CAREER: Computing Information in Image Processing and Stochastic Differential Equations

本研究的目标是开发新的方法和数学理论的问题在两个领域：数字图像处理，和随机扰动微分方程在物理和工程系统。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。在图像处理中，研究人员和他的同事开发了一种新的基于交叉通道信息的多通道图像去噪策略。几个高层次的数学工具，包括几何偏微分方程（PDE），多分辨率谐波分析和变分法与一些统计方法和计算机视觉理论，如颜色空间，以消除图像中的噪声，同时保留突出的几何特征，如边缘和角落。研究者和合作者还研究了图像处理中基于小波的PDE技术的严格误差分析理论。在随机微分方程中，研究人员及其同事分析了包括模数转换（ADC）模型在内的电子振荡器的相位噪声和时间抖动。其关键是使用基于矢量束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，该方程将确定性统计特性（如均值和方差）与随机性分离。基于Fokker-Planck方程设计了一种计算Shannon熵的新方法，该方法可以用来评价振荡器的性能。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，在获取时被来自环境（如空气、水、照明条件和透镜上的灰尘）的噪声污染，或者在诸如无线通信的传输过程中被损坏。图像去噪是去除图像中的噪声，是图像处理中最重要的任务之一.本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，这些信息通常很难与噪声分离。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。例子包括复合材料，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系统建模。相位噪声是无线通信中引起信道干扰的主要因素之一，也是设计振荡器时必须考虑的重要因素。本研究的目的之一是发展数学理论和方法来分析和计算有用的统计信息，从随机过程中的振荡器，使他们可以用来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相结合。

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.