Theory And Application Of Two-dimensional Non-separable Linear Canonical Transform

As a generalized form of linear canonical transform,the two-dimensional non-separable linear canonical transform has more adjustable parameters and greater flexibility.It has a wide range of applications in signal and image processing,optics,pattern recognition and other fields,and the necessary premise of these applications is to have an accurate and efficient fast algorithm.Fast algorithms for linear canonical transform have been studied in depth.However,due to the complex form of two-dimensional non-separable linear canonical transform,the development of its fast algorithm is slow,and the existing algorithms can not well meet the requirements of short calculation time,high accuracy and low complexity.Therefore,this paper studies the fast algorithm of two-dimensional non-separable linear canonical transform by referring to the numerical calculation method of linear canonical transform.In the wide application of two-dimensional non-separable linear canonical transform,besides numerical calculation,some properties of the transform are also needed to provide theoretical support for some applications.For example,Wigner distribution can detect and analyze non-stationary signals,convolution and correlation theorems can provide simple and quick theoretical basis for signal sampling theorem and filter design,etc.Therefore,the effective realization of the related applications of the transform cannot be separated from the support of these important theories.In this paper,the Wigner distribution,ambiguity function,convolution and correlation theories based on two-dimensional non-separable linear canonical transform are presented,and the fast algorithm of the transform is studied,which provides theoretical support for the application and development of two-dimensional non-separable linear canonical transform in the engineering field.The specific work is summarized as follows:The first part is to give the Wigner distribution and ambiguity function based on twodimensional non-separable linear canonical transform.Firstly,the definition of two dimensional Wigner distribution and ambiguity function related to two-dimensional non-separable linear canonical transform are proposed.Secondly,we deduced the related properties,including conjugation symmetry property,marginal properties,shift properties,energy distribution and related properties,and Moyal formula.Furthermore,the affine transformation relationship between the new definition and two-dimensional non-separable linear canonical transform is discussed,and it is found that the traditional definition is one of the special cases of the new definition.Finally,the proposed Wigner and ambiguity functions are applied to the detection of one and bi-component two-dimensional chirp signals,and the effectiveness of the new definition is proved by several examples.In the second part,the convolution and correlation of two-dimensional non-separable linear canonical transform domain and its application are studied.Firstly,the product theorem of two-dimensional non-separable linear canonical transform domain and two equivalent convolution definitions,as well as two equivalent correlation definitions,are proposed to pave the way for convolution and correlation theorems.Secondly,the convolution theorem and correlation theorems of two-dimensional non-separable linear canonical transform domain are derived.Furthermore,the design of the multiplicative filter in two-dimensional nonseparable linear canonical transform domain is studied,and the method of time domain convolution is proposed to construct the multiplicative filter,which can greatly improve filtering efficiency.Finally,the sampling theory of two-dimensional non-separable linear canonical transform domain is proposed,and the calculating formulas of 2D uniform sampling and low-pass reconstruction are derived respectively.Both applications show the feasibility of the proposed theory.In the third part,we study the fast algorithm of two-dimensional non-separable linear canonical transform based on matrix factorization.Firstly,a new method of four-stage parameter matrix decomposition is proposed.According to the idea of matrix decomposition,the parameter matrix of two-dimensional non-separable linear canonical transform is decomposed into its special form of cascade.Secondly,the advantages and disadvantages of the proposed algorithm and the previous algorithm in terms of operation time and precision are compared and analyzed by simulation when different parameter matrices and input functions are used.Simulation results from different angles confirm the advantages of the proposed algorithm.In addition,the performance of additivity and reversibility of two-dimensional non-separable linear canonical transform is simulated and discussed,and it is proved that the two properties are better under the action of the proposed algorithm.Finally,a new optical system composed of two-dimensional inverse Fourier transform and elliptic gradient index media is constructed,and the proposed algorithm is applied to the system.The simulation results show the powerful advantages of the proposed algorithm again.