Picture

ڿ (, Won-Kwang Park)

Official Information

  • Full Professor
  • Department of Information Security, Cryptology, and Mathematics
  • College of Science and Technology, Kookmin University.
  • 77, Jeongneung-ro, Seongbuk-gu, Seoul, 02707, Korea.
  • Personal Web Page: https://home1.kookmin.ac.kr/~parkwk/
  • Office: D-714, Building E4, Kookmin University
  • Scientific Computing Lab: 206, Building N6, Kookmin University
  • E-mail: parkwk@kookmin.ac.kr.
  • Tel: (+82) 02 910 5748
  • Fax: (+82) 02 910 4739
  • Curriculum vita: view.

  • Education & Experiment

  • B.S., Department of Mathematical Education, Kookmin University, August 2000.
  • M.S., Department of Mathematics, Yonsei University, August 2004.
  • - Thesis title: Partial differential equations in image processing.
    - Thesis advisor: Jin Keun Seo.
    - Jury members: Hi Jun Choe and Jeehyun Lee.
  • Ph.D., Centre de Mathématiques Appliquées, École Polytechnique in cooperation with Laboratoire des Signaux et Systèms, École Supérieure d'Électricité (Supélec), February 2009.
  • - Thesis title: Diffraction inverse par des inclusions minces et des fissures (Inverse scattering from two dimensional thin inclusions and cracks).
    - Thesis advisors: Habib Ammari and Dominique Lesselier.
    - Jury members: Elena Beretta, Oliver Dorn, François Jouve, Roman Novikov, and Knut Sølna.
  • Post Doc., Institute for Mathematics and Scientific Computing, Karl Franzens University of Graz, December 2009.
  • - Research subject: Multi-frequency imaging of thin electromagnetic inclusions buried within a half-space affected by random noise.
    - Advisor: Karl Kunisch.

    Research Area

  • Inverse problems, Microwave imaging, Non-destructive evaluations, Electromagnetics, Scientific computing.

  • Team members

  • Former Ph.D. student
  • - Young-Deuk Joh, Mathematical analysis of MUSIC and subspace migration for imaging of perfectly conducting cracks and thin electromagnetic inclusions (received excellent thesis prize).
  • Former M. S. students
  • - Joo Young Huh, Mathematical analysis of subspace migration imaging function and its improvement.
    - Young Mi Kwon, Mathematical analysis of subspace migration in full- and limited-view inverse scattering problems (received excellent thesis prize).
  • Former B. S. students
  • - Sangwoo Kang, Génie électrique et électronique de Paris (GeePs), CentraleSupelec, Université Paris-Sud.
    - Jung Ho Park, Department of Mathematical Sciences, Seoul National University (received outside achievement award).
    - Kyungrok Lee, Department of Mathematics, Graduated school, Yonsei Univerisity.
    - Hyeoncheol Jo, Department of Computational Science & Engineering, Yonsei Univerisity.
    - Taekyung Ki, Department of Mathematics, Graduated school, Yonsei Univerisity.

    Selected publications

    Below is a list of selected publications of mine. Complete list can be found in my curriculum vita.

    1. Won-Kwang Park and Dominique Lesselier, MUSIC-type imaging of a thin penetrable inclusion from its far-field multi-static response matrix, Inverse Problems, 25, Article No. 075002, 2009. [Abstract]

      The imaging of a thin inclusion, with dielectric and or/magnetic contrast with respect to the embedding homogeneous medium is investigated. A MUSIC-type algorithm operated at a single time-harmonic frequency is developed in order to map the inclusion (that is, to retrieve its supporting curve) from scattered field data collected within the Multi-Static Response (MSR) matrix. Numerical experiments carried out for several types of inclusions (dielectric and/or magnetic ones, straight or curved ones), mostly single inclusions but also two of them close-by as a straightforward extension, illustrate the pros and cons of the proposed imaging method.

    2. Hyundae Lee and Won-Kwang Park, Location search algorithm of thin conductivity inclusions via boundary measurements, ESAIM: Proceedings, 26, 217-229, 2009. [Abstract]

      We propose an algorithm for retrieving the end points of thin, rectangular inclusions of finite conductivity in a homogeneous medium. It is based on an appropriate asymptotic formula for steady state voltage potentials in the presence of thin inclusions. Numerical experiments exhibit the proposed algorithm is fast, effective and stable.

    3. Won-Kwang Park and Dominique Lesselier, Reconstruction of thin electromagnetic inclusions by a level set method, Inverse Problems, 25, Article No. 085010, 2009. [Abstract]

      In this contribution, we consider a technique of electromagnetic imaging (at a single, non-zero frequency) which uses the level set evolution method for reconstructing a thin inclusion (possibly made of disconnected parts) with either dielectric or magnetic contrast with respect to the embedding homogeneous medium. Emphasis is on proof of concept, the scattering problem at hand being so far based on a two-dimensional scalar model. To do so, two level set functions are employed; the first one describes location and shape, and the other one connectivity and length. Speeds of evolution of level set functions are calculated via the introduction of Fréchet derivatives of a least-square cost functional. Several numerical experiments on noiseless and noisy data as well illustrate how the proposed method behaves.

