A Computational Method for Electrical Impedance Imaging

©

(:op~Tighl 11-':\( . (:(ll111'ol !It l) i.... lrilHlll'd ParaIlll'llT S\' ~ 1 l' 1Il S, l'l' rpi g'll :lll. F r;tllct'. I ~ ' K~j

A COMPUTATIONAL METHOD FOR ELECTRICAL IMPEDANCE IMAGING F. Santosa",·1 and M. Vogelius",*·2 "' li l'l)((rllll"1I1 "l ,\I !1Ii1I'1I1!1liml Slil'lIl'I',I. ['lIi"I'n ily of IJl'lml 'I/ /{'. ,Yl'
DL 1,)"/1 (j. l '.\4

""'D I'IJllrllll"111 of ,\I l/lill'IlI!llirs . l ' IIi1'l'nily of '\/!I/I' /I/I/II . Co/ll'gl' PI/rl!.

,lID 207-/2. [ 'SA

ABSTRACT Electrical impedance imaging is a technique for determ ining the internal impedance (or conducti\'ity) of a medium based on e lectrostat ic boundary measurements. Applications for this imaging method include medical tomography and nondestruct ive inspection, In this pape r. we describe an aparatus for data generation and a backprojection algorithm for imagin g devised by D.e . Barber and B,H, Brown. We explain how their algori t hm fit s within the fram e work of inversion of gene ralized Radon transforms. A new computational method which impro\'es upon 13arber and 13rown's a lgorithm is proposed. The effectiveness of t he new met hod is d emo nstrated in numerica l silllulat ion s .

INTROD UCT ION ,r(8,1/2..,;)

In electrical impedance imaging one see ks to reconstruct the internal conductiv it v (or impedance) profil e o f an object from boundary measurements of volta:;es and corrcsponding C1lrrent fiuxes. Significant advances on the practical aspects of t his prob· lem has been made in rece nt \'ears , The reader is refered to 13arber and Brown (1986). Brown. Barber and J ossinet (1988) and Isaacso n (\986) . On the theoretical s ide. there has also

" ·1' 1

bccn significant ad\'anccs cOllcerning ulliquC'lI cSS and contin uous

dependence; we refer to Alessandrini ( \ 91'8) . Friedman and Vogeli us (1989). I\ o h n and Vogelius (l!J85). Nac hm an (1988) and Syl vester a nd Uh ll1l ann (19S,), In t heir 1986 paper. BarbN and Brown proposed a s imp le meas urement apparatus for il1laging th0 inlNior of a human chest region . Th eir device consists of a circular affa,' of electrodes wh ich are attached to the skin. In Figure 1. we give a sc hemat ical picture of the de\'ice, The region whose elect rical impedance is to he im aged is represe nled b,' th" int.0rior of the circle. A sequence of 0xperil1lenls is cOllducted, Current fiows between a pai r of elect rod es 1 and "2 (call"d the d ri \"(' 1' pai 1') wh ile voltages are measured at all 16 electrodes (1I10re preci sc1,\': measu rements are only taken al \-1 electrodes. the voltage, on the 2 dri\'e r electrocles are obtained b,'interpolation), In the next experim e nt , the dri\'er pair is electrodes Land :l, This is repeated until we haw cycled through all possihle dri\'er pairs, III their work. Barber and Brown developed a backprojec t ion algorithm which we desrrilw in the next sect ion. \\' hat is particu larl\' intNest in g about I his algorithm is that it is surprisingl:, effect ive and efficient. Impedance images are returned in realtime usi ng a d es ktop computer. T hi s pa per cons is ts of a su m III a of ou r work (San tosa and Vogeli us (1988)) and lIew resullS which ha\'e not ,'et been re· ported, \\'e begin wi t h all explanalion of ho\\' the backproject i0n method fits \\'ilhin tl10 franlf'\\'ork of generalized Radon tra nsfor ms, s tudi ed \)\' Ih',\'l kin (11);Q) , \ ex l. \\'p propose a new method wh ic h nses the backprojeclion o\wrator in an iterative

{:

\ '"

