Planar and cylindrical active microwave temperature imaging: numerical simulations

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 11. NO. 1, DECEMBER 1992 457 Plana and Cylind ical Ac i e Mic owa e Tempe a u e Imaging: Nume ical Simula ions J. M. Rius, S uden Membe , IEEE, C. Picho , L. Jo e, Membe , IEEE, J. C. Bolomey, N. Joachimowicz, A. B oque as, Membe , IEEE, and M. Fe ando, Membe , IEEE Abs uc - Ac i e mic owa e imaging sys ems o biomedical appli ca ions ha e been gaining a en ion in ecen yea s. The need o i de nal s uc u e in o ma ion imposes he use o di ac ion omog aphy app oaches. Two di e en geome ies, plana and cylinci ical, ha e been used o his pu pose. In his pape we p esen a compa a i e s udy a 2.45 GHz conce ning bo h measw emen and econs uc ion pa ame e s o he wo con igu- a ions. Fo he sake o compa ison, a nume ical model consis ing o w 3 nonconcen ic cylinde s is conside ed and econs uc ed using bo h geome ies om simula ed expe imen al da a. The sca e ed ields and econs uc ed images allow o ex ac e y use ul in o ma ion abou dynamic ange, sensi i i y, esolu ion and quan i a i e image accu acy o he choice o he con igu a- ion ,in a pa icula applica ion. Th l geome ies chosen a e plana and cylind ical a ay con- igu ai ions. Bo h geome ies can measu e o wa d and backwa d sca e ed ields. The backsca e ing measu emen imp o es he image esolu ion and econs uc ion in lossy mediums, bu , on he o he hmd, has se e al dynamic ange di icul ies. This adeo be ween only o wa d and o wa d-backwa d ield measu emen is analyzed using a plana a ay o o wa d sca e ing and a cylind icaE one o o wa d-backwa d. As di e en ial empe a u e imaging is a weakly sca e ing p oblem, Bo n app oxima ion algo i hms can be used. The sim- plici y o Ba n econs uc ion algo i hms and he use o FFT make hem e y a ac i e o eal- ime biomedical imaging sys ems [19]. I. INTRODUCTION IFFERENT ypes o adia ion a e ac ually used o non D in asi e es ing o biological bodies. Each o hem has i s own speci ic a ibu es, and he econs uc ed image depends on he way in which he adia ion in e ac s wi h ma e [l] The mic owa e egion o he elec omagne ic spec um o - e s he access o an in o ma ion which is ela ed o he elec ic p ope ies o he ma e . These p ope ies, usually exp essed as Manusc ip ecei ed Deembe 3, 1990; e ised Ap il 27, 1992. This wo k was suppo ed by CAICYT (Spanish Commi ee o Scien i ic and Technical Resea ch unde G an 116S-84), FISS (Spanish Na ional Ins i u e o Heal h unde G an 84/21 12), Spanish-F ench Coope a ion P og am (G an 301135) and Spanish-B i ish Coope a ion P og am (G an 171173). This wo k was also suppo ed by he Eu opean Commission o he CEE in he ame o he Collabo a ion COMAC-BME P ojec “Op imiza ion o Hype he mia Technology and Assessmen o Clinical E icacy in T ea men o Cance .” J.M. Rius, L. Jo e, A. B oque as, and M. Fe ando a e wi h E.T.S.E. Telecommunicaci6, Uni e si a Poli kcnica de Ca alunya, Apdo. co eos 30002, 08080 Ba celona, Spain. C. Picho , J. C. Bolomey, and N. Joachimowicz, a e wi h Labo a oi e des Signaux e Sys emes (CNRSIESE), Ecole SupC ieu e d’Elec ici e, Pla eau de Moulon, F91192 Gi -su -Y e e Cedex, Pa is, F ance. IEEE Log Numbe 9202263. a complex pe mi i i y (pe mi i i y and conduc i i y), ansla e in mac oscopic e ms he ine s uc u e o he issues a he molecula le el, and he e o e depend on i s physical o physiological ac o s as well as he in e oga ion equency [2]. The mic owa e p opaga ion in biological issues shows some speci ic p ope ies. -The a enua ion is high, in pa icula o he issues wi h a high wa e con en as muscles, li e , kidney. Howe e , so issues such as a , lungs and bones p esen lowe mic owa e abso p ion [3]. I is in hose so issues whe e ul asound ene gy su e s he g ea es a enua ion. -A he same equency, he wa eleng h in he issues a e educed up o eigh imes compa ed o he ai [4]. -Unlike X- ay imaging, whe e he wa eleng h is e y small as compa ed o he