Parameterized Discriminant Analysis Methods

Parameterized Discriminant Analysis Methods

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.ch005
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Abstract

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 linear discriminant analysis, weighted nullspace linear discriminant analysis and weighted LDA in the range of within-class scatter matrix. We offer a brief summery of the chapter in Section 5.4.
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Parameterized Direct Linear Discriminant Analysis

Introduction

Direct LDA (D-LDA) (Yu & Yang, 2001) is an important feature extraction method for SSS problems. It first maps samples into the range of the between-class scatter matrix, and then transforms these projections using a series of regulating matrices. D-LDA can efficiently extract features directly from a high-dimensional input space without the need to first apply other dimensionality reduction techniques such as PCA transformations in Fisherfaces (Belhumeur, Hespanha, & Kriengman, 1997) or pixel grouping in nullspace LDA (N-LDA) (Chen, Liao, Ko, Lin, & Yu, 2000), and as a result has aroused the interest of many researchers in the field of pattern recognition and computer vision. Indeed, there are now many extensions of D-LDA, such as fractional D-LDA (Lu, Plataniotis, & Venetsanopoulos, 2003a), regularized D-LDA (Lu, Plataniotis, &Venetsanopoulos, 2003b; Lu, Plataniotis, & Venetsano-poulos, 2005), kernel D-LDA (Lu, Plataniotis, & Venetsanopoulos, 2003c), and boosting D-LDA (Lu, Plataniotis, Venetsanopoulos, & Li, 2006).

But there nonetheless remain some questions as to its usefulness as a facial feature extraction method. First, as been pointed out in Lu, Plataniotis and Venetsanopoulos (2003b; Lu, Plataniotis, & Venetsanopoulos, 2005), D-LDA performs badly when only two or three samples per individual are used. Second, regulating matrices in D-LDA are either redundant or probably harmful. The second drawback of D-LDA has not been seriously addressed in previous studies.

In this section, we present a new feature extraction method—parameterized direct linear discriminant analysis (PD-LDA) for SSS problems (Song, Zhang, Wang, Liu, & Tao, 2007). As an improvement of D-LDA, PD-LDA inherits advantages of D-LDA such as “direct” and “efficient”. Meanwhile, it greatly enhances the accuracy and robustness of D-LDA.

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Direct Linear Discriminant Analysis

The Algorithm of D-LDA

Let and denote the between- and the within-class scatter matrices respectively. The calculation procedure of D-LDA is as follows:

Step 1. Perform eigenvalue decomposition on the between-class scatter matrix

Let be the eigenvalue matrix of in decreasing order and be the corresponding eigenvector matrix. It follows that

. (1)

Let be the rank of the matrix . Let and , and we have

. (2)

Step 2. Map each sample vector to get its intermediate representation using the projection matrix

Step 3. Perform eigenvalue decomposition on the within-class scatter matrix of the projected samples, which is given by

. (3)

Let be the eigenvalue matrix of in ascending order and be the corresponding eigenvector matrix. It follows that

. (4)

Step 4. Calculate the discriminant matrix and map each sample to The discriminant matrix of D-LDA is given by

. (5)

Complete Chapter List

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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