Figure L A diagram showing Barber a nd Brown 's ~xp erimen­ tal device , The electrodes are placed on thrt boundary of the circular domain, In t hi s example. the driver pair is electrodes 1 and 2,

BACKPROJECTION AND RADON TRANSFORM The elect rostatic problem is modeled by the equa tion Y'('fYU)=O

1','

procpciun:' based on th e (onjugc)t{' r('sidlJa l algorithm.

in

r!.

where u(x) is th e yoltage potentia l. and ',(,t') denotes the medium conduct ivity, Henceforth. we will tak e th e regio n r! to be the unit ball. r! = {x, I.TI:S I}. To model the in put curre nt of the imaging device. the driver electrodes are rep resented by a dipole source at a point on the boundar,·. Henc e .

This is

followed b\' a discu ss ion of results from onr numerical s imul a tion with the n('\\' sc henH>. which silo\\':'; that its JH'rforrnance is superior to that of th e f3ar\wr and Browu III et hod, Finalh'. we describe modificalion s to the imaging melhod II'h"n Ih e bou ndar\' of the regiou is a pol\'gou, i'relimin,n\' resu lts of imp edauce imaging of regions of arhit rary :-;l1al)(1 ,HP pl'f:'s('nt('{1. 1

C(:,..;)=U(,r , w)=sj

'"r

ou an

=

-1r

a aT Ow

on

or! .

where oON/OT is the counter-clockwise tangential derivative of a Dirac del t a-function. wEar! , The il> verse problem is to find ',( .r) given voltage measurement s !lloo caused by dipoles located at a set of different w.

ow.

Researc h partly support ed by a gran1- from O\,R and an NSF-AFOSR grant. Research p<'l.rtly ~l1pp()rtl~d h:- <1.11 O\,R cont-rao alld a grant from \"SF .

IHi

F. Santosa and 1\1. Vogelius

188

We will consider the linearized problem. Assume

where 6, is a small perturbation. Then correspondingly we have u(x) = U(x)

+ 6U(x),

with U(x) being the background voltage potential and 6U(x) being the perturbation. The field 6U satisfies 6.6U

-VCI/'VU

8(6U)

o

---a;;-

on

in I! ,

(1 )

ol!.

(It is further assumed that 6, == 0 near the boundary). The background potential has a closed-form solution given by U

where

=

w.L

X' 1

with x;

x? + x'i

= (-W2,WJ!

= ",.L . x,

x;

=1-

",. x,

(2)

Formula (6) has a simple geometric interpretation (cf. Figure 1): given a point x to be imaged, and a dipole location w , consider the equipotential circular arc {z : U(z,w) = U(x,w) = s} which originates at wand passes through x. The point where this equipotential arc intersects the boundary is x(s , 1/2, w). We read the quantity (fc6V/fcV)(x(s,1/2,w) ,w) from the data. The approximate value of 6"1 at x is the weighted sum of these quantities. Our work connects the result above with generalized Radon transforms. In particular, we show that Barber and Brown's choice for the function cli(x,w) is the correct weight to use when the inverse problem is viewed in the context of Radon transform inversion. We proceed with a definition of the generalized Radon transform. The function (x,O = I~I V(x , UIW,

x E

n,

~ E m2 \ {O}.

is positive-homogeneous of degree one:

is the 1f/2 rotate of the dipole location

w = (WI,W2). The linearized inverse problem is to find the con-

(m 2

ductivity perturbation 6, given pert urbational boundary voltages 6Ul 30 corresponding to a set of different background fields El (depending on dipole location ",).

it is also infinitely often differentiable in I!

We can now describe Barber and Brown 's method for finding an approximate solution to this linearized problem.The function

The function defines a family of arcs (parts of circles) to be used for the generalized Radon transform

H s .w

= {x

= s},

En: (x,w)

X

sE m,

\

{O}) and

1",1 = 1.

