Research On Riemannian Manifold Based Image Classification

In various tasks in the field of machine learning and computer vision,the require-ments for data representation methods are gradually increasing.People hope to use a compact and highly discriminative model to represent data containing huge amounts of information.At the same time,this representation method should be as robust as possi-ble to various changes.In this context,researchers have gradually shifted their attention from traditional Euclidean space to nonlinear manifolds.In recent years,the computer vision community's attention to Riemannian manifolds has greatly increased,and a large number of important applications have emerged,including face recognition,motion recog-nition,clustering,visual tracking,and motion grouping and segmentation.The increasing popularity of Riemannian manifolds is because its special nonlinear spatial structure is of great significance for image feature representation.Among many different types of Rie-mannian manifolds,the symmetric positive definite matrix manifold has become a natural and reasonable method for representing images because of its complete geometric prop-erties and metric calculations.This paper takes the symmetric positive definite matrix manifold as the research center to carry out related research on the image classification problem based on Riemannian manifold.Calculating the covariance matrix of the target data conforming to the manifold ge-ometry of the symmetric positive definite matrix is currently the most commonly used data representation method based on Riemannian manifolds,and we analyze several shortcom-ings of the conventional covariance matrix in the image classification problem and propose different The improved method to achieve efficient representation of image features.The main work carried out in this article can be summarized as follows:(1)A covariance matrix of image set based on two-dimensional image representation is proposed.Compared with the covariance matrix calculated from the one-dimensional image vector,we designed two methods to take full advantage of the two-dimensional im-age representation.The covariance dimension of the improved image set is significantly reduced,thereby saving a lot of computing time and alleviating the under-definite covari-ance matrix problem to a large extent.We applied this method to image set classification problems such as face recognition and object classification and verified its effectiveness.(2)A method of low-dimensional vector estimation for an infinite-dimensional sym-metric positive definite matrix using specific feature mapping is proposed.Infinite-dimensional symmetric positive definite matrices often have better performance in im-age classification problems than low-dimensional symmetric positive definite matrices,but there are also problems such as application difficulties and high computational com-plexity.We obtain the low-dimensional approximation vector of the infinite-dimensional covariance matrix by directly estimating the infinite-dimensional symmetric positive def-inite matrix manifold kernel,and verify its effectiveness in image classification problems such as material recognition,virus classification,and face recognition.(3)On the basis of modularization,the manifold representation of the image set is replaced by a single covariance matrix with a set of low-dimensional sub-covariance ma-trices,and then a novel block-diagonal covariance is constructed matrix.We adopt two methods to calculate the sub-covariance matrix of the image set for different types of image features,and the new block-diagonal covariance matrix has obvious advantages compared with the conventional covariance matrix in terms of discriminability and computation-al complexity.To verify the effectiveness of this method,we conducted a comparative analysis in object classification,scene recognition,face recognition,and other image set classification problems and finally got satisfactory results.(4)The dimensionality reduction method of symmetric positive definite matrices is to enhance the discriminativeness of symmetric positive definite matrices by learning a mapping from a high-dimensional manifold to a low-dimensional manifold,and the pro-cess of finding this mapping can be regarded as a Grassmann manifold An optimization problem on the above.According to the related theory of non-parametric estimation,we propose a symmetric positive definite matrix dimensionality reduction method based on probability distribution.By estimating the probability density function on the mani-fold,we calculate a relationship function for describing the structure information of the manifold,so as to achieve The best manifold dimensionality reduction effect.Finally,we verified the effectiveness of this method in object classification,texture recognition,face recognition,scene classification,and other issues.