magni ude o in e es ing objec de ails, o mic owa es, e ec s o wa e p opaga ion can no be explained in e ms o ays, and di ac ion e ec s ha e o be conside ed [5]. Ac i e mic owa e imaging sys ems ha e no been de el- oped o use in biosys ems due o a numbe o appa en limi a ions ela ed wi h he esolu ion and pene a ion. Al- hough he e was a i s a emp a he beginning o he 70’s [6], in ac hey we e he i s expe imen s o La sen and Jacobi [7], [8], a he end o he decade, ha show he po en iali y o he me hod. The basic p inciple employed consis s o imme sing he a ge in wa e and wo king in he lowe egion o he mic owa e spec um (1-10 GHz.). The mos signi ican poin s o he p ocedu e a e he ollowing. a) Wa e , being a medium wi h dielec ic cons an close o hose o human body, imp o es he coupling and pene a ion o he elec omagne ic adia ion in o he body. b) The esolu ion o di ac ion-limi ed imaging sys ems is de e mined by he wa eleng h o he in e oga ion adia ion. When ope a ing in wa e , wi h a ela i e pe mi i i y close o 75, he e is a con ac ion in wa eleng h and he e o e in esolu ion close o 9. c) The high a enua ion o wa e a hese equencies a oids he mul ipa h p opaga ion ex e nal o he in e oga ed body. Wa e is no he only possible ma ching medium. Fo ins ance, a issues could be be e ma ched using lowe pe mi i i y liquids [9]. The choice he ope a ing equency is a adeo be ween esolu ion and dep h o pene a ion. The equi emen o pen- e a ion wi h deep-lying o gans a gues o a lowe equency 0278-0062/92$(13.00 0 19’92 IEEE 458 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 1 I, NO. 4, DECEMBER 1992 o ope a ion. In con as , di ac ion-limi ed imaging sys ems equi e high ope a ing equencies o enhanced esolu ion. The 1 o 6 GHz egion appea o be op imum o mic owa e imaging o he human body [lo]. The i s s udies we e de o ed o p ojec ion imaging [7], [ 111, [ 121. Bu , apidly, omog aphic econs uc ion p ocesses we e conside ed. A i s app oach (131, close o X- ays CAT, ailed o s ongly inhomogeneous media. Howe e , he di ac ion e ec s ha e o be aken in o accoun [14], [15] in o de o ob ain good quali y images. The use o mic owa e adia ion in imaging biological s uc u es has been gaining a en ion ecen ly. The in e es ely on i s low cos and low complexi y echnology, as well as in i s capaci y o disc imina e di e en issues (pe mi i i ies anges om 5 o 50, compa ed o he ew pe cen o densi ies in so issues) and o de ec physiological changes, as solu e concen a ion, pe usion, blood low a e and empe a u e [ 161-[19]. Mo eo e , he pho on ene gy o mic owa es is su icien ly low o p e en any ioniza ion o occu in biological issue a o dina y in ensi y le els. The illumina ion and measu emen geome y depend on he pa icula applica ion and on he echnology a ailable. In his pape , by using a compu e model, we compa e he mos signi ican pa ame e s o he wo basic con igu a ions ha ha e mainly been used in ac i e mic owa e imaging: he plana geome y [18] and he cylind ical geome y [20]. The model consis s o wo nonconcen ic cylinde s wi h di e en pe mi i i y alues, simula ing a gene al biological body. Sec ion I1 ou lines he basics o he o mula ion o he wo geome ies. An e o has been made ying o enhance he common s eps o bo h solu ions. The hi d sec ion is de o ed o compa e he di e en mea- su emen and econs uc ion pa ame e s o he wo geome ies on he basis o a common simula ed s uc u e. 11. TOMOGRAPHIC FORMULATION Imaging echniques consis o illumina ing he objec wi h an inciden adia ion, measu ing he di ac ed ields, and ob aining an image o he dis ibu ion o he p ope ies o he objec in e ac ing wi h he illumina ing ene gy. Two kinds o mic owa e omog aphic sys ems will be o mula ed, di e ing on he geome y o inciden wa e and he measu emen an enna. Tomog aphy wi h di ac ing ene gy, such as mic owa es, mus be modelled wi h a second-o de linea di e en ial equa ion, he inhomogeneous wa e equa ion, which desc ibes he elec omagne ic ield dis