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图像反问题中的数学与深度学习方法

董彬   

  1. 北京大学北京国际数学研究中心, 北京, 100871
  • 收稿日期:2019-09-29 出版日期:2019-12-15 发布日期:2019-11-16
  • 作者简介:董彬,北京大学北京国际数学研究中心长聘副教授、主任助理,北京大数据研究院深度学习实验室研究员、生物医学影像分析实验室副主任.2003年本科毕业于北京大学数学科学学院、2005年在新加坡国立大学数学系获得硕士学位、2009年在美国加州大学洛杉矶分校数学系获得博士学位.博士毕业后曾在美国加州大学圣迭戈分校数学系任访问助理教授、2011——2014年在美国亚利桑那大学数学系任助理教授,2014年底入职北京大学.主要研究领域为应用调和分析、反问题计算、深度学习及其在图像和数据科学中的应用.在理论上,与合作者一起将图像领域独立发展近30年的两个数学分支(PDE/变分方法和小波方法)建立深刻的联系,改变了领域内对这两类方法的一些既定认识.应用上,以数学理论为指导思想,为来源于医学影像、计算机视觉、深度学习等领域中的重要问题提供行之有效的解决方案.在国际重要学术期刊和会议上发表论文60余篇,现任期刊《Inverse Problems and Imaging》编委.于2014年获得香港求是基金会颁发的求是杰出青年学者奖.

董彬. 图像反问题中的数学与深度学习方法[J]. 计算数学, 2019, 41(4): 343-366.

Dong Bing. MATHEMATICAL AND DEEP LEARNING METHODS IN IMAGE INVERSE PROBLEMS[J]. Mathematica Numerica Sinica, 2019, 41(4): 343-366.

MATHEMATICAL AND DEEP LEARNING METHODS IN IMAGE INVERSE PROBLEMS

Dong Bing   

  1. Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China
  • Received:2019-09-29 Online:2019-12-15 Published:2019-11-16
我们生活在数字的时代,数据已经成为了我们生活中不可或缺的一部分,而图像无疑是最重要的数据类型之一.图像反问题,包括图像降噪,去模糊,修复,生物医学成像等,是图像科学中的重要领域.计算机技术的飞速发展使得我们可以用精细的数学和机器学习工具来为图像反问题设计有效的解决方案.本文主要回顾图像反问题中的三大类方法,即以小波(框架)为代表的计算调和分析法、偏微分方程(PDE)方法和深度学习方法.我们将回顾这些方法的建模思想和一些具体数学形式,探讨它们之间的联系与区别,优点与缺点,探讨将这些方法有机融合的可行性与优势.
We live in the digital age, and data has become an essential part of our lives. Images are undoubtedly one of the most important types of data. Image inverse problems, including image denoising, deblurring, restoration, biomedical imaging, etc., are important areas in imaging science. The rapid development of computer technology has enabled us to use sophisticated mathematics and machine learning tools to design effective algorithms for image inverse problems. This paper mainly reviews three types of methods in image inverse problem, namely, applied and computational harmonic analysis method (represented by wavelets and wavelet frames), partial differential equation (PDE) method and deep learning method. We will review the modeling philosophies of these methods, explore the connections and differences among them, their advantages and disadvantages, and further discuss the feasibility and benefit of the integration of these methods.

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