newton kantorovitch theorem

8/8/2019 Newton Kantorovitch Theorem

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32 IEEE TR ANSAC TIONSNN T E N N A SNDR OPAGATION, VOL. AP-29,O . 2 , MAR C H 1981Newton-Kantorovitch Algorithm Applied to an Electromagnetic Inverse Problem

ANDREROGER

Abstract-An efficient algorithm is presented to compute the shapeperfectly conducting cylinders via knowledge of scattering cross

choice of scattering data avoids the diftlculties linkedor reconstructions. The mathematicalperformed n erms of operatorand unctions,and heof heproblem is demonstrated.Then herestored by means of a Tikhonov-Miller regularization.

efficiency of themethod is outlined by numericalexamples.me algorithm applies to

electromagnetic inverse problems, especially to gratings.

I . INTRODUCTIONED E A Lwithaprof i le econs truct ionproblem,wherethe scat ter ing obje ct is descr ib ed by a funct ion or prof i leof one var iab le . The p rof i le of t he unknow n s ca t t e r e r i s torecons tru cted v ia knowledge of data on t he f a r s ca t t e r edTechniques based on th e near - f ie ld econs truct ionbycont inuat ion [ 1 1 , [ 21 aredescribed n he itera-

Thesemethods equire one t ok n o w h e a rs c a t t e r e dmplitu de and phase , wh ich raises substantial experi-or heoreticaldiff iculties .Generalizationof he Gel-ans quantum inverse theory [3 ] has been bui l t up ,ve ry d i f f i cu l t t o imp lemen t inuse of the s ca t t e r ing ma t r ix a t a llThis paper wi ll p resent an ef f ic ien t and pract ica l a lgor i thm

ven only scattering cross sections, which are related t o th eof th e arscat tered ie ldand huss implymeas u r -This algorithm is based on t h e use of the Newton -Kan-h m e t h o d [ 4 ] . I t a l l o w s one t o s tudy theo re t i ca l ly thebehavior of he operator which inks he shape of he

to the scat ter ing cross sect ion , and it b r ings to light

11. D I R E C T P R O B L E M OF T H E P E R F E C T L YC O N D U C T I N G C Y L I N D E R

L etuscons ideraper fect lyconduct ingcyl inder Fig . 1 )across ec t iondescr ibed , in polarcoord inates in t h e

y plane,by heequat ion p = F(B)(O < 0 < 27~ ) .L etanplane wave with imedependence actorx p (-iwt) ! inc iden tupon he y l inde r u r f ace ,wi thk para ll e l t o th e Ox ax is . Fo r the s ak e of simpli-the e lec t r ic f ie ld is assumed to b e para l le l to th e O z axis

Ed be, respect ively , the com-Oz ax i s o f the to t a l e l ec t r i c f i e ld , o f the in -

andof hediffracte d ield; he ollow ing rela-

E&, 0 ) = exp ikx = exp ( ikp COS e )Manuscript received April 14, 1980 ; December 2, 1980.The author is with the Laboratoue dOptique Electromagnetique,des Sciences et Techniques, Centre de St-JeiBrne, 13397 Mar-13, France.

I

Fig. 1. Geometry of scatterer.

E ( P ,6 ) = 0 )+ E& , e).Since the Hanke l f unc t ion - ( i /4 ) ffo ( ) (kp ) i s the Green func -t iono f h i sp rob lem, hedif f racted ie ldm aybeexp res s eds imply in terms of a s ingle layer poten t ia l51 :

with

w h e r e @(e) s the normal der ivat ive of E above the cy l ind r i ca lsur face . As o pp ose d to the c lass ica l expan s ion of Ed n t e rmsof Bessel orH a n k e l u n c t i o n s , this r ep res en ta t ion is valideverywhere outs ide he scat terer . The boundary condi t ion a tthe s u r f ace o f the s ca t t e r e r s t a te s tha t the to t a l f i e ld m us t bezero , and th is y ie lds an in tegra l equat ion for@(e) [ S I :-:n ~ p ~ ) ~ o ( l ) [ k r ( B , ) ] de = - exp ( ikF(e ) cqs e )

(2)wi th @ , e ) = ro(F(e) , e , e ) .The r ep res en ta t ion (1 ) l ea d s t o t h e a s y m p t o t i c f o r m o f Edfor largevalues of p :

and thus the b i s t a t i c s ca t t e r ing c ro s s s ec t ions given b y

0018-926X/81/0300-0232$00.75 0 1 9 8 1 I E E E


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ROGER: NEWTON-KANTOROVICH ALGORITHMw i th

