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

ATD：协作研究：逆源问题和信号分析的多尺度随机方法

• 批准号: 1042958
• 负责人: Peijun Li
• 金额: $ 13.89万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-10-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1042958&HistoricalAwards=false
• 关键词: ATD; Collaborative; Research; Multiscale; Stochastic

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.

拟议工作的主要目标是为遥感及其在化学和生物威胁探测中的应用开发新的模型、计算方法和相关的数学理论。在这些应用中，数据通常由光学传感器收集，然后进行处理，以重建或分析源的性质，如化学或生物羽流。由于面临许多挑战，这些任务是困难的。例如，问题是不适定的，源函数通常是随机的，数据是噪声和不完整的。PI和合作者建议研究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反随机源问题。这些问题广泛存在于光学、遥感和医学成像等领域。他们的目标是开发新的有效策略，从不完整的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，他们将基于小波的多尺度方法与基于偏微分方程的非局部平均方法相结合，用于三维激光雷达图像的去噪和信息提取。该方法综合了几何偏微分方程组、多尺度小波变换和变分等高级数学工具，结合激光雷达图像的一些特殊性质，以快速的计算取得了较好的结果。在数据分析阶段，PI和合作者研究了一种新的基于希尔伯特变换和经验模式分解(EMD)的非线性解混方法用于信号分析。EMD被设计用于处理非线性和非平稳信号，而传统的基于小波或傅立叶的方法不能很容易地处理这些信号。通过使用EMD，它们可以通过瞬时频率分析等技术提取有用但隐藏的信息。由于国土安全、军事、地球科学、医学和工程等多个学科的新挑战，遥感技术得到了前所未有的关注。特别是，它们已经成为在遥不可及、不友好或危险的环境中收集数据的主要工具之一。例如，化学或生物威胁检测技术的最新进展是使用激光束和光学传感器来收集来自目标的信号，例如气溶胶羽流。然后对收集的数据进行处理，以识别有害物质。成功的关键一步是从收集的数据集中确定来源的材料性质，例如是否存在某些化学或生物制剂。这就需要解决所谓的逆问题。在实践中，由于许多问题，它们是具有挑战性的。采集的数据往往是不完整的、随机的和有噪声的，气溶胶羽流太厚而看不到中心部分的信号，有害物质和正常气溶胶粒子的特征混合在一起，很难分离。在这项提案中，私人投资机构重点研究在化学和生物威胁探测中应用的遥感技术的最新进展的三个方面。他们的目标是发展新的、稳健和高效的计算方法和相关的数学理论，以解决不完整数据集中的逆随机源问题，从信号中去除噪声，并分离不同气溶胶颗粒的特征，以便从信号中容易地识别有害物质。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。