ibu ion in he objec o be imaged. Since compu ing he objec image om he sca e ed ield measu emen s is an ex emely di icul p oblem o sol e, a i s -o de app oxima ion has o be used, so ha only weakly sca e ing objec s can be econs uc ed wi h easonable e o . When inciden adia ion is concen a ed on a slice o he objec , and wi h objec slowly a ying in he hi d dimension, a bidimensional model can be used: dielec ic objec s and elec omagne ic ields will be assumed o ha e p ope ies a ying only o e he 2-y plane c oss sec ion. Using e ically pola ized an ennas in he z-axis di ec ion, all elec ic ield and cu en ec o s will be z-di ec ed, and acco dingly we can use scala ield equa ions. The wo dimensional model is alid only i he objec is ac ually uni o m in he z-di ec ion inside he slice in which illumina ing ene gy is concen a ed. The econs uc ed objec will be he (T. y) image o his slice. A. Spec al Fo niula ion o Di ac ion P oblems The physical p ope ies o he objec which in e ac wi h mic owa e adia ion a e he elec ical pe mi i i y and con- - duc i i y. Since di ac ion heo y is based on cohe en iel.ds, phaso o mula ion o ields and cu en s will be used wi h he ime dependence pJw . Complex pe mi i i y and wa enumbe a e hen de in ed as ollows: -complex dielec ic pe mi i i y o he objec unde . es : -complex dielec ic pe mi i i y o ex e nal medium: -complex wa enumbe in he ex e nal medium: = wm, whe e po is he acuum magne ic pe meabili y. Le E(?') ep esen he o al elec ic ield and E'( F) he inciden ield, ha is o say, he ield in he absence o he objec . The sca e ed ield is de ined as 4.'); 0(F); ES(F) = E(?') - I?(?') 1) and can be conside ed as he ield gene a ed by he eqni alen elec ic cu en adia ing in he ex e nal medium: whe e G(F- 7) is he G een's unc im, and (3) he equi alen induced elec ic cu en . en is de ined h ough he equa ion The bidimensional Fou ie ans o m o he :qui alen cu - (4) I he objec complex pe mi i i y is dose o ha o he su ounding medium, he sca e ed ield is negligible in on o he inciden one (weak sca e ing) and he Bo n app oxima ion can be applied: (F) N €0 -- E(?') N I?(?). (5) When he objec is illumina ed by a plane wa e inciden a an angle 00, using a uni ec o 00 in his di ec ion, we ha e (6) E";) = e-3Ko& 7. Equa ion (4), e alua ed a I I? I = KO, can be exp essed as: ql') = -jw oC -( I? - K0ijO 11-1 K = K" (7) RlUS e U/.: MICROWAVE TEMPERATURE IMAGING: NUMERICAL SIMULATIONS 459 Fig. 1. Fou ie domain o an inciden plane wa e. Unde Bo n app oxima- ion, he bidimensional Fou ie ans o m o he con as p o ile is ob ained in he Fou ie domain o e a ci cum e ence shi cd on he di ec ion opposi e o he inciden plane wa e. whe e C(g - Kndo) FT[C(F)] C(~) 'I(I;.---oH ,) i(jF K = KO (8) =U 1-1 is he bidimensional Fou ie ans o m o he con as p o ile de ined as (F) C(F) = 1 - ~ 0 (9) e alua ed o e ci cum e ences o adius Kc, and cen e -KoHo in he Fou ie space (Fig. 1). When he su ounding medium is lossy, he wa enumbe KO is complex, and he Fou ie ans o m becomes a Laplace one. Howe e , in o de o use he FFT algo i hm he complex wa enumbe mus be app oxima ed by i s eal pa . This in oduces an a enua ion o he highe spa ial equencies, which p oduces "high- equency" noise in he econs uc ed image. Fo ha eason, some low-pass il e ing o he econ- s uc ed image is necessa y when backsca e ing in o ma ion is p ocessed, because i is associa ed o he highe spa ial equencies in he objec spec um. As he bidimensiona: Fou ie ans o m o he equi alen cu en e alua ed a I K I= KO can be easily ob ained om sca e ed ield measu emen s, he con as p o ile o a weakly sca e ing objec will be econs uc ed h ough (7). The econs uc ion (7) gi es in o ma ion o he 2-D Fou ie ans o m o he con as p o ile only o e a ci cum e ence shi ed in di ec ion opposi e o he p opaga ion o he inciden ield. Thus, di e en di ec ions o incidence mus be consid- e ed in o de o ob ain a bidimensional sampling inside a ci cle in he spec al domain (Fig. 2). Acco dingly, he econs uc ed Fig. 2. Mul i iew Fou ie domain. I mul iple iews a e aken o di e en di ec ions o illumina ion, he bidimensional Fou ie ans o m o he con as p o ile can be ob ained inside a ci cle o adius 21io cen e ed a he o igin o he Fou ie domain inciden plane wa e Fig. 3. Plana sys em measu emen geome y ( ansmisison con igu a ion). On plana geome y sys ems, he objec is illumina ed by a plane wa e and he sca e ed ield is measu ed along a p obing line pa allel o he inciden wa e on . I ano he p obing line is loca ed a -,I,", bo h o wa d and backsca e ed ields can he measu ed. con as p o ile ob ained h ough in e sion o C(k) will be a low-pass il e ed e sion o he o iginal one. R. Plana Geome y A omog aphic sys em wi h plana geome y uses a plane wa e inciden ield Ei (80; F) p opaga ing on di ec ion 00, o simplici y Ho = Oo, and he o al ield E(z,.y) is measu ed along he line z = icg pa allel o he inciden wa e on (Fig. 3). Mechanical o a ion o he objec a ound z-axis p o ides di e en di ec ions o incidence ela i e o he objec . The Fou ie ans o m o he equi alen cu en can be ob ained om he measu ed o al ield h ough he Fou ie ans o m o (2) along y-di ec ion. A e some manipula ion, see [ 181, he ollowing exp ession can be ob ained: E"(: ;o. K,y) = -jw,aoG( J-) ,'( JW, K,) (10) 460 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 1 I, NO. 4, DECEMBER 1992 whe e i”(Ky) and a e he angula spec um o he sca e ed ield and he G een’s unc ion, espec i ely, Ula) (1lb) being K, and Ky he Ca esian componen s o he eal app oxima ion o he p opaga ion ec o ela ed o each o he plane wa e componen s o he angula spec um I? = (K,.K,) K, = KocosQ K, KO sin 0. Unde Bo n app oxima ion, he 2-D Fou ie ans o m o he con as p o ile is ob ained h ough (7), (lo), and (11): When he plana omog aphic sys em measu es only he o wa d sca e ed ield h ough he objec , only he componen s o he angula spec um o he sca e ed ield p opaga ing owa ds posi i e z(Kx > 0) a i e a he measu e line. These spec al componen s a e associa ed o he igh hal o he ci cle (K, > 0) in he spec al domain (Fig. 4). Thus, o a ion o he objec p o ides he 2-D Fou ie ans o m o he con as p o ile inside o ci cle o adius iK0 (Fig. 5) [18], [19]. I ano he p obing line is loca ed a -20 (Fig. 3), bo h o wa d and backsca e ed ields can be measu ed. In his case, which is no analyzed in his pape , o a ion o he objec p o ides 2-D Fou ie ans o m o he con as p o ile inside a ci cle o adius 2K0 as indica ed in Fig. 2. The simplici y o he econs uc ion algo i hm and he use o FFT make i e y a ac i e o eal- ime imaging sys ems [ 191. I 4- -KoOo Fig. 4. Fou ie domain o 1 iew plana sys em. I only o wa d sca e ed ield is measu ed on plana geome y sys ems, he bidimensional Fou ie ans o m o he con as p o ile is ob ained only o e hal o he shi ed ci - cum e ences. I backsca e ed ield is also measu ed, he whole ci cum e ence is ob ained. -JZK C. Cylind ical Geome y In cylind ical omog aphic sys ems he measu emen is pe - o med no by a s aigh line, bu by a ci cula a ay o iso opic an ennas (Fig. 6). Di e en di ec ions o illumina ion can be achie ed by ansmi ing wi h each one o he omnidi ec ional elemen s, so ha mechanical o a ion o he objec is a oided. Al hough he inciden ield is no a plane wa e, bu a cylind ical one, and he measu e an enna is a ci cula a ay, he spec al o mula ion de eloped o plana geome y imaging sys ems can be applied i a syn he ic ape u e app oach is made. As he ci cula a ay can measu e, in p inciple, bo h he o wa d and backwa d sca e ed ield, he Fou ie ans o m o he con as p o ile is ob ained o e he en i e shi ed ci cle in he spec al domain, and be e esolu ion is achie ed wi h o wa d-backsca e ing algo i hms due o a less igo ous low- pass il e ing han wi h o wa d sca e ing sys ems. Howe e , in p ac ice, backsca e ed ields a e di icul o measu e nea Fig. 5. Fou ie domain o 8 iew ansmission plana sys em. When mul i- ple iews a e