Linear discriminant analysis : a detailed tutorial

Linear Discriminant Analysis (LDA) is a very common technique for dimensionality reduction problems as a preprocessing step for machine learning and pattern classification applications. At the same time, it is usually used as a black box, but (sometimes) not well understood. The aim of this paper is to build a solid intuition for what is LDA, and how LDA works, thus enabling readers of all levels be able to get a better understanding of the LDA and to know how to apply this technique in different applications. The paper first gave the basic definitions and steps of how LDA technique works supported with visual explanations of these steps. Moreover, the two methods of computing the LDA space, i.e. class-dependent and class-independent methods, were explained in details. Then, in a step-by-step approach, two numerical examples are demonstrated to show how the LDA space can be calculated in case of the class-dependent and class-independent methods. Furthermore, two of the most common LDA problems (i.e. Small Sample Size (SSS) and non-linearity problems) were highlighted and illustrated, and state-of-the-art solutions to these problems were investigated and explained. Finally, a number of experiments was conducted with different datasets to (1) investigate the effect of the eigenvectors that used in the LDA space on the robustness of the extracted feature for the classification accuracy, and (2) to show when the SSS problem occurs and how it can be addressed

Learning in high dimensions with projected linear discriminants

The enormous power of modern computers has made possible the statistical modelling of data with dimensionality that would have made this task inconceivable only decades ago. However, experience in such modelling has made researchers aware of many issues associated with working in high-dimensional domains, collectively known as `the curse of dimensionality', which can confound practitioners' desires to build good models of the world from these data. When the dimensionality is very large, low-dimensional methods and geometric intuition both break down in these high-dimensional spaces. To mitigate the dimensionality curse we can use low-dimensional representations of the original data that capture most of the information it contained. However, little is currently known about the effect of such dimensionality reduction on classifier performance. In this thesis we develop theory quantifying the effect of random projection - a recent, very promising, non-adaptive dimensionality reduction technique - on the classification performance of Fisher's Linear Discriminant (FLD), a successful and widely-used linear classifier. We tackle the issues associated with small sample size and high-dimensionality by using randomly projected FLD ensembles, and we develop theory explaining why our new approach performs well. Finally, we quantify the generalization error of Kernel FLD, a related non-linear projected classifier

Processing of image sequences from fundus camera

Cílem mé diplomové práce bylo navrhnout metodu analýzy retinálních sekvencí, která bude hodnotit kvalitu jednotlivých snímků. V teoretické části se také zabývám vlastnostmi retinálních sekvencí a způsobem registrace snímků z fundus kamery. V praktické části je implementována metoda hodnocení kvality snímků, která je otestována na reálných retinálních sekvencích a vyhodnocena její úspěšnost. Práce hodnotí i vliv této metody na registraci retinálních snímků.The aim of my master's thesis was to propose a method of retinal sequence analysis which will evaluate the quality of each frame. In the theoretical part, I will also deal with the properties of retinal sequences and the way of registering the images of the fundus camera. In the practical part the method of evaluating image quality is implemented. This algorithm is tested on real retinal sequences and its success is assessed. This work also evaluates the impact of proposed method on the registration of retinal images.

SPARSE RECOVERY BY NONCONVEX LIPSHITZIAN MAPPINGS

In recent years, the sparsity concept has attracted considerable attention in areas of applied mathematics and computer science, especially in signal and image processing fields. The general framework of sparse representation is now a mature concept with solid basis in relevant mathematical fields, such as probability, geometry of Banach spaces, harmonic analysis, theory of computability, and information-based complexity. Together with theoretical and practical advancements, also several numeric methods and algorithmic techniques have been developed in order to capture the complexity and the wide scope that the theory suggests. Sparse recovery relays over the fact that many signals can be represented in a sparse way, using only few nonzero coefficients in a suitable basis or overcomplete dictionary. Unfortunately, this problem, also called `0-norm minimization, is not only NP-hard, but also hard to approximate within an exponential factor of the optimal solution. Nevertheless, many heuristics for the problem has been obtained and proposed for many applications. This thesis provides new regularization methods for the sparse representation problem with application to face recognition and ECG signal compression. The proposed methods are based on fixed-point iteration scheme which combines nonconvex Lipschitzian-type mappings with canonical orthogonal projectors. The first are aimed at uniformly enhancing the sparseness level by shrinking effects, the latter to project back into the feasible space of solutions. In the second part of this thesis we study two applications in which sparseness has been successfully applied in recent areas of the signal and image processing: the face recognition problem and the ECG signal compression problem