    4. Won-Kwang Park and Dominique Lesselier, Electromagnetic MUSIC-type imaging of perfectly conducting, arc-like cracks at single frequency, Journal of Computational Physics, 228 (21), 8093-8111, 2009. [Abstract]

      We propose a non-iterative MUSIC (MUltiple SIgnal Classification)-type algorithm for the time-harmonic electromagnetic imaging of one or more perfectly conducting, arc-like cracks found within a homogeneous space . The algorithm is based on a factorization of the Multi-Static Response (MSR) matrix collected in the far field at a single, nonzero frequency in either Transverse Magnetic (TM) mode (Dirichlet boundary condition) or Transverse Electric (TE) mode (Neumann boundary condition), followed by the calculation of a MUSIC cost functional expected to exhibit peaks along the crack curves each half a wavelength. Numerical experimentation from exact, noiseless and noisy data shows that this is indeed the case and that the proposed algorithm behaves in robust manner, with better results in the TM mode than in the TE mode for which one would have to estimate the normal to the crack to get the most optimal results.


    5. Habib Ammari, Hyeonbae Kang, Hyundae Lee and Won-Kwang Park, Asymptotic imaging of perfectly conducting cracks, SIAM Journal on Scientific Computing, 32 (2), 894-922, 2010. [Abstract]

      In this paper, we consider cracks with Dirichlet boundary conditions. We first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack. Based on this formula, we design a non-iterative approach for locating a collection of small cracks. In order to do so, we construct a response matrix from the boundary measurements. The location and the length of the crack are estimated, respectively, from the projection onto the noise space and the first significant singular value of the response matrix. Indeed, the direction of the crack is estimated from the second singular vector. We then consider an extended crack with Dirichlet boundary conditions. We rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack. To reconstruct an extended crack from many boundary measurements, we develop two methods for obtaining a good guess. Several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave.

    6. Won-Kwang Park, On the imaging of thin dielectric inclusions buried within a half-space, Special issue on electromagnetic inverse problems: emerging methods and novel applications, Inverse Problems, 26, Article No. 074008, 2010. [Abstract]

      Motivated from the application area of imaging of anti-personnel mines completely embedded in the homogeneous medium, the problem of non-iterative imaging of thin dielectric inclusions buried within a dielectric half-space is considered. For that purpose, a non-iterative imaging algorithm operated at several frequencies is proposed. It is based on the asymptotic expansion formula of the scattering amplitude in the presence of the inclusions. Various numerical examples illustrate how the method behaves.

    7. Won-Kwang Park, On the imaging of thin dielectric inclusions via topological derivative concept, Progress in Electromagnetics Research, 110, 237-252, 2010. [Abstract]

      In this paper, we consider the imaging of thin inclusions with dielectric contrast with respect to the embedding homogeneous domain. To image such inclusion from boundary measurements, topological derivation concept is adopted. For that purpose, an asymptotic expansion of the boundary perturbations that are due to the presence of a small inclusion is considered. Applying this formula, we can design only one iteration procedure for imaging of thin inclusions by means of solving adjoint problem. Various numerical experiments without and with some noise show how the proposed techniques behave.


    8. Habib Ammari, Josselin Garnier, Hyeonbae Kang, Won-Kwang Park and Knut Sølna, Imaging schemes for perfectly conducting cracks, SIAM Journal on Applied Mathematics, 71 (1), 68-91, 2011. [Abstract]

      We consider the problem of locating perfectly conducting cracks and estimating their geometric features from multi-static response matrix measurements at a single or multiple frequencies. A main objective is to design specific crack detection rules and to analyze their receiver operating characteristics and the associated signal-to-noise ratios. In this paper we introduce an analytic framework that uses high-frequency asymptotic methods in combination with a hypothesis test based formulation to construct specific procedures for detection of perfectly conducting cracks. A central ingredient in our approach is the use of random matrix theory to characterize the signal space associated with the multi-static response matrix measurements. We present numerical experiments to illustrate some of our main findings.

    9. Won-Kwang Park and Dominique Lesselier, Fast electromagnetic imaging of thin inclusions in half-space affected by random scatterers, Special issue on imaging in complex media, Waves in Random and Complex Media, 22 (1), 3-23, 2012. [Abstract]

      We consider an inverse scattering problem for finding penetrable thin electromagnetic inclusions completely embedded in a half-space affected by random scatterers. A non-iterative algorithm for retrieving the shape of such inclusions is presented. It is based on the fact that collected Multi-Static Response (MSR) matrix data can be modeled via a rigorous asymptotic expansion formula of the scattering amplitude in the presence of the inclusions. Various numerical implementations exhibit that presented algorithm performs satisfactorily for single and multiple thin inclusions, even with a fair amount of random scatterers.

