Research On Image Processing And Classification Based On Generalized Linear Algebra

Image processing and computer vision have applications in various industriesand play a crucial role in certain domains.With the increasing use of color images,efective image processing has become a critical part of artifcial intelligence.Hence,we see rapid development of many algorithms and applications due to the improvements in image processing and computational power of computers.Reviewing history,we can see that changes in image color refect the passage of time.For many years,color images were mostly processed as individual channels,i.e.,Color images were converted into multiple frame sof single gray scale images for study.The problem could be some what resolved by representing color images with generalized matrices because the yallow comprehensive processing of color images in an integrated space.However,this also increases the additional high-dimensional space,leading to increased additional costs for image processing and storage.In this thesis,we use generalized scalar number store present pixels in higher-order images.Generalized scalars consider the correlation between pixels and convert higher-order images into a generalized matrix model.The main research contents of this thesis include:1.Firstly,this thesis applies an eighborhood expansion strategy of the second-order array in this study,called the ”inception” strategy.By using this strategy,we can combine the higher-order singular value decomposition algorithm of lower-order color image swith the ”t-product” convolution algebra under the generalized commutative algebra model,further developing the generalized high-order singular value decomposition algorithm.This thesis uses the generalized high-order singular value decomposition algorithm to process higher-order images(higher-order matrices),and by low-rank approximation processing of singular value matrices,we obtain the reconstructed images of the original images.In mathematical theory,the principle of applying the generalized high-order singular value decomposition algorithm to low-rank approximation to obtain reconstructed images can be explained through the block matrix model of generalized matrix algebra.This study also uses the traditional block matrix model to represent the generalized matrix algebra model base donthe commutative algebra ring.After analyzing the experimental data,we found that the image quality obtained by using the generalized high-order singular value decomposition algorithm to obtain low-rank approximation is superior to the results obtained using the traditional generalized singular value decomposition method.2.This thesis combines finite-dimensional commutative algebra and uses the Neighborhood expansion strategy of traditional matrices to extend the higher-order singular value decomposition algorithm of higher-order matrices to the generalized highorder singular value decomposition algorithm base don the generalized matrix model.On top of the basic concept of traditional tensors,this thesis combines the generalized matrix theory to extend it to generalized tensors,and on this basis,it uses the generalized highorder singular value decomposition algorith mtheory.Experimental data analysiss hows that applying the generalized high-order singular value decomposition algorithm to hand written digit classifcation yields better classifcation results than the high-order singular value decomposition algorithm of traditional tensors.The experimental data also show the theoretical research based on commutative algebra has an advantage in performance over the traditional tensor array’s algorithm theoretical research.3.This study explores the classifcation research of the Grassmann manifold under the frame work of generalized high-order singular value decomposition.Using the matrix of generalized commutative algebra,this thesis optimizes the classifcation method of medical images based on the Grassmann manifold metric.This thesis uses the generalized subspace included angle distance to describe the shortest geodesic distance between two points on the generalized Grassmann manifold and applies this generalized Grassmann manifold distance measurement method to the supervised classifcation.between pixel points in medical images.To further realize the intrinsic dimension estimation of the generalized subspace,this thesis conducteda large number of experiment and compared the classifcation accuracy rates derived from the three generalized subspace included angle vector norms,thus optimizing the Grassmann manifold distance classifcation method.Furthermore,this thesis utilizes the concept of commutative algebra matrix defnition and generalized subspace(sub-module),generalized.Grassmann manifold model,and generalized vector norm,and describes the process of constructing compound pixel susing high order medical image neighborhood pixels that can also be represented as generalized vectors.Toverify the accuracy and effciency of the generalized Grassmann manifold distance measurement classifcation method,this thesis separately compared the classifcation results of generalized Grassmann manifold measurement based on generalized high-order singular value decomposition and traditional Grassmann manifold measurement base d on traditional high-order singular value decomposition.Experimental data show that the classifcation method of generalized Grassmann manifold measurement based on the generalized high-order singular value decomposition algorithm has a signifcant advantage inmedical image classifcation.