Two dimensional inverse scattering problems of small and extended scatterers

TWO-DIMENSIONAL INVERSE SCATTERING PROBLEMS OF SMALL AND EXTENDED SCATTERERS KRISHNA AGARWAL (B. Tech, Indian School of Mines, Dhanbad, India) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2010 ii Acknowledgements गुरु गोव द िं दोऊ खड़े, काके ऱागिं पािंए | बलऱहारी गुरु, आपने गोव द िं ददयो लिऱाये || Behold, the Teacher and the Truth on the same altar, who I respect first! To you indeed, my Teacher! You introduced the Truth to me. ‒ Kabirdas In this proud moment, when I am writing the acknowledgement for my PhD dissertation, I first pay my humble respects to my teachers. Specifically, I recall the teachers who left indelible marks in my life: my parents, the Master Ji in Jawad, Mrs. N. Ojha, Mrs. P. P. Mathur, Mrs. S. Bafna, Mrs. A. Tondon, Mr. J. Abraham, Mr. Varghese, Mr. T. Prasad, Mr. Hosur, Mr. Ravi, Mr. D. Prasad, Dr. B. S. R. Shastry, Dr. A. Bhattacharya, my parents-in-law, Dr. X. Chen, and Dr. D. Srinivasan (in the chronological order of their first impact in my life). Life is a great teacher, and I salute to the provider of life. I especially thank Mr. Varghese, Dr. A. Bhattacharya, and Dr. X. Chen, for mentoring a stubborn person as me. While Mr. Varghese created the spark of research, Dr. Bhattacharya fanned it, and Dr. Chen brought it to full flames. I am extremely grateful to Dr. Chen for accepting the supervision of a lacking research candidate as I. While all Ph.D. students have supervisors, I had the unique privilege of having SUPERvisor. He patiently and persistently guided me, helped me, and mentored me during the past four years. He is a true mentor. He helped me to understand technical as well as non-technical aspects of contemporary research. He has corrected me on several of my technical mistakes, very gently and very effectively. He also helped me resolve iii some of my philosophical conflicts regarding the research methodologies. He is bestowed with a super vision, which enables him to understand the troubles, doubts, misgivings, and incorrect perceptions of the students. In addition to all these, he has been an excellent friend and the most compassionate boss I ever had. Thank you Sir. I take this opportunity to thank National University of Singapore and Ministry of Education for providing me with the necessary infrastructure and funding for my doctoral studies. I thank Dr. M. S. Leong, Dr. L. W. Li, Late Dr. B. L. Ooi, Dr. K. Mouthan, Dr. A. Popov, Dr. D. Srinivasan, Dr. A. O. Adeyeye, and Dr. Y. Wu for conducting exceptional graduate modules. I thank Dr. Dipti Srinivasan for being a constant source of inspiration for me. I thank Mr. Sing and Ms. Guo Lin for providing me with a good and hassle-free laboratory environment, except for the sunlight. I thank T. J. Lu for providing me with my first experience of mentoring and academic supervision. I had the privilege of having great friends in my life, who make the life a rich and happy experience. I begin with the friends who helped me directly or indirectly with my doctoral studies. I thank Dilip and Ananya for motivating and helping me to join the doctoral program. I thank Dr. Zhong Yu, Davood and Roger for being and remaining very good friends in odd and even times. I treasure many technical discussions with them. Other friends in my research group and research lab, who deserve special mention, are Dr. Shen, Xiuzhu, Pan Li, Ebrahim, Tianjian, and Meysam. My Ph.D. life would not have been as much fun without Chen Ling (my best friend in Singapore), Amy (Li Li Ying), Cissy (Gu Qian), Maple (Kong Yi), Chi, Balaji, Vishal, Dwarika, Shaleen, Kalpesh, Vijay, Fazal, Kabita bhabhi, Vignesh, and fellow ISMites in Singapore. Special thanks to Mimi for giving me lively, child-like, and pure moments of joy. I also thank my friends and ex-colleagues, who are too many to count. In a non-exclusive list, I mention Chaitali, iv Jana, Dubi, K. Prasad Sir, Manjunath, Anil, Mdm Thilaxi, Sarvanan, and Ramnik. I recall fondly Arnica, Pratibha, Kajal, Bilkis, Parvin, Vineet, Vinay, Deepika, and others. I convey love and gratitude to my three little sisters, Poonam, Vidhi, and Komal, for livening up my life. Komal, the dearest among the three, will hopefully write a doctoral thesis too some day. I also thank my sisters-in-law, Kaushalya and Sangita for making me an inseparable part of the family. I thank my parents and parents-in-law for bearing with my foolhardiness and supporting me in my decisions even if they had different opinions. I thank Poonam Agarwal and Kaushalya Gupta for taking care of my family in the difficult times and ensuring that my parents and parents-in-law not miss my presence. I fall short of words when trying to acknowledge my love, Dilip. I know him in more than a thousand capacities, and he has excelled and helped me in each capacity. I thank him for existing in my life. I dedicate this important document of my life to my parents and parents-in-law. To you, my love, I dedicate every breath of mine. I wish I had more to dedicate to you. v Table of Contents Acknowledgements ii Table of Contents . v Summary viii List of Figures . ix List of Tables . xiii List of Publications . xiv List of Abbreviations xvi Chapter 1: Introduction 1.1 Outline 10 1.2 Mathematical notations 11 1.3 Inverse problems: mathematical framework 12 1.4 Two dimensional scattering model 14 1.4.1 Region of interest and measurement setup . 