ATD: Collaborative Research: Multiscale and Stochastic Methods for Inverse Source Problems and Signal Analysis

拟议工作的主要目标是发展新的模型、计算方法和有关的数学理论，以便遥感应用于化学和生物威胁探测。在这些应用中，数据通常由光学传感器收集，然后进行处理以重建或分析源的特性，例如化学或生物羽流。由于许多挑战，任务很困难。例如，问题是不适定的，源函数通常是随机的，数据是有噪声的和不完整的。pi和合作者建议调查一些新出现的遥感技术的三个不同但密切相关的方面。在数据采集方面，他们研究亥姆霍兹方程的逆随机源问题。这些问题在光学科学、遥感和医学成像的广泛应用中都存在。他们的目标是开发新颖有效的策略，从不完全边界数据测量中重建随机源函数的分布，并进行不确定性评估。在数据处理方面，他们开发了基于小波的多尺度方法，结合基于PDE的非局部平均方法，用于三维激光雷达图像的图像去噪和信息提取。该方法将几何偏微分方程（PDE）、多尺度小波变换、变分法等高级数学工具与激光雷达图像的一些特性相结合，计算速度快，效果好。在数据分析阶段，pi及其合作者研究了一种基于希尔伯特变换和经验模态分解（EMD）的非线性混频分析方法。EMD设计用于处理非线性和非平稳信号，而传统的小波或基于傅里叶的方法难以处理这些信号。通过使用EMD，他们可以通过瞬时频率分析等技术提取有用但隐藏的信息。由于在国土安全、军事、地球科学、医学和工程等许多学科中面临的新挑战，遥感技术得到了前所未有的重视。特别是，它们已成为在难以到达、不友好或危险环境中收集数据的主要工具之一。例如，化学或生物威胁探测技术的最新进展使用激光束和光学传感器从目标（如气溶胶羽流）收集信号。然后对收集到的数据进行处理，以确定有害物质。成功的关键一步是确定来源的材料特性，例如是否存在某些化学或生物制剂，从收集的数据集。这需要解决所谓的逆问题。在实践中，由于一些问题，它们具有挑战性。收集到的数据往往不完整、随机和有噪声，气溶胶羽流太厚而无法从中心部分“看到”信号，有害物质和正常气溶胶颗粒的特征混合在一起，难以分离。在这一建议中，pi将研究重点放在遥感技术在化学和生物威胁探测方面的最新进展的三个方面。他们的目标是开发新颖，稳健和高效的计算方法和相关的数学理论来解决不完整数据集的逆随机源问题，从信号中去除噪声，并分离不同气溶胶颗粒的特征，以便从信号中轻松识别有害物质。此外，另一个主要目标是通过研讨会和课程将研究活动与本科生，研究生和博士后的教育和培训结合起来。

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.