Kernel PCA and the Nyström method

This thesis treats kernel PCA and the Nystrom method. We present a novel incre- ¨ mental algorithm for calculation of kernel PCA, which we extend to incremental calculation of the Nystrom approximation. We suggest a new data-dependent ¨ method to select the number of data points to include in the Nystrom subset, ¨ and create a statistical hypothesis test for the same purpose. We further present a cross-validation procedure for kernel PCA to select the number of principal components to retain. Finally, we derive kernel PCA with the Nystrom method ¨ in line with linear PCA and study its statistical accuracy through a confidence bound

Discriminant feature pursuit: from statistical learning to informative learning.

Lin Dahua.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 233-250).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- The Problem We are Facing --- p.1Chapter 1.2 --- Generative vs. Discriminative Models --- p.2Chapter 1.3 --- Statistical Feature Extraction: Success and Challenge --- p.3Chapter 1.4 --- Overview of Our Works --- p.5Chapter 1.4.1 --- New Linear Discriminant Methods: Generalized LDA Formulation and Performance-Driven Sub space Learning --- p.5Chapter 1.4.2 --- Coupled Learning Models: Coupled Space Learning and Inter Modality Recognition --- p.6Chapter 1.4.3 --- Informative Learning Approaches: Conditional Infomax Learning and Information Chan- nel Model --- p.6Chapter 1.5 --- Organization of the Thesis --- p.8Chapter I --- History and Background --- p.10Chapter 2 --- Statistical Pattern Recognition --- p.11Chapter 2.1 --- Patterns and Classifiers --- p.11Chapter 2.2 --- Bayes Theory --- p.12Chapter 2.3 --- Statistical Modeling --- p.14Chapter 2.3.1 --- Maximum Likelihood Estimation --- p.14Chapter 2.3.2 --- Gaussian Model --- p.15Chapter 2.3.3 --- Expectation-Maximization --- p.17Chapter 2.3.4 --- Finite Mixture Model --- p.18Chapter 2.3.5 --- A Nonparametric Technique: Parzen Windows --- p.21Chapter 3 --- Statistical Learning Theory --- p.24Chapter 3.1 --- Formulation of Learning Model --- p.24Chapter 3.1.1 --- Learning: Functional Estimation Model --- p.24Chapter 3.1.2 --- Representative Learning Problems --- p.25Chapter 3.1.3 --- Empirical Risk Minimization --- p.26Chapter 3.2 --- Consistency and Convergence of Learning --- p.27Chapter 3.2.1 --- Concept of Consistency --- p.27Chapter 3.2.2 --- The Key Theorem of Learning Theory --- p.28Chapter 3.2.3 --- VC Entropy --- p.29Chapter 3.2.4 --- Bounds on Convergence --- p.30Chapter 3.2.5 --- VC Dimension --- p.35Chapter 4 --- History of Statistical Feature Extraction --- p.38Chapter 4.1 --- Linear Feature Extraction --- p.38Chapter 4.1.1 --- Principal Component Analysis (PCA) --- p.38Chapter 4.1.2 --- Linear Discriminant Analysis (LDA) --- p.41Chapter 4.1.3 --- Other Linear Feature Extraction Methods --- p.46Chapter 4.1.4 --- Comparison of Different Methods --- p.48Chapter 4.2 --- Enhanced Models --- p.49Chapter 4.2.1 --- Stochastic Discrimination and Random Subspace --- p.49Chapter 4.2.2 --- Hierarchical Feature Extraction --- p.51Chapter 4.2.3 --- Multilinear Analysis and Tensor-based Representation --- p.52Chapter 4.3 --- Nonlinear Feature Extraction --- p.54Chapter 4.3.1 --- Kernelization --- p.54Chapter 4.3.2 --- Dimension reduction by Manifold Embedding --- p.56Chapter 5 --- Related Works in Feature Extraction --- p.59Chapter 5.1 --- Dimension Reduction --- p.59Chapter 5.1.1 --- Feature Selection --- p.60Chapter 5.1.2 --- Feature Extraction --- p.60Chapter 5.2 --- Kernel Learning --- p.61Chapter 5.2.1 --- Basic Concepts of Kernel --- p.61Chapter 5.2.2 --- The