Inverse Scattering Level Set Algorithm for Retrieving the Shape and Location of Multiple Targets

Transcription

1 EuCAP Convened Papers Inverse Scatterng Level Set Algorm for Retrevng e Shape and Locaton of Multple Targets Mohammad R. Hajhashem and Magda El-Shenawee Unversty of Florda, Ganesvlle, USA mhreza@ufl.edu Unversty of Arkansas, Fayettevlle magda@uark.edu Abstract The level set technque s an nverse scatterng problem soluton concern e shape of unknown targets. In s work, e level set algorm wll demonstrate e capablty of reconstructng D and 3D PEC and delectrc targets. In addton to ts robustness, e level set technque has e advantage of retrevng several objects from a sngle ntal guess w no a pror nformaton. I. INTRODUCTION In nverse scatterng, e goal s to nfer e characterstcs of unknown targets llumnated by electromagnetc waves. The collected scattered felds at e recever ponts are processed to retreve e nformaton about e scatterng targets. Inverse scatterng solutons are applcable n several areas such as see-rough-wall magng, medcal magng, ground penetratng radar, non-destructve testng etc. (e.g. [1]- []). The nverse problems are non-unque and non-lnear, whch make em challengng, n many applcatons. Retrevng e shape and locaton of targets s e focus of s work w e assumpton at er electrcal propertes are a pror known. The level set meod has proven to be topologcally flexble and robust, among e several shape reconstructon technques. In s work, several algorms based on level set technque are developed for retrevng e shape and locaton of multple complex targets n two- and ree-dmensons (D and 3D). The level set algorm has proven to be successful n reconstructng multple targets mmersed n ar or hdden behnd a wall. The transverse electrc (TE) and transverse magnetc (TM) exctatons for e D cases are nvestgated. Successful shape reconstructon results are demonstrated n bo cases even when e synetc data s corrupted w Gaussan nose w SNR as low as 5dB. II. METHODOLOGY In e level set meod, e evolvng nterface s represented as e zero level of a hgher dmensonal functonφ. Therefore at each tme t, we have e followng expresson for e evolvng nterface [3]: { φ } Γ () t = r ( r,) t = 0 (1) Where r = ( x, y) s e D poston vector, when e movng nterface s a contour (1D), and r = ( x, y, z) s e 3D poston vector, when e movng nterface s a surface (D). Dfferentatng (1) w respect to e evolvng tme t, yelds [3]: φ + φ V( r) = 0 t The symbol of V ( r) = dr dt represents e velocty vector. Eq. () can be rewrtten as: () φ + F( r) φ = 0, φ0 = φ( rt, = 0) (3) t The symbol F( r ) represents e component of e velocty vector n e normal drecton to e evolvng surfaces. The level set functon Φ 0 n (1) s ntalzed to e sgned dstance functon correspondng to e chosen ntal guess of unknown objects. The objectve of e algorm s to mnmze e error between e scattered felds of e evolvng objects and ose of unknown object(s). As a result, e forward scatterng problem s solved many tmes durng e reconstructon algorm for calculatng e cost functon and calculaton of e deformaton velocty. The Meod of Moments (MoM) s used n s work as e forward solver. The cost functon s defned as [4] Nnc Nmeas s nc meas s nc meas t = θ θj meas θ θj (4) = 1 j= 1 FC ( ) E(, ) E (, ) nc Where θ s e ncdent angle and meas j θ s e measurement drecton angle when e ncdent angle sθ nc. j 3979

