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

退化图像的复原是图像处理中的一个基础性和挑战性问题。这个问题是不适定的。全变分正则化及其变种常被用来转化为适定问题。由此得到的正则化模型通常具有不可微的目标泛函，再加上基础图像的巨大维度，使得最小化在理论上和数值上都是困难的。这种最小化的典型数值处理是间接的，因为这些方法是为原始模型的平滑或对偶模型开发的。在这个项目中，主要研究人员使用凸分析中的工具，直接在统一的框架下求解图像恢复模型。PI解决了与所获得的定点公式相关的更一般的数学挑战和计算困难。该项目为具有最小二乘和最大范数保真项并结合全变差正则项的模型的解提供了不动点刻画。这项研究考虑了被高斯噪声、脉冲高斯噪声和泊松噪声污染的图像，这些噪声都与不同的应用相关。在各种应用中需要从现有数据恢复图像，包括计算机断层成像；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人脸识别。先进的数学模型和高效的计算算法是解决这一问题的关键。所开发的数值格式支持针对这些应用的改进的自动图像恢复。此外，这些项目产生的跨学科方法丰富了高级本科生和研究生的课程开发和教学活动。

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.