are iagonalize by the sae rotation atrix ; to be ore precise, the coon set o eigenvectors is speciie by the sei axes o the ellipse. Consequently, the orientation o the ellipse can be recovere ro knowlege o the eigenvectors o. The our reaining unknown quantities are the conuctivities,, an the lengths an o the sei axes o the ellipse. An equivalent set o paraeters are,,, an. The backgroun conuctivity is in practice either a-priori known, or can easily be estiate ro the hoogeneous potential ata or the inuce input current. Inee, in this case /3K is the associate voltage easureent, ro which one can estiate. This eans that actually three quantities reain to be ientiie, naely,, an the area o the unknown ellipse. However, the eigenvalues o give only two pieces o ata. Thereore, we nee urther inoration about the inhoogeneities, in the or o a-priori knowlege about their shape, size, or conuctivity. For exaple, i is known, then ro (5.2) we can copute an by i i an i] (5.3) In any practically relevant situations the conuctivity contrast will be rather high, i.e., we will have or. As a sipliication we ight assue that, i is positive einite, an we ight assue that ä, i is negative einite. Using these extree values in (5.3) we obtain % "M an (5.4) The lengths o the sei axes o the ellipse can then be expresse as ollows, % an (5.5)

18 Q Martin Brühl, Martin Hanke, Michael S. Vogelius phanto ata reconstructe ata backgroun conuctivity position scale polarization tensor eigenvalues rotation angle lengths o sei axes conuctivity in ellipse Table 5.. Reconstruction o an ellipse. (assuption) We illustrate the reconstruction proceure by a nuerical exaple. As a phanto we choose a sall ellipse within the unit isk, with a uch lower conuctivity than that o the backgroun. The geoetry o this ellipse is speciie in Table 5.. Synthetic easureents are generate using a stanar bounary eleent etho. In the irst step we locate the position o the inhoogeneity by eploying the technique introuce in Section 3. The position is oun as the point, where! S attains its axiu. Aterwars the scale polarization tensor is recovere ro (5.). A spectral ecoposition o yiels the orientation o the ellipse. Finally, the lengths o the sei axes are obtaine ro (5.4) an (5.5). In two urther nuerical experients we choose phantos o ierent shape. The sae algorith as beore yiels an ellipse with a scale polarization tensor as given by the ata (that ellipses are suicient to it the ata is a well known act, see [6]). I the original shape is close to an ellipse then we can hope that this reconstruction provies a goo approxiation. This is illustrate in Figure 5.2, where we have chosen phantos in the shape o a kiney (let) an a booerang (right); the conuctivity contrast or the calculation o the synthetic ata was taken to be &9 in both cases. The original phanto is rawn with a thick soli line, an a thin soli line is use or the reconstructe ellipse in the absence o noise. The ajacent boxes zoo in on the region o interest. We also perore experients with noisy ata, where the easureents were perturbe by aing I relative noise. The corresponing reconstructions are shown in ashe lines. A Polarization tensor or an ellipse Accoring to [6] the entries o the polarization tensor corresponing to a sall inclusion centere at K, o relative shape, an

19 O i ' 5 " A irect ipeance toography algorith 9 Fig Reconstruction o non-elliptic objects without noise (soli ellipse) an with noise (ashe ellipse). with conuctivity are given by i % _ o 65 *-, 2 Here, the unctions _ are the solutions to the ollowing proble: G in an in (A.a) K o 65 o on (A.b) 65 K ji M _ G on (A.c) where the an i superscripts enote the liits on an interior o, respectively. It is not iicult to calculate that i (A.) ro the exterior is a two-iensional isk, then the associate polarization tensor is a ultiple o the ientity atrix, naely o o (A.2) + Here we will calculate in the case equals, an ellipse with ocal interval i on the -axis an eccentricity! O, or in other wors, an ellipse whose sei-ajor axis is o length! O Ò M an lies on the -axis, an whose sei-inor axis is o length an lies on the -axis. First we have to solve the bounary value probles (A.) in orer to eterine the unctions an. In orer to o this, we introuce elliptic coorinates,!! O O M Ò M VX W