2 Nnc The unknownn object s llumnated by ncdent drectons. The symbol N meas represents e number of scatterng drectons under ncdence. The symbol s nc meas E ( θ, θj ) represents e smulated scattered feld and e s nc meas symbol E meas( θ, θj ) represents e measured scattered feld. The cost functon s mnmzed usng data at all ncdent and scatterng drectons and frequences. In order to acheve convergence, e deformaton velocty s chosen such at e cost functon has a negatve dervatve w respect to e evolvng tme (decreasng functon). The approprate form of e deformaton velocty s obtaned usng e recprocty eorem and e work of Roger et al [5]. Generally, e deformaton velocty depends on forward and adjont scatterng problems [6]. B. Reconstructon of fve complex targets usng full and half measurements data. We have proposed a new measurement system where scatterng data are collected only n one sde of e targets, π π accordng to θnc < θsc < θnc +. The symbols θnc and θsc represent e ncdent and scatterng drectons, respectvely. The reconstructon results for fve complex targets are gven n [4]. The obtaned results showed at collectng e scattered felds only at locatons half space around e target (half measurements) produced better reconstructon results compared w e case when e datat are collected from all around e target (full measurements). More detals are gven n [4]. A. Reconstructon n of two targets In e frst example, e levell set algorm s employed to retreve a D gun-shapetargets are assumed to be nfntely conductng cylnders w gun-shape cross secton. TM-polarzed plane waves are used to llumnate e targets where e electrc feld s target w nose contamnated data. The parallel to e cylnders axes. Two levels of nose are nvestgated, correspondng to SNR=10 db and SNR=5 db. The obtaned results after 4740 teratons are shown n Fg. 1 and Fg., respectvely. Accordng to e obtaned results, e level set has satsfactory performance even w nosy data w SNR=5 db. x-axs (cm) x-axs (cm) Fg. 3 Half measurements versus full measurements system [4] Fg. 1 Reconstructon of a gun-shaped target w SNR= =10 db. C. Comparson between TM and TE polarzatons In anoer example, e performance of e algorm s nvestgated under two dfferent polarzatons; TM versus TE polarzaton. The reconstructon under e TE polarzaton s more challengng compared w att usng TM polarzaton [7]. The cost functon and e retreved profles for reconstructon of two rectangular cylnders are shown n Fg. 4 and Fg. 5, respectvely. It s observed at after 000 teratons, two separate targets are retreved under TM polarzaton whle t s stll evolvng under TE polarzaton. Fg. Reconstructon of a gun-shaped target w SNR= =5 db. Fg. 4 Cost functon for reconstructon of two rectangular targets n TM and TE polarzatons 3980

3 Fg. 5 Reconstructon of two rectangular targets n TM and TE Polarzatons [7] C. Parallelzaton of e level set algorm To speed-up e algorm, e MPI parallelzaton technques are used to scale down e reconstructon tme from several hours to a few seconds, e detals of parallelzaton are gven n [8]. The reconstructon results for magng a star-shaped PEC target and e speedup versus e number of processors are shown n Fg. 6 and Fg. 7, respectvely. The obtaned results show a speedup of ~ 84 X s acheved usng 56 processors on e San Dego super computer. Fg.8. Normalzed cost functon for reconstructon of e defected ppe reconstructon versus e dstance to e wall was dscussed n [9]. wall Intal guess True object x-axs (cm) Fg. 6 Reconstructon of star-shaped target at 500MHz, 1GHz, and 10GHz, usng e MPI parallelzed algorm n [8] Fg. 9 Reconstructon of a defected ppe hdden behnd a dalect wall of ckness 0 cm, conductvty S/m and permttvty of 4.5. The reconstructon was converged after ~10000 teratons at 9 GHz, usng e level set algorm [9] E. Reconstructon of 3D PEC target In anoer example, e results level set algorm demonstrates retrevng e shape of a 3D perfectly conductng concal frustum as shown n Fg. 10. The cost functon and e results usng noseless and nosy data are depcted n Fg. 10-1, respectvely. The detals of e reconstructon algorm are gven n [10]. Fg. 7 Speed-up versus e number of processors D. Reconstructon of e defected ppe behnd a delectrc wall The level set algorm s modfed to retreve e shape of a defected ppe located behnd a delectrc wall [9]. The statonary phase approxmaton s used to evaluate sommerfeld ntegrals e Green s functons n stratfed meda. The cost functon and e fnal reconstructed profle are shown n Fg. 8 and Fg. 9, respectvely. The obtaned results show at e level set algorm s capable of retrevng small features n hdden targets behnd a delectrc wall. The Fg. 10. Normalzed cost functon for reconstructon of a concal frustum 3981

