We present a method for imaging small scatterers in a homogeneous medium from polarization measurements of the electric field at an array. The electric field comes from illuminating the scatterers with a point source with known location and polarization. We view this problem as a generalized phase retrieval problem with data being the coherency matrix or Stokes parameters of the electric field at the array. We introduce a simple preprocessing of the coherency matrix data that partially recovers the ideal data where all the components of the electric field are known for different source dipole moments. We prove that the images obtained using an electromagnetic version of Kirchhoff migration applied to the partial data are, for high frequencies, asymptotically identical to the images obtained from ideal data. We analyze the image resolution and show that polarizability tensor components in an appropriate basis can be recovered from the Kirchhoff images, which are tensor fields. A time domain interpretation of this imaging problem is provided and numerical experiments are used to illustrate the theory.
We introduce a new approach for narrow band array imaging of localized scatterers from intensity-only measurements by considering the possibility of reconstructing the positions and reflectivities of the scatterers exactly from only partial knowledge of the array data, since we assume that phase information is not available. We reformulate this intensity-only imaging problem as a non-convex optimization problem and show that we can have exact recovery by minimizing the rank of a positive semidefinite matrix associated with the unknown reflectivities. Since this optimization problem is NP-hard and is computationally intractable, we replace the rank of the matrix by its nuclear norm, the sum of its singular values, which is a convex programming problem that can be solved in polynomial time. Numerical experiments explore the robustness of this approach, which recovers sparse reflectivity vectors exactly as solutions of a convex optimization problem.   Abstract We consider the problem of determining a potential V(x) in the one-dimensional Schrodinger equation, given as data the reflectivity r(k) = | R(k) | 2 , where R(k) denotes the usual quantum mechanical reflection coefficient. It is well known that in the absence of phase information, there can be a considerable degree of nonuniqueness, which is closely connected to the presence of zeros of R(k) in the upper half of the complex plane. Some earlier work of the authors showed that this ambiguity can be resolved by providing a small amount of extra information about the potential. In this article we develop a computational technique, based on an optimization approach to the problem of locating the zeros of R(k) . Some numerical examples are given. We will construct new classes of Parseval frames for a Hilbert space which allow signal reconstruction from the absolute value of the frame coefficients. As a consequence, signal reconstruction can be done without using phase or its estimation. This verifies a longstanding conjecture of the speech processing community. Two related problems are considered: (i) the inverse scattering problem for a potential V(x) supported on the half‐line x≥0 , when the given data is ‖R−(k)‖, the amplitude of the reflection coefficient and (ii) determination of a function g(t) supported on the half‐line t≥0 when the given data is ‖g(k)‖, the amplitude of the Fourier transform of g. Under certain conditions on V or g, uniqueness theorems are proved and computational methods are developed. A numerical example of recovery of V(x) from ‖R−(k)‖ is given. An algorithm is presented for the rapid solution of the phase of the complete wave function whose intensity in the diffraction and imaging planes of an imaging system are known. A proof is given showing that a defined error between the estimated function and the correct function must decrease as the algorithm iterates. The problem of uniqueness is discussed and results are presented demonstrating the power of the method.  AbstractUniqueness is proven for two non-overdetermined 3-d inverse problems of the determination of the spatially distributed sound speed in the frequency dependent acoustic PDE. The main new point is the assumption that only the modulus of the scattered complex valued wave field is measured on a certain set. The phase is unknown.