Reproducing Kernel Map --- p.62Chapter 5.2.3 --- The Mercer Kernel Map --- p.64Chapter 5.2.4 --- The Empirical Kernel Map --- p.65Chapter 5.2.5 --- Kernel Trick and Kernelized Feature Extraction --- p.66Chapter 5.3 --- Subspace Analysis --- p.68Chapter 5.3.1 --- Basis and Subspace --- p.68Chapter 5.3.2 --- Orthogonal Projection --- p.69Chapter 5.3.3 --- Orthonormal Basis --- p.70Chapter 5.3.4 --- Subspace Decomposition --- p.70Chapter 5.4 --- Principal Component Analysis --- p.73Chapter 5.4.1 --- PCA Formulation --- p.73Chapter 5.4.2 --- Solution to PCA --- p.75Chapter 5.4.3 --- Energy Structure of PCA --- p.76Chapter 5.4.4 --- Probabilistic Principal Component Analysis --- p.78Chapter 5.4.5 --- Kernel Principal Component Analysis --- p.81Chapter 5.5 --- Independent Component Analysis --- p.83Chapter 5.5.1 --- ICA Formulation --- p.83Chapter 5.5.2 --- Measurement of Statistical Independence --- p.84Chapter 5.6 --- Linear Discriminant Analysis --- p.85Chapter 5.6.1 --- Fisher's Linear Discriminant Analysis --- p.85Chapter 5.6.2 --- Improved Algorithms for Small Sample Size Problem . --- p.89Chapter 5.6.3 --- Kernel Discriminant Analysis --- p.92Chapter II --- Improvement in Linear Discriminant Analysis --- p.100Chapter 6 --- Generalized LDA --- p.101Chapter 6.1 --- Regularized LDA --- p.101Chapter 6.1.1 --- Generalized LDA Implementation Procedure --- p.101Chapter 6.1.2 --- Optimal Nonsingular Approximation --- p.103Chapter 6.1.3 --- Regularized LDA algorithm --- p.104Chapter 6.2 --- A Statistical View: When is LDA optimal? --- p.105Chapter 6.2.1 --- Two-class Gaussian Case --- p.106Chapter 6.2.2 --- Multi-class Cases --- p.107Chapter 6.3 --- Generalized LDA Formulation --- p.108Chapter 6.3.1 --- Mathematical Preparation --- p.108Chapter 6.3.2 --- Generalized Formulation --- p.110Chapter 7 --- Dynamic Feedback Generalized LDA --- p.112Chapter 7.1 --- Basic Principle --- p.112Chapter 7.2 --- Dynamic Feedback Framework --- p.113Chapter 7.2.1 --- Initialization: K-Nearest Construction --- p.113Chapter 7.2.2 --- Dynamic Procedure --- p.115Chapter 7.3 --- Experiments --- p.115Chapter 7.3.1 --- Performance in Training Stage --- p.116Chapter 7.3.2 --- Performance on Testing set --- p.118Chapter 8 --- Performance-Driven Subspace Learning --- p.119Chapter 8.1 --- Motivation and Principle --- p.119Chapter 8.2 --- Performance-Based Criteria --- p.121Chapter 8.2.1 --- The Verification Problem and Generalized Average Margin --- p.122Chapter 8.2.2 --- Performance Driven Criteria based on Generalized Average Margin --- p.123Chapter 8.3 --- Optimal Subspace Pursuit --- p.125Chapter 8.3.1 --- Optimal threshold --- p.125Chapter 8.3.2 --- Optimal projection matrix --- p.125Chapter 8.3.3 --- Overall procedure --- p.129Chapter 8.3.4 --- Discussion of the Algorithm --- p.129Chapter 8.4 --- Optimal Classifier Fusion --- p.130Chapter 8.5 --- Experiments --- p.131Chapter 8.5.1 --- Performance Measurement --- p.131Chapter 8.5.2 --- Experiment Setting --- p.131Chapter 8.5.3 --- Experiment Results --- p.133Chapter 8.5.4 --- Discussion --- p.139Chapter III --- Coupled Learning of Feature Transforms --- p.140Chapter 9 --- Coupled Space Learning --- p.141Chapter 9.1 --- Introduction --- p.142Chapter 9.1.1 --- What is Image Style Transform --- p.142Chapter 9.1.2 --- Overview of our Framework --- p.143Chapter 9.2 --- Coupled Space Learning --- p.143Chapter 9.2.1 --- Framework of Coupled Modelling --- p.143Chapter 9.2.2 --- Correlative Component Analysis --- p.145Chapter 9.2.3 --- Coupled Bidirectional Transform --- p.148Chapter 9.2.4 --- Procedure of Coupled Space Learning --- p.151Chapter 9.3 --- Generalization to Mixture