Reconstruction of 3D Anisotropic Objects by VIE and Model-Based Inversion Methods

Transcription

1 Pogess In Electomagnetics Reseach C, Vol. 8, 97, 08 Reconstuction of D Anisotopic Objects by VIE and Model-Based Invesion Methods Lin E. Sun, * and Mei Song Tong Abstact A model-based invesion algoithm combined with the cul-confoming volume integal equation method is pesented fo the econstuction of D anisotopic objects. The fowad algoithm utilizes the cul-confoming volume integal equation method. The invesion algoithm is based on the Gauss-Newton method. The appoach is applied to the econstuction of the pemittivities of D anisotopic objects. Moeove, sensitivity analysis of the data fom diffeent polaizations of tansmittes and eceives to the anisotopic popeties is pefomed. Numeical examples show the effectiveness of the invesion algoithm and demonstate the sensitivities of data fom diffeent tansmitte and eceive pais to the anisotopy.. INTRODUCTION Invese scatteing methods have been used in the econstuction of objects in vaious aeas such as subsuface sensing, biomedical imaging and nondestuctive evaluation fo decades. The scatte popeties such as conductivity, pemittivity and pemeability, and location and shape ae econstucted fom the scatteed electomagnetic fields. Thee ae mainly two categoies of invesion methods. One is the pixel-based method. In this method, the inveted paametes ae the mateial popeties, such as the pemittivity and conductivity in each pixel. It is vey suitable fo objects with highly inhomogeneous media [, ]. The othe categoy of method is model-based method. In this method, the invesion domain is divided into egions. The shapes of diffeent egions ae descibed using geometical paametes. The inveted paametes ae the shapes of egions and the mateial popeties fo each egion. Compaed to the pixel-based method, model-based method is often moe efficient since the numbe of inveted paametes is highly educed, and the apioiinfomation is used [, 8, 9]. Analysis of scatteing fom anisotopic mateials has been of geat inteest due to its wide application in subsuface sensing, metamateials, etc. Among vaious numeical methods, a culconfoming volume integal equation (VIE) method has been poposed to model the scatteing fom anisotopic objects [0]. Although much wok has been done on the invesion of anisotopic objects using ti-axial tansmittes and eceives in the subsuface sensing aea [, 7], invesion of anisotopic objects in fee space is still limited. In the liteatue wok fo the econstuction of D dielectic objects in fee space, most of the pevious wok handles the isotopic objects only. In this pape, we popose a combined method of VIE and model-based invesion method to solve the econstuction of D anisotopic objects. The fowad method employs the cul-confoming VIE method fo anisotopic objects poposed in [0]. The invesion method is based on the multiplicatively egulaized Gauss-Newton Method []. It is well known that the invese scatteing poblem is nonlinea and ill-posed. Hence, to solve the invese scatteing poblem fo the anisotopic objects, it is essential to pefom the sensitivity analysis fo the measuements. Finally, in the numeical esults pat, we Received 0 Januay 08, Accepted Mach 08, Scheduled Apil 08 * Coesponding autho: Lin E. Sun (lsun@ysu.edu). Depatment of Electical and Compute Engineeing, Youngstown State Univesity, Youngstown, OH, USA. Depatment of Electonic Science and Technology, Tongji Univesity, Shanghai, China.

