Subspace based inversion methods for solving electromagnetic inverse scattering problems

Subspace-based Inversion Methods for Solving Electromagnetic Inverse Scattering Problems Zhong Yu (M. Eng., B. Eng., Zhejiang University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 Acknowledgements My deepest gratitude goes first and foremost to Dr. Chen Xudong, my supervisor, for his constant warming encouragement and professional guidance. Without his consistent and illuminating instruction, this thesis could not have reached its present form. I would like to thank the National University of Singapore for providing scholarship to support me to pursue my doctoral degree in electromagnetic inverse problems. I also own my gratitude to Prof. Ran Lixin from Zhejiang Univeristy, who, as my supervisor when I was in Zhejiang University, introduced me into the world-class electromagnetic research area. I would like to thank staff from Microwave and RF research group in the Department of Electrical and Computer Engineering, especially Prof. Leong Mook Seng, Prof. Li Le-wei, Prof. Ooi Ban Leong, Dr. Koen Mouthaan, Mr. Sing Cheng Hiong, and Ms. Guo Lin for teaching me the fundamentals of electromagnetics and providing their kind assistance during my doctoral study. I would like to express my appreciation to my fellow team mates from microwave research lab and MMIC lab, especially Krishna Agarwal, for her always helpful discussion and her selflessness of maintaining the computing instruments, Wang Ying, Zhang Yaqiong, Tang Xinyi, Nan Lan, Chen Ying, and Zhong Zheng, for their friendliness to share their most genuine happiness with me all these years. Last but not least, I would like to present my heartfelt gratitude to my parents. Without their decades’ support and sacrifice, I would not be able to pursue my dream and reach the place where I am now. I would like to dedicate this thesis to them, especially my father. i Contents Acknowledgements i Contents ii Summary v List of Figures vi List of Acronyms ix List of Publications xi Introduction 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Original contributions and overview of the thesis . . . . . . . . . . . . Preliminaries 2.1 Forward problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Inversion methods for point-like scatterers . . . . . . . . . . . 17 2.2.2 Inversion methods for extended scatterers . . . . . . . . . . . 21 ii iii Subspace-based inversion methods for small scatterers 3.1 3.2 3.3 A robust non-iterative method for retrieving scattering strength . . . 27 3.1.1 The least squares retrieval method . . . . . . . . . . . . . . . 28 3.1.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 MUSIC imaging method for small anisotropic scatterers . . . . . . . . 33 3.2.1 Formulas for the forward problem of the multiple-scattering small anisotropic spheres . . . . . . . . . . . . . . . . . . . . . 34 3.2.2 Inverse scattering problem . . . . . . . . . . . . . . . . . . . . 39 3.2.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 43 3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 MUSIC imaging method with enhanced resolution . . . . . . . . . . . 50 3.3.1 Forward scattering problem . . . . . . . . . . . . . . . . . . . 52 3.3.2 The MUSIC algorithm with enhanced resolution . . . . . . . . 53 3.3.3 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Subspace-based inversion methods for extended scatterers 4.1 4.2 26 64 SOM and nested SOM . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.1.1 The subspace-based optimization method . . . . . . . . . . . . 65 4.1.2 The nested SOM . . . . . . . . . . . . . . . . . . . . . . . . . 69 Twofold SOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2.1 The twofold SOM . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.2 Computational test . . . . . . . . . . . . . . . . . . . . . . . . 77 iv 4.2.3 4.3 Discussion and summary . . . . . . . . . . . . . . . . . . . . . 82 Improved SOM and its implementation in three-dimensional inverse scattering problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.1 Three-dimensional SOM . . . . . . . . . . . . . . . . . . . . . 85 4.3.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 90 4.