2 nB ( 0 ) = e x p [--i,w(e) co s (0 - ) ] $(e> e . (4)The di rec t problem is to co mp ute th e bis ta t ic c ross sec t ion(@w h e n F ( 0 ) is given. This can be achieved by calculating firstth eso lu t i on Q of (2), thenca lcula t ing B b y (4), a n d h e n

N ow i t is i n te r es t ing t o s t udy t he e x i s t e nc e o f t he so lu t i onof (2). This existence is guaranteed by he physical meanin gof Q.However , since (2 ) i s a Fredho lm integra l eq ua t ion of t h ef irst kin d, a typ e of equation which usually involves diff icul-t ies , i t i s impor tant to unders tand why i t a lways has a solut ionin this case. This poin t is developed in [ 181 : t he e x i s t e nc e o fa solut ion Q o f ( 2 ) follows essential ly f rom he regular i ty oft he i gh tm e m b e ro f (2). O n he o the r ha n d , if t h e p r o f i l eF ( 0 ) is smooth , i . e . , w i thou t sha r p e dge , t he un ique ne ss o f Qfollows directly f rom general results 61 .

o(e) ( 3 ) .

111. NEWTO N-KAN TORO VITCH hlETH ODI t i s c onve ni e n t t o s c he m a t i z e t he d i r e c t p r ob l e m w i th a n

o p e r a t o r 0, a s soc i a t i ng a n ou tpu t f unc t i on a(0) (bistat ic crosss e c t i o n ) t o a n i n p u t f u n c t i o n( 0 ) (prof i le of the sca t te re r ) :a = 0 - F .

2330 - A I

E qua t ions (2)-(4) s h o w t h a t t h e o p e r a t o r o i s r a t h e r c o m p l i -ca ted. I t is a nonl inear opera tor , and, moreover , i t cannot bewr i t ten xpl ic i t ly incehe a lcula t ion of o(0) f r om F ( 0 )r e qu i r e s t he f i nd ing o f a n i n t e r me d ia t e f unc t i onQ(0)which isthe solut ion of an in tegra l equa t ion. Thus , the use of a com-pu te r i s absolute ly ecessary t o solve he i r ec t roblem.Solving th e inverse problem is deduc ing the input func t ion Ff r o m h eo u t p u t u n c t i o n u. Obviously , heprocedure de-ve lope d f o r t he d i r e c t p r ob l e m is no t r e ve rs ib l e: f o r e xa mple ,i t i s c lea rly no t poss ible to ded uce di rec t ly B ( 0 ) f r om a(0) ifn o o t h e r p ie ce of i n f o r ma t ion is available (see (3)) . I n o r d e r t oc o m p u t e F , w eshallmake use of he Newton-Kantorovi tchalgori thm, which is a generalization of t he w e l l- know n N e w tonm e t h o d . L e t us consid er the rea l equa t ion :

a = O Fwhere E , u,a n d 0 ow d enote , r esp ec t ive ly , tw o eal variablesand a known rea l func t ion. 00 being a given va lue ofu,w ha t i sthe cor responding va lueF o o f F , s u c h t h a t

0 . Fo = 0 0 ?T he N e w ton a lgor i t hm i s a s t e p by s t e p p r oc e dur e , w h ic h , i fconverging, ends owards an e xa c t o lu t i on F o , O ne t a r t sf rom a f i r s t es t imate F l o f F o , a nd one c ompute s a be t t e r e s t i -m a t e F 2 (Fig. 2 ) by inve r t i ng t he l i ne a r e qua t i on :

w i th

Z)F is the differen tial of 0 a t h e p o i n t F , Le., the inearf unc t i on t a nge n t t o 0 t t h e p o i n t F . In this case, i t is e le-

Fig. 2. Scheme of Newton method.

