Collaborative Research: Proximity Algorithms for Optimization Problems Arising from Image Processing

协作研究：图像处理优化问题的邻近算法

• 批准号: 1115469
• 负责人: Charles Micchelli
• 金额: $ 14.5万
• 依托单位: SUNY at Albany
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115469&HistoricalAwards=false
• 关键词: Collaborative; Research; Proximity; Algorithms; Optimization

The restoration of degraded images is a fundamental and challenging problem in image processing. This problem is ill-posed. The total-variation regularization and its variants are commonly used to convert to a well-posed problem. The resulting regularized model usually has a non-differentiable objective functional, which together with the large dimension of the underlying image makes the minimization theoretically and numerically difficult. Typical numerical treatments for this minimization are indirect in the sense that the methods are developed for a smoothed or dual model of the original model. With this project, the principal investigators use tools from convex analysis to find the solution of the image restoration models directly under a unified framework. The PIs address more general mathematical challenges and computational difficulties associated with the obtained fixed-point formulation. This project provides a fixed-point characterization for the solutions of models with least squares and max norm fidelity terms combined with the total variation regularization term. The study considers images corrupted by Gaussian noise, impulsive Gaussian noise and Poisson noise, which are all of relevance for different applications. Restoring images from available data is required in a variety of applications including computer tomography; natural resources and pollution control via satellite imaging in environmental sciences; and fingerprint and face recognition in security identification. Advanced mathematical models and efficient computational algorithms for solving this problem are essential. The developed numerical schemes support improved automatic image restoration for these applications. Furthermore, interdisciplinary approaches resulting from the projects enrich upper level undergraduate and graduate curriculum development and teaching activities.

退化图像的复原是图像处理中的一个基本而又具有挑战性的问题。这个问题是不适定的。全变分正则化及其变体通常用于转换为适定问题。由此产生的正则化模型通常具有不可微的目标泛函，这与底层图像的大维度一起使得理论和数值上的最小化变得困难。这种最小化的典型的数值处理是间接的意义上说，该方法是为一个平滑或双模型的原始模型。在这个项目中，主要研究人员使用凸分析工具，直接在统一的框架下找到图像恢复模型的解决方案。PI解决了更一般的数学挑战和计算困难与所获得的固定点配方。 该项目提供了一个固定点的特点与最小二乘和最大范数保真度项与总变差正则化项相结合的模型的解决方案。该研究考虑了被高斯噪声、脉冲高斯噪声和泊松噪声破坏的图像，这些噪声都与不同的应用相关。 从现有数据恢复图像在各种应用中是必需的，包括计算机断层扫描;环境科学中通过卫星成像进行自然资源和污染控制;以及安全识别中的指纹和面部识别。先进的数学模型和有效的计算算法来解决这个问题是必不可少的。开发的数值方案支持这些应用程序的改进的自动图像恢复。此外，项目产生的跨学科方法丰富了高层次的本科生和研究生课程开发和教学活动。