As a measure on each arc H s •w we take is the harmonic conjugate to - U on I!. Indeed x ~ (- U, 11) (the so-called Poincare map) conformally maps n onto the upper half plane P = {11 > 1/2}. The problem (1) simplifies in the (U,1I) coordinates to read

_ 0(6,)

6.6U

in P

(3)

8V

0(6V)

o

DV

on 8P

Following the notation in Beylkin (1984) the generalized Radon transform is defined to be

= {V = 1/2}.

Note that the function 6, = 6,( x( V, V, w)) is now a function of U, 11 and w. The extra data being used to reconstruct 6, is the function 6Vlv=I/2 (a function of V and ""). For a single fixed dipole location Wo, a consistent conductivity increment 6"10 is given by

as follows easily from (3). Note that this increment violates the assumption that 6, == 0 near the boundary. Barber and Brown suggest the weighted average

1.

1 0 B(x) = -2:(8V6Ulv=I/2)(S,,,,·)ls=U(x.w)cli(x,w)dSw, 1f Iwl=1 (4)

as a rough approximation to the conductivity increment 6, (iJ is of course not in general consistent with any dipole measurements). The weight was chosen with particular care to achieve uniformity in the reconstruction. They arrived at cli(x , "')

= 211(."",) -

where da denotes standard arclength; we let a denote the amplitude (9)

(Ru)(s , w)

for any U E CO'(I!). Beylkin provides a recipe for the approximate inversion of transforms like that in (10). His main assertion is that R* 11' R

8

= id + T,

( 11)

where T is compact: L 2(1!, compact) ~ L2(1! , LocI and R* is the so-called backprojection ( R * v )( x) =

1.

h(x ,w) -(--)v(s.w)ls=~(.rw)dSw. Iwl=lax , w .

(12)

11' consists in convolution with the generalized kernel k(s) =

- -1 -2

2(21l')

Joo Irle,esdr. - 00

(13)

'

The function h is given by

We will show that the data

-(

.~ 6C/ 88T U)(x( 8.1/2.",).w')

or

crudely approximates 21l' j, RO"I' To this end we consider the Green's function G(Uoyo)( U. V) , for > 1/2, which solves

la

Thus , the final formula is

( 8T 6U /

(10)

fH". u(·')lv ,..p(x,w)lrla

(S)

1.

fH' ~ u(x)a(.r.",)d,l

I.

It is easy to show that the quantity being averaged is related to the data by

B(x)= -1 21f Iwl=1

=

8

or [')( x( s. 1/2. ",). '" )IS=['(.T.W)

(1- 2\'(x.w))dS~.

(6)

8

6.G(Uoyo)

-8UIi( Uo,\'o)

8G(Uo.vo)

o

av-

on 8P

in

p=mx(I/2 , cx,)

= {V = 1/2}.

189

Electrical Impedance Imaging This Green's function is known in closed form. It follows from this and (3) that

100 1.1/2 G(U',v')(U, V)81(U', V',w)dV'dU' , 00

bU(U, V,w) =

-00

of the ball are not illuminated. Henceforth, we let 81 represent the discretized conductivity perturbation. The data set for the inverse problem as described in Figure 1 consists of 16 experiments , with 16 measurements per experiment. The rescaled perturbation in the data is

and conseq uen tly -( 8~' 8Ulv=I/2 )(s , w)

_100 1.1/2 ~G(U',v')(s, 1/2)81(U', V',w)dV'dU' 8U 00

-00

~ 100

2"

1.00

-00

(1/2 - V'? - (s - U')2 8 .(U' V' w)dV'dU'. 1/2 [Cs - U')2 + (1/2 - V')2]2' , , (14)

Fourier transformation of (l4) with respect to s leads to -(

~6Ulv=I/2)"(r, w) = ~Irl 2

8U

1.

00

1~

e- W'-1/2) lrI 8,"(r, V',w)dV'. (15 )

If we replace e-(V'-1/2)l r l by its value at r then (15) reads -(

o (or

V'

1/2)

8~6Ulv=I/2)"(r,W) "" ~Irl (~~ h(·. V',w)r/V')"(r) ,

or equivalently

'

1.

