The main goal of the proposed work is to develop new models, computational methodologies and related mathematical theory for remote sensing with applications in chemical and biological threat detections. In those applications, data is usually gathered by optical sensors and then processed to reconstruct or analyze the properties of sources, such as chemical or biological plume. The tasks are difficult due to a number of challenges. For examples, the problems are ill-posed, the source functions are often random in nature and the data is noisy and incomplete. The PIs and collaborators proposed to investigate three different, but closely related, aspects of some newly emergent remote sensing techniques. In data acquisition, they work on inverse random source problems for the Helmholtz equation. Such problems exist in a wide range of applications in optical science, remote sensing and medical imaging. They aim to develop novel and efficient strategies to reconstruct the distributions of random source functions from incomplete boundary data measurements and perform uncertainty assessments. In data processing, they develop wavelet based multiscale methods in conjunction with the PDE based non-local mean methods for image denoising and information extraction of 3-D Lidar images. The methods integrate several high level mathematical tools, such as geometrical partial differential equations (PDE's), multiscale wavelet transforms and calculus of variation, together with some special properties of the Lidar imagery to achieve better results with fast computations. In data analysis stage, the PIs and collaborators study a novel nonlinear de-mixing method based on Hilbert transform and empirical mode decomposition (EMD) for signal analysis. EMD are designed to handle nonlinear and non-stationary signals, which cannot be easily processed by the traditional wavelet or Fourier based methods. By using EMD, they can extract useful but hidden information through techniques such as instantaneous frequency analysis.
Remote sensing techniques have gained unprecedent attentions due to new challenges in many disciplines including homeland security, military, geosciences, medical science and engineering. Specially, they have become one of the primary tools for data collections in unreachable, unfriendly or hazardous environments. For instance, a most recent advance in chemical or biological threat detection technology uses laser beams and optical sensors to collect signals from targets, such as aerosol plumes. Then the gathered data is processed to identify harmful agents. A key step to succeed is to determine the material properties of the sources, such as whether there exist certain chemical or biological agents, from the collected data sets. This requires solving the so called inverse problems. In practice, they are challenging due to a number of issues. The collected data is often incomplete, random and noisy, the aerosol plume is too thick to ``see'' signals from the center parts , the signatures of the harmful agents and normal aerosol particles are mixed and hard to be separated. In this proposal, the PIs focus their studies in three aspects of the most recent advances in remote sensing techniques with applications in chemical and biological threat detections. They aim to develop novel, robust and efficient computational methods and related mathematical theory to solve the inverse random source problems from incomplete data sets, to remove noise from the signals, and to separate the signatures of different aerosol particles so that harmful agents can be easily identified from the signals. In addition, another major objective is to integrate the research activities with education and training of undergraduate, graduate students and postdocs through seminars and courses.
Scattering problems concerns with the effect that an inhomogeneous medium has on an incident wave. If the total field is viewed as the sum of an incident field and a scattered field, the direct scattering problem is to determine the scattered field from a knowledge of the incident field and the differential equation governing the wave motion; the inverse scattering problem is to determine the nature of the inhomogeneity, such as geometry, location, and material property, from a knowledge of the scattered field. These problems are basic in many scientific areas such as radar and sonar, geophysical exploration, and medical imaging. The major goals of the project are to develop mathematical models, examine mathematical issues, and design computational methods for random source and related scattering problems. This is part of a long-term research effort to develop effective mathematical models and novel computational algorithms for new and important classes of direct and inverse scattering problems that arise from wave propagation in complex and random environments. Specifically, the mathematical studies and numerical methods mainly addressed two classes of direct and inverse scattering problems: (1) a random source scattering problem; (2) a multiple multiscale scale scattering problem. The inverse source scattering problem is largely motivated by many medical applications such as reflection tomography, diffusion-based optical tomography, and fluorescence microscopy. The PI developed a novel method to reconstruct the statistical properties the random source functions, such as the mean and variance, which help to characterize the uncertainties involved in the scattering sources. To model the wave propagation in a cluttered medium, the PI considered the scattering of a time-harmonic plane wave incident on a two-scale heterogeneous medium, which consists of point scatterers and extended scatterers. The PI developed an efficient generalized Foldy-Lax formulation to fully capture the multiple scattering among all the scatterers, and developed a simple direct imaging method to simultaneously reconstruct both the point scatterers and the extended scatterers. The research results in efficient modeling and computational techniques, suitable for qualitative and quantitative study for wave propagation in complex and random environments. The developed computational models and tools could provide an inexpensive and easily controllable virtual prototype of the structures in the design and fabrication of optical and electromagnetic devices. During the life of the award, two female graduate students were supported and two new advanced graduate courses were developed. The research results were published in peer-reviewed journals and were presented in invited seminars, workshops, and conferences.
