We consider the acoustic source imaging problems in using multiple frequency data at sparse directions/points. Using at most n observation directions/points, we show that the size and location of the source can be recovered. A non-iterative method is then proposed to image the source. The method is simple to implement and extremely fast since it only computes an indicator function on the domain of interest using matrix vector multiplications. Numerical examples are presented to validate the effectiveness of the method.

Detection of human activities in complex environments such as through wall by ultrawideband radar has many important applications in security, vital rescue, and so on. It is much more difficult to detect vital signs of moving human beings than static ones. In this letter, we build a model for moving targets and apply the time domain finite element method to simulate single-input multiple-outputs (SIMO) radar data. Human respiration is modeled by changing body size and physical parameters. The background removal is performed for radar data. Then, we use the back projection to reconstruct the consecutive target locations, which constitute the moving path, leading to a curve carrying vital signs in the radar image. Since SIMO radar data are multivariate, we use multivariate empirical mode decomposition (MEMD) and fast Fourier transform to separate and extract the respiratory characteristic frequencies. The reconstructed frequency coincides with that in the original model. The result shows that the combination of SIMO radar and MEMD can effectively identify the moving path of the human being behind the wall and extract vital signs.

We present an iterative method to compute the Maxwell’s transmission eigenvalue problem which has importance in non-destructive testing of anisotropic materials. The transmission eigenvalue problem is first written as a quad-curl eigenvalue problem. Then we show that the real transmission eigenvalues are the roots of a nonlinear function whose value is the generalized eigenvalue of a related self-adjoint quad-curl eigenvalue problem which is computed using a mixed finite element method. A secant method is used to compute the roots of the nonlinear function. Numerical examples are presented to validate the method. Moreover, the method is employed to study the dependence of the transmission eigenvalue on the anisotropy and to reconstruct the index of refraction of an inhomogeneous medium.