00

- (8U8Ulv=I/2)(S,W) 8 "" 2"K( 1/2 h(', V',w)dV')(s),

(16)

where J( represents convolution with the generalized kernel k (cf. (13)). By a simple change of variables

1.

h(s , V' , w)dV' = ( h(x)lV'xU(x,w)ldu = R(h)(s,w). JH"w

Using this and (5) in (16) , we arrive at the approximate relation -

~(: 8U/: U)(x(s, 1/2 , w),w) "" 27r uT uT

J{R(01)(S,W).

(17)

Accepting this approximation, we can use Beylkin's recipe to write down an inversion formula for 81. A straight forward calculation shows that the weight h(.r,w) in the Radon inversion formula (12) is related to the function V(x.w) by h(x , w)/a(x.w)

= 2V(x,w) -

1,

x E

n, Iwl = 1.

Inserting this in the operator R* and applying R* to both sides of (17). w'e get

1

where,E is a matrix which represents the linearized forward map, described in (1) and (2). The backprojection (6) is discretized according to the specifications above. We denote it by the matrix E. Thus, E acts on wand produces an approximate image representing 81. In their work, Barber and Brown (1986) found that when reconstructing point images with E, the resolution of the reconstruction depended on the position of the point image. In order to focus the recovered image, they designed a posi tion dependen t filter. The construction of this filter is purely heuristic, and some of the parameters are arrived at by experimentation. However, we cannot overlook its effectiveness and so we have included it in our implementation. We denote the filter matrix by F. Premultiplying E by F gives us the filtered Barber and Brown backprojection, denoted here by B= FE.

00

1/2

2"

where Xi is the ith electrode location, and Wj is the jth dipole location. Recall that the driver pair with electrodes 1 and 2 generates an approximate dipole whose location is a point midway between the two electrodes. The relation between hand w is given by E81 = w,

1

8

Iwl=1

8.

1

("8U/,,U)(x(s.- , w),w)I,=u(x.w) (1-2V(x.w))dS w uT uT 2 ""R*J(Ro,

"" h· The lefthand side is the Barber and Brown backprojection of the
Thus given a data vector w, we find a rough reconstruction through (I8) 61"" Bw. Numerous simulated reconstructions using the matrix B can be found in Barber and Brown (1986). The linearized inverse problem is to find 81 given w in Eh = w.

(19a)

Notice that E is in general not square and that actual measurements w need not be in the range of E. It is therefore natural to solve (19a) in the least squares sense (e.g. by solving the normal equations) (l9b) Numerical simulations with the Barber and BroJn backprojection suggest that B is a crude approximate inverse to E. With this in mind , we are led to consider an alternate problem BEh = Bw.

(20)

Notice that (20) represents a linear combination of the system of equations in (19a). It is not clear that solutions to (20) satisfy (19a). In our numerical studies, however , we always found that the solution to (20) was indeed an acceptable solution to the original problem in spite of the illposedness. We summarize the properties of E an BE gained through numerical studies. (i) BE is a square matrix. The symmetric part of BE has only one large negative eigenvalue whose eigenvector corresponds to h which is nonzero near the boundary of the unit ball. (ii) The matrix BE is better conditioned than the matrix ET E.

NUMERICAL IMPLEMENTATION AND ITERATIVE REFINEMENT

(iii) The eigenvectors of ET E with large eigenvalues correspond to 01 which are non zero only near the boundary. Thus for (19b) we expect stable determination of pixel values for pixels near the boundary.

In our numerical work. we co\'er the domain fl (unit ball) with a 16-by-16 array of square pixels. Pixels \\'hich lie outside

(iv) Some of the eigenvectors of BE with large eigenvalues correspond to 81 which are nonzero near the center. Thus

