北大中文核心期刊
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运筹学学报 ›› 2017, Vol. 21 ›› Issue (4): 69-83.doi: 10.15960/j.cnki.issn.1007-6093.2017.04.005

所属专题： 第五届中国运筹学会科学技术奖获奖者专辑

• 运筹学 • 上一篇    下一篇

图像处理中全变差正则化数据拟合问题算法回顾

杨俊锋1,*   

  1. 1. 南京大学数学系, 南京  210093
  • 收稿日期:2017-07-27 出版日期:2017-12-15 发布日期:2017-12-15
  • 通讯作者: 杨俊锋
  • 基金资助:

    国家自然科学基金(Nos. 11371192, 11771208, 11671195), 中央高校基本科研业务费专项资金

An algorithmic review for total variation regularized data fitting problems in image processing

YANG Junfeng1,*   

  1. 1. Department of Mathematics, Nanjing University, Nanjing 210093,  China
  • Received:2017-07-27 Online:2017-12-15 Published:2017-12-15

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摘要：

全变差正则化数据拟合问题产生于许多图像处理任务, 如图像去噪、去模糊、图像修复、磁共振成像、压缩图像感知等. 近年来, 求解此类问题的快速高效算法发展很快. 以最小二乘、最小一乘等为例简要回顾求解此类问题的主要算法, 并讨论一个全变差正则化非凸数据拟合模型在脉冲噪声图像去模糊问题中的应用.

关键词: 全变差, 最小二乘, 最小一乘, 图像处理, 收缩算子, 快速傅里叶变换, 梯度下降, 阈值算法, 分裂罚算法, 交替方向乘子法

Abstract:

Total variation regularized data fitting problems arise from a number of image processing tasks, such as denoising, deconvolution, inpainting, magnetic resonance imaging, and compressive image sensing, etc. Recently, fast and efficient algorithms for solving such problems have been developing very rapidly. In this paper, we focus on least squares and least absolute deviation data fitting and present a brief algorithmic overview for these problems. We also discuss the application of a total variation regularized nonconvex data fitting problem in image restoration with impulsive noise.

Key words: total variation, least squares, least absolute deviation, image processing, shrinkage operator, fast Fourier transform, gradient descent, thresholding method, splitting and penalty method, alternating direction method of multipliers