RUI: New Applications of Curvature in Image Processing

研究者、学生和合作者将开发和建立图像处理新模型的数学基础，实现、测试和比较这些新模型与现有模型，并直接将这些模型应用于图像融合、医学成像和视频处理等问题。拟议的工作有几个总体目标。首先是一种新的图像去噪框架的开发和数学分析，该框架以一种新的有效的方式利用了图像曲率的丰富信息。第二是更好地理解如何在学习的、结构化的和过完备的字典中建模几何特征，以处理和融合退化的数据。在这项工作中提出的新模型将在变分和基于补丁的框架中制定，并将主要解决受噪声和线性退化影响的数据。数字图像现在几乎应用于每一个科学技术领域。本计划开发的模型将用于解决现实世界的问题，包括图像融合、视频处理和医学成像等问题。然而，这些模型将以足够的普遍性来表述，以潜在地应用于科学领域的广泛应用。在此资助下开发的软件将向公众开放。该项目还将支持本科生研究人员实施、测试和比较新的和现有的图像处理模型，确定适当的数值方案，直接与科学家合作，将这些模型应用于现实世界的问题，并在地方和国家会议上展示他们的成果。调查员定期教授图像处理课程，并为中学生和高中女性和少数民族举办图像处理讲习班。因此，这项工作将应用于科学问题，使在直接和间接相关领域工作的其他人也能接触到这项工作，促进青年科学家的培训，并为代表性不足的群体提供教育机会。

The investigator, students, and collaborators will develop and establish the mathematical foundations of new models for image processing, implement, test and compare these new models with existing ones, and directly apply these models to problems in image fusion, medical imaging, and video processing. The proposed work has several over-arching goals. The first is the development and mathematical analyses of a new framework for image denoising that exploits the rich information of the curvature of an image in a new and effective way. The second is a better understanding of how to model geometric features in learned, structured, and overcomplete dictionaries for processing and fusing degraded data. The new models proposed in this work will be formulated in both the variational and patch-based frameworks, and will mainly address data that have been compromised by noise and linear degradations.Digital images are now used in almost every area of science and technology. The models developed in this project will be used to solve real world problems, including problems in image fusion, video processing, and medical imaging. However, the models will be formulated in enough generality to potentially be applied to a wide array of applications in the sciences. Software developed under the auspices of this grant will be made publicly available. This project will also support undergraduate researchers who will implement, test and compare new and existing image processing models, determine appropriate numerical schemes, work directly with scientists to apply these models to real world problems, and present their results at local and national meetings. The investigator regularly teaches courses on image processing, and leads workshops on image processing for middle school students and high school women and minorities. Thus this work will be applied to problems in the sciences, be accessible for others working in directly and indirectly related fields, promote the training of young scientists, and provide educational opportunities for underrepresented groups.