RUI: New Variational Models for Denoising, Decomposition, and Deblurring

研究人员、合作者和学生将开发新的图像处理模型，证明这些模型在数学上是可靠的，确定其解的几何性质，开发准确有效的数值方案，并将这些模型直接应用于科学问题。这些模型将解决图像处理中的四个基本问题：边缘保持去噪、分解、去模糊和图像融合。所提出的公式将基于变分方法和偏微分方程，这为研究模型的数学基础和物理解释提供了合适的框架，包括包含凸线性增长泛函、Besov半范数和负Sobolov范数的模型，这些模型可以保留理想的几何图像属性，如边缘和纹理，同时避免引入虚假伪影。这里解决的挑战之一将是找到连续问题的适当离散表示，这些表示保留了理想的几何特征并且易于处理。数字图像现在几乎应用于科学和技术的每一个领域。该项目开发的模型将用于解决医学成像和材料科学等领域的现实世界问题。然而，这些模型的制定将具有足够的普遍性，可能会应用于科学中的广泛应用。在这笔赠款中开发的软件将公开提供。该调查员定期教授图像处理课程，并为中学生、高中妇女和少数民族组织图像处理研讨会。该项目还将支持本科生研究人员。因此，这项工作将促进对青年科学家的培训，并为代表性不足的群体提供教育机会。

The investigator, collaborators and students will develop new models for image processing, show the models are mathematically sound, determine geometric properties of their solutions, develop accurate and efficient numerical schemes, and directly apply these models to problems in the sciences. The models will address four fundamental problems in image processing: edge-preserving denoising, decomposition, deblurring, and image fusion. The proposed formulations will be based on variational methods and partial differential equations, which provide an appropriate framework for studying the mathematical foundations and physical interpretation of the models.These include models involving convex linear growth functionals, the Besov semi-norm, and negative Sobolov norms, which can retain desirable geometric image properties such as edges and textures, while avoiding the introduction of false artifacts. One of the challenges addressed here will be finding appropriate discrete representations of the continuous problem that retain desirable geometric features and are tractable.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 in areas such as medical imaging and material science. However, the models will be formulated in enough generality to potentially be applied to a wide array of applications in the sciences. Software developed in this grant will be made publicly available. The investigator regularly teaches courses on image processing, and leads workshops on image processing for middle school students and high school women and minorities. This project will also support undergraduate researchers. Thus this work will promote the training of young scientists, as well as provide educational opportunities to underrepresented groups.