aken on a ansmission plana geome y sys em, he bidimen- sional Fou ie ans o m o he con as p o ile is ob ained inside a ci cle o adius &lio ins ead o 21;”. he ansmi ing elemen , and esolu ion is no as good as heo e ically expec ed. Using a ecip oci y heo em o mula ion, he bidimensional Fou ie ans o m o he equi alen cu en can be easily ob ained om he sca e ed ield measu ed along an a bi a ily shaped cylind ical an enna, p o iding ha plane wa es in any di ec ion can be syn hesized by cu en s on he ecei ing an enna, ac ing as ansmi e . The ecip oci y heo em s a es ha RIUS e al.: MICROWAVE TEMPERATURE IMAGING: NUMERICAL SIMULATIONS ~ 46 1 Y ecei e ansmi e ' e& Fig. 6. Cylind ical geome y sys em. Fo cylind ical geome y sys ems, he measu emen is achie ed wi h a ci cula a ay o omnidi ec ional an ennas. Di e en di ec ions o illumina ion can be achie ed by ansmi ing wi h each one o he omnidi ec ional elemen s. whe e and 51 a e he elec ic cu en s exis ing in olumes 71, and 'ub, which adia 2elec ic ields and ,!?h, espec i ely. I we assume ha Jh is he elec ic equi alen cu en; in he objec , induced by he inciden ield E' (e Ho), and J,, is he elec ic cu en on he an enna ac ing as ansmi e ha adia es a plane wa e izld p opaga ing along he uni Xec o 8, hen we ha e ha -7, adia es he sca e ed ield E"(?), which is measu ed along he an enna, and ha .xL adia es he ield e-3~oo.~. F om (3) and (9) we ge subs i u ing unde Bo n app oxima ion in (13), and conside ing (7) and (8), we ob ain C(Ko(8 - Fo a ci cula an enna o adius R, cen e ed in he :E-y plane, (15) can be exp essed as C(K"(Q - whe e 1(0 - a) is he ampli ude o he cu en dis ibu ion a angula posi ion 0 along he an enna, ha , when ac ing as ansmi e , adia es inside he sou ce ing a plane wa e p opaga ing a an angle 0 in he di ec ion 8 When he inciden ield is a cylind ical wa e, we can expand he plane wa e E'(< 0") as a supe posi ion o cylind ical wa es E'(< a") gene a ed by line sou ces I(& - 00) loca ed as posi ions 00 along he ci cula an enna. Then i ollows ha Ez (< 8,) = 1 I(& - a")E'(( '; 0o)R dao (18) and he e o e he ield sca e ed om an inciden plane wa e, measu ed along he ci cula an enna can be exp essed as E"(c ;&) = i I(& - ao)E"(c~:o~)Rdc~ 2. 2T (19) whe e E"(a; 00) is he sca e ed ield measu ed a posi ion c , gene a ed by an inciden cylind ical wa e p oduced by a uni sou ce loca ed a posi ion 00 (Fig. 5). Finally, om (16) and (19) we ob ain . 1(0 - a)I(0o - 00) d0dao. (20) Equa ion (20) has he ollowing physical in e p e a ion: E"(0: 00) is con ol ed wi h he cu en ha , exis ing along a ci cula a ay, gene a es a plane wa e. This con olu ion ans o ms he ci cula geome y in o a Ca esian one, o bo h ansmi e and ecei e a iables o and 00. As wi h plana con igu a ion algo i hm, his equa ion can be easily implemen ed by means o Fas Fou ie T ans o ms (FFT), which allows eal- ime econs uc ions [ 191, [20]. Fu he in o ma ion abou he algo i hm heo y and implemen a ion can be ound in [21]. 111. NUMERICAL RESULTS This sec ion compa es he ela i e pe o mance o he o wa d and o wa d-backsca e ing measu emen using, e- spec i ely, plana and cylind ical geome y econs uc ion algo i hms. The sca e ed ields ha e been compu ed in bo h cases by momen me hod, sol ing he linea sys em o equa- ions by conjuga e-g adien i e a i e algo i hm. The so wa e used o compu e sca e ed ields, o bo h geome ies, and o econs uc he objec image o plana geome y ha e been de eloped a Labo a o ie des Signaux e Sys kmes, Ecole SupC ieu e d'Elec ici C, Gi -su -Y e e, while he cylind ical geome y econs uc ion p og am has been de eloped a AMR g oup, Depa men o Signal Theo y and Communica ions, Poly echnic Uni e si y o Ca alonia, Ba celona. Pos p ocessing algo i hms ha e been applied o imp o e image quali y when necessa y. -Ge chbe g-Papoulis algo i hm has been applied o plana geome y econs uc ions when he numbe o iews is small. -Cylind ical geome y econs uc ions mus be low-pass il e