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Conclusion 96 Bibliography 101 A Derivation of the Cramer-Rao bound 113 Summary This thesis studies several methods for solving electromagnetic inverse scattering problems, all of which are on the basis of the concept of subspace. The original contributions of this thesis can be cataloged into two folds: Firstly, we not only apply the multiple signal classification (MUSIC) method to locate small anisotropic scatterers, dimensions of which are much less than the wavelength, but also propose a new MUSIC algorithm that improves resolution and in the meanwhile is able to deal with small degenerate scatterers; Secondly, we propose a new series of subspacebased optimization methods (SOM) to solve the inverse scattering problems for extended scatterers, including the nested SOM, twofold SOM, and improved SOM. Based on the concept of subspace, we actually utilize the most stable part of the measured scattered fields, thus, methods proposed in this thesis not only converge fast but also are quite robust against noise. Various numerical simulations have been carried out and validate the proposed algorithms. v List of Figures 3.1 Comparison of the result obtained by least squares retrieval method and that given in [1] for the case that the scatterers have same scattering strengths, τm = 1, m = 1, 2, 3, 4. The errors are averages over 1000 repetitions. The CRB of the estimation is also shown. . . . . . . 30 3.2 Normalized percentage of the estimation errors for the case that the scatterers have different scattering strengths, τm = m, m = 1, 2, 3, 4. The errors are averages over 1000 repetitions. The CRB of the estimation is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Normalized percentage of the estimation errors for the case that the number of the transceivers is 31 and the scatterers have same scattering strengths, τm = 1, m = 1, 2, 3, 4. The errors are averages over 1000 repetitions. The CRB of the estimation is also shown. . . . . . . 33 3.4 The definition of the rotation angles φn,m , θn,m and ϕn,m , where en,m is the lth electric (n = E) or magnetic (n = H) principle axis of the mth scatterer, l = 1, 2, and m = 1, 2, . . . , M . . . . . . . . . . . . . . 35 3.5 Pseudo-spectrum image and the accuracy of the retrieval of the scattering strength tensors for two small anisotropic spheres located at (0, 0) and (0, λ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.6 Pseudo-spectrum image and the accuracy of the retrieval of the scattering strength tensors for four small anisotropic spheres located at (0, 0), (0, λ/12), (λ/12, 0) and (λ/12, λ/12). . . . . . . . . . . . . . . . 48 3.7 Singular values and pseudo-spectrum obtained by the standard MUSIC algorithm in noise free case. (a) The 10 base logarithm of the singular values of the MSR matrix (j = 1, 2, . . . , 48). (b), (c) and (d) are the 10 base logarithm of the pseudo-spectrum in y = x + 0.112λ plane obtained by the standard MUSIC algorithm with test dipoles in x, y and z directions, respectively. . . . . . . . . . . . . . . . . . . 57 (l) vi vii 3.8 Pseudo-spectrum obtained by the proposed MUSIC algorithm in noise free case. (a), (b), (c) and (d) are the 10 base logarithm of the pseudospectrum in y = x + 0.112λ plane obtained by the proposed MUSIC algorithm corresponding to the L = 4, 5, and cases, respectively. . 58 3.9 Pseudo-spectrum obtained by the proposed MUSIC algorithm in noise free case when the test dipole is constrained to be real. (a), (b), (c), (d), (e) and (f) are the 10 base logarithm of the pseudo-spectrum in y = x + 0.112λ plane obtained by the proposed MUSIC algorithm corresponding to the L = 4, 5, 6, 7, and cases, respectively. . . . . 59 3.10 Singular values and pseudo-spectrum obtained by the standard MUSIC algorithm in noise-contaminated case (30dB). (a) The 10 base logarithm of the singular values of the MSR matrix (j = 1, 2, . . . , 48). (b), (c) and (d) are the pseudo-spectrum in y = x + 0.112λ plane obtained by the standard MUSIC algorithm with test dipoles in x, y and z directions, respectively. . . . . . . . . . . . . . . . . . . . . . . 60 3.11 Pseudo-spectrum obtained by the proposed MUSIC algorithm in noisecontaminated case (30dB). (a), (b), (c), (d), (e) and (f) are the pseudo-spectrum in y = x + 0.112λ plane obtained by the proposed MUSIC algorithm corresponding to the L = 4, 5, 6, 7, and cases, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.12 Pseudo-spectrum obtained by the proposed MUSIC algorithm in noisecontaminated case (30dB) when the test dipole is constrained to be real. (a), (b), (c), (d), (e) and (f) are the pseudo-spectrum in y = x + 0.112λ plane obtained by the proposed MUSIC algorithm corresponding to the L = 4, 5, 6, 7, and cases, respectively. . . . . 62 4.1 The original dielectric profile of the ’Austria’ structure. . . . . . . . . 71 4.2 Reconstruction result obtained using 20 × 20 mesh grid after 100 iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Reconstruction result obtained using 30 × 30 mesh grid after 100 iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Reconstruction result obtained using 30 × 30 mesh grid after 150 iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5 Reconstruction result obtained using 40 × 40 mesh grid after 1000 iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 viii 4.6 Reconstruction result obtained using 64 × 64 mesh grid after iterations with result in Fig. 4.2 as the initial guess. . . . . . . . . . . . . 