Model --- p.152Chapter 9.3.1 --- Coupled Gaussian Mixture Model --- p.152Chapter 9.3.2 --- Optimization by EM Algorithm --- p.152Chapter 9.4 --- Integrated Framework for Image Style Transform --- p.154Chapter 9.5 --- Experiments --- p.156Chapter 9.5.1 --- Face Super-resolution --- p.156Chapter 9.5.2 --- Portrait Style Transforms --- p.157Chapter 10 --- Inter-Modality Recognition --- p.162Chapter 10.1 --- Introduction to the Inter-Modality Recognition Problem . . . --- p.163Chapter 10.1.1 --- What is Inter-Modality Recognition --- p.163Chapter 10.1.2 --- Overview of Our Feature Extraction Framework . . . . --- p.163Chapter 10.2 --- Common Discriminant Feature Extraction --- p.165Chapter 10.2.1 --- Formulation of the Learning Problem --- p.165Chapter 10.2.2 --- Matrix-Form of the Objective --- p.168Chapter 10.2.3 --- Solving the Linear Transforms --- p.169Chapter 10.3 --- Kernelized Common Discriminant Feature Extraction --- p.170Chapter 10.4 --- Multi-Mode Framework --- p.172Chapter 10.4.1 --- Multi-Mode Formulation --- p.172Chapter 10.4.2 --- Optimization Scheme --- p.174Chapter 10.5 --- Experiments --- p.176Chapter 10.5.1 --- Experiment Settings --- p.176Chapter 10.5.2 --- Experiment Results --- p.177Chapter IV --- A New Perspective: Informative Learning --- p.180Chapter 11 --- Toward Information Theory --- p.181Chapter 11.1 --- Entropy and Mutual Information --- p.181Chapter 11.1.1 --- Entropy --- p.182Chapter 11.1.2 --- Relative Entropy (Kullback Leibler Divergence) --- p.184Chapter 11.2 --- Mutual Information --- p.184Chapter 11.2.1 --- Definition of Mutual Information --- p.184Chapter 11.2.2 --- Chain rules --- p.186Chapter 11.2.3 --- Information in Data Processing --- p.188Chapter 11.3 --- Differential Entropy --- p.189Chapter 11.3.1 --- Differential Entropy of Continuous Random Variable . --- p.189Chapter 11.3.2 --- Mutual Information of Continuous Random Variable . --- p.190Chapter 12 --- Conditional Infomax Learning --- p.191Chapter 12.1 --- An Overview --- p.192Chapter 12.2 --- Conditional Informative Feature Extraction --- p.193Chapter 12.2.1 --- Problem Formulation and Features --- p.193Chapter 12.2.2 --- The Information Maximization Principle --- p.194Chapter 12.2.3 --- The Information Decomposition and the Conditional Objective --- p.195Chapter 12.3 --- The Efficient Optimization --- p.197Chapter 12.3.1 --- Discrete Approximation Based on AEP --- p.197Chapter 12.3.2 --- Analysis of Terms and Their Derivatives --- p.198Chapter 12.3.3 --- Local Active Region Method --- p.200Chapter 12.4 --- Bayesian Feature Fusion with Sparse Prior --- p.201Chapter 12.5 --- The Integrated Framework for Feature Learning --- p.202Chapter 12.6 --- Experiments --- p.203Chapter 12.6.1 --- A Toy Problem --- p.203Chapter 12.6.2 --- Face Recognition --- p.204Chapter 13 --- Channel-based Maximum Effective Information --- p.209Chapter 13.1 --- Motivation and Overview --- p.209Chapter 13.2 --- Maximizing Effective Information --- p.211Chapter 13.2.1 --- Relation between Mutual Information and Classification --- p.211Chapter 13.2.2 --- Linear Projection and Metric --- p.212Chapter 13.2.3 --- Channel Model and Effective Information --- p.213Chapter 13.2.4 --- Parzen Window Approximation --- p.216Chapter 13.3 --- Parameter Optimization on Grassmann Manifold --- p.217Chapter 13.3.1 --- Grassmann Manifold --- p.217Chapter 13.3.2 --- Conjugate Gradient Optimization on Grassmann Manifold --- p.219Chapter 13.3.3 --- Computation of Gradient --- p.221Chapter 13.4 --- Experiments --- p.222Chapter 13.4.1 --- A Toy Problem --- p.222Chapter 13.4.2 --- Face Recognition --- p.223Chapter 14 --- Conclusion --- p.23