ed in o de o emo e high- equency noise due o backsca e ed ield measu emen s. A. Nume ical Model One o he possible applica ions o mic owa e omog aphy conce ns medical imaging. As human body is composed mainly o wa e , and he complex dielec ic pe mi i i y o 462 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 1 I, NO. 4, DECEMBER 1992 5 cm - SCATTERED FIELD > X Fig. 7. Simula ed model o biological objec . Fo compu e simula ion pu pose, a e y ough biological objec is modeled by a ci cula egion o wa e o adius 4 cm a 37OC imme sed in wa e a 25OC. A local empe a u e change inside he body is simula ed by a sma!le . nonconcen ic cylinde o wa e o adius 1 cm a 38OC o 41OC. TABLE I Complex pe mi i i y and con as p o ile o wa e Complex pe mi i i y Con as p o ilc (2.jo) = 77.27 - ,j9.29 ~(37") = 73.49 - jG.35 ~(38") = 73.13 - ,jG.35 ~(41") = 72.34 - j.j.88 IC(2Y)J = 0 (C(37")I = 0.0591 lC'(3s")I = 0.OG4G IC(4l")i = 0.07G5 wa e a ies linea ly wi h empe a u e, empe a u e a ia ions inside human body can be de ec ed and localized. This sug- ges s an applica ion o hype he mia moni o ing and umo de ec ion. A comple e se o compu e simula ions has been pe o med o es he capabili y o bo h algo i hms o de ec localized empe a u e a ia ions inside wa e . A e y ough biological objec is modelled by a ci cula egion o wa e a 37°C imme sed in wa e a 25°C. A local empe a u e change inside he body is simula ed by a smalle , nonconcen ic, cylinde o wa e a 38°C o 41°C (Fig. 7). We ha e used an ope a ing equency o 2.45 GHz [19], [20], wi h a wa eleng h o 1.39 cm in wa e a 25"C, which is a good adeo be ween a enua ion and esolu ion. 1) Model Pa ame e s: Ou model is a weakly sca e ing objec , because i s con as p o ile is low and i s dimension no oo la ge compa ed o he wa eleng h. Thus, o al ield is simila o inciden ield, and Bo n app oxima ion can be applied. The complex pe mi i i y and con as p o ile o wa e a 2.45 GHz o di e en empe a u es a e lis ed in Table I. In his ange, he pe mi i i y sensi i i y wi h empe a u e is almos linea , equal o a€'/€; - = -3.9876 10-"c-' AT 2) Plana Geome y Simula ion Pa ame e s: Leng h o measu emen line: Numbe o ecei ing an ennas: 67 Numbe o iews: 64 Spacing be ween ecei ing an ennas: 31.6 cm 4.7 mm = X/3 - -Inc. y axis deg CYLINDRICAL Fig. 8. Sca e ed ield. Magni ude o he p obing sca e ed ield in dB o bo h plana and Cylind ical geome y o local empe a u e change inside he cylinde o 4'. Two di e en di ec ions o incidence ha e been conside ed, along . -axis (0') and !/-axis (OOO). A minimum dynamic ange o 30 dB o he cylind ical case and 45 dB o he plana one is needed o de ec he di e ence be ween he wo iews. Dis ance o he cen e o he objec : 5 cm Dimension o econs uc ed image ma ix: 67 X 67 3) Cylind ical Geome y Simula ion Pa ame e s: Numbe o ansmi ing an ennas: 64 Numbe o ecei ing an ennas: 64 Radius o he ci cula a ay: 12.5 cm 5.625" = 1.23x 64 x 64 Spacing be ween ecei ing an ennas: Dimension o econs uc ed image ma ix: B. Sca e ed Field Fo compa ison pu pose he inciden ield a he cen e o he objec is ixed: 0.5 V/m in ampli ude, ze o deg ee in phase. Fig. 8 shows he magni ude o he p obing sca e ed ields o bo h geome ies. Two di e en iews o he objec ha e been plo ed o di e en di ec ions o incidence along he x-axis (0") and y-axis (90"). Fo he plana case he wo cu es o e lap, because he p obing line is always in he same ela i e posi ion o he inciden wa e, while o he cylind ical case he wo plo ed cu es a e simila in shape, al hough shi ed 90". Fig. 9 shows he magni ude o he di e en ial ields, i.e. he di e ence be ween he sca e ed ields o a empe a u e change o 4°C and he sca e ed ields o no empe a u e change. 