2 98 Sun and Tong demonstate the sensitivity analysis of diffeent measuements to the anisotopic pemittivities. We also show the econstuction esults of anisotopic pemittivities. It is poved that the poposed fowad and invesion methods ae capable and efficient fo the econstuction of D anisotopic objects.. FORWARD SCATTERING PROBLEM Conside a D inhomogeneous and anisotopic object in fee space with elative pemittivity ɛ () and elative pemeability μ (). The object is illuminated by eithe a plane wave o a field excited by a dipole souce indicated as T in Fig.. The eceives denoted as R ae located in a plane. Fo each incident field, the total field E() is solved fom the following volume integal equation [0] E() = E inc ()+k0 G(, ) [ɛ ( ) I ] E( )d V + g(, ) [ μ ( ) I ] E( )d () V + Hee, V + epesents the volume that is slightly lage than the volume of the object V. G(, )is the dyadic Geen s function fo the unbounded and homogeneous medium. It satisfies the following equation G(, ) k0 G(, )=Iδ( ) () and given by ( G(, )= I + ) k0 g(, ) () whee g(, ) is the scala Geen s function, and k 0 isthewavenumbeinfeespace. z x y Figue. Configuation of the invese poblem and measuement setup. To solve the volume integal equation in Eq. () by method of moments, we need to convet it into a set of linea algebaic equations. Fist, we discetize the volume of the object into a sum of tetahedons, and each tetahedon is specified by nodes and edges. Next, we use the fist-ode edge basis defined on each edge of the tetahedons to discetize the unknown fields []. Hence, the electic field E() can be expanded into discetized foms as N e E() = I i N i (), V () Inseting () into (), we have the discetized fom of Equation () N e N e E inc () = I i N i ()+k0 I i G(, ) [ɛ ( ) I ] N i ( )d V + i= N e i= i= i= I i V + g(, ) [ μ ( ) I ] N i ( )d ()

3 Pogess In Electomagnetics Reseach C, Vol. 8, Using the Galekin s method and testing Eq. () using edge bases, we can convet the discetized volume integal equation to a linea matix equation [e inc ] = [ Z ] [I] () whee the matix element is given by ( Z )ji = ( Z ) i ii + ( Z ɛ ) ji + ( Z μ (7) )ji ( Z i) = N j(), N i () (8) ii ( ɛ) Z Nj (), G(, ) [ɛ ( ) I ], N i ( ) (9) ji = k 0 ( Z μ)ji = N j (), g(, ) [ μ ( ) I ], N i ( ) (0) By solving matix Equation () using the iteative method GMRES, the total electic field in the whole solution domain can be obtained. The scatteed electic field measued at R in Fig. is given by E sca ( j ) = k0 G( j, ) [ɛ ( ) I ] E( )d V + G( j, ) [ μ ( ) I ] E( )d () V + and the total electic field in this equation is solved using MOM method intoduced above. Hee j epesents the position of the j-th eceive, j =,,...,M,andM is the numbe of eceives.. INVERSION ALGORITHM The invese poblem is govened by Eqs. () and (). In ode to solve fo (ɛ ( ) I) in Eq. (), we conside the invese scatteing poblem as an optimization poblem, whee it can be solved by minimizing the following cost function C(x) = { } W d E sca (x) E d + λwm (x x p ) () Hee, the fist tem is the diffeence of the simulated scatteed fields and the measued data, which epesents the data misfit. In the above, E sca (x) denotes the vecto of simulated scatteed field, and E d denotes the vecto of measued data. The second tem is the diffeence of the model paametes and thei pescibed values. Hee, x denotes the vecto of model paametes (ɛ ( ) I). x p denotes the pescibed model paametes. Since the a pio infomation about the model paametes is assumed unknown in this wok, x p is chosen as the model paametes in the pevious step. W d is the data weighting matix, which is chosen as the identity in this wok. Note that when the noises fo diffeent measuements ae diffeent fom each othe, the diagonal tems of W d may need to take diffeent values based on the noises. W m is the model weighting matix, which is chosen as the identity as well in this wok. In both tems of the cost function, L nom is assumed. In the second tem, λ is the egulaization facto defined as a function that is popotional to the data mismatch λ = E α sca (x) E d () Hee α is a constant mainly detemined fom test. In this wok, it is chosen as 0.0. It is also noted that when thee is noise in the data, the data mismatch tem in the cost function will convege to a cetain value depending on the noise level. Hence, lage values fo α will be needed. Upon minimizing the above cost function, we use the multiplicatively egulaized Gauss-Newton method and obtain the linea equation as below [] H k δx k = g k () Hee, H k denotes the Hessian matix given by H k = J T k W T d W d J k + λw T m W m ()

