项目名称: 非光滑凸优化问题的快速算法及其在图像分析中的应用
项目编号: No.91330105
项目类型: 重大研究计划
立项/批准年度: 2014
项目学科: 数理科学和化学
项目作者: 孔德兴
作者单位: 浙江大学
项目金额: 65万元
中文摘要: 非光滑的凸优化问题及其快速算法是计算科学与应用数学领域中的前沿主流研究课题,在高新技术中具有十分重要的作用。本项目拟对一类具有广泛应用的非光滑凸优化问题发展其新的数学理论及最佳的数值解法。该优化问题的一个重要应用是医学影像分析与处理,譬如部分并行成像(PPI)技术中的多对比度的磁共振图像的同时重构:部分并行成像技术是一种多线圈并行数据采集技术,其优点是扫描时间短、价格低、能大大减少患者的痛苦。本项目将着重研究下述几个方面的问题:1、对上述非光滑的凸优化问题,研究具有最佳收敛速率的修正ADMM算法以及具有变化步长的Bregman算子分裂算法,发展相应的数学理论;2、设计合理的回朔策略以加速算法的收敛速度、减少计算时间,从而提高所提出的算法的实际性能;3、把上述快速算法应用于部分并行成像中的多对比度的磁共振图的联合重构问题。这些问题的解决无论是在理论上还是在应用方面均具有十分重要的科学价值。
中文关键词: 非光滑凸优化;修正ADMM算法;Bregman算子分裂算法;;最佳收敛速率;图像重构
英文摘要: Non-smooth convex optimization and its accelerated algorithms are the frontier key research topics in computational science and applied mathematics, they play an important role in high and new technologies. This proposal aims at developing novel theories and optimal numerical methods for solving a class of non-smooth convex optimization problems with wide applications. A typical application of this class of non-smooth convex optimization problems is in medical image analysis and processing, e.g., simultaneous multi-contrast MR image reconstructions in partially parallel imaging. This project will investigate the following important problems: (1) Develop accelerated modified ADMM algorithms and Bregman operator splitting algorithm (with variable stepsizes) with the optimal rate of convergence for solving the class of non-smooth convex problems under consideration; (2) Develop backtracking strategies to speed up the convergence, reduce computational time and improve the performance of the proposed algorithms in practice; (3) Apply our proposed accelerated algorithms to simultaneous multi-contrast magnetic resonance (MR) image reconstruction in partially parallel imaging. Clearly, the solution of these problems has great scientific significance in both theoretical and applied aspects.
英文关键词: Non-smooth Convex Optimization;modified ADMM algorithms;Bregman operator splitting;Optimal convergence rate;Image reconstruction