mentar i ly re la ted to the der iva t ive of ow i th re sp ec t to F . T h e none ppl ies he ameprocess again t o f ind a equence ofestimates F , , F ,, .-, F,, w hic h unde r some c ond i t i ons t e ndstow a r ds F o . The Newton-Kantorovi tch a lgor i thm [41 i s qui tes imi la r t o t he N e w to n a lgor i t hm ; now the t h r e e symb ol s E , 0 ,a nd O de no te ; r e spe c t i ve ly , w o r ea l f unc t ions o f r ea l w i a -bles)andanopera tor .T heopera torcanbemathemat ica l lydefi ned as th e Frechet differential of 0 [ 71 . T he imple me n ta -t i on of the Newton-Kan torovi tch a lgor i thm requi res tha t wef ind a f i r s t es t imate F , . Then the cor responding bis ta t ic c rosssection a1 mu st be com puted ; th is i s what we ca l led the di rec tp r ob l e m.The n he Frechet differential Z)F has to b ec o m -puted. F ina l ly , (5) h a s t o b e i n v e r t e d . I t is w o r t h n o t i n g t h a tth is equa t ion, which was e lementary in the Newton a lgor i thm,now be c ome s a f unc t i ona l e qua t i on .T he N e w ton- K a n to rov i t c h a lgor it hm y ie lds no i n f o r m a t io na b o u t t h e e x i s t e n c e a n d u n i q u e n e s s o f t h e s o l u t i o n o f t h e i n -ver se p r ob l e m . O n the o the r ha nd , i t a pp e a r s o be very adapt -able to m any prac t ica l problems. In p ar t icula r , it i s poss ible tochoose as Sca t te r ing da ta a func t ion of t h e i n t e n s i t y o f t h e f a rsc a t t e r ed f i e ld , suc h a s t he b i s t a t i c s c a t te r i ng c r os s s e c t i on . I tis no t ne ce s sa ry t o me a su r e o r t o c om pute t he pha se o f t he d i f -f rac ted f ie ld , and th is suppresses the di f f icul t phase problems.By compar isonwi thure lyumer ica lradientme thods ,theNewton-Kantorovi tch me thodpresents woma ina dva n-tages. First , i t is possible , a t least in th e cases we hav e studied,t o g ive aigorous xpression of the rcchet if ferential ,near lynlosedor m.hushercche tif ferential ,which is thegenera l iza t ion of the onc ept of gradient toth e case of inf ini te d imension a l spaces , can b e as ily ca lcula tedwi th very l i t t le increase of the co mpu ta t ion t im e . Th e pro gramso lv ing t he d i r e c t p r ob l e m i s use d on ly tw ic e pe r s t e p o f t healgori thm, instead of (n + 1) t i m e s w h e n o n e c o m p u t e s n u m e r -ically thegradientwi th espec t o n parameter s .Se c ond ly ,us ing heNewton-Kantorovi tcha lgor i thm,onecanana lyzethe p r ob l e m d i r e c t l y i n t e r ms o f f unc t i ons a nd ope r a to r s . O ned o e s n o t c h o o s e a priori n parameter s which could be redun-d a n to r useless. Inpar t icula r ,knowing he inear pproxi -ma t ion o f he ope r a to r 0 a l lowso ne to s tudy he oc a l be -h a v i o r o f h i s o p e r a t o r a n d o u n d e r s t a n d h e origin o f h e ins tabi l i ty .O necandemonst ra te [4 ] t ha t i f t h e f i rs test i-ma te is c l ose e nough to a so lu ti on F o , t h e s e q u e n c e o f t h e ,converges owards F o , a nd he r e is n oo t h e r o l u t i o n in ane ighbor hood o f F o . However , there may exis t severa l i sola tedsolut ions which a re obta ined by s ta r t ing the Newton-Kantoro-vi tch process wi th di f fe rent es t imates .

I V . I N V E RSE PRO BL E M O F T H E P E R F E C T L YC O N D U C T I N G C Y L I N D E RT hebis ta t icc rosssec t ion o0(f?) i s know n, a nd he c o r r e -

spond ingprof i le F o ( f ? ) s to bec ompute dby heN e w ton-Kantorovi tch lgor i thm. i r s t ,hei r s t s t imate F l ( 0 ) ofF o ( 8 ) canbe ,fo re xa mple , hec i rcula rcyl indero fradius awhich gives the same tota l sca t te r ing c ross sec t ion as the cy l in-


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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-29, NO. 2 , MARCH 1981

e r of profile Fo( 0 ) . The o ta lscat ter ingcrosssect ionofaa in th e pr+esence of a T E polarizedk ( k = I k I ) is given [ 81, if ka isrge enough, by the approxim at ive formula:

