卷积视角下循环矩阵对角化证明新方法及其应用  

A New Method for Diagonalization of Circulant Matrices from the Convolution Perspective and Its Application

作　　者：唐毅鋆 陈颖频 陈惠 乔嘉琪 陈振雕 李一凡 TANG Yi-yun;CHEN Ying-pin;CHEN Hui;QIAO Jia-qi;CHEN Zhen-diao;LI Yi-fan(School of Electronic Information,Zhangzhou Institute of Technology,Zhangzhou 363000,China;School of Physics and Information Engineering,Minnan Normal University,Zhangzhou 363000,China)

机构地区：[1]漳州职业技术学院电子信息学院,福建漳州363000 [2]闽南师范大学物理与信息工程学院,福建漳州363000

出　　处：《长春师范大学学报》2023年第6期67-74,共8页Journal of Changchun Normal University

基　　金：漳州市自然科学基金项目“仿人类视觉注意力的相关滤波视频目标跟踪研究”(EZZZZ2023J010015);福建省大学生创新创业训练计划项目“面向闽南地区古建筑照片的超分辨率重建关键技术研究”(S202210402038);福建省中青年教师教育科研项目(社科类)“5G时代基于区块链的新媒体版权保护研究”(JAS21614);闽南师范大学校级教研课题“‘图像工程’虚拟教研室探索与实践”(202211)。

摘　　要：循环矩阵可被离散傅里叶变换矩阵对角化,该性质成为空域信号与频域信号之间的桥梁,被广泛应用于图像恢复、视频目标跟踪技术中。传统的循环矩阵对角化证明方法需要大量的计算,且不易于被理解和掌握。本文提出一种卷积视角下的列循环矩阵对角化证明方法,巧妙地避开传统证明方法中的数学运算,相比于传统的证明方法更易理解。此外,利用列循环矩阵与行循环矩阵的转置关系进一步证明了行循环矩阵的对角化性质,并提供该性质的工程应用实例。The discrete Fourier transform matrix can diagonalise the column circulant matrix.This property becomes the bridge between the spatial and frequency domain signals,which have been widely used in image restoration and video target tracking technology.The traditional method of diagonalizing a column-circular matrix requires many mathematical skills and is not easy to understand and master.This paper proposes a diagonalization proof method for column circulant matrices from a convolution perspective,which cleverly avoids the calculation in traditional proof methods and is easier to understand than traditional proof schemes.In addition,we further prove the diagonalization property of the row circulant matrix by using the transposition relationship between the column circulant matrix and the row circulant matrix.Finally,we provide an application based on the diagonalization property of the row circulant matrix.

关 键 词：循环矩阵 离散傅里叶变换 对角化 卷积 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]