14 1.4.2 Forward model 15 1.4.3 Discretization: coupled dipole model . 17 1.4.4 Important operators . 19 1.4.5 Transverse magnetic and transverse electric cases . 20 1.4.6 Comparison with three dimensional inverse scattering problems 26 1.5 Singular value decomposition 27 Chapter 2: Inverse scattering problems of small scatterers . 30 2.1 Multistatic response matrix 31 vi 2.1.1 Multistatic response matrix for the TM case 31 2.1.2 Multistatic response matrix for the TE case . 31 2.2 Locating small scatterers using multiple signal classification . 32 2.2.1 MUSIC for the TM case . 33 2.2.2 MUSIC for the TE case 42 2.2.3 Summary . 52 2.3 Non-iterative retrieval of the polarization tensors . 53 2.3.1 Retrieval of the polarization tensors in the TM case 53 2.3.2 Retrieval of the polarization tensors in the TE case . 56 2.3.3 A practical example - retrieval of radii of wire cables . 58 2.3.4 Summary . 61 Chapter 3: Inverse scattering problems of extended scatterers 62 3.1 Multipole based linear sampling method . 62 3.1.1 Introduction to the linear sampling method 62 3.1.2 Expansion of scattered far field in terms of multipole radiation 64 3.1.3 Multipole based linear sampling method 66 3.1.4 A comprehensive example 70 3.1.5 More numerical examples 78 3.2 Subspace based optimization method and an initial guess scheme . 82 3.2.1 Subspace based optimization method . 82 3.2.2 Choice of L and the termination condition 84 3.2.3 Scheme for initial guess 85 3.2.4 Numerical example . 87 vii 3.3 SOM for the TE case 91 3.3.1 Ill-posedness and directional probing . 91 3.3.2 Computational complexity . 92 3.3.3 Non-linearity . 92 3.3.4 Numerical example . 94 3.4 Summary 96 Chapter 4: Inverse scattering problems of anisotropic scatterers 98 4.1 MUSIC for small anisotropic scatterers . 100 4.1.1 Noise-free scenario . 102 4.1.2 Noisy Scenario 103 4.1.3 Numerical examples . 104 4.2 SOM for extended anisotropic scatterers . 107 4.2.1 SOM for anisotropic scatterers with known optical axes . 108 4.2.2 Numerical examples . 109 4.3 4.3.1 4.4 SOM for extended anisotropic scatterers with unknown optical axes . 113 Numerical examples . 114 Summary 115 Chapter 5: Conclusion 117 5.1 Summary of contributions 117 5.2 Discussion 119 References 126 viii Summary This thesis studies two-dimensional inverse scattering problems of small scatterers, extended scatterers, and anisotropic scatterers. For small scatterers, the impact of the choice of test source and signal subspace on the performance of multiple signal classification (MUSIC) is studied. A non-iterative two-step least squares based method is proposed for retrieving the polarization tensors of small scatterers. For extended scatterers, a new multipole based linear sampling method (MLSM) is proposed, which uses a radiation model for linear sampling method and a physical regularization scheme. MLSM is used to generate an initial guess for subspace based optimization method (SOM), a fast and robust method for the reconstruction of extended scatterers. Further, SOM is extended for the vectorial inverse scattering problem. For anisotropic scatterers, a modified MUSIC algorithm is proposed, which computes the optimal test direction at each point non-iteratively. The application of SOM is extended for the reconstruction of extended anisotropic scatterers. ix List of Figures Fig. 1-1: The experimental setup. 15 Fig. 1-2: Variation of the polarization tensor of a pixel of size  200 with the relative permittivity in the TM and TE cases 22 Fig. 2-1: Nature of the multipoles induced in a small cylinder in the TM case . 33 Fig. 2-2: Selection of the signal subspace and detection of a cylinder using pseudospectrums  z (r ) and  x (r ) in the noise-free scenario. . 37 Fig. 2-3: Selection of the signal subspace and detection of a cylinder using pseudospectrums  z (r ) and  x (r ) in the noisy scenario. . 38 Fig. 2-4: A hypothetical example to demonstrate the definition of resolution. Since (2.15) is satisfied, it is considered that the scatterers are resolved . 40 Fig. 2-5: Resolution of MUSIC in the TM case . 41 Fig. 2-6: Example of imaging of multiple cylinders (in Table 2-1) using MUSIC in the presence of 20 dB SNR. . 42 Fig. 2-7: Nature of the multipoles induced in a small cylinder in the TE case. . 42 Fig. 2-8: MUSIC for the TE case in the noise-free scenario. 46 Fig. 2-9: Selection of the signal subspace and detection of an isotropic cylinder using pseudospectrums  x (r ) and  z (r ) in the noisy scenario. 47 x Fig. 2-10: Resolution of MUSIC in the TE case. . 