Research On Image Denoising And Reconstruction Algorithms Based On Finite-Dimensional Commutative Algebra

Image approximation and denoising are two important research directions in computer vision and pattern recognition.With the development of information technology,data dimensions are gradually increasing,and insufficient storage space has become a problem.For such problems,low-quality approximation of data can effectively solve storage issues.However,the acquired images often contain noise or outliers in practical applications.Principal component analysis(PCA)has some effects on dimensionality reduction and denoising.Still,it is affected by outliers when measuring data similarity,and its projection direction is easily disturbed,which makes it less robust to such noise.Researchers proposed robust principal component analysis(RPCA)based on matrix operations and tensor robust principal component analysis(TRPCA)based on third-order tensor operations to solve this problem.However,these methods often ignore the correlation between image pixels,and the neighborhood information between image pixels has not been fully utilized.Some scholars have proposed a generalized matrix model of finite-dimensional commutative algebra(t-algebra),effectively solving this problem.The t-matrix model extends scalars to generalized scalars using the neighborhood information of images and further expands generalized matrices and generalized tensors with generalized scalar elements.This thesis will discuss how to generalize traditional algorithms for image approximation reconstruction and image denoising on the generalized matrix model,forming generalized algorithms,and propose corresponding algorithms and models.By applying these methods to practical image processing problems,we can fully utilize the neighborhood information of images,improve the effects of dimensionality reduction and denoising,and achieve better performance in computer vision and pattern recognition.The four main contributions of this thesis are as follows:1.Based on the generalized matrix model,a neighborhood expansion strategy is adopted to process the matrix images,extending the traditional singular value algorithm to a generalized singular value algorithm.Optimizing the linear interpolation of generalized scalars improves the continuity between generalized scalar elements.The optimized algorithm is called the generalized linear interpolation singular value algorithm.Experimental results show that,in terms of low-rank approximation of images,the generalized linear interpolation singular value algorithm outperforms the reconstruction results of the generalized singular value algorithm under certain conditions and is entirely superior to the traditional singular value algorithm.2.Based on the generalized matrix model,the neighborhood of third-order tensor images is expanded,extending the higher-order singular value algorithm to the generalized higherorder singular value algorithm.The generalized linear interpolation of generalized scalars is obtained by optimizing the linear interpolation of the higher-order singular value algorithm.Experimental results show that,in terms of low-rank approximation of third-order tensor images,the generalized linear interpolation higher-order singular value algorithm outperforms the reconstruction results of the generalized higher-order singular value algorithm under certain conditions and is completely superior to the higher-order singular value algorithm.3.For the issue of insufficient neighborhood constraint information in images for robust principal component analysis,matrices are expanded in the neighborhood,and robust principal component analysis is extended to generalized robust principal component analysis on the generalized matrix model.Unlike the traditional singular value algorithm solution model,this paper solves the corresponding data by means of a generalized matrix singular value threshold algorithm.This paper validates the generalized robust principal component analysis model through numerical experiments,combining the constraints between the low-rank characteristics of matrices and sparse matrix data to verify the results of recovered data satisfying certain conditions.The excellence and validity of generalized robust principal component analysis in different data sets is also verified through public experimental data sets for denoising in different data sets,and the peak signal-to-noise ratio is compared with that of robust principal component analysis.4.Although tensor robust principal component analysis effectively utilizes the spatial structure of the tube fibers of third-order tensors based on the "t-product" model and Kilmer’s TSVD,the spatial structure of each frontal slice of the third-order tensor has not been well utilized.For this problem,this thesis expands the neighborhood of all frontal slices.It extends tensor robust principal component analysis to generalized tensor robust principal component analysis on the generalized matrix model.The generalized tensor singular value thresholding algorithm is applied to solve the corresponding data.In this paper,we validate the generalized tensor robust principal component analysis model by numerical experiments,and verify the recovered data results under certain conditions by combining the constraint relationship between tensor Tubal rank and sparse tensor data.The superior performance of generalized tensor robust principal component analysis on different data sets is also verified by publicly experimenting the data set for denoising in different data sets.