4 Fg. 13. Normalzed cost functon for reconstructon of two ellpsods (d) Fg. 11 a-d. Reconstructon of a concal frustum at dfferent stages, ntal guess, after 10 teratons at 1 GHz after 1770 teratons at GHz, (d) after 1770 teratons at GHz (top vew). Fg. 14. The reconstructon of two ellpsods at dfferent teratons ntal guess, 1GHz, 3GHz [11] Fg. 1 a-b. Fnal reconstructon result of a concal frustum after 1770 teratons at GHz usng SNR=10 db, after 1770 teratons at GHz usng SNR=10 db (top vew). F. Reconstructon of two ellptcal delectrc targets In s example, e level set shows t capablty to reconstruct delectrc spherods as presented n Fg. 13. The formulaton of e deformaton velocty s dfferent for delectrc targets w detals gven n [10]. The dmensons of each ellpsod are a=8cm and b=c=cm. The delectrc materal of e objects has permttvty of ε r = 5.0. The cost functon and e retreved surfaces after several teraton numbers are shown n fg. 13 and Fg. 14, respectvely. G. Reconstructon of two delectrc targets w dfferent permttvty values In e last example, we have used two level set functons to reconstruct e shape of two delectrc targets havng dfferent permttvty values [10]. The two objects are a delectrc ellpsod w e permttvty of ε r 1 = 5.0 and loss tangent of tan( δ 1) = and a delectrc cube w e permttvty ε r =. and loss tangent of tan( δ ) = The cost functon and e reconstructon results are shown n Fg. 15 and Fg. 16, respectvely. In all results, e frequency hoppng was employed whch explans e jumps n e cost functon n all results of s secton. Moreover, we beleve at e oscllatons shown n e cost functons at occur towards e end of each frequency are numercal artefacts and can be easly elmnated upon usng careful stoppng crtera n e algorm. Ongong research to mprove e accuracy of e level set algorm for bo e D and 3D confguraton s underway. The level set algorm stll has a space for mprovements especally regardng ts speed-up where all results show excessve 398

5 Fg. 15. Normalzed cost functon for e ellpsod and e cube number of teratons. We beleve s drawback can be mproved even when runnng e algorm on a sngle CPU. (d) REFERENCES [1] A. T. Voulds, C. N. Kechrbars, T. A. Manats, K. S Nkta and N. K. Uzunoglu, Investgatng e enhancement of ree-dmensonal dffracton tomography by usng multple llumnaton planes, J. Optcal Socety of Amerca, vol., no. 7, pp , July 005. [] R. E. O'Malley and M. Cheney, Scatterng and Inverse Scatterng n Pure and Appled Scence, SIAM Revew, vol. 45, no. 3,pp , September 003. [3] J. A. Sean, Level Set Meods and Fast Marchng Meods, CambrdgeUnversty Press, [4] M. R. Hajhashem and M. El-Shenawee, Shape Reconstructon Usng he Level Set Meod for Mcrowave Applcatons, IEEE Antennas and Wreless propagaton letters, vol. 7, pp. 9-96, 008. [5] A. Roger, Recprocty eorem appled to e computaton of functonal dervatves of e scatterng matrx, Electro-Magnetcs, vol., pp , 198. [6] Ralph Ferrayé, Jean-Yves Dauvgnac, and Chrstan Pchot, An Inverse Scatterng Meod Based on Contour Deformatons by Means of a Level Set Meod Usng Frequency Hoppng Technque, IEEE Trans. on Antenn. Propag., vol. 51, no. 5, May 003. [7] M. R. Hajhashem and M. El-Shenawee, TE versus TM for e Shape Reconstructon of -D PEC Targets usng e Level-Set Algorm, IEEE Tran. Geosc. Remote Sensng, vol. 48, no. 3, March 010. [8] M. R. Hajhashem and M. El-Shenawee, "Hgh performance computng for e level-setreconstructon algorm," J. Parallel and Dstrbuted Computng, vol. 70, no. 6,pp , June 010 [9] M. R. Hajhashem and M. El-Shenawee, "The Level Set Shape Reconstructon Algorm Appled to D PEC Targets Hdden Behnd a Wall," Progress n Electromagnetcs Research B, vol. 5, pp , 010 [10] M. R Hajhashem, Inverse scatterng level set algorm for retrevng e shape and locaton of multple targets, PhD dssertaton, Unversty of Arkansas, 010. [11] M. R. Hajhashem and M. El-Shenawee, The Level Set Technque for Mcrowave Imagng of 3D Delectrc Objects, Proc. IEEE Internatonal Symposum on Antennas and Propagaton and USNC/URSI Natonal Rado Scence Meetng, Toronto, Canada, July 11-17, 010. Fg. 16. Reconstructon of e ellpsod and e cube after dfferent teraton numbers, ntal guess, after 500 teratons at 0.5 GHz, after 1000 teratons at 0.5GHz, (d) after500 teratons at GHz. III. CONCLUSIONS Several examples are presented to demonstrate e capablty of e level set algorm n nverse scatterng shape reconstructon problems. The examned cases nclude D and 3D conductng and delectrc targets mmersed n ar or hdden behnd a delectrc wall. The frequency hoppng plays an mportant role n assurng teh convergence of e algorm. We observed at n general, e lower frequences help to retreve e locatons of e targets and er general profles whle e hgher frequences help retreve e fner detals of er shapes. The level set algorm was tested on expermental data as dscussed n [10]. 3983