    10. Won-Kwang Park, Topological derivative strategy for one-step iteration imaging of arbitrary shaped thin, curve-like electromagnetic inclusions, Journal of Computational Physics, 231 (4), 1426-1439, 2012. [Abstract]

      In this manuscript, a fast imaging of thin, curve-like electromagnetic inclusions completely hidden in the homogeneous domain with smooth boundaries is considered. By creating an electromagnetic inclusion of a small diameter and applying the asymptotic expansion formula in the existence of such an inclusion, the topological derivative is successfully derived. Based on this derivative, a one-step iteration imaging algorithm is designed by solving an adjoint problem. Various numerical experiments of single and multiple inclusions demonstrate the viability and limitation of the designed algorithm.


    11. Won-Kwang Park and Taehoon Park, Multi-frequency based direct location search of small electromagnetic inhomogeneities embedded in two-layered medium, Computer Physics Communications, 184 (7), 1649-1659, 2013. [Abstract]

      In this paper, we consider a problem for finding the locations of electromagnetic inhomogeneities completely embedded in homogeneous two layered medium. For this purpose, we present a filter function operated at several frequencies and design an algorithm for finding the locations of such inhomogeneities. It is based on the fact that the collected Multi-Static Response (MSR) matrix can be modeled via a rigorous asymptotic expansion formula of the scattering amplitude due to the presence of such inhomogeneities. In order to show the effectiveness, we compare the proposed algorithm with traditional MUltiple SIgnal Classification (MUSIC) algorithm and Kirchhoff migration. Various numerical results demonstrate that the proposed algorithm is robust with respect to random noise and yields more accurate location than the MUSIC algorithm and Kirchhoff migration.

    12. Won-Kwang Park, Multi-frequency topological derivative for approximate shape acquisition of curve-like thin electromagnetic inhomogeneities, Journal of Mathematical Analysis and Applications, 404 (2), 501-518, 2013. [Abstract]

      In this paper, we investigate a non-iterative imaging algorithm based on the topological derivative in order to retrieve the shape of penetrable electromagnetic inclusions when their dielectric permittivity and/or magnetic permeability differ from those in the embedding (homogeneous) space. The main objective is the imaging of crack-like thin inclusions, but the algorithm can be applied to arbitrarily shaped inclusions. For this purpose, we apply multiple time-harmonic frequencies and normalize the topological derivative imaging function by its maximum value. In order to verify its validity, we apply it for the imaging of two-dimensional crack-like thin electromagnetic inhomogeneities completely hidden in a homogeneous material. Corresponding numerical simulations with noisy data are performed for showing the efficacy of the proposed algorithm.

      Note: This paper is dedicated to professor Dominique Lesselier for his 60th birthday.

    13. Young Mi Kwon and Won-Kwang Park, Analysis of subspace migration in limited-view inverse scattering problems, Applied Mathematics Letters, 26 (12), 1107-1113, 2013. [Abstract]

      An analysis of subspace migration occurs in the limited-view inverse scattering problem is considered herein. Based on the structure of singular vectors associated with the nonzero singular values of Multi-Static Response (MSR) matrix, we establish a relationship between subspace migration imaging function and Bessel functions of integer order of the first kind. The revealed structure and numerical examples answer that why subspace migration is applicable for imaging of small scatterers in the limited-view inverse scattering problems.


    14. Won-Kwang Park, Analysis of a multi-frequency electromagnetic imaging functional for thin, crack-like electromagnetic inclusions, Applied Numerical Mathematics, 77, 31-42, 2014. [Abstract]

      Recently, a non-iterative multi-frequency subspace migration imaging algorithm was developed based on an asymptotic expansion formula for thin, curve-like electromagnetic inclusions and the structure of singular vectors in the Multi-Static Response (MSR) matrix. The present study examines the structure of subspace migration imaging functional and proposes an improved imaging functional weighted by the frequency. We identify the relationship between the imaging functional and Bessel functions of integer order of the first kind. Numerical examples for single and multiple inclusions show that the presented algorithm not only retains the advantages of the traditional imaging functional but also improves the imaging performance.

    15. Young-Deuk Joh, Young-Mi Kwon and Won-Kwang Park, MUSIC-type imaging of perfectly conducting cracks in limited-view inverse scattering problems, Applied Mathematics and Computation, 240, 273-280, 2014. [Abstract]

      Although standard MUltiple SIgnal Classification (MUSIC) algorithm has been considered a promising non-iterative imaging technique for cracks, its application is restricted to the full-view inverse scattering problems. Many experimental results revealed that MUSIC can be applied to the imaging of perfectly conducting cracks with small length in limited-view inverse scattering problem. On the other hand, MUSIC is not applicable for imaging of extended cracks in limited-view problems but the reason behind this restricted application has not been theoretically investigated. This gave an impetus for this study to attempt to identify the structure of MUSIC-type algorithm that appears in the limited-view inverse scattering problems by establishing a relationship between MUSIC and Bessel functions of integer order of the first kind. Some numerical experiments are illustrated to support the identified structure of MUSIC.