F. Salllosa and ;\1. \ 'ogclills

190

for (2 0 ) we expect b etter determination o f pixel values for pixels near the center . We refer th e reader to our paper, Santosa and Vogelius (1988) for details. In general (when the perturbatio n is not around a constant conductivity) it is not practical to compute the matrix E: instead we may calcu late the action of E by a fini te e lem ent method. This rules out direct methods for the solution of (19) or (20). Prop ert y (i) suggests that a good way t o solve (20) is to use th e conjugate residual method (see for instance Eisenstat . Elman a nd Schultz (1983)). W e d o need to ta ke care to ma ke certain that the iterates b"'l remain suppo rted away from t he boundary to avoid the "bad" directio n corresponding to t he negati\'e eigenvalue of the symmet ri c part o f BE. Properties (ii) ind icates that problem (20) is bet t e r co ndition ed t han problem ( 19b ). Fin ally. based o n properties (iii ) and (iv) . we expect that when 6, is supported away from the boundary then t he conj ugate residual method applied to (20) will give a good approx imate solut ion more quickly than the same method app li ed to (19 b ). At this point it sho uld be mentioned that we were un able to find an it e rative method which would co n verge for (19a) without imJlli(itl~' going through the nor mal equations (1 9b) .

Figure 2b Reconst ruct ion bac k projection.

us ing t he filt e red

Barber-Drown

NUME RI CAL SIM ULAT IO N In the previous section, we mad e a p rediction that of the two problems, ETE 6') = ETw (PI)

BE 6"'1 = Bw.

(P2 )

the latter is more well- beha\·ed. By this we meant t hat the iterates of the conju gate res idu a l a lgorith m applied to eq uat ion (P2) s hou ld con verge faster and ultimately get closer to a cons is tent profi le t han ite rates of the same a lgorithm a ppli ed to (PI). To verify our prediction we s how the results of computations with a representative test profile. The da ta is gene rated by a multiplication of the tes t profil e by the matr ix E, i.e., by model ing perfect dipoles at the proper locat ions, a nd solving the perturbational equations (1) th roug h t he use of the Green's fun ction and numerical quadrature. T he data is in the ra nge of E a nd no iseless (to roundoff e rrors) . The test the profi le 6"'1 used is s hown in Figure 2a. It. is in the form of two rings. o ne twice as high as t he other. In Fig ure 3a we d isplay the relative problem residu a l

liE hi - wll/llwll

Fig ure 2c Reconstruction based on (PI) at t he 30th iteration o f the co nj uga t e r('s id ua l a lp;orit h m.

versus numb er of iteration s of t he con ju gat e residual a lgori t hm for both equation (PI) and ( P2 ). 11 11 denotes t he Euclidian norm . It is clear that the residuals in the case of (P2) a re sma ll e r than in the case o f ( P 1) (by about a fac to r of 1/3 at t he 30th iterate) . T his is also reR ected in how well t he iterates match the

Figure 2d ll. ,'co ll s t rnctioll has('(1 Oil ( P2) at th e :lOth it<'ration or thf' conju~a l p 1'('sidllai algorithlll.

Figure 2a Th e test profile.

Elcct rica l O, tO

J

J

0,08

~

~

1

::::, ~

1

-,

0,06

j

;::

'<>

,i

"'-

.,

0,04

J

'if. ~

I

~

"0

0,02

1

(P2) 0,00 0

10

20

191

1111 pcdallcc Il11agillg

30

LI:\EARIZED I:\VERSE PROI3LDI FOR A POL'{G01\AL DO~IAI:\

We now briefly discuss an examp le of the determination of a conductivity distribution ins id e a polvgonal region. As in the previous case. we attach a number of electrodes to the boundary. Current is allowed to flow between a pair of adjacent electrodes and \'oltage potentia Is at all other electrodes are measured. This procedure is repeated until we ha\'e c~'d"d through all possible pairs of electrodes. The rescaled perturbation in data. u·. has components 1L'iJ where the ind ex i denotes the locatio n of an electrode and ind ex j denotes the locations of the driver pair through which current has been passed. After lin earization of the in\'erse problem. we ha\'e a background field [i J(x) for each dri\'er pair j s atisf~'ing Laplace's equation with :\eumann boundary condition

iteration number i

ou j

Figure 3a Reduction in the relative problem residual versus number of iterations,

;

0,6 ~

-,::

r

:2.

S J;'

;f ~

~

05

- - = j J on

Oil

6M,' J

~

D(Ui ) )

on

-v6"(, vl, i 0

on

in

n.