4 00 Sun and Tong and g k denotes the gadient vecto given by ( g k = J T k W T d W d E sca (x k ) E d) + λw T m W m (x k x p ) () In the above, J k is the Jacobian matix at the k-th iteation. The mn-th element in the Jacobian is computed using the deivative of the simulated data E sca m with espect to the model paametes x n J mn = Esca m = Esca m (( + δ)x n) E sca m (x n) (7) x n δx n Hee, x n is the n-th component of the model paamete vecto x. E sca m is the m-th component of the simulated data vecto s. δ is a small value chosen as 0.0. In Equation (), δx k denotes the step vecto at the k-th step. Fom the solution fo δx k,we can solve the model paamete x k+. This pocess continues until the solution conveges. The iteation pocess will be teminated as one of the thee conditions occus fist: i) The numbe of iteations exceeds a pescibed maximum; ii) The diffeence between the cost function at two successive iteates is within the toleance 0 ; iii) The diffeence between the model paametes at two successive iteates is within the toleance 0.. NUMERICAL RESULTS Two anisotopic cubes as shown in Fig. ae excited by an electic diploe. The centes of the two cubes ae located at (0., 0., 0.) m and (0., 0.9, 0.) m. The side length of each cube is (.0, 0.,.0) m. Each cube is discetized into 8 tetahedons and, edge bases. The dipole is located at (0.,., 0.) m. The eceives ae also electic dipoles, and they ae in the x-z plane at y =.0m, along the x and z axes fom.0 m to.0 m at evey 0. m. The total numbe ( of eceives ) is ɛ 0 0. The fequency is 0. GHz. The elative pemittivities ae uniaxial tensos with ɛ = 0 ɛ ɛ z (a) (b) (c) (d) (e) (f) Figue. Scatteed electic field of y-x polaization fo diffeent anisotopy atios. (a) ɛ z =,(b) ɛ z =,(c)ɛ z =8,(d)ɛ z =, (e) ɛ z =, (f) ɛ z = 0.

5 Pogess In Electomagnetics Reseach C, Vol. 8, Example : Sensitivity Analysis Fo cube, ɛ =.0, ɛ z =.0. Fo cube, ɛ =.0. Fo the examination of the sensitivity of measuements to the anisotopy, ɛ z ae chosen as,, 8,,, 0, espectively. Then the scatteed fields in the eceive plane ae plotted fo each case. Fig. shows that the scatteed fields fo the tansmitte ae y-polaized, and eceives ae x-polaized. Fig. shows that the scatteed fields fo the tansmitte ae z-polaized, and eceives ae still x-polaized. We can see that the z-x polaized measuements ae moe sensitive to the change of the atio ɛ z /ɛ. (a) (b) (c) (d) (e) (f) Figue. Scatteed electic field of z-x polaization fo diffeent anisotopy atios. (a) ɛ z =,(b) ɛ z =,(c)ɛ z =8,(d)ɛ z =, (e) ɛ z =, (f) ɛ z = 0... Example : Reconstuction of Isotopic Pemittivities In this example, we conside that both cubes ae isotopic dielectics. The elative pemittivity of the left cube is ɛ =, and the elative pemittivity of ight cube is ɛ =. Fist, the eceives ae placed on the same side of the tansmitte at y =.0m. The tansmitte is polaized in y and z diections, and the eceives ae polaized in x, y and z diections, espectively. Next, we will show the numeical esults fo both the noiseless and noisy data. Fo the noisy data, db (. pecents) signal-to-noise atio (SNR) is applied to the data. The initial guess fo the contast of the elative pemittivity is.0. Figue shows the econstucted pemittivities of the two cubes using the magnitude of the scatteed electic field fo diffeent tansmitte-eceive polaizations fo both the noiseless and noisy cases. Next, we place the eceives at y =.0 m, on the diffeent side of the tansmitte. Fig. shows the inveted esults. Fom Figs. and, we can see that all the esults convege to actual values within steps fo the isotopic pemittivities fo both the noiseless and noisy data. Next, we plot the cost function () and data mismatch (DM) fo diffeent cases. Hee the data mismatch (DM) is defined as the atio of the nom of the diffeence of econstucted data and measuements with the nom of the measuements. E sca (x) E d DM = E d (8)

6 0 Sun and Tong (a)... 0 (d). 0 (b) (e). 0 (c) (f) (g) 0 (h). 0 (i) (j) 0 (k). 0 (l) Figue. Inveted elative pemittivities fo isotopic cubes fom diffeent tansmitte-eceive polaizations when the eceives ae at y =.0 m and the tansmitte is at (0.,., 0.) m: solid line actual values, dotted line inveted values, blue line ɛ, ed line ɛ. The fist two ows: noiseless data; the last two ows: noisy data. (a) y-x, (b)y-y, (c)y-z, (d)z-x, (e)z-y, (f)z-z, (g)y-x, (h)y-y, (i) y-z, (j) z-x, (k) z-y, (l) z-z.