    16. Chi Young Ahn, Kiwan Jeon, Yong-Ki Ma and Won-Kwang Park, A study on the topological derivative-based imaging of thin electromagnetic inhomogeneities in limited-aperture problems, Inverse Problems, 30 (10), Article No. 105004, 2014. [Abstract]

      The topological derivative-based non-iterative imaging algorithm has demonstrated its applicability in limited-aperture inverse scattering problems. However, this has been confirmed through many experimental simulation results, and the reason behind this applicability has not been satisfactorily explained. In this paper, we identify the mathematical structure and certain properties of topological derivatives for the imaging of two-dimensional crack-like thin penetrable electromagnetic inhomogeneities that are completely embedded in a homogeneous material. To this end, we establish a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on the derived structure, we discover a necessary condition for applying topological derivatives in limited-aperture inverse scattering problems, and thus confirm why topological derivatives can be applied. Furthermore, we analyze the structure of multi-frequency topological derivative, and identify why this improves the single-frequency topological derivative in limited-aperture inverse scattering problems. Various numerical simulations are conducted with noisy data, and the results support the derived structure and exhibit certain properties of single- and multi-frequency topological derivatives.

    17. Young-Deuk Joh and Won-Kwang Park, Analysis of multi-frequency subspace migration weighted by natural logarithmic function for fast imaging of two-dimensional thin, arc-like electromagnetic inhomogeneities, Computers & Mathematics with Applications, 68 (12A), 1892-1904, 2014. [Abstract]

      The present study seeks to investigate mathematical structures of a multi-frequency subspace migration weighted by the natural logarithmic function for imaging of thin electromagnetic inhomogeneities from measured far-field pattern. To this end, we designed the algorithm and calculated the indefinite integration of square of Bessel function of order zero of the first kind multiplied by the natural logarithmic function. This is needed for mathematical analysis of improved algorithm to demonstrate the reason why proposed multi-frequency subspace migration contributes to yielding better imaging performance, compared to previously suggested subspace migration algorithm. This analysis is based on the fact that the singular vectors of the collected Multi-Static Response (MSR) matrix whose elements are the measured far-field pattern can be represented as an asymptotic expansion formula in the presence of such inhomogeneities. To support the main research results, several numerical experiments with noisy data are illustrated.


    18. Won-Kwang Park, Multi-frequency subspace migration for imaging of perfectly conducting, arc-like cracks in full- and limited-view inverse scattering problems, Journal of Computational Physics, 283, 52-80, 2015. [Abstract]

      Multi-frequency subspace migration imaging techniques are usually adopted for the non-iterative imaging of unknown electromagnetic targets, such as cracks in concrete walls or bridges and anti-personnel mines in the ground, in the inverse scattering problems. It is confirmed that this technique is very fast, effective, robust, and can not only be applied to full- but also to limited-view inverse problems if a suitable number of incidents and corresponding scattered fields are applied and collected. However, in many works, the application of such techniques is heuristic. With the motivation of such heuristic application, this study analyzes the structure of the imaging functional employed in the subspace migration imaging technique in two-dimensional full- and limited-view inverse scattering problems when the unknown targets are arbitrary-shaped, arc-like perfectly conducting cracks located in the two-dimensional homogeneous space. In contrast to the statistical approach based on statistical hypothesis testing, our approach is based on the fact that the subspace migration imaging functional can be expressed by a linear combination of the Bessel functions of integer order of the first kind. This is based on the structure of the Multi-Static Response (MSR) matrix collected in the far-field at nonzero frequency in either Transverse Magnetic (TM) mode (Dirichlet boundary condition) or Transverse Electric (TE) mode (Neumann boundary condition). The investigation of the expression of imaging functionals gives us certain properties of subspace migration and explains why multi-frequency enhances imaging resolution. In particular, we carefully analyze the subspace migration and confirm some properties of imaging when a small number of incident fields are applied. Consequently, we introduce a weighted multi-frequency imaging functional and confirm that it is an improved version of subspace migration in TM mode. Various results of numerical simulations performed on the far-field data affected by large amounts of random noise are similar to the analytical results derived in this study, and they provide a direction for future studies.