(22)

an.

By the linearized fOI'\\'ard map E we mean the map with compon ent s

(P2)

a (" j ( " .; ) ' - cha M' i ( .ri )/ ()r whNe ." ; are the electrode. location s ., The in,",'rse ~rohl em can

t

0,2 1 0

(21 )

wh e re j j represents the curre nt dis tribution on the boundary caused by driver pair j. j j is modeied b, positi\'e and negative .. triangular hat functions" of area one. peaked at the centers of th e driver electrodes. The purturbational fLeld DU' a ssociated with th e driver pair j solves

~

:T

on.

hf' stated as one of detcrmllllng /)"', ~1\'CIl I hat 10

20

'

30

iteration number i

Figure 3b Relative L2 error in the recovered profile \'('rsus number of iterations. "correct" profile O"{. In Figure 3b we displa~' the relative error

\'ers us number of iterations of the conjugate residual algorithm for both equations (PI) and (P2) . Al the 30th iteration the error in the case of equation (PI) is about 61% whereas in the case of equation (P2) this has been reduced to around 28% (a s li ghtl~' s maller reduction than for the residuals). Most impressive to observe is how much faster the conjugate residual iterates for (P2) converge during the first 6 steps when compared to those for (PI). For further comparison we examine the reconstrncted profiles at the 30th iteration, For reference we s how the filtert'd RarberBrown backprojection (initial guess for the iterati\'e sclwmes ) in Figure 2b, The 30th iterates for (PI) and (P2) are shown in Figures 2c and 2d respect ively. Notice that the holes (the areas of low conductivity inside the rings) are reco\'ered in the case of ( P2) (in fact they are alreadv vi s ible in the ,th itNale). while they are still invisible in the ca se of (PI), This test. and severa l others we have performed. indicate that B acts as a reasonably good preconditioner for the original problem, In summary we conclude that the conjugate residual algorithm applied to the equation BEb; = Eu' gi\'es an itera tive method which allows us to s ignificantly improve upon the filtered Barber-Brown backprojection. while outperforming the conjugate gradient/res idual method to solve the "output leas t squares" formulation of EOA/ = ll/.

\\-e will assume that 1 he dri\"('r elect wdes MP s uffi cien tl." close to each other so that the field (" .I (.r) neated bv j) is \\'ell approximated by a dipole solution. It is now poss ible to construct il backprojection operator /J w hic h p;i\"('s a ("l"ud" approximation to h The construction is b." mean s o f a conformal Illappinp;. Sin ce n is a pol~'gon. we can usp a Schwarz- Christolfel map to t ake n to the interior of the unit cirrlC', Ou the IlIlit cirrl e we can 1I ~(, tlt p salll(' cOllstruction "lS ill th t' f'arii('r part of thi. paper. .-\nalop;ou s to the prohlenl in (20 ) for tl\(, circular domain is the prohkm
BEh = Ill/".

(21 )

\\-1' will study this problem in numerical s imulalious. In ollr nU111erical implelll(,lltation. we use a finite ei(>ment

soh'er to evaluate the map F. Although in this case we could e\'aluale E by conforma l mapping (and use of Green 's function). we wanted to build in th e extra fl e xibility of beinp; able to find E fo r conductivity perturbation about a "OIlCOIlS/lIllt background ( i.e. A; = A' Q + b; ). The package PLT~IG developed h,' Bank ( 1988 ) ha s heen ext e ns i\'el,' modifi ed and used in conjuct ion wilh th e Yal e Sparse ~Iatrix Packap;e for thi s purpose. PLT'I IG '''1'5 piecewi se linear triangles as ba s is function s. In our implemenla I io n. f,) is piecew ise con s ta n t on eac h t ria ngle. For the conformal map need "d in co mpuling B. we used se1'_,\("1, de \'eloped b,' Trefethen (la .~O ) . The operalor [] cons is ts o f inlerpolalion. backproject ion o\'e r circula r arcs. and filtering. Th e filter used is nearlv identical to Ihe OM used ea rli er for circul a r domains. A study to ga in deeper understand ing of the role of th e filter and finding wa\'s 10 impro\'e it is currently und e r \\"ay.