7 Pogess In Electomagnetics Reseach C, Vol. 8, (a) (b). 0 (c) (d)... 0 (e)... 0 (f) (g)... 0 (j). 0 (h)... 0 (k) 0 (i) (l) Figue. Inveted elative pemittivities fo isotopic cubes fom diffeent tansmitte-eceive polaizations when the eceives ae at y =.0 m and the tansmitte is at (0.,., 0.) m: solid line actual values, dotted line inveted values, blue line ɛ, ed line ɛ. The fist two ows: noiseless data; the last two ows: noisy data. (a) y-x, (b)y-y, (c)y-z, (d)z-x, (e)z-y, (f)z-z, (g)y-x, (h)y-y, (i) y-z, (j) z-x, (k) z-y, (l) z-z.

8 0 Sun and Tong Figue shows the as the function of iteation step when the tansmitte and eceives ae on the same side. Fig. 7 shows the as the function of iteation step when the tansmitte and eceives ae on the opposite sides. Fo both cases, the cost functions convege to vey small values. As fo the DM, it is shown that the DM value can convege to vey small values fo the noiseless (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figue. Cost function and data mismatch using data fom diffeent tansmitte-eceive polaizations when the eceives ae at y =.0 m and the tansmitte is at (0.,., 0.) m fo isotopic cubes: blue line, ed line DM (%). Fist two ows: noiseless data; last two ows: noisy data. (a) y-x, (b) y-y, (c) y-z, (d) z-x, (e) z-y, (f) z-z, (g) y-x, (h) y-y, (i) y-z, (j) z-x, (k) z-y, (l) z-z.

9 Pogess In Electomagnetics Reseach C, Vol. 8, (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figue 7. Cost function and data mismatch using data fom diffeent tansmitte-eceive polaizations when the eceives ae at y =.0 m and the tansmitte is at (0.,., 0.) m fo isotopic cubes: blue line, ed line DM (%). Fist two ows: noiseless data; last two ows: noisy data. (a) y-x, (b) y-y, (c) y-z, (d) z-x, (e) z-y, (f) z-z, (g) y-x, (h) y-y, (i) y-z, (j) z-x, (k) z-y, (l) z-z.

10 0 Sun and Tong case. While fo the noisy data, due to the noise added in the measuement data, the DM value cannot convege to vey small values. Fo most of the cases, it conveges to about %, which is the pecent of noise added to the data (d) (g) (a) (b) 0 (e) (h) 0 (c) (f) (i) (j) 0 (k) 0 (l) Figue 8. Inveted elative pemittivities fo anisotopic cubes fom diffeent tansmitte-eceive polaizations when the eceives ae at y =.0m and the tansmitte is at (0.,., 0.) m: solid line actual values, dotted line inveted values, blue line ɛ, geen line ɛ z, ed line ɛ, puple line ɛ z. Fist two ows: noiseless data; last two ows: noisy data. (a) y-x, (b) y-y, (c) y-z, (d) z-x, (e) z-y, (f) z-z, (g)y-x, (h)y-y, (i)y-z, (j)z-x, (k)z-y, (l)z-z.

11 Pogess In Electomagnetics Reseach C, Vol. 8, Example : Reconstuction of Anisotopic Pemittivities In this example, the elative pemittivities of the two cubes ae chosen to be anisotopic. The pemittivity tensos ae ɛ =.0,ɛ z =.0 andɛ =.0,ɛ z =.0 fo the left and ight cubes, espectively. We still conside two cases, which ae when eceives ae on the same side and diffeent (a) (d) (g) (j) (b) 0 (e) (h) (k) (c) 0 (f) (i) (l) Figue 9. Inveted elative pemittivities fo anisotopic cubes fom diffeent tansmitte-eceive polaizations when the eceives ae at y =.0 m and the tansmitte is at (0.,., 0.) m: solid line actual values, dotted line inveted values, blue line ɛ, geen line ɛ z, ed line ɛ, puple line ɛ z. Fist two ows: noiseless data; last two ows: noisy data. (a) y-x, (b) y-y, (c) y-z, (d) z-x, (e) z-y, (f) z-z, (g)y-x, (h)y-y, (i)y-z, (j)z-x, (k)z-y, (l)z-z.

