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.