Th e pol,'gon. shown in ri g ure ~. ha s been chosen for our example. Approximate dipol es a re se t a t the numbe red loca tio ns. \'oltage potentials are rec o rd ed ,t a ll localion s eac h time a d ipole is fired. The tes t profil e is dis pla,'ed ill Fil\lIre .~a. In Figure 5 b. we display t he profil e o btained by the backprojectio n of the data. This profile is used as an initial guess in the

F. Santosa and 1\1. \'ol-(dius 13

12

11

CONCLUSION

10 9

14

8

1" 7

2

3

4

5

6

Figure 4. The region to be imaged. Approximate dipoles are genera ted at the numbered locations. Voltage potential measurements are recorded at all locations each time a dipole is fired.

We studied the backprojection method of Barber and Brown for solvi ng a lillearized electrical impedance tomography problem. We show that their method is related to generalized Radon transforms. A new iterative method which uses the backprojection as a preconditioner is proposed. It is shown that this method provides much better resolution at the cost of a modest decrease in compu ting efficiency. Modifications of the method for the case where the region is no longer circular are described. Se\'eral numerical examples are given.

REFERE NCES

Alessandrini , G. (1988). Stable determination of cond uctivity by boundary measurements. Applicable Analysis, 27, 153-1 /2. Bank, R. (1 988). PLTMG User's Guide - Edition 5.0. University of California, San Diego, Technical Report. Barber, D. and B. Brown (1986). Rec ent developments in applied potential tomography - APT. In S. Bacharach (Ed.), Information Processing in Medical Imaging , Nijhoff, 106 -12 l. I3cylkin. G. (1984). The inversion problem and applications of the generalized Radon transform. Comm Pure. Appl. Math. 37 .580-599.

Figure 5a. The t<'st profile.

Brown, \3., D. Barber, and J. Jossinet (19S8). Electrical impedance tomography - Applied potenti a l tomography. Clinical Physics and Physiological Measurement, 9, Supplement A.3-147. Eisenstat , S., H. Elman and M. Schultz (1983). Variationaliterati\'e methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 20 , 345-357. Isaacson, D. (1986). Distinguis hability of conductivities by electrical current computed tomography. IEEE Trans. Medical Imaging, MI-5 , 91-95. Figure 5b. Recons tru cted profile from back projection of sim ulal<'d data.

Friedman. A. and M. Vogelius (1989). Determining cracks by boundary measurements. To a ppear in Indiana Univ. Math. J. l\ohn . R. and M.. Vogelius (1985). Dete rmining conductivity by boundary measurements 11. Interior results. Comm. Pure AppJ. Math. 38, 643-667. :\acitman. A. (1988). Reconstructions from boundary measurements. Annals of Math.,128 , .531-.576. Santosa . F. and Ill. Vogelius ( 1988) . A backprojection algorithm for electrica l impedance imaging, C'niversit)' of :'Iaryland. To appear in SIAM J. AppJ. Math.

Figure 5c. Resu lting profile after 10 iterations of conjugate ['0sidual on the problem in (23).

conjugate res idual scheme for (23). The resulting profile after 10 iteration s is shown in Figure ·5c. The rin g-s haped profile is c\p(lrly mOJ"(: \'isihlp in Figure

,')c.

\ lu ch \I'ork u('<'Cls to he done to arri\'e at a more complet e understanding of th e structure of probl em (23). As an example. \I'e ha,'c Hot ,'et in\,estigated the effect on the reconstructed image of changes in th e dipole locations. It will be also be extremel,' iHteresting to se .. if our it erati\'e scheme can be used in connection with the solution of the original nonlinear problem.

Sd\'ester. J. and G. Uhlma nn (1987). A global uniqueness theorem for an inverse boundary \'alue problem. Annals of l\1ath. 125. 153-169. Trefethen . L. (1980). \'umerical Computation of the SchwarzChri st offel Transformation. SIAM J. Sci. Stat. Comput. 1": 82-1 02.