Tensor Principal Component Analysis And Its Applications

Robust principal component analysis(RPCA)that aims to extract low rank and sparse components in data has been widely used in matrix analysis.So far,different convex models and algorithms have been proposed to solve this problem.RPCA can be applied in many signal processing problems,such as image alignment,background modeling for surveillance videos and illumination normalization for face images.The traditional RPCA methods should flatten or vectorize the tensor data such as videos so as to solve the problem in the matrix.This way doesn't use the structural feature of the data effectively since the information loss involves in the operation of matricization.Tensors are used for high-dimensional data,so it's better to achieve principal component analysis by means of tensor methods,which can take advantage of structural information of multi-dimensional data.In recent years,tensor principal component analysis(TPCA)has been developed and utilizing convex optimization models of matrix data in TPCA is a hot research topic.TPCA methods are based on different tensor decomposition models,such as traditional canonical polyadic decomposition and higher-order singular value decomposition.In this thesis,tensor singular value decomposition(t-SVD)model was studied that has been proposed recently and used in different tensor models widely.This thesis studies tensor robust principal component analysis(TRPCA)that is based on t-SVD model.The main innovations and contributions include follow three aspects:1.Block TRPCA model has been proposed.In general TRPCA models,the tensor data has been decomposed into many blocks and we extract low rank component in blocks parallelly.The sparse component can be separated in the convex optimization model.2.Improved nuclear norm for TRPCA model has been proposed.We study the tensor nuclear norm and make improvement in nuclear norm model,which has been applied in TRPCA model.In the improved TRPCA model,we use alternating direction method of multipliers decompose low rank and sparse component.The effect of this model has been demonstrated in two applications that are removing sparse noise of color images and background modeling in videos.3.We apply the above two TRPCA methods in removing rain for videos.In this application,the effectiveness of the two models has been proved.