Tensor Subspace On Manifold For Face Recognition

Manifold learning is one of the hottest topics, since two papers about manifoldlearning were published in science journal2000. It assumes that high dimensional data,such as face images, lie in or close to a low dimensional manifold embedding a highdimensional Euclidean space. It aims at unfold these manifold for dimensionality re-duction.Other hand, these high dimensional data, such as face images, can naturally berepresented by tensor, namely mutil-dimensional array. The advantages of tensor rep-resentation include preserving natural structure of input data, avoiding small samplesize problem and being able to deal with with massive data. In this thesis, I make athorough study on manifold based tensoral subspace analysis. Finally I demonstrate theeffectiveness of those methods by some real world applications including face recog-nition, color space selection etc. More concretely, the main contributions of this thesisinclude:1. I represent a color image as a third-order tensor and present the tensor discrimi-nant color space (TDCS) model. The model can keep the underlying spatial structure ofcolor images. With the definition of n-mode between-class scatter matrices and within-class scatter matrices, TDCS constructs an iterative procedure to obtain one color spacetransformationmatrixandtwodiscriminantprojectionmatricesbymaximizingtheratioof these two scatter matrices. And I propose a sparse tensor discriminant color space(STDCS)modelwhichrepresentsacolorimageasathird-ordertensorinthispaper. Themodel can not only keep the underlying spatial structure of color images but also en-hance robustness and give intuitionistic or semantic interpretation. STDCS transformsthe eigenvalue problem to a series of regression problems. Then one spare color spacetransformation matrix and two sparse discriminant projection matrices are obtained byapplying lasso or elastic net on the regression problems. 2. I propose the fusion tensor subspace analysis framework, a novel idea wheredifferent transformation strategies are implemented on different modes of tensor. Un-der the framework, we propose two algorithms:(1) fusion tensor color space model forfacerecognitionand(2)fusiondiscriminantcorrelationtensoranalysisforactionrecog-nition. The experimental results show that the performances of the two algorithms arebetter than the existing tensor subspace analysis algorithms.3. I propose a second-order discriminant tensor subspace analysis (DTSA) algo-rithm to extract discriminant features from the intrinsic manifold structure of the tensordata. DTSA combines the advantages of previous methods with DI, the tensor methodspreservingthespatialstructureinformationoftheoriginalimagematrices, andtheman-ifold methods preserving the local structure of the samples distribution. DTSA definestwo similarity matrices, namely within-class similarity matrix and between-class simi-larity matrix. The within-class similarity matrix is determined by the distances of pointpairs in the same class, while the between-class similarity matrix is determined by thedistances between the means of each pair of classes. Using these two matrices, the pro-posed method preserves the local structure of the samples to fit the manifold structureof facial images in high dimensional space better than other methods. Moreover, com-pared to the2D methods, the tensor based method employs two-sided transformationsrather than single-sided one, and yields higher compression ratio. As a tensor method,DTSA uses an iterative procedure to calculate the optimal solution of two transforma-tion matrices. In this paper, we analyzed DTSA's connections to2D-DLPP and TSA,theoretically. The experiments on the ORL, Yale and YaleB facial databases show theeffectiveness of the proposed method.4. IproposeamethodcalledSp-TensortoextendTensorFacesbyapplyingthesub-pattern technique. Advantages of the proposed method include:(1) a portion of spatialstructure and local information of facial images is preserved;(2) dramatically reducethe computation complexity than other existing methods when building the model. Theexperimental results demonstrate that Sp-Tensor has better performance than the origi-nal TensorFaces and Sp-PCA1, especially for facial images with un-modeled views andlight conditions. 5. I propose a Two-dimensional locality preserving projection based on maximumscatter difference (2D-DLPP/MSD).2D-LPP/MSD use additive principle to preservethe locality by maximizing the between-class scatter and within-class scatter instead ofusing multiplicative principle of2D-DLPP. Theoretically, we also discuss the influenceof balance factor αon performance and reveal the relations between2D-LPP/MSD and2D-DLPP. Experimental results on the ORL and Yale face databases show the effec-tiveness of the proposed2D-DLPP/MSD.6. As a general framework, Laplacian embedding, based on a pairwise similaritymatrix, infers low dimensional representations from high dimensional data. However,it generally suffers from three issues:1) algorithmic performance is sensitive to thesize of neighbors,2) the algorithm encounters the well-known small sample size (SSS)problem, and3) the algorithm de-emphasizes small distance pairs. To address theseissues, here we propose a general exponential framework by using matrx exponential.In the framework, the matrix exponential can be roughly interpreted by the randomwalk over the feature similarity matrix, and thus is more robust. The positive definiteproperty of matrix exponential deals with the SSS problem. The behavior of the decayfunction of Exponential Embedding is more significant to emphasize small distancepairs. Under this framework, we apply matrix exponential to extend many popularLaplacian embedding algorithms, e.g., LPP, UDP and MFA. Experiments conducted onthe roll, UCI, and Georgia Tech face database show that the proposed new frameworkcan well address the issues mentioned above.