48 Fig. 2-11: MUSIC imaging for multiple cylinders (in Table 2-1) in the presence of 30 dB SNR. . 49 Fig. 2-12: Imaging isotropic elliptic cylinder using MUSIC. 51 Fig. 2-13: The value of error TM for various values of permittivity. 55 Fig. 2-14: Error TM for the example of multiple cylinders (Table 2-1) for various values of SNR. . 55 Fig. 2-15: The value of error  TE for various values of permittivity. . 58 Fig. 2-16: Error  TE for the example of multiple cylinders (Table 2-1) for various values of SNR. . 58 Fig. 2-17: Example of metallic cylinders (wire lines) of various diameters. 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Stability of the solution and non-linearity are the issues of main concern in solving the inverse scattering problems In this thesis, inverse scattering problems for three categories of scatterers are considered These three categories are small scatterers, extended scatterers, and anisotropic scatterers For the inverse scattering problems in each category, one qualitative reconstruction approach and one... it was used for detecting scatterers in a scalar two- dimensional scattering setup (which corresponds to the homogeneous acoustic scattering or the electromagnetic transverse magnetic scattering problems) The work by [4, 8-16, 44] extended the application of MUSIC for two and three -dimensional scattering problems and further developed the theory of MUSIC for inverse scattering problems Work in [5, 9,... robustness of SOM and accelerating the convergence As an extension of SOM, we adapt SOM for the vectorial inverse scattering problem We compare SOM for the TM and TE cases in terms of the computational complexity, ill- 9 posedness, and non-linearity This adaptation is also useful in extending SOM for anisotropic scatterers Inverse scattering problems of anisotropic scatterers: The inverse scattering problems. .. stability of the inverse scattering problem is the most difficult issue The instability of the inverse problems can be understood as the sensitivity of the solution x to small changes in the data y This property is attributed to the ill-conditioned nature of the operator and the topologies of the spaces X and Y [140] In inverse problems, such sensitivity of the data occurs often due to the discretization of. .. more demanding than the qualitative imaging problem Inverse scattering problems of small scatterers: In the case of small scatterers, typically the number of independent sources induced on the scatterers is less than the number of detectors Thus, there is an injective mapping between the currents induced on the 3 scatterers and the scattered fields measured at the detectors Due to this, these problems. .. of cost function, optimization scheme, and the initial guess play an important role 1.4 Two dimensional scattering model In this section, we introduce a generic experimental setup describing a two- dimensional forward and inverse scattering experiment, the physical quantities involved, and explain the mathematical model of scattering 1.4.1 Region of interest and measurement setup Let us consider a two- dimensional. .. presents the inverse scattering problem of small scatterers Section 2.1 presents the multistatic response matrix, which is the fundamental operator used in Chapter 2 Section 2.2 presents and discusses various aspects of MUSIC when applied for two dimensional inverse scattering problem Special attention has been given to the impact of the choice of test source and signal subspace on the performance of MUSIC... discussed in Section 3.2 and Section 3.3 Section 3.2 presents a scheme for generating an initial guess for SOM We use MLSM and a two- step least squares approach for generating an estimate of the permittivity distribution of the region of interest In Section 3.3, we extend SOM for two- dimensional vectorial inverse scattering problem (TE case) 11 Inverse scattering problems of anisotropic scatterers are discussed... applicable to the case of extended scatterers The primary reason is that MUSIC exploits the injectivity of the operators for small scatterers, and this property of injectivity is lost in the case of extended scatterers [15, 47] Though some work has been done to apply MUSIC to detect extended scatterers [48-50], the work is still in preliminary stage Other qualitative methods used for extended scatterers include... fields in the case of small scatterers is utilized and the problem of reconstruction is split into two problems, each of which is linear Since the nature of sources induced in the two- dimensional scenario is different in the transverse magnetic (TM) and transverse electric (TE) cases, the two- dimensional scenario needs additional attention Specifically, the nature of polarization tensors and the induced . TWO- DIMENSIONAL INVERSE SCATTERING PROBLEMS OF SMALL AND EXTENDED SCATTERERS KRISHNA AGARWAL (B. Tech, Indian School of Mines, Dhanbad, India) . India) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2010