Tensor Linear Discriminant Analysis

Tensor Linear Discriminant Analysis

David Zhang (Hong Kong Polytechnic University, Hong Kong), Fengxi Song (New Star Research Institute Of Applied Technology, China), Yong Xu (Harbin Institute of Technology, China) and Zhizhen Liang (Shanghai Jiao Tong University, China)
DOI: 10.4018/978-1-60566-200-8.ch009
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Abstract

Linear discriminant analysis is a very effective and important method for feature extraction. In general, image matrices are often transformed into vectors prior to feature extraction, which results in the curse of dimensionality when the dimensions of matrices are huge. In this chapter, classical LDA and its several variants are introduced. In some sense, the variants of LDA can avoid the singularity problem and achieve computational efficiency. Experimental results on biometric data show the usefulness of LDA and its variants in some cases.
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Introduction

Linear discriminant analysis is a popular technique for feature extraction, which has been successfully applied in many fields such as face recognition and character recognition. Linear discriminant analysis seeks to find the direction which maximizes between-class scatter and minimizes the within-class scatter. Based on linear discriminant analysis, Foley and Sammon (1975) proposed optimal discriminant vectors for two-class problems. Duchene and Leclercq (1988) further presented a set of discriminant vectors to solve multi-class problems. Although Foley-Sammon optimal discriminant vectors (FSODV) are orthogonal and perform well in some cases, the features which are obtained by optimal orthogonal discriminant vectors are statistically correlated. To avoid this problem, Jin, Yang, Hu, and Luo (2001) proposed a new set of uncorrelated discriminant vectors(UDV) which is proved to be more powerful than that of optimal orthogonal discriminant vectors in some cases. Then Jing, Zhang, and Jin (2003) further stated improvements on uncorrelated optimal discriminant vectors. Subsequently, Xu, Yang, and Jin (2003) studied the relationship between the Fisher criterion values of FSODV and UDV. Then Xu, Yang, and Jin (2004) developed a new model for Fisher discriminant analysis, which applies the maximal Fisher criterion and the minimal statistical correlation between features. Since the methods mentioned above are based on vectors rather than matrices, these methods face the computational difficulty when the dimension of data is too huge. To overcome this problem, Liu, Cheng, and Yang (1993) firstly proposed a novel linear projection method, which performs linear discriminant analysis in terms of image matrices. However, feature vectors using Liu’s method could be statistically correlated. In order to effectively deal with this problem, Yang, Yang, Frangi, and Zhang (2003) proposed a set of two-dimensional (2D) projection vectors which satisfy conjugate orthogonal constraints. Most importantly, feature vectors obtained by Yang’s method are statically uncorrelated. Then Liang, Shi, and Zhang (2006) proposed a new technique for 2D Fisher discriminant analysis. In their algorithm, the Fisher criterion function is directly constructed in terms of image matrices. Then they utilize the Fisher criterion and statistical correlation between features to construct an objective function. Then discriminant vectors are obtained in terms of the objection function. At the same time, they theoretically analyze that the proposed algorithm is equivalent to uncorrelated two-dimensional discriminant analysis in some condition. In Xiong, Swam and Ahmad (2005), one-sided 2DLDA is developed for classification tasks. Ye, Janardan, Park, and Park (2004) further developed generalized 2DLDA, which can overcome the singularity problem and achieves the computational efficiency. In Liang (2006; Yan et al., 2005), a multilinear generalization of linear discriminant analysis is discussed and an iterative algorithm is developed for solving multilinear linear discriminant analysis. Meanwhile, a non-iterative algorithm is also proposed in Liang (2006). In addition, multilinear LDA provides a unified framework for classical LDA and 2DLDA.

Basic Algorithms

Notations: Let denote a set of images, . Each image belongs to exactly one of c object class. The number of images in class is denoted by and . Let ,where vec denotes the vector operator which can convert the matrix by stacking the column of the matrix. .

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Table of Contents
Acknowledgment
Chapter 1
Overview  (pages 1-23)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
A biometric system can be regarded as a pattern recognition system. In this chapter, we discuss two advanced pattern recognition technologies for... Sample PDF
Overview
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Chapter 2
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
This chapter is a brief introduction to biometric discriminant analysis technologies — Section I of the book. Section 2.1 describes two kinds of... Sample PDF
Discriminant Analysis for Biometric Recognition
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Chapter 3
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
As mentioned in Chapter II, there are two kinds of LDA approaches: classification- oriented LDA and feature extraction-oriented LDA. In most... Sample PDF
Discriminant Criteria for Pattern Classification
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Chapter 4
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we first give a brief introduction to Fisher linear discriminant, Foley- Sammon discriminant, orthogonal component discriminant... Sample PDF
Orthogonal Discriminant Analysis Methods
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Chapter 5
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we mainly present three kinds of weighted LDA methods. In Sections 5.1, 5.2 and 5.3, we respectively present parameterized direct... Sample PDF
Parameterized Discriminant Analysis Methods
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Chapter 6
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we introduce two novel facial feature extraction methods. The first is multiple maximum scatter difference (MMSD) which is an... Sample PDF
Two Novel Facial Feature Extraction Methods
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Chapter 7
Tensor Space  (pages 135-149)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we first give the background materials for developing tensor discrimination technologies in Section 7.1. Section 7.2 introduces... Sample PDF
Tensor Space
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Chapter 8
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
Tensor principal component analysis (PCA) is an effective method for data reconstruction and recognition. In this chapter, some variants of... Sample PDF
Tensor Principal Component Analysis
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Chapter 9
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
Linear discriminant analysis is a very effective and important method for feature extraction. In general, image matrices are often transformed into... Sample PDF
Tensor Linear Discriminant Analysis
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Chapter 10
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we describe two tensor-based subspace analysis approaches (tensor ICA and tensor NMF) that can be used in many fields like face... Sample PDF
Tensor Independent Component Analysis and Tensor Non-Negative Factorization
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Chapter 11
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In this chapter, we describe tensor-based classifiers, tensor canonical correlation analysis and tensor partial least squares, which can be used in... Sample PDF
Other Tensor Analysis and Further Direction
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Chapter 12
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
In the past decades while biometrics attracts increasing attention of researchers, people also have found that the biometric system using a single... Sample PDF
From Single Biometrics to Multi-Biometrics
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Chapter 13
Feature Level Fusion  (pages 273-304)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
This chapter introduces the basis of feature level fusion and presents two feature level fusion examples. As the beginning, Section 13.1 provides an... Sample PDF
Feature Level Fusion
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Chapter 14
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
With this chapter we aims at describing several basic aspects of matching score level fusion. Section 14.1 provides a description of basic... Sample PDF
Matching Score Level Fusion
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Chapter 15
Decision Level Fusion  (pages 328-348)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
With this chapter, we first present a variety of decision level fusion rules and classifier selection approaches, and then show a case study of face... Sample PDF
Decision Level Fusion
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Chapter 16
Book Summary  (pages 349-358)
David Zhang, Fengxi Song, Yong Xu, Zhizhen Liang
With the title “Advanced Pattern Recognition Technologies with Applications to Biometrics” this book mainly focuses on two kinds of advanced... Sample PDF
Book Summary
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About the Authors