    19. Jung Ho Park and Won-Kwang Park, Localization of small perfectly conducting cracks from far-field pattern with unknown frequency, Applied Mathematics Letters, 43, 25-32, 2015. [Abstract]

      In inverse scattering problem, it is well-known that subspace migration yields very accurate locations of small perfectly conducting cracks when applied frequency is known. In contrast, when applied frequency is unknown, inaccurate locations are identified via subspace migration with wrong frequency data. However, this fact has been examined through the experimental results so, the reason of such phenomenon has not been theoretically investigated. In this paper, we analyze mathematical structure of subspace migration with unknown frequency by establishing a relationship with Bessel function of order zero of the first kind. Identified structure of subspace migration and corresponding results of numerical simulation answer that why subspace migration with unknown frequency yields inaccurate location of cracks and gives an idea of improvement.

    20. Won-Kwang Park, Asymptotic properties of MUSIC-type imaging in two-dimensional inverse scattering from thin electromagnetic inclusions, SIAM Journal on Applied Mathematics, 75 (1), 209-228, 2015. [Abstract]

      The main purpose of this paper is to study the structure of well-known non-iterative MUltiple SIgnal Classification (MUSIC) algorithm for identifying the shape of extended electromagnetic inclusions of small thickness located in the two-dimensional homogeneous space. We construct a relationship between MUSIC-type imaging functional for thin inclusions and Bessel functions of integer order of the first kind. Our construction is based on the structure of left-singular vectors of collected Multi-Static Response (MSR) matrix whose elements are measured far-field pattern and asymptotic expansion formula of in the existence of thin inclusions. Some numerical examples are shown to support constructed structure of MUSIC.

    21. Chi Young Ahn, Kiwan Jeon and Won-Kwang Park, Analysis of MUSIC-type imaging functional for single, thin electromagnetic inhomogeneity in limited-view inverse scattering problem, Journal of Computational Physics, 291, 198-217, 2015. [Abstract]

      This study analyzes the well-known MUltiple SIgnal Classification (MUSIC) algorithm to identify unknown support of thin penetrable electromagnetic inhomogeneity from scattered field data collected within the so-called multi-static response matrix in limited-view inverse scattering problems. The mathematical theories of MUSIC is partially discovered, e.g., in the full-view problem, for an unknown target of dielectric contrast or a perfectly conducting crack with the Dirichlet boundary condition (Transverse Magnetic--TM polarization) and so on. Hence, we perform further research to analyze the MUSIC-type imaging functional and to certify some well-known but theoretically unexplained phenomena. For this purpose, we establish a relationship between the MUSIC imaging functional and an infinite series of Bessel functions of integer order of the first kind. This relationship is based on the rigorous asymptotic expansion formula in the existence of a thin inhomogeneity with a smooth supporting curve. Various results of numerical simulation are presented in order to support the identified structure of MUSIC. Although \textit{a priori} information of the target is needed, we suggest a least condition of range of incident and observation directions to apply MUSIC in the limited-view problem.


    22. Won-Kwang Park, Performance analysis of multi-frequency topological derivative for reconstructing perfectly conducting cracks, Journal of Computational Physics, 335, 865-884, 2017. [Abstract]

      This paper concerns a fast, one-step iterative technique of imaging extended perfectly conducting cracks with Dirichlet boundary condition. In order to reconstruct the shape of cracks from scattered field data measured at the boundary, we introduce a topological derivative-based electromagnetic imaging functional operated at several nonzero frequencies. The structure of the imaging functionals is carefully analyzed by establishing relationships with infinite series of Bessel functions for the configurations of both symmetric and non-symmetric incident field directions. Identified structure explains why the application of incident fields with symmetric direction operated at multiple frequencies guarantees a successful reconstruction. Various numerical simulations with noise-corrupted data are conducted to assess the performance, effectiveness, robustness, and limitations of the proposed technique.

      Note: This paper is dedicated to professor Chang Bum Kim on the occasion of his retirement.

    23. Won-Kwang Park, Certain properties of MUSIC-type imaging functional in inverse scattering from an open, sound-hard arc, Computers & Mathematics with Applications, 74 (6), 1232-1245, 2017. [Abstract]

      This paper concerns mathematical formulation of well-known MUltiple SIgnal Classification (MUSIC)-type imaging functional in the inverse scattering problem by an open sound-hard arc. Based on the physical factorization of so-called Multi-Static Response (MSR) matrix and the structure of left-singular vectors liked to the non-zero singular values of MSR matrix, we construct a relationship between imaging functional and Bessel function of order 0, 1, and 2 of the first kind. We then expound certain properties of MUSIC and present numerical results for a number of differently chosen smooth arcs.

    24. Won-Kwang Park, Hwa Pyung Kim, Kwang-Jae Lee, and Seong-Ho Son, MUSIC algorithm for location searching of dielectric anomalies from S-parameters using microwave imaging, Journal of Computational Physics, 348, 259-270, 2017. [Abstract]

      Motivated by the biomedical engineering used in early-stage breast cancer detection, we investigated the use of MUltiple SIgnal Classification (MUSIC) algorithm for location searching of small anomalies using S-parameters. We considered the application of MUSIC to functional imaging where a small number of dipole antennas are used. Our approach is based on the application of Born approximation or physical factorization. We analyzed cases in which the anomaly is respectively small and large in relation to the wavelength, and the structure of the left-singular vectors is linked to the nonzero singular values of a Multi-Static Response (MSR) matrix whose elements are the S-parameters. Using simulations, we demonstrated the strengths and weaknesses of the MUSIC algorithm in detecting both small and extended anomalies.