463 RIUS e al.: MICROWAVE TEMPERATURE IMAGING. NUMERICAL SIMULATIONS DIFFERENTIAL FIELD (41 C - 37 C) bo -10 -20 53 4 I/,', ,, , , , , , ,, , ,, , , , , ,, -18 -12 a 4 o 4 8 12 is -SI PLANAR cm 50 o 60 120 en 240 m 360 a ' ,,,.,' ..I. m... I ,,I CYLINDRICAL d=g Fig. 9. Di e en ial sca e ed ield (4loC-37OC). Magni ude o he di e - ence be ween he sca e ed ield measu ed o a 4OC empe a u e change and he sca e ed ields o no empe a u e change. Two iews along .I' and axis a e shown. A minimum dynamic ange o 35 dB is needed o he cylind ical case, and 60 dB o he plana one. The la e can be educed down o 40 dB wi h a sho e p obing line. Fo p ac ical pu poses, only he p obing ield co esponding o a ce ain dynamic ange, i.e., 30 dB, has a clea in luence on he econs uc ed image. Because o he di e en beha io o he wo measu ed pa e ns, he use ul angula ex end o he measu ed ield o he cylind ical geome y will be la ge han he equi alen one o he plana case. This can be explained as ollows. The di e en ial ield can be conside ed as he ield sca e ed by he empe a u e change. Fo he plana case, he empe a u e change loca ion is close o he cen e o he measu e line and a away om he ends. The maximum di e ence o dis ances is abou 13 cm, leading o an a enua ion di e ence o abou 30 dB [Fig. lO(a)]. In he cylind ical case, he empe a u e change loca ion is nea he cen e o he ci cula a ay, and he e o e he dis ances o each one o he ecei ing elemen s a e simila . The g ea es dis ance di e ence is abou 4 cm, and he a enua ion di e ence is abou 9 dB [Fig. 10(b)]. On he o he hand, he la ge numbe o elemen s in he cylind ical a ay which measu e ields wi hin he speci ied dynamic ange, p o ide a la ge amoun o in o ma ion o be p ocessed, which esul s in a sligh ly be e spec al padding and image quali y, a he p ice o a longe p obing line. C. Recons uc ed Images I) Absolu e Recons uc ions: Fig. 11 shows he econ- I ( h) Fig. 10. Sca e ed ield a enua ion. The dis ance om he empe a u e change loca ion o each one o he ecei ing elemen s o he measu e a ay is app oxima ely cons an in he cylind ical geome y sys em [Fig. 10(b), bu no in he plana one [Fig. lO(a)]. Thus, he sca e ed ield a i ing a he elemen s close o he ends o he measu e line is s ongly a enua ed compa ed o he ield a he cen e o he a ay. Fo ha eason, he plana geome y sys em equi es a dynamic ange o abou 20 dB g ea e han he cylind ical geome y one. s uc ed con as p o iles o empe a u e changes o 0, 1, and 4°C. Plana geome y econs uc ion and cylind ical one a e e y simila . I mus be no iced ha cylind ical con igu a ion has a la ge spec um co e age due o he back-sca e ed ield measu emen : ci cle o adius 2Ko while only iKo o he plana one. Acco dingly, in heo y he spa ial esolu ion o an ideal backsca e ing sys em is be e by a i ac o . Howe e , images ob ained by he cylind ical con igu a ion om o wa d and backwa d sca e ing ield measu emen s a e no so be e [see Fig. 111 o he ollowing easons. -The backsca e ed ield a elemen s e y close o he ansmi ing one canno be measu ed, and is app oxima ed by ze o. As backsca e ed ields co espond o high equencies in he Fou ie domain o he econs uc ed objec (Figs. 1 and 2), some spa ial esolu ion is los . -When he FFT algo i hm is used o compu e he 2-D in e se Fou ie ans o m o he objec spec um C (K) , he wa enumbe I? is assumed o be eal alued. I he medium is lossy, some e o is in oduced in he highe equencies o he objec spec um. As he backwa d ield measu emen s a e associa ed wi h his highe spa ial equencies, a low-pass image il e mus be applied and some esolu ion is los again. This il e ing is no necessa y when only o wa d sca e ing measu emen is pe o med. 464 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 11, NO. 4, DECEMBER 1992 m 75 74 73 CONTRAST PROFILE ~ 23 0.018 L 0.018 1 ;', 72 71 0.06 0.05 0.M 0.m 0.m 0.01 0 $1 $1 '" ''- PLANAR an -10 a B 4 I o z 4 e e 10 CYLINDRICAL an CONTRAST PROFILE Fig. 11. Recons uc ion o con as p o ile o empe s u e changes o OOC, l0C, and 4OC. Plana geome y and cylind ical geome y econs uc ions a e e y simila -al hough he cylind ical geome y should be in p inciple be e by a ac o & due o he backsca e ed ield measu emen . ..