A Direct Imaging Method Using Far Field Data * Songming Hou † Knut Solna ‡ Hongkai Zhao ‡ January 23, 2007 Abstract We present a direct imaging algorithm for extended targets using far field data generated by incident plane waves. The algorithm uses a factorization of the response matrix for far field data that derives from physical considerations, moreover, a res- olution based regularization. The algorithm is simple and efficient since no forward solver or iteration is needed. Efficiency and robustness of the algorithm with respect to measurement noise are demonstrated. Keywords: far field, singular value decomposition, response matrix, MUSIC. 1 Introduction Probing a medium using incident plane waves and recording far field pattern is a classical inverse scattering problem that has been studied in depth. The objective is to find the location and geometry of the targets using the far field pattern of the scattering operator, that is, using the relation between incident plane waves and scattered outgoing plane waves. There are essentially two type of methods that have been presented to solve such an inverse problem: iterative methods and direct methods. Iterative methods treat the inverse problem as a nonlinear optimization problem. Usually, for each iteration, an adjoint forward problem needs to be solved. Direct method gives a characterization/visualization of the geometry by designing an imaging function that peaks near the target boundary. For example, the MUltiple SIgnal Classification (MUSIC) algorithm [6, 8, 10, 14, 15] is a direct imaging function which can locate small targets using an array of transducers that can send and receive signals. The MUSIC algorithm is generalized in [9] to image the shape of extended targets for near field data. In this paper, we generalize the MUSIC algorithm to solve the inverse scattering problem for far field data. This method is a direct method that is very efficient and robust. It can be parallelized easily since the evaluation at different search points are independent. The linear sampling method, first proposed in [4], is also a direct imaging algorithm for the inverse scattering problem. The method is based on a characterization of the range of the scattering operator for the far field pattern. It is shown that the far field pattern of a point source located inside the object is in the range of the scattering operator. Kirsch presents a factorization of the scattering operator in [11] and uses this factorization for imaging. The relation between the MUSIC and the linear sampling method is studied in [2, 12]. A good * The research is partially supported by ONR grant N00014-02-1-0090, DARPA grant N00014-02-1-0603, NSF grant 0307011 and the Sloan Foundation. † Dept of Math, Michigan State Univ., East Lansing, MI, 48824, mickey@math.msu.edu ‡ Dept of Math, Univ. of Cal. at Irvine, CA, 92697. ksolna, zhao@math.uci.edu 1