二维线性正则变换理论及应用研究进展  

Research Progress of the Two-Dimensional Linear Canonical Transform Theory and Its Application

作　　者：宋玉娥[1] 卜红霞[2] 李炳照 SONG Yu-e;BU Hong-xia;LI Bing-zhao(School of Electrical and Information Engineering,Beijing Polytechnic College,Beijing 100042,China;College of Physics,Hebei Normal University,Shijiazhuang 050024,China;School of Mathematics and Statistics,Beijing Institute of Technology,Beijing 102488,China)

机构地区：[1]北京工业职业技术学院电气与信息工程学院,北京100042 [2]河北师范大学物理学院,河北石家庄050024 [3]北京理工大学数学与统计学院,北京100081

出　　处：《数学的实践与认识》2019年第18期222-234,共13页Mathematics in Practice and Theory

基　　金：国家自然科学基金(61671063);2018年北京工业职业技术学院重点科课题(BGZYKY201820Z);河北师范大学自然科学科研基金(L2016B06)

摘　　要：线性正则变换作为经典傅里叶变换和分数阶傅里叶变换的广义形式,拥有更大的灵活性,是分析和处理非平稳信号的有力工具.同样,二维线性正则变换在处理和分析二维信号时具有良好性能.首先系统地总结了近年来二维线性正则变换的发展历程和理论研究成果,重点阐述了二维不可分离的线性正则变换的最新基础理论,包括其重要性质、采样和离散理论、快速算法、不确定性原理、特征函数等;然后介绍了二维线性正则变换在滤波器设计、图像处理等领域中的最新应用成果;最后对二维线性正则变换的发展前景做出展望.对研究者全面了解二维线性正则变换具有很好的参考价值,可以进一步促进其工程应用.As a generalized form of classical Fourier transform and fractional Fourier transform,linear canonical transform(LCT)has greater flexibility and is a powerful tool for analyzing and processing non-stationary signals.In the same way,the two-dimensional LCT has good performance in processing and analyzing two dimensional signals.We first review recent developments processand the research results oftwo-dimensional LCT systematically,mainly focusing on the latest theory of two-dimensional non-separate LCT,such as important properties,sampling and discrete theory,fast algorithm,Heisenberguncertainty principle,eigen-functions theory.Based onthe summary of its theory,we summarize various and latest applications of the two-dimensionalLCT in many fields,including filter design,image processing,etc.As well,the applicationof two-dimensional LCT for estimatingtwo-dimensional linear frequency modulated signal is analyzed in deeply.Finally,we make a development prospect of two-dimensional LCT.It is of great referencevalue for researchers to fully understand the two-dimensional LCT and can further promote its engineering applications.

关 键 词：二维线性正则变换 采样 快速算法 应用 

分 类 号：O174.2[理学—数学]