OT(Q)= 4 ~ ( l 0 . 4 9 8 0 7 ( k ~ ) - ~ ' ~.0 1 1 7 ( k ~ ) - ~ ' ~ ) .h e total scatteringcrosssectionof hecylinderofprofileis easily calculated:

a i s determined by so lv ing numer ical ly the equat ionO d Q ) = D O T .

as to be so lved for a cy l inder fFl(e) ofis a c i r cu la r p ro f i l e , t he o the r e s t ima tes F z , F 3 : .-,ob ta ined a t the fo l lowing s t ep s a r e a rb i t r a ry and o longer

is solved using the form -as developed in Sect ion 11.T h i r d l y , w e m u s t c o m p u t e t h e o p e r a t o r D F , hich links a6F(B) of the p ro f i l e to a s m a l l va r i a t ion So(6)thebis taticcross ec t ion . In the es tof hepaper , he6 A will mean a sm al l var ia t ion of the quant i ty A , d u ea small variation 6 F of th e prof ile . Let us rewrite (2)-(4) inform:

G-@J=s (2')G i s t he in t eg ra l ope ra to r o f ke rne l :

s(e) = -exp [ k F ( 6 ) CO S 8 I1o =- IBIZ4 k

S,(e') = exp [ w ( e ' ) CO S (e - e ' ) ] ;1 ) deno tes the s ca la r p roduc t :

2 n( F , 1 F 2 )= 1 ~ , ( e f y ~ ( e r ) e '.

(3') a nd ( 4 ' ) , o n e o b t a i n s

S O =- B 6 B + B G )14 k

T h e t e r m (6Se I #d is easily expressed in terms of 6 F , since6se(B' ) = ik CO S (e - ' )w(e ' ) e x p [ k F ( 0 ' ) cos (e -e ' ) ] .

O n t h e o t h e r h a n d , t h e t e r m ( S o I&$) i s very d i f f icu l t to ex -press in terms of 6 F . Differentiating (2 ' ) yields

Let us i n t ro juce hea d j o i n tope ra to r G* (whosekernel isG*(O, e ' ) = G ( e ' , e ) ) and the s o lu t ionQs o f t h e e q u a t i o n

Thus we get

T h e t e r m ( # e 16s) s easily expressible in termsof 6F . N o w w emus t ca lcu la te the las t term. The var ia t ion of the kernelf G is

a n d w e o b t a i n

T h e s y m b o l "vp" i s the Cauch y pr incipal value . Indeed , heexpress ions under the in tegra l have a s ingular i ty of t h e f o r ml / (# - 6 " ) w h e n 8' + e", and thus thes e in t eg ra l s mus t beconsideredasCauchypr incipal values. In o the r wo rds , bo thI G (F + 6 F ) - @) nd I G(F) - @) r e g iven by impr ope r bu tconvergent n tegra ls ,bu twhen hese n tegralsarecombinedso as to ca lcu la teexplicitly 16 G - @J> = I ( G ( F + 6 F ) -G ( F ) ) * @) n t e rms o f SF, i t i s found that th is las t express ionis given by an integral on 6 F with a kernel being i tse l f an in te-gral on @ wi th a C auchy s ingu la r i ty a t = 8 ' ; t he C auchy p r in -cipal value follows.


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ROGER: NEWTON-KANTOROVICH ALGORITHMDespite the presence of a Cauchy pr incipal value, one cande mons t r a t e [ 9 ] tha t he orders of n tegra t ion may be nte r -changed, so tha t the fol lowing re la t ion holds :

F ( e " ) - ( e ' ) cos (e " - ') * H l ( ' ) ( k r @ , @ ' ) .rTh us the Fre 'chet d i f fe rent ia l can be wr i t ten

with

q e ' ) - e " ) (e" - ' ) H ( ' ( k r )r@,(e") + COS (e - ' )

- e x p [--ikF(B') COS (e - ')]-

1- k& ( OJ) [: vp de''2 a

q e ' ) - e " ) cos (e" - ') H , ( ' ) ( k r )Y

- @(e")+ CO S 8' e x p [ k F ( 0 ' ) CO S e ' ]1where &(e ' ) i s t he so lu t i on o f

. 2 n- + [ N o ( * ) ( k r ( e ' , ">)&,(e"> de"

= e x p [ -ikF(O') COS (e - ')].-G,(O'> can be comp uted wi th very l i t t le increase of the com-pu ta t i on t ime , be c a use it s t he so lu t i o n o f a n i n t e g r a l e pa t i o nw hic h ha s t he s a me ke r ne l a s t ha t w h ic hives @(e).Fina l l y , a f t e r c omput ing t he ope r a to r D F , he imple me n ta -t ion fheNewton-K antorovi tch equi resonver t qua-t ionss imi la r to ( 6 ) , i .e ., Fredho lm ntegra l equa t ion s of hef i r st k ind. This leads t o fundamenta l m athemat ica l d i f f icul t ies ,which a re discussed n the n ext sec t ion.