    25. Won-Kwang Park, Appearance of inaccurate results in the MUSIC algorithm with inappropriate wavenumber, Journal of Inverse and Ill-Posed Problems, 25 (6), 807-817, 2017. [Abstract]

      MUltiple SIgnal Classification (MUSIC) is a well-known non-iterative location detection algorithm for small, perfectly conducting cracks in inverse scattering problems. However, when the applied wavenumbers are unknown, inaccurate locations of targets are extracted by MUSIC with inappropriate wavenumbers, a fact that has been confirmed by numerical simulations. To date, the reason behind this phenomenon has not been theoretically investigated. Motivated by this fact, we identify the structure of MUSIC-type imaging functionals with inappropriate wavenumbers by establishing a relationship with Bessel functions of order zero of the first kind. This result explains the reasons for inaccurate results. Various results of numerical simulations with noisy data support the identified structure of MUSIC.

    26. Won-Kwang Park, A novel study on subspace migration for imaging of a sound-hard arc, Computers & Mathematics with Applications, 74 (12), 3000-3007, 2017. [Abstract]

      In this study, the influence of a test vector selection used in subspace migration to reconstruct the shape of a sound-hard arc in a two-dimensional inverse acoustic problem is considered. In particular, a new mathematical structure of imaging function is constructed in terms of the Bessel functions of the order 0, 1, and 2 of the first kind based on the structure of singular vectors linked to the nonzero singular values of a Multi-Static Response (MSR) matrix. This structure indicates that imaging performance of subspace migration is highly related to the unknown shape of arc. The simulation results with noisy data indicate support for the derived structure.


    27. Won-Kwang Park, Topological derivative-based technique for imaging thin inhomogeneities with few incident directions, Inverse Problems in Science & Engineering, 26, 1490-1508, 2018. [Abstract]

      Many non-iterative imaging algorithms require a large number of incident directions. Topological derivative-based imaging techniques can alleviate this problem, but lacks a theoretical background and a definite means of selecting the optimal incident directions. In this paper, we rigorously analyze the mathematical structure of a topological derivative imaging function, confirm why a small number of incident directions is sufficient, and explore the optimal configuration of these directions. To this end, we represent the topological derivative based imaging function as an infinite series of Bessel functions of integer order of the first kind. Our analysis is supported by the results of numerical simulations.

      Note: This paper is dedicated to professor Jae-Ryong Kim on the occasion of his retirement.

    28. Won-Kwang Park, Direct sampling method for anomaly imaging from scattering parameter, Applied Mathematics Letters, 81, 63-71, 2018. [Abstract]

      In this paper, we develop a fast imaging technique for small anomalies located in homogeneous media from scattering parameter data measured at dipole antennas. Based on the representation of scattering parameters when an anomaly exists, we design a direct sampling method (DSM) for imaging an anomaly and establishing a relationship between the indicator function of DSM and an infinite series of Bessel functions of integer order. Simulation results using synthetic data at f=1GHz of angular frequency are illustrated to support the identified structure of the indicator function.

    29. Sangwoo Kang, Marc Lambert, and Won-Kwang Park, Direct sampling method for imaging small dielectric inhomogeneities: analysis and improvement, Inverse Problems, 34 Article No. 095005, 2018. [Abstract]

      The direct sampling method (DSM) has been introduced for non-iterative imaging of small inhomogeneities and is known to be fast, robust, and effective for inverse scattering problems. However, to the best of our knowledge, a full analysis of the behavior of the DSM has not been provided yet. Such an analysis is proposed here within the framework of the asymptotic hypothesis in the 2D case leading to the expression of the DSM indicator function in terms of the Bessel function of order zero and the sizes, shapes and permittivities of the inhomogeneities. Thanks to this analytical expression the limitations of the DSM method when one of the inhomogeneities is smaller and/or has lower permittivity than the others is exhibited and illustrated. An improved DSM is proposed to overcome this intrinsic limitation in the case of multiple incident waves. Then we show that both the traditional and improved DSM are closely related to a normalized version of the Kirchhoff migration. The theoretical elements of our proposal are supported by various results from numerical simulations with synthetic and experimental data.

    30. Won-Kwang Park, Reconstruction of thin electromagnetic inhomogeneity without diagonal elements of a Multi-Static Response matrix, Inverse Problems, 34, Article No. 095008, 2018. [Abstract]

      This paper aims to shape the identification of thin inhomogeneities with different dielectric/magnetic properties from a two-dimensional homogeneous background. The shapes are identified through subspace migration without requiring the diagonal elements of the collected multi-static response matrix. To understand why subspace migration without diagonal elements can retrieve the shape of a thin inhomogeneity, we carefully investigate the relations between the imaging function and Bessel functions of orders 0 and 1. This analysis exploits the fact that when a thin homogeneity exists, the measured far-field pattern can be represented as an asymptotic expansion formula. The investigated relation is supported in numerical experiments with noisy data.

