CAREER: Computing Information in Image Processing and Stochastic Differential Equations

这项研究的目标是为两个领域的问题发展新的方法和他们的数学理论：数字图像处理，以及物理和工程系统中的随机摄动微分方程。共同的主题是计算和提取嵌入或隐藏在问题中的所需信息，并将其用于应用程序。在图像处理方面，研究人员和他的同事们提出了一种新的基于跨通道信息的多通道图像去噪策略。将几何偏微分方程(PDE)、多分辨率调和分析和变分等高级数学工具与一些统计方法和计算机视觉理论(如颜色空间)相结合，在去除图像噪声的同时保留边缘和角点等显著的几何特征。研究人员和合作者还研究了图像处理中基于小波的偏微分方程技术的严格误差分析理论。在随机微分方程中，研究人员和同事分析了电振荡器的相位噪声和时间抖动，包括模数转换(ADC)模型。关键是使用基于矢丛理论的移动坐标系来完全分解相位噪声和幅度噪声，然后研究相关的Fokker-Planck方程，该方程将确定性的统计特性如均值和方差从随机性中分离出来。基于Fokker-Planck方程设计了一种新的数值方法来计算Shannon熵，该方法可以用来评估振荡器的性能。计算信息已经成为许多科学和技术领域发展最快的方面之一。例如，数字图像处理从数字图像中分析和提取有用的信息。许多图像，包括卫星、雷达或声纳图像和医学图像，在获取时都会受到空气、水、照明条件和镜头上的灰尘等环境噪声的污染，或者在无线通信等传输过程中受到损坏。图像去噪是应用中最重要的任务之一。这项研究的一个关键目标是设计新的策略来去除图像中的噪声，同时恢复重要和有用的信息，如边缘和形状，这些信息通常很难从噪声中分离出来。随机微分方程常用于描述具有不确定性的复杂物理或工程系统。例如复合材料、湍流、电路设计和光学。例如，电振荡器是许多电子设备(如天线)中使用的关键电路，并且通常由随机微分方程组建模。在无线通信中，引起信道干扰的相位噪声是振荡器设计的重要因素之一。这项研究的目的之一是发展数学理论和方法来分析和计算振荡器中随机过程的有用统计信息，以便用于设计或评估振荡器的性能。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 goal of this research is to develop new methodologies and their mathematical theory for problems in two areas: digital image processing, and stochastically perturbed differential equations in physical and engineering systems. The common theme is computing and extracting desired information embedded or hidden in the problems and use it in the applications. In image processing, the investigator and his colleagues develop a novel multi-channel image denoising strategy based on cross channel information. Several high level mathematical tools including geometrical partial differential equations (PDE's), multiresolution harmonic analysis and calculus of variation are integrated together with some statistical methods and computer vision theory such as color spaces to remove noise from images while retaining salient geometrical features such as edges and corners. The investigator and collaborators also study a rigorous error analysis theory for wavelet based PDE techniques in image processing. In stochastic differential equations, the investigator and colleagues analyze the phase noise and time jitter for electric oscillators including an Analog Digital Conversion (ADC) model. The key is to use a moving coordinate system based on a vector bundle theory to completely decompose the phase noise and amplitude noise, and then study the associated Fokker-Planck equations which separate the deterministic statistical properties such as mean and variance from the randomness. A novel numerical method based on the Fokker-Planck equations is designed to compute Shannon's entropy which can be used to evaluate the performance of the oscillators.Computing information has become one of the fastest growing aspects in many areas of science and technology. For instance, digital image processing analyzes and extracts useful information from digital images. Many images, including satellite, radar or sonar images and medical images, are polluted by noise from the environments like air, water, lighting conditions and dust on lens when they are acquired, or damaged during transmission processes such as wireless communications. Image denoising, which removes the noise, becomes one of the most important tasks in applications. A key objective in this research is to design new strategies removing noise in images while restoring important and useful information such as edges and shapes, which are often hard to be separated from the noise. Stochastic differential equations are commonly used to describe complicated physical or engineering systems with uncertainties. Examples include composite materials, turbulence, circuit design and optics. For instance, electric oscillators are the key circuits used in many electric devices such as antennas, and are often modeled by systems of stochastic differential equations. Phase noise, which causes channel interference in wireless communications, is one of the most important factors for designing oscillators. One objective of this study is to develop mathematical theory and methods to analyze and compute useful statistical information from random processes in oscillators so that they can be used in designing or evaluating the performance of oscillators. 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.