Robust Face Recognition based on Color and Depth Information

One of the most important advantages of automatic human face recognition is its nonintrusiveness property. Face images can sometime be acquired without user's knowledge or explicit cooperation. However, face images acquired in an uncontrolled environment can appear with varying imaging conditions. Traditionally, researchers focus on tackling this problem using 2D gray-scale images due to the wide availability of 2D cameras and the low processing and storage cost of gray-scale data. Nevertheless, face recognition can not be performed reliably with 2D gray-scale data due to insu_cient information and its high sensitivity to pose, expression and illumination variations. Recent rapid development in hardware makes acquisition and processing of color and 3D data feasible. This thesis aims to improve face recognition accuracy and robustness using color and 3D information.In terms of color information usage, this thesis proposes several improvements over existing approaches. Firstly, the Block-wise Discriminant Color Space is proposed, which learns the discriminative color space based on local patches of a human face image instead of the holistic image, as human faces display different colors in different parts. Secondly, observing that most of the existing color spaces consist of at most three color components, while complementary information can be found in multiple color components across multiple color spaces and therefore the Multiple Color Fusion model is proposed to search and utilize multiple color components effectively. Lastly, two robust color face recognition algorithms are proposed. The Color Sparse Coding method can deal with face images with noise and occlusion. The Multi-linear Color Tensor Discriminant method harnesses multi-linear technique to handle non-linear data. Experiments show that all the proposed methods outperform their existing competitors.In terms of 3D information utilization, this thesis investigates the feasibility of face recognition using Kinect. Unlike traditional 3D scanners which are too slow in speed and too expensive in cost for broad face recognition applications, Kinect trades data quality for high speed and low cost. An algorithm is proposed to show that Kinect data can be used for face recognition despite its noisy nature. In order to fully utilize Kinect data, a more sophisticated RGB-D face recognition algorithm is developed which harnesses theColor Sparse Coding framework and 3D information to perform accurate face recognition robustly even under simultaneous varying conditions of poses, illuminations, expressionsand disguises

Sparse Dimensionality Reduction Methods: Algorithms and Applications

Ph.DDOCTOR OF PHILOSOPH