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

退化图像的恢复是图像处理中的一个基本问题，也是一个具有挑战性的问题。这个问题是不适定的。全变分正则化及其变体通常用于将问题转化为适定问题。得到的正则化模型通常具有不可微的目标泛函，加上底层图像的大尺寸，使得最小化在理论上和数值上都很困难。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光滑模型或对偶模型开发的。在这个项目中，主要研究者使用凸分析的工具直接在统一的框架下找到图像恢复模型的解。pi解决了与获得的不动点公式相关的更一般的数学挑战和计算困难。该项目提供了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解的不动点表征。该研究考虑了高斯噪声、脉冲高斯噪声和泊松噪声对图像的破坏，它们都有不同的应用。从可用数据恢复图像需要在各种应用程序，包括计算机断层扫描；利用环境科学中的卫星成像技术控制自然资源和污染；以及指纹和人脸识别在安全识别中的应用。为了解决这一问题，先进的数学模型和高效的计算算法是必不可少的。开发的数值方案支持这些应用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级本科和研究生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.