2 3 5V. F R E D HO L M I N T E G R AL E Q U A T IO N S O F T H EFI RST K I N D - RE G U L A RI Z A T I O N

Thiskindof ntegra lequa t ion i s br ie f lys tud i e d , o rex -ample , in [ 101. We dea l wi th a Fredholm integra l equa t ion ofthe f i r s t k ind tha t we wr i te in an ope ra tor form :

A . P = Q . ( 7 )P a n d Q belong to Hi lber t spaces of square in tegrable func t ionson finite intervals. Let A* be t he a d jo in t ope r a to r o f A . A * Ais a se l f -adjoint and pos i t ive opera tor ( for the sake of simpli-c i t y , he c a se of zeroeigenvalues wil l beexclu ded ). We calla n 2 the real posit ive eigenvalues and P, t he no r ma l i z e d o r tho -gona l e igenfunc t ions of the opera torA * A :

A * A P, = an2P, .A nothe rse t of or thogona lnormal ized unc t ions Q, ca nb edef ined by

A - P, = a nQn .I t i s wo r th ot ing h a t A* Q, = a,P, a n d A A* Q, =an2&, . T h e Q , are s imply henormal ized or thogona l e igen-f unc t i ono f A A * . The ealposit ivevalues a, ar eca l led thesingular values of he ope r a t i on A. Classical theoremsa l lowo n e o u s e h ese tso fe igenfunc t ions P, a n d Qn asHilber tbases:

P = PnP, ( 8 )n

Q = 4 n Q n .nInser t ing (8 ) a nd ( 9 ) into ( 7 ) , we ge t

4nan

p n = - . ( 1 0 )A square in tegrable solut ion P of (7) pill exist if and onl y ifthe ser ies X, IP, I * converges ,butsince A * A i s a com pac tope r a to r , a, t e nds t ow a r ds z e r o w he n t e nds t ow a r ds i n f i n i t y .T h u s t h e e x i s t e n c e o f , which involves the conv ergence of theseries Z I4 , 1 2 , doe s no t i nvo lve t he e x i s t e nc e o f P.We shallob t a in a so lu t i on P only if the coeff icients q , decrease rapidlye n o u g h o w a r d sz e r ow i t h e s p e c t o h e a,: th i sc ond i t i onimpl i e s ha t he unc t i ons Q w hic hprovideasolut ion P of( 7 ) belong t o a restr icted class of fun ction s "regular" in som esense . The asymptot ica l behavior of t h e a, i s de te rm ined bythe p r ope r t i e s o f r e gu l a r i t y f t he ke r ne l A ( $ , t ) f the in tegra lope r a to r A [ 1 1 .Fo r he inverse problem ,node mons t r a t i ono fe x i s t e nc esimilar to t ha to f hedi rec tp r ob l e mca nb ea c h i e ve d : hef unc t i on Q i s now equa l to 6o,(e) = ao(Q - o,(O), a nd t h i sf unc t i on is a priori arbi t ra ry because of he exper imenta l orrounding e r ror s on OO(0). T he e x i s t e nc e o f a so lu t i on o f (6)i s notat all ens ured . In thec a sew he r e heope r a to r A * Aadmits e igenfunc t ions Pn0 f o r a n eige nva lue e qua l t o z e r o , t heso lu t i on o f ( 7 ) i s not unique . The se t of a ll solut ion s i s ob-


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3 6 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL.AP-29, NO. 2, MARCH 1981byadd ing o an arb i t rarysolu t ionof (7) any inearP n o . This property will be illus-

f the cy l inder (Sect ion VI) .Another d i f f icu l ty arises in Fredholm equat ions of the f i r s tP wi th re s pect to the da ta[ 121. Let 6 P be a small variation of P and 6 Q t he co r r e -