12 08 Sun and Tong (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figue 0. Cost function and data mismatch using data fom diffeent tansmitte-eceive polaizations when the eceives ae at y =.0 m and the tansmitte is at (0.,., 0.) m fo anisotopic cubes: blue line, ed line DM (%). Fist two ows: noiseless data; last two ows: noisy data. (a) y-x, (b)y-y, (c)y-z, (d)z-x, (e)z-y, (f)z-z, (g)y-x, (h)y-y, (i)y-z, (j)z-x, (k)z-y, (l)z-z. sides of the tansmitte. And we will still show the numeical esults fo both the noiseless and noisy data. Fo the noisy data, db SNR is applied again. The initial guess fo the contast pemittivity tenso of each cube is chosen as the unit dyad I. Figue 8 ae the esults when the tansmitte and eceives on the same side. Fig. 9 shows the

13 Pogess In Electomagnetics Reseach C, Vol. 8, inveted esults when the tansmitte and eceives on diffeent sides. We can see that when the tansmitte and eceives ae on the same side, the invesion using the y-x polaized data stops in 9 steps fo noiseless data and in steps fo noisy data, and the invesion does not convege. The eason (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figue. Cost function and data mismatch using data fom diffeent tansmitte-eceive polaizations when the eceives ae at y =.0 m and the tansmitte is at (0.,., 0.) m fo anisotopic cubes: blue line, ed line DM (%). Fist two ows: noiseless data; last two ows: noisy data. (a) y-x, (b) y-y, (c) y-z, (d) z-x, (e) z-y, (f) z-z, (g) y-x, (h) y-y, (i) y-z, (j) z-x, (k) z-y, (l) z-z.

14 0 Sun and Tong fo the stop of the iteation is that the diffeence of data mismatches in the two successive steps is within the defined toleance. Similaly fo the esults when the tansmitte and eceives ae on diffeent sides, the y-y polaized invesion fo both the noiseless and noisy cases stops in about steps due to the small change of the data mismatch in two successive steps. Fom these obsevations, we can conclude that the y-x and y-y polaized measuements lack the sensitivity to the anisotopic pemittivities. We also find that when the tansmitte is z-polaized, the invesion esults convege in 0 steps no matte the eceives ae on the same side o diffeent sides of the tansmitte (Except that in Fig. 9, the invesion fom the z-y polaization does not convege fo the noisy case). Hence, it can be summaized that y-x and y-y measuements have weak sensitivity to the anisotopy when the anisotopic pemittivity is isotopic in the x-y plane while changes in the z diection. In this case, z-polaized measuements such as y-z, z-x, z-y, z-z ae moe sensitive to anisotopy. Next, we plot fo diffeent cases. Fig. 0 shows the as a function of iteation step when tansmitte and eceives ae on the same side. Fig. show the esults when tansmitte and eceives ae on diffeent sides. As shown, some of the tansmitte-eceive pais lack sensitivity to the anisotopy, and cost function does not change as iteation poceeds fo these cases. Also simila to the isotopic case, the DM conveges to vey small values fo the noiseless data. While fo the noisy data, the values cannot convege to the ones below the noise level... Example : Reconstuction of Anisotopic Objects In the this example, we conside two cubes with centes at (0, 0., 0) and (0, 0., 0) and side lengths of m. The measuement data ae geneated by one of the cubes centeed at (0, 0., 0) only with elative pemittivity of ɛ =andɛ z =. The fequency is 0. GHz. We then assume two cubes in the oiginal model and invet thei elative pemittivities. The tansmittes ae electic dipoles located at pointsat(±.0, 0, 0), (0, ±.0, 0), (0, 0, ±.0) and polaized in the x, y, z diections at each point. The eceives ae polaized in the z diection only and placed aound the objects with a distance of m fom the oigin. The azimuthal angle (φ) vaies fom 0 to 0 with step of 90, and the pola angle (θ) vaies fom 0 to 80 with step of 0. Fig. shows the invesion esults. We see that the elative pemittivity of the fist cube conveges to an identity since the fist cube does not exist in the synthetic model. The pemittivity of the second cube is well econstucted.... inveted actual z inveted z actual 7 inveted actual z inveted z actual (a) (b) Figue. Inveted esults of elative pemittivities. (a) Cube, (b) cube.. CONCLUSION The model-based invesion algoithm is applied to the econstuction of anisotopic pemittivities in fee space. The invesion esults ae compaed with isotopic pemittivities. The esults show that the invesion conveges well fo isotopic pemittivities using the tansmitte-eceive pais with diffeent polaizations. While fo anisotopic pemittivities, diffeent polaized tansmitte-eceive pais have diffeent sensitivities to the anisotopy. Some tansmitte-eceive pais ae moe sensitive to the