20 K g O O O + 20 Martin Brühl, Martin Hanke, Michael S. Vogelius " in which the ellipse is given by.c ; H. Separation o variables yiels a general solution o the Laplace equation o the or! Ò M! O M in i an in the exterior o (with ierent sets o coeicients). For the solution in i to exten to a haronic unction in we ust urtherore require that i an S i i h (A.3) The coeicients o the speciic solutions an are now eterine by the conitions (A.3) an the bounary conitions (A.b) (A.). The result is % o! an % o with " " or or an or or Using this it turns out that the polarization tensor takes iagonal or, naely ' o % o o % (A.4) o where is the area o the ellipse. Note that or the special case o a isk we have an then (A.4) reuces to (A.2). For an arbitrary ellipse whose sei axes are not aligne with the coorinate axes, one can in an orthogonal transoration such that, where is o the above or. The polarization tensor corresponing to is then given by ", c. [, Section 6]. Ò M Reerences. Aari, H., Moskow, S., Vogelius, M. S.: Bounary integral orulae or the reconstruction o electric an electroagnetic inhoogeneities o sall volue. Subitte. 2. Aari, H., Vogelius, M. S., Volkov, D.: Asyptotic orulas or perturbations in the electroagnetic iels ue to the presence o inhoogeneities o sall iaeter II. The ull Maxwell equations. To appear in J. Math. Pures Appl., IX. Sér.

21 A irect ipeance toography algorith 2 3. Borcea, L., Berryan, J. G., Papanicolaou, G. C. (996): High-contrast ipeance iaging. Inverse Probl. 2, Brühl, M. (200): Explicit characterization o inclusions in electrical ipeance toography. SIAM J. Math. Anal. 32, Brühl, M., Hanke, M. (2000): Nuerical ipleentation o two noniterative ethos or locating inclusions by ipeance toography. Inverse Probl. 6, Ceio-Fengya, D. J., Moskow, S., Vogelius, M. S. (99): Ientiication o conuctivity iperections o sall iaeter by bounary easureents. Continuous epenence an coputational reconstruction. Inverse Probl. 4, Cheney, M.: The linear sapling etho an the MUSIC algorith. To appear in Inverse Probl.. Cheney, M., Isaacson, D., Newell, J. C. (999): Electrical ipeance toography. SIAM Rev. 4, Devaney, A. J.: Super-resolution processing o ulti-static ata using tie reversal an MUSIC. To appear in J. Acoust. Soc. A. 0. Follan, G. B. (976): Introuction to Partial Dierential Equations. Princeton University Press, Princeton.. Friean, A., Vogelius, M. (99): Ientiication o sall inhoogenities o extree conuctivity by bounary easureents: a theore on continuous epenence. Arch. Ration. Mech. Anal. 05, Gisser, D. G., Isaacson, D., Newell, J. C. (990): Electric current copute toography an eigenvalues. SIAM J. Appl. Math. 50, Kato, T. (966): Perturbation Theory or Linear Operators. Springer, Berlin. 4. Kirsch, A. (99): Characterization o the shape o the scattering obstacle using the spectral ata o the ar iel operator. Inverse Probl. 4, Mirana, C. (970): Partial Dierential Equations o Elliptic Type. Springer, Berlin, 2n e. 6. Movchan, A. B., Serkov, S. K. (997): The Pólya-Szegö atrices in asyptotic oels o ilute coposites. Eur. J. Appl. Math., Vogelius, M. S., Volkov, D. (2000): Asyptotic orulas or perturbations in the electroagnetic iels ue to the presence o inhoogeneities o sall iaeter. RAIRO, Moélisation Math. Anal. Nuér. 34,