-. E 0 > DIELECTRIC PERMllTlVlTY - --? P :: ,'.,,, I/ - ' x (an) Fig. 12. Recons uc ion o dielec ic pe mi i i y on cylind ical sys em. As a conclusion, he image esolu ion ob ained wi h he o wa d sca e ing plana con igu a ion and he cylind ical one a e e y simila , al hough he la ge spec um co e age due o he backsca e ing measu emen in he cylind ical case. Cylind ical geome y econs uc ion is pe o med aking in o accoun he lossy medium wi h complex KO in (ZO), so ha he co ec alue o he objec spec um 6' is ob ained and quan i a i e alues o he con as p o ile a e ob ained wi h e y good app oxima ion. The complex pe mi i i y o he image can be easily de i ed om (9) (see Fig. 12), and a empe a u e map image can be ob ained om he app oxima ely linea ela ionship be ween con as p o ile and empe a u e (Fig. 13). The esul s o Fig. 11 ha e been ob ained by he plana algo i hm using K, eal alued in (12). Thus, he highly a enua ed ampli ude o he sca e ed ield due o he lossy medium is supposed o be caused by a less di ac ing objec and he econs uc ed alue o he con as p o ile is lowe han he o iginal. The e o e, he plana geome y algo i hm wi h K, eal in (12) p o ides only a quali a i e image o he con as p o ile (Fig. 11). 2) Loss Co ec ion in Plana Geome y Algo i hm: When a quan i a i e econs uc ion in a lossy medium is desi ed, a loss co ec ion mus be pe o med in plana geome y algo i hm. This is done by aking K, complex alued in (12). Wi h his co ec ion, each componen o he angula spec um o he measu ed sca e ed ield is ampli ied by a exponen ial ac o equal o he a enua ion in he lossy medium. This may be a majo p oblem in an expe imen al sys em, because he noise ela ed o signals a i ing a bo h ends o he a ay is ampli ied RlUS e al.: MICROWAVE TEMPERATURE IMAGING: NUMERICAL SIMULATIONS 465 78- n- 78- 75 74 73- 72,o TEMPERATURE MAP COMPLEX PERMITTIVITY model -30 c - ~ -'I " .a d 4 -2 0 2 4 6 8 10 -104 6 4 -2 0 2 4 8 e 10 CYUNDRICAL cm Fig. 13. Recons uc ion o empe a u e map on cylind ical sys em. As he dielec ic pe mi i i y o wa e a ies linea y wi h empe a u e, empe a u e maps ob ained wi h he cylind ical sys em. by he loss co ec ion exponen ial ac o . In gene al, he loss co ec ion is e y sensi i e o he ampli ica ion o noise o he highe spa ial equencies. Figs. 14 and 15 show he econs uc ed images wi h bo h he plana and cylind ical geome y algo i hms, o a empe a u e change o 1"C, loca ed a z = 0 cm. A loss co ec ion has been pe o med o he plana case, while o he cylind ical one he loss co ec ion is implici aking complex KO in (20). Dielec ic pe mi i i y is econs uc ed p ope ly wi h bo h algo i hms, bu he plana geome y image has been dis o ed by losses co ec ion, as shown in Fig. 14. A empe a u e map o bo h geome ies is also shown in Fig. 15. The main eason o he be e esul s achie ed wi h he cylind ical algo i hm compa ed o he plana one wi h loss co ec ion is ha in lossless media he o wa d sca e ed ield is much la ge han he backsca e ed, bu in lossy media he ela i e impo ance o he backsca e ed ield inc eases wi h he loss angen , so ha he cylind ical algo i hm akes ad an age o e he plana one. This poin has been alida ed by nume ical simula ion: i in he cylind ical algo i hm only he o wa d sca e ing in o ma ion is used, he esul s a e e y simila o hose o he plana wi h losses co ec ion. As a conclusion, i can be no iced in Figs. 14 and 15 ha he backsca e ed ield measu emen in lossy media imp o es he image quali y when a quan i a i e econs uc ion is ob ained. 3) Di e en ial Recons uc ions: An image o a localized empe a u e change in he objec can be ob ained sub ac ing he image o he objec om he image o he objec wi h he 72 -loa 8 4 -2 o 2 4 e 8 10 PIANAR w'kh LOSSES CORRECTION cm (a) CONTRAST PROFILE plana wi h losses co ec ion c lind ical -.z. P ( b) Fig. 14. Recons uc ion o dielec ic pe mi i i y wi h losses co ec ion. Dielec ic pe mi i i y is econs uc ed p ope ly wi h bo h algo i hms bu plana geome y image has been dis o ed by loss co ec ion.