    31. Won-Kwang Park, Direct sampling method for retrieving small perfectly conducting cracks, Journal of Computational Physics, 373, 648-661, 2018. [Abstract]

      In this paper, direct sampling method is considered for determining the location of a set of small, linear perfectly conducting cracks from the collected far-field data corresponding to an incident field. To show the feasibility of the direct sampling method, this study proves that the indicator function of the direct sampling method can be represented by the Bessel function of order zero and the crack lengths. The results of the numerical simulations are shown to support the fact that the imaging performance is highly dependent on the crack lengths. To explain the fact that the imaging performance is highly dependent on the rotation of the cracks, the direct sampling method is further analyzed by establishing a representation using Bessel functions of orders zero and one. Based on the derived representation of indicator function, we design improved direct sampling methods by applying incident fields with multiple directions and multiple frequencies. Corresponding analysis of indicator functions and simulation results are shown for demonstrating the effectiveness and improvements.


    32. Won-Kwang Park, Real-time microwave imaging of unknown anomalies via scattering matrix, Mechanical Systems and Signal Processing, 118, 658-674, 2019. [Abstract]

      We consider an inverse scattering problem to identify the locations or shapes of unknown anomalies from scattering parameter data collected by a small number of dipole antennas. Most of researches does not considered the influence of dipole antennas but in the experimental simulation, they are significantly affect to the identification of anomalies. Moreover, opposite to the theoretical results, it is impossible to handle scattering parameter data when the locations of the transducer and receiver are the same in real-world application. Motivated by this, we design an imaging function with and without diagonal elements of the so-called scattering matrix. This concept is based on the Born approximation and the physical interpretation of the measurement data when the locations of the transducer and receiver are the same and different. We carefully explore the mathematical structures of traditional and proposed imaging functions by finding relationships with the infinite series of Bessel functions of integer order. The explored structures reveal certain properties of imaging functions and show why the proposed method is better than the traditional approach. We present the experimental results for small and extended anomalies using synthetic and real data at several angular frequencies to demonstrate the effectiveness of our technique.

    33. Sangwoo Kang, Marc Lambert, and Won-Kwang Park, Analysis and improvement of direct sampling method in the mono-static configuration, IEEE Geoscience and Remote Sensing Letters, 16 (11), 1721-1725, 2019. [Abstract]

      The recently introduced non-iterative imaging method entitled direct sampling method (DSM) is known to be fast, robust, and effective for inverse scattering problems in the multi-static configuration but fails when applied to the mono-static one. To the best of our knowledge no explanation of this failure has been provided yet. Thanks to the framework of the asymptotic and the far-field hypothesis in the 2D scalar configuration an analytical expression of the DSM indicator function in terms of the Bessel function of order zero and sizes, shapes and permittivities of the inhomogeneities is obtained and the theoretical reason of the limitation identified. A modified version of DSM is then proposed in order to improve the imaging method. The theoretical results are supported by numerical results using synthetic data.

    34. Seong-Ho Son, Kwang-Jae Lee, and Won-Kwang Park, Application and analysis of direct sampling method in real-world microwave imaging, Applied Mathematics Letters, 96, 47-53, 2019. [Abstract]

      In this paper, a direct sampling method (DSM) is designed for a real-time detection of small anomalies from scattering parameters measured by a small number of dipole antennas. Applicability of the DSM is theoretically demonstrated by proving that its indicator function can be represented in terms of an infinite series of Bessel functions of integer order, Hankel function of order zero, and the antenna configurations. Experiments using real-data then demonstrate both the effectiveness and limitations of this method.

    35. Won-Kwang Park, Fast imaging of short perfectly conducting cracks in limited-aperture inverse scattering problem, Special issue on microwave imaging and its application, Electronics, 8 (9), Article No. 1050, 2019. [Abstract]

      In this contribution, we consider the application and analysis of subspace migration technique for a fast imaging of a set of perfectly conducting cracks with small length in two-dimensional limited-aperture inverse scattering problem. In particular, an imaging function of subspace migration with asymmetric multi-static response matrix is designed and its new mathematical structure is constructed in terms of an infinite series of Bessel functions and the range of incident and observation directions. This is based on the structure of left and right singular vectors linked to the nonzero singular values of MSR matrix and asymptotic expansion formula due to the existence of cracks. Investigated structure of imaging function indicates that imaging performance of subspace migration is highly related to the range of incident and observation directions. The simulation results with synthetic data polluted by random noise are exhibited to support investigated structure.


