图像配准中方向场正则化模型的适定性和收敛性

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图像配准中方向场正则化模型的适定性和收敛性郑晓俊¹, 郇中丹², 刘君²1. 海南师范大学初等教育学院海口 571158;
2. 北京师范大学数学科学学院北京 100875 Well-posedness and Convergence of the Vector Field Regularization Model in Image Registration Xiao Jun ZHENG¹, Zhong Dan HUAN², Jun LIU²1. Elementary Education School, Hainan Normal University, Haikou 571158, P. R. China;
2. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China

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摘要图像配准是图像处理的一个重要方面.方向场正则化模型是现有配准方法中效果相对突出的模型.然而它仍然无法正确对齐所有感兴趣的区域.因此，本文从理论角度研究方向场正则化模型，希望寻找模型设计可能存在的问题.由于模型中有一个直接变量和一个由直接变量通过常微初值问题确定的间接变量，所以它是数学上的一类新颖的正则化模型.方向场正则化模型定义为min_v{α||v||_H²+ρ（T（y^v（τ）），S）}，其中T和S分别是模板图像和参考图像，y^v（τ）：x?y^v（τ；0，x）是由初值问题dy/ds=v（s，y），y（0）=x的解y^v（s；0，x）定义的变换，ρ是相似性泛函，α>0是正则化参数，H是希尔伯特空间.本文首先证明了方向场正则化模型有稳定解，然后证明了其收敛性.结合y^v（τ）与v的收敛关系和正则化问题的经典理论可得上述结论.然而，在现有理论下，ρ，S和T需满足较强的条件.本文通过充分利用y^v（τ）的性质，提出了关于ρ，S和T的相对弱的条件.此外，我们还验证了配准常用的3个相似性泛函都满足所提条件.

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收稿日期: 2020-03-03

MR (2010):

O177.92

基金资助:海南省自然科学基金资助项目（118QN023）

作者简介: 郑晓俊,E-mail:xjzheng88@163.com;郇中丹,E-mail:zdhuan@bnu.edu.cn;刘君,E-mail:jliu@bnu.edu.cn

引用本文:

郑晓俊, 郇中丹, 刘君. 图像配准中方向场正则化模型的适定性和收敛性[J]. 数学学报, 2021, 64(3): 385-404. Xiao Jun ZHENG, Zhong Dan HUAN, Jun LIU. Well-posedness and Convergence of the Vector Field Regularization Model in Image Registration. Acta Mathematica Sinica, Chinese Series, 2021, 64(3): 385-404.

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