RUI: Variational and PDE based methods for image processing

本项目将开发、分析和应用新模型来解决图像处理中的三个基本问题：(1)边缘保持噪声去除；(2)将图像分解为物体加纹理；(3)通过图像插值（或“图像补绘”）恢复丢失的信息。这种新型模型的主要特点是它们能够在自然场景中隔离和检测被噪声、纹理或其他遮挡物体阻挡的目标物体，而不会在此过程中产生错误或误导性的特征。识别和/或开发假目标边界一直是边缘保持图像处理模型的一个挑战；这个项目试图找到解决这个问题的通用方法。这些新模型是基于变分方法和偏微分方程的。研究者将建立他们的数学有效性，确定他们的解决方案的性质，为他们的实施制定有效和准确的数值方案，并直接应用这些模型来解决科学和工程中的关键问题。通过与医学成像、材料科学、地质学、制药和光学字符识别等领域的研究人员的现有合作，研究者和本科生将使用这些新模型来解决科学和技术中的关键问题。这些问题包括去除噪声，同时隔离高度退化的磁共振图像中的关键医学特征，识别和分析纳米级材料的颗粒结构以优化高科技金属，以及识别遥感图像中野火或洪水等高风险的土地覆盖区域，这些区域的边界被道路或地形等纹理所阻碍。目前，处理这些问题的唯一可靠方法依赖于手绘对象边界。对于大型数据集来说，这是非常耗时的，因此自动化边界检测过程将大大提高技术水平。错误的目标检测在这些应用中的任何一个都可能是毁灭性的，因此现有的自动化方法不能直接应用。该项目旨在找到理论上合理的方法，在消除障碍物和准确识别目标物体的同时避免这一缺点。

This project will develop, analyze, and apply new models for solving three fundamental problems in image processing: (1) edge-preserving noise removal; (2) image decomposition into objects plus textures; (3) recovery of lost information through image interpolation (or 'image inpainting'). The main feature of this new class of models is their ability to isolate and detect in natural scenes target objects that are obstructed by noise, textures, or other occluding objects, without generating erroneous or misleading features in the process. The identification and/or development of false object boundaries has long been a challenge for edge-preserving image processing models; this project seeks to find a universal approach for solving this problem. These new models are based on variational methods and partial differential equations. The investigator will establish their mathematical validity, determine properties of their solutions, develop efficient and accurate numerical schemes for their implementation, and directly apply these models to solve critical problems in the sciences and engineering.Through existing collaborations with researchers in medical imaging, materials science, geology, pharmaceuticals, and optical character recognition, the investigator and undergraduate students will use these new models to solve key issues in science and technology. These problems include removing noise while isolating key medical features in highly degraded magnetic resonance images, identifying and analyzing the grain structure of nanoscale materials for optimizing high technology metals, and identifying land cover regions that are at high risk for hazards such as wildfires or flooding in remotely sensed images where the boundaries of these regions are obstructed by textures such as roads or topography. Currently, the only reliable methods for treating each of these problems depend on hand-drawn object boundaries. This is prohibitively time consuming on large data sets, so automating the boundary detection process will greatly enhance the state of the art. False object detection can be devastating in any one of these applications, so existing automated methods cannot be directly applied. This project seeks to find theoretically sound approaches that avoid this drawback while removing obstructions and accurately identifying target objects.