Q. Equat ions s imilar o ( 8 ) , 9 ) , a n d ( 10)

6 P = x p n P nn

n

6 p n = -.qnan, if th e tw o pos it ive rea l numb ers E a nd IM are given, E

M arbitrarily large, i t is possible to f i ndP u c h t h a t

can be achieved by se t t ing 6 p n = 0, e x c e p t f o rn = N , Nchos en s uch ha t an < E / M . Thusanarbitrarily smal l6 Q of Q can co r r e s pond to an arbitrarily large varia-6 P of P . The s o lu t ion P o f (7), f i t actually exis ts , is un-

respect t o sm al l var ia t ions of the r igh t-hand mem-r Q . The more r egu la r the ke rne lA (3, t ) s , t he m ore r ap id lye e igenvalues an2 decrease owards zero when n increases,s the s o lu t ionP . An i l lustration of

in t he cas e o f the pe r f ec t ly conduc t ing cy l inde r3 . For a c i rcu lar prof i le , we have computed theo f t he Frkche t d i f f e r en t i a l , and then the co r r e s pond ingPn andeigenvalues a n 2 . Fig. 3 s how s he e f-6 P = t P ~ o :he re la t ive var ia t ion ofeprofileof hecyl inder is about wenty- f ivepercent ,a l-ean r e l a t ive va r i a t ion o f the b i s t a t ic c ro s s s ec t ionon ly one pe r cen t . Thus , fo r so lv ing he n teg ra l equa t ionsof the cy l inder , i t s neces-

y to res tore in some way the ex is tence and s tab i l i ty of theThis can be achieved by using the so-called regulari-[ 1 2 1 , [ 131. The problem of so lv ing (7) is re-

var ia t ional problem: f ind the funct ionwhich minimizesM ( ~ ) = I I A ~ P - Q I I 2 + r l l R ~ ~ 1 I 2 , r > o .a n d r are , espect ively, th e regular iz ing opera to rand heparameter . This var ia t ional problem leads o th e

( A * A + r R * R ) P = A * * Q . (1)Q. The regular iza t ion parameter r c a n b e

pr ior knowledge. For ns tance , if the me an

Y t

Fig. 3. Illustration of instability of inverseproblem.Full line corre-sponds to circularcylinder of radius 1 and its bistatic cross sec-tion. Dotted line shows distorted profile nd orrespondingbi-static cross section Wavelength is h = 6.28.exper imenta l precis ion AQ o n t h e f u n c t i o n Q i s k n o w n , r canbe chos en s uch tha t

This i s ca lled the res idue m ethod [ 121. In fac t , the regular iza-t ion is a me thod fo r choos ing the most phys ical so lu t ion of(7), t ak ing into accoun t the even tua l e r ro r on the da taQ . T h efunc t ions P which are to o i rregular ( t oo large or too oscil-la t ing) cor respond to large values of R P and thus a r e e l imi -na ted when min imiz ing the func t iona lM(P).

VI. NUMER IC AL A PPLIC ATIONT hedirectproblem of theper fect lyconduct ingcyl inderhas been solved using (2)-(4). All the func tion s bein g per iodi cin 8, the s imple rec tangular ru le which has been employed forthe n tegrat ions g ives very good accuracy . The n tegral equa-t ion (2 ) has been s o lved by the po in t ma tch ing me thod , wh icha l lows on e to t ak e rec i s e ly in to accoun t the s ingu la r ityf t h ekernel [ 141. The program has been tested by classical numeri-cal tes ts : op t ica l theorem [5, q. 12 .351, rec iproci ty theorem,and convergence when increas ing the numberf points .Thenverse roblem as een o lved yheNewton -Kantorovi tchalgor i thm.T heFredholm ntegralequat ions ofthe irs tkindhavebeen egularizedby heTikhonov-Millerme thod , and th i s l eadso equa t ions s imi la r to ( 1 1 ) :

The regular iz ing operator is determined by numer ical and the-oretical reasons [ 15] , [ 161 . E q u a t i o n ( 1 2 )s e x p a n d e d o n t h eFou r ie r bas i s , becaus e the ke rne l of VF ha s no ingu la ri ty andis very regular; hischoice esu l ts in muchsmal lermatr icesthan thos e ob ta ined by a po in t ma tch ing me thod . The compu-ta t ion of the Cauchy principal values of he kernel D ( 0 , e )requires adequate numer ical methods [ 1 4 1 . T h e c o m p u t a t i o nt ime i s abou t s ix s econds on a C DC 7600 fo r the r econs t ruc -t ion of a cy l inder prof i le .A par t icu lar proper ty of the b is ta t ic scat ter ing cross sect ionmus t be t aken in to accoun t : t he inva r i ance in a t r an s la t ion o fthe cy l inder in the Ox o r Oy d i r ec t ion . I n f ac t , t h e unknown