15 Pogess In Electomagnetics Reseach C, Vol. 8, 08 anisotopy than the othes. Sensitivity analysis fo diffeent tansmitte-eceive pais ae pefomed. The invesion esults show that when thee is enough sensitivity in the measuements, the anisotopic pemittivities ae econstucted well. On the othe hand, fo less sensitive measuements, the invesion does not convege. Results fom both the noiseless and noisy data ae pesented. The numeical esults show good potential fo the invesion of the geomety fo the anisotopic objects using the model-based invesion method, which will be futhe investigated in the futue. ACKNOWLEDGMENT The authos would like to thank Pof. W. C. Chew of Pudue Univesity fo his valuable instuction to the wok. REFERENCES. Chew, W. C. and Y. M. Wang, Reconstuction of two-dimensional pemittivity distibution using the distoted bon iteative method, IEEE Tans. on Medical Imaging, Vol. 9, No., 8, Jun Chew, W. C., Waves and Fields in Inhomogeneous Media, IEEE Pess, 99.. Li, F., Q. H. Liu, and L.-P. Song, Thee-dimensional econstuction of objects buied in layeed media using bon and distoted bon iteative methods, IEEE Geoscience and Remote Sensing Lettes, Vol., No., 07, Ap Habashy, T. M. and A. Abubaka, A geneal famewok fo constaint minimization fo the invesion of electomagnetic measuements, Pogess In Electomagnetics Reseach, Vol.,, Sep Abubaka, A. and P. M. van den Beg, Thee-dimensional nonlinea invesion in coss-well electode logging, Radio Sci., Vol., , Jul. Aug Omeagic, D., L. E. Sun, V. Polyakov, Y.-H. Chen, X. Cao, T. Habashy, T. Vik, J. Rasmus and J.-M. Denichou, Chaacteizing teadop invasion in hoizontal wells in the pesence of boundaies using LWD diectional esistivity measuements, th Annual Society of Petophysicists and Well Log Analysts (SPWLA) Symposium, Jun., Hu, Y., G. L. Wang, L. Liang, and A. Abubaka, Estimation of esevoi paametes fom invesion of tiaxial induction data constained by mud-filtate invasion modeling, IEEE Jounal on Multiscale and Multiphysics Computational Techniques, Vol., 8, Fioozabadi, R. and E. L. Mille, A shape-based invesion algoithm applied to micowave imaging of beast tumos, IEEE Tans. Antennas Popagat., Vol. 9, No. 0, 79 79, Oct Li, M., A. Abubaka, and T. M. Habashy, A thee-dimensional model-based invesion algoithm using adial basis functions fo micowave data, IEEE Tans. Antennas Popagat., Vol. 0, No. 7, 7, Jul Sun, L. E. and W. C. Chew, A novel fomulation of the volume integal equation fo electomagnetic scatteing, Waves in Random and Complex Media, Vol. 9, No., 80, Feb Jin, J. M., The Finite Element Method in Electomagnetics, John Wiley & Sons. Inc., New Yok, 00.