72 4.7 Singular values of GS and GD . . . . . . . . . . . . . . . . . . . . . . 76 4.8 The recalculated objective function values while choosing M0 = 50, 100, 200 and 500. The original SOM’s curve is also presented. . . . . . . . . . 78 4.9 Reconstruction results using M0 = 50, 100, 200 and 500 with 10% additive noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.10 Reconstruction result after iterations using M0 = 500 with the initial guess generated by using M0 = 50 after 30 iterations with 10% additive noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.11 Reconstruction results in the first and second step of TSOM on scattering data with 30% additive noise. . . . . . . . . . . . . . . . . . . . 80 4.12 Reconstruction results in the first and second step of TSOM on scattering data with 50% additive noise. . . . . . . . . . . . . . . . . . . . 81 4.13 Reconstruction results of inhomogeneous scatterers in the first and second step of TSOM on scattering data with 10% additive noise. . . 82 4.14 A coated cube with its inner edge length a = 0.6λ and outer edge length b = 1.6λ. The relative permittivity of the inner cube is ǫr1 = 1.6 and the relative permittivity of the outer layer is ǫr2 = 1.3. . . . . 90 3D 4.15 Singular values of GS . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.16 The objective function values within 60 iterations when L = 1, 5, 10, 30 and 60. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.17 The real part of the retrieval result of the dielectric profile for the domain of interest after 60 iterations. The real part of the relative permittivity of the inner cube and outer layer are 1.6 and 1.3, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.18 The imaginary part of the retrieval result of the dielectric profile for the domain of interest after 60 iterations. The imaginary part of the relative permittivity of the inner cube and outer layer are both 0. . . 94 List of Acronyms 2D Two-dimensional 3D Three-dimensional AD Analog to digital CDM Coupled dipole method CG Conjugate gradient CRB Cramer-Rao bound CSI Contrast source inversion DDA Discrete dipole approximation DOA Direction of arrival DORT Decomposition of the time reversal operator EFIE Electric field integral equation EISP-ES Electromagnetic inverse scattering problems for extended scatterers EISP-PLS Electromagnetic inverse scattering problems for point-like scatterers FFT Fast Fourier transform LM Levenberg-Marquardt MOM Method of moment MSE Multiple scattering effect MSR Multi-static response MUSIC Multiple signal classification ix 100 Having applied the SOM to solve the three-dimensional inverse scattering problems, one may expect to extend the twofold SOM to the three-dimensional case so as to obtain better reconstruction results in a more efficient manner. However, such extension is not straightforward. When we use twofold SOM to solve the two-dimensional problem, we firstly need to obtain the spectral information of the scattering operator that maps the induced current to the scattered field inside the domain of interest, which is accomplished by the singular value decomposition (SVD) of the operator. As we know, the computational burden of the SVD on the operator is very costly. For example, if we are dealing with an M × M mesh grid for a two-dimensional domain of interest, the size of the scattering mapping mentioned above is M × M . The computational cost of the SVD on this operator is O(M ). Such computational cost is still manageable when we are dealing with medium size two-dimensional problems. However, when dealing with threedimensional inverse scattering problem, the size of the scattering operator becomes very large, which directly results in an even costly computational burden for the SVD operation. 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(C.5) and (C.7), we have  ∂K (t) ∂τj     =      ∂K (t) ∂τj ∂K (t) ∂τj . ∂K Nr Nt (t) ∂τj           (A.4) = ξ(t)RS [g0,r (Xj ) + G0,r · O · H −1 · g00 (Xj )] · gt (Xj )T , where RS {·} is a reshape operation which changes an arbitrary matrix A = A1 , A2 , . . . , AN into its vectorized form B as follows  B = RS A = RS A1 , A2 , . . . , AN and            g00 (Xj ) =                =     A1   A2   ,  .    AN (A.5) T G0 (X1 , Xj )   G0 (X2 , Xj )     .    G0 (Xj−1 , Xj )   ,     G0 (Xj+1 , Xj )     .   G0 (XM , Xj ) (A.6) 115 while G0,r , O, g0,r , gt , H and ξ(t) have same definitions with those in [68]. Thus, by using σ = K 2F , Nr Nt 10SN R/10 the lower bound of the normalized errors of the estimation of the scattering strengths is obtained via tr {I −1 } = 100 |O′ | ECRB = 100 K 2F tr 2Nr Nt 10SN R/10 |O′ |2 where tr {·} is the trace of a matrix, · ′ F ∗ ℜ ∂K · ∂K −1 , (A.7) is the Frobenius norm, |·| is the L2 norm of a vector, O = diag {O}, and SN R is in dB. [...]