    36. Chi Young Ahn, Seongje Chae, and Won-Kwang Park, Fast identification of short, sound-soft open arcs via orthogonality sampling method in limited-aperture inverse scattering problem, Applied Mathematics Letters, 109, Article No. 106556, 2020. [Abstract]

      The orthogonality sampling method (OSM) is a recently developed non-iterative technique for imaging and identifying targets in inverse scattering problems. In this paper, a set of short sound-soft open arcs is obtained by OSM in the limited-aperture inverse scattering problem. We introduce an indicator function of OSM and explore its mathematical structure. In this formulation, the identification largely depended on the arc lengths, the range of observation directions, and the propagation direction of the plane-wave incident field. The explored structure was verified in numerical results on synthetic data corrupted by random noise.

    37. Sangwoo Kang, Marc Lambert, Chi Young Ahn, Taeyoung Ha, and Won-Kwang Park, Single- and multi-frequency direct sampling methods in limited-aperture inverse scattering problem, IEEE Access, 8, 121637-121649, 2020. [Abstract]

      Although the direct sampling method (DSM) has demonstrated its feasibility and robustness for imaging of small inhomogeneities, mathematical analyses of DSM have been conducted only on the full-aperture inverse scattering problem. Numerous studies have shown that DSM can also be applied in the limited-aperture inverse scattering problem, but most of its applications are still heuristic. This study considers an application, mathematical analysis, and improvement of DSM with a single-incident field only in the limited-aperture inverse scattering problem. First, we introduce a traditional indicator function of DSM at a single frequency, establish its mathematical structure, and examine its inherent limitation. To demonstrate the theoretical result, various results of numerical simulations with synthetic and experimental data are presented. Next, we consider the multi-frequency indicator function of DSM with a single-incident direction to improve imaging performance. For this, we design a multi-frequency indicator function of MDSM, analyze its mathematical structure, and theoretically explain the improvement of the imaging of single inhomogeneity and the limitation on the identification of multiple inhomogeneities. Various numerical simulations with synthetic and experimental data are presented to validate our results.

    38. Seongje Chae, Chi Young Ahn, and Won-Kwang Park, Localization of small anomalies via orthogonality sampling method from scattering parameters, Electronics, Special issue on photonic and microwave sensing developments and applications, 9 (7), Article No. 1119, 2020. [Abstract]

      We investigate the application of orthogonality sampling method (OSM) in microwave imaging for a fast localization of small anomalies from measured scattering parameters. For this purpose, we design an indicator function of OSM defined on Lebesgue space to test the orthogonality relation between the Hankel function and the scattering parameters. This is based on an application of the Born approximation and the integral equation formula for scattering parameters in the presence of a small anomaly. We then prove that the indicator function consists of a combination of an infinite series of Bessel functions of integer order, an antenna configuration, and material properties. Simulation results with synthetic data are presented to show the feasibility and limitations of designed OSM.

    39. Chi Young Ahn, Taeyoung Ha, and Won-Kwang Park, Direct sampling method for identifying magnetic inhomogeneities in limited-aperture inverse scattering problem, Computers & Mathematics with Applications, 80 (12), 2811-2829, 2020. [Abstract]

      The direct sampling method (DSM) in limited-aperture inverse scattering problems for transverse electric (TE) polarization is considered. Based on the asymptotic expansion formula in the presence of small targets, we demonstrate that the indicator function of DSM can be represented by an infinite series of Bessel functions of integer order and correspondingly examine various properties of DSM. Simulation results with noisy data are created to support theoretical results. Based on the identified structure of the indicator function, we design a new indicator function of DSM, analyze its mathematical structure, and generate simulation results to demonstrate its effectiveness and improvement.


    40. Won-Kwang Park, Application of MUSIC algorithm in real-world microwave imaging of unknown anomalies from scattering matrix, Mechanical Systems and Signal Processing, 153, Article No. 107501, 2021. [Abstract]

      In this contribution, we consider MUltiple SIgnal Classification (MUSIC)-type algorithm for a non-iterative microwave imaging of small and arbitrary shaped extended anomalies located in a homogeneous media from scattering matrix whose elements are scattering parameters measured at dipole antennas. In order to explain the feasibility of MUSIC in microwave imaging, we investigate mathematical structure of MUSIC by establishing a relationship with an infinite series of Bessel function of integer order and antennas setting. This is based on the representation formula of scattering parameters in the presence of small anomalies and the application of Born approximation. Simulation results using real-data at f=925MHz of angular frequency are exhibited to show the feasibility of designed algorithm and to support investigated structure of imaging function.

      Note: This paper is dedicated to Professor Jin Keun Seo on the occasion of his 60th birthday.


    Miscellaneous

    1. You can find list of publications of mine in MathSciNet.
    2. You can find my articles including preprints in arXiv.
    3. You can find my information in Google Scholar and ORCID.
    4. You can see My Mathematical Genealogy.

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