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ROGER: NEWTON-KANTOROVICH ALGORITHM 2 3 7func t ion i s the p rof i le of the cy l inder o wi thin a t r ans la tion inth e x0 y plane. This ambiguity can be supp ressed by a par t i-cula r choice of the origin 0. For i ns t a nc e , one c a n c hoose t hebarycente r of a ll the poin ts of the prof i le . The prof i le of thescatte rer is thus descr ibed , in polar coordinates, by a functionp = F @ ) , w i th t he c ond i t i ons :

q e ) os e de = l 2 ' F W sin e de = 0. (13)The f i r s t es t imate F , ( e ) is a circular cylind er of c e n t e r 0 a n dthus ver if ies ( 1 3). In or de r t ha t t he se c ond i t i ons ho ld a t t hefol lowingsteps of theNewton-Kantorovi tchalgori thm , t isenough to res t r ic t heexpansion bas is used fo r (1 2) o heFour ie r bas is wi th the except ionof the func t ions s in0 a n d c o s8 . This is not a physical restr ict ion, since i t co rresponds sim plyto a par t icular choice of the or igin 0. ( I n o the r w or ds , one c a nde mon s t r a t e t ha t t he f unc t i ons s i n 8 a n d c o s 0 are eigenfunc-t i ons o f t he ope r a to rDF*l&, with a zero eigenvalue.)The program devoted to the inverse problem has been tes tedon imula tedda ta ,obta inedbycomput ingnumer ica l ly hebis ta t ic sca t te r ing c ross sec t ion of a known c i rcula r sur face andadding a random er ror in ordero s imula te exper imenta l e r ror s .The inverse program works very accurately for simple prof i lessuchascircular or elliptic profiles. Th ec onve r ge nc eof heprocess requi res genera l ly four or five steps. Fig. 4 s h o w s t h ereconst ruc t ion ofa complicated prof i le , for var ious wavelengthsof the inc ident wave. The random er ror has an ampl i tud e ofone percent of thescatter ing crosssection. When th e wave-lengthdecreases, the l lumin ated face of t he c y l i nde r is stillvery well r econst ruc ted, b ut the op posi te f ace s not well com-puted: clear ly, i t is more and mo re diff icult to de t e c t t he de -tailsof heprofi le n heshadow eg ion; hisphenomenoncould be expec ted f rom a physica l point of view! On the otherhand, the sca t te r ing cyl inder must not be too smal l wi th re-spec t to the wave length h. T he c omp ute d p r o f i l e ( l ine 2) be-com es very imprecise if the cylinde r s smaller han X/3, al-thou gh i t st i l l gives a bistat ic scatter ing cross section equal toth e des ired one .

The di f f icul ty of the shadow region is removed if the scat-ter ing data is the backscattered cross section ub(6i) for a vari-able angle o f nciden ce e,, ins teadof hebistat icscatter ingcrosssec t ion ora ixed nc idence .T hebacksca t te redcrosssection is given by

2nB ' ( e i ) = e x p [ w ( e ' ) os (ei- ' ) ] $'(ei, e ' ) de'

where @'(ei , ' ) i s the solut io n of th e in tegra l e qua t ion:

= - xp [ k F ( 0 ) C O S (e - e,)]an d th e Fre'chet differential is

Fig. 4. Reconstruction of scatterer from its bistatic scatteringcrossreconstructed pro fie for A = 25.13. 3: A = 6.28.section. 1: true profde and reconstructed profiie for A = 12.57. 2:

'

Fig. 5. Reconstruction of scatterer from its backscattering cross sec-tion. 1: true profile. 2: reconstructed profiie for A = 1256. 3: A =25.13. 4: A = 3.14.with

A comp uter program has been wr i t ten on the bas is of theseform ulae , and gives good results . Fig. 5 show s t he r e c ons t r uc -t ion of a square cyl inder for var ious wave lengths . This i s , ofcourse , a f ea t much more di f f icul t t o per form because of thepresence of the edges .VII. CONCLUSION