... techniques are discussed, for both point-like scatterers and extended scatterers First, the MUSIC that is used in solving acoustic inverse 7 scattering problems and its preliminary usage in solving electromagnetic inverse scattering problems for point-like scatterers is presented Second, the inversion technique based on EFIE for solving the electromagnetic inverse scattering problems for extended scatterers... accepted as Yu Zhong and Xudong Chen, “A Nested Subspace- Based Algorithm for Solving Inverse Scattering Problem,” International Conference on Inverse Problems, Wuhan, China, accepted 5 The content in Chapter 4, Section 2 has been published as Yu Zhong and Xudong Chen, “Twofold subspace- based optimization method for solving inverse scattering problems, ” Inverse Problems, Vol 25, ID:085003 xi xii (11pp),... an annular object [37] The subject of this thesis is in two folds: First, to investigate MUSIC methods for solving electromagnetic inverse scattering problems for point-like scatterers, so as to obtain a better resolution; Second, to investigate methods for solving electromagnetic inverse scattering problems for extended scatterers, which makes the optimization converge faster and obtain satisfactory... fast methods to solve various inverse scattering problems However, 3 because of the intrinsic nonlinearity and (or) ill-posedness, the inverse scattering problems usually can only be solved within some precision and the computational cost is usually quite large The electromagnetic inverse scattering problems studied in this thesis can be divided into two types: electromagnetic inverse scattering problems. .. overview of the thesis The original contributions of the thesis consist two parts: methods for solving EISPPLS and methods for solving EISP-ES, both of which are on the basis of the concept of subspace The subspace- based methods for solving EISP-PLS are introduced in Chapter 3, while those for EISP-ES are introduced in Chapter 4 Before these two chapter, in Chapter 2, some preliminaries are given, and the... good resolving ability, MUSIC method was mainly used in solving direction of arrival (DOA) problem in signal processing society [19–26] and was only recently introduced into acoustical society for solving acoustic inverse scattering problems The transplant of MUSIC method from acoustic inverse scattering problem to electromagnetic inverse scattering problem is not so straightforward due to electromagnetic. .. stable subspace of the MSR matrix and the polarization characteristic of the electromagnetic wave, a new MUSIC method is proposed to improve the resolving ability, which is also able to deal with small degenerate scatterer whose scattering strength tensor is rank deficient or almost rank deficient In Chapter 4, based on the concept of subspace, several methods for solving electromagnetic inverse problems for. .. based optimization method,” Progress in Electromagnetic Research, Vol 102, pp 351-366, 2010 Chapter 1 Introduction This thesis deals with inversion methods for solving electromagnetic inverse scattering problems, by which we use electromagnetic wave to probe the location, shape, and physical characteristics of scatterers Methods studied in the thesis include those for small scatterers, dimensions of which... mention its counterpart, the forward problem In our topic, the forward problem is the electromagnetic scattering problem, which has been studied for a long time Thus, in the first part of this chapter, a forward problem solver based on the integral equation solution of the Maxwell equations is presented Such forward problem solver is used in solving the inverse scattering problems in the rest part of... After introducing the forward problem solver, those methods mentioned in the previous chapter for solving the electromagnetic inverse scattering problems for both point-like scatterers and extended scatterers will be introduced 2.1 Forward problem As we know, the Maxwell equations consist of four equations, Faraday’s law, Ampere’s law, Gauss’ law for magnetic fields and Gauss’ law for electric fields, . investigate MUSIC methods for solving electromagnetic inverse scattering problems for point-like sca t te r er s, so as to obtain a better resolution; Second, to investigate methods for solving elec- tromagnetic. Subspace-based Inversion Methods for Solving Electromagnetic Inverse Scattering Problems Zhong Yu (M. Eng., B. Eng., Zhejiang University, China) A THESIS SUBMITTED FOR THE DEGREE. in solving electromagnetic inverse scattering pr o b l em s for point-like scatterers is presented. Second, the inversion techniqu e based on EFIE for solving t h e electromagneti c inverse scattering