图像处理问题的快速数值方法

项目名称： 图像处理问题的快速数值方法

项目编号： No.60872129

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 杨余飞

作者单位： 湖南大学

项目金额： 34万元

中文摘要： 本项目对几类图像处理问题提出新的理论和快速数值方法。 研究非光滑二阶正则化方法及其在图像去噪问题中的应用，提出了解LLT模型的半光滑牛顿型方法并证明了其Q-超线性收敛性。对图像去噪问题中带两个L1正则项的极小化问题，分别提出了非线性多重网格方法和新的投影方法。为了减轻图像去噪过程中的阶梯效应或边缘模糊，提出了基于对偶策略的两步模型，并利用增广Lagrange策略，提出了解此模型的投影梯度法。基于LLT图像恢复模型，提出了修正不动点迭代算法，该算法不需计算逆矩阵，从而能加快收敛并减少舍入误差。此外，还提出了求解图像恢复问题的本原-对偶有效集算法、基于分裂Bregman迭代的投影方法及基于修改LOT模型的分裂Bregman迭代方法。并给出了上述所有方法的收敛性分析。 研究图像放大的非局部全变差正则化技术，提出了相应的分裂Bregman迭代算法。利用对偶策略、LOT模型、TV-Stokes模型及分裂Bregman迭代，提出了两种用于图像放大的两步方法。并给出了上述几种方法的收敛性分析。利用TV-Allen-Cahn方程，提出了求解图像分割问题的改进的对偶算法，数值实验结果验证了算法的有效性。

中文关键词： 图像恢复；图像分割；图像放大；数值方法；收敛性分析

英文摘要： This project presented new theories and fast numerical methods for several kinds of image processing problems. We studied the nonsmooth second-order regularization method and its application in image denoising. Then we proposed a semismooth Newton method for solving the LLT model and proved its Q-superlinearly convergence. We respectively proposed a nonlinear multi-grid method and a new projection method for solving the minimization problem with two L1 regularization terms in image denoising. In order to alleviate the staircase effect or the edge blurring in the course of the image denoising, we proposed a two-step model based on the duality strategy. Following the augmented Lagrangian strategy, we proposed a projection gradient method for solving this model. Based on the LLT image restoration model, we proposed a modified fixed point iterative algorithm without computing inverse matrices so that it can speed up the convergence and reduce the roundoff error. Furthermore, we presented the primal- dual active set algorithm, projection method based on the split Bregman iterative and split Bregman iterative method based on the modified LOT model for solving image restoration. At the same time, we gave convergence analysis of all of the above methods. We studied the nonlocal total variation regularization technique for image zooming and proposed the related split Bregman iterative algorithm. By applying the dual strategy, the LOT model, the TV-Stokes model and the split Bregman iteration, we presented two kinds of two-step methods for image zooming. At the same time, we gave the convergence analysis of the aforementioned methods. By applying the TV-Allen-Cahn equation, we proposed an improved dual algorithm for solving image segmentation, numerical experimental results showed the effectiveness of the proposed algorithm.

英文关键词： Image restoration; image segmentation; image zooming; numerical method; convergence analysis