We have descr ibed and implemented an e f f ic ient and prac t i -c a l a l go r i thm f o r c omput ing t he sha pe o f pe r f e c t ly c onduc t ingcylinders via knowledgeof catter ing ross ections in T Epolar iza t ion.T h eme tho d of Newton-Kan torovitch we havee mploye d a l l ow s one t o a na lyz e a nd t o c ope w i th t he f unda -mental nstabil i ty of this nverseproblem. t anbe easilyappl ied omany nverseproblemsof lec t romagnet ics , orins tance , to he per fec t ly condu c t ing cyl inders in Tivl polari-za t ion. In severa l previous papers , the same a lgor i thm has beenused success fully or erfectly ondu cting rat ings [ 1 7 1 ,[ 181. On t he o the r ha nd , t he e xpr e s s ion of t he F r gc he t d i f -ferential is very useful for the synthesis, i .e ., to f in d scat tere rswith previously sp ecif ied Scatter ing propert ies [ 191. Generali-


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8 IEEE TRANSACTIONS ON ANTENNASND PROPAGATION, VOL. AP-29,O. 2, MARCH 1981of the metho d to meta l l ic scat terers of f in i te conduc-

to de te rmine the s hape o f the g rooves[20 . So

REFERENCESR.Mittra. ComputerTechniques f o r Electromagnetics. Oxford.England : Pergam on. 1973. pp. 352-359.V . H. WestonndW .M. oerner. An inversecatteringtechnique for electromagnetic bistatic scattering. C a n . 1. Phvs . .vol. 47, pp. 1177-1 184. 1969.V . H . Weston.Inverse cattering or hewaveequation. J .L. Y.antorovitch.Onewto ns method forunctionalequations. Dokl . Akad. Nauk. SSSR.vol. 59. pp. 1237-1240. Jan .1948.VanBladel. ElectromagneticFields. New York:McGraw-Hill.1964, pp. 368-393.D. S . Jones. The Theory of Electromagnetism. New York:Pergamon. 1964. ch . 9 .N .Dunfordan d J. T . Sc hwar tz. LinearOperators. New York:Interscience.1966.p. 92 .W . P. King and T. . Wu. The Scattering andDiffraction of Waves .Cambridge , MA: Harvard Univ. Press. 1959. p. 66.F. D. Gakhov. Boundary Value Problems. New York: Pergamon.1966. p. 46.R.Courant. D. Hilbert. Methods of Marhemarical Phvsics. Ne wYork: nterscience.1965.pp. 159-160.E. Hille and J . D. Tamarkin. On the characteristic valuesof linearintegral equations. Acra Ma th. . vol. 57. pp 1-76. Apr. 1931.A. ikhonovndV.Arsgnine. M6thodes .d e RPsolution deProblkmes Mal Poses. Moscou: Mir. 1976 French ranslation).K. M iller. Least squares methods for ll-posed problems with a

M a t h . P h v s . , V O I . 13, pp . 1952-1956. 1973 .

prescribed bound. SIAM 1.Math. A nal . . vol. 1. pp. 52-74. F eb.1970.R.Petit . ElectromagneticTheory of Grarings. Berlin.Germany:Springer, 1980.A. Roger. D. Maystre. and M. Cadilhac. On a problem of inversescattering in optics:Thedielectric nhomog eneous medium. J .O p t . . vol. 9. pp. 83-90. Feb. 1978.M. ertero. C. de Mol . and G . A. Viano, Restoration of opticalobjects using regularization. Opt. Letr . . vol. 3. pp. 51-53. Aug.1978.A. Roger and D . Maystre. The perfectly conducting grating fromthe point of view of inverse diffraction. O p t . Acra. vol. 26. pp.447-460.1979.-. Inversescattering method in electro mag neticoptics:Ap-plication to diffraction gratings, J . Opt. SOC.Amer. , vel. 70, PP.1483-1495, Dec. 1980 .A .Roger. Gratingprofileoptim ization s by inverse catteringmethods. O p t . C om m un. . vol. 32. pp. 11-13. Jan. 1980 .A. Roger and M. reidne. Grating profile reconstruction by aninverse catteringmethod, Opr. C om m un. , vol.35, PP. 299-301, Dec.1980

AndreRoger was born in Marseilles. France. onMay 29. 195 2. He receivedheAgregationdegree romheEcole Normal Sup6rieure eSaint-Cloud in 1975 nd is presentlyworkingtoward he Ph.D. degreeat heLaboratory ofElectromagneticOpt ics. Faculte Saint-Jerbme.Marseilles .His reas of research re ratingtheory and electromay netic inverse scattering.

newton kantorovitch theorem

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