Key features:

• Explains the main theoretical ideas behind each method (which are augmented with a rigorous mathematical derivation of the formulas), their implementation (in C++) and demonstrated working in real applications.
• Places an emphasis on tensor and statistical based approaches within object detection and recognition.
• Provides an overview of image clustering and classification methods which includes subspace and kernel based processing, mean shift and Kalman filter, neural networks, and k-means methods.
• Contains numerous case study examples of mainly automotive applications.
• Includes a companion website hosting full C++ implementation, of topics presented in the book as a software library, and an accompanying manual to the software platform.

Boguslaw Cyganek received his M.Sc. degree in electronics in 1993, then in computer science in 1996 from the AGH University of Science and Technology, Krakow, Poland. He obtained his Ph.D. degree cum laude in 2001 with a thesis on correlation of stereo images, and D.Sc. degree in 2011 with a thesis on methods and algorithms of object recognition in digital images. During the recent years, Dr. Boguslaw Cyganek has been cooperating with many scientific centers in development of computer vision systems. He has also gained several years of practical experience working as a Software Development Manager and a Senior Software Engineer both in the USA and Poland. He is currently a researcher and lecturer at the Department of Electronics, AGH University of Science and Technology. His research interests include computer vision, pattern recognition, as well as development of programmable devices and embedded systems. He is an author or a co-author of over eighty conference and journal papers and four books including An Introduction to 3D Computer Vision Techniques and Algorithms published by Wiley. Dr. Cyganek is a member of the IEEE, IAPR and SIAM.

Preface 7

Acknowledgements 8

Notations and Abbreviations 9

1. Introduction 13

1.1. A Sample of Computer Vision 14

1.2. Overview of Book Contents 17

2. Tensor Methods in Computer Vision 19

2.1. Abstract 19

2.2. Tensor – a Mathematical Object 19

2.2.1. Main Properties of linear spaces 20

2.2.2. Concept of a Tensor 20

2.3. Tensor – a Data Object 21

2.3.1. IMPLEMENTATION Tensor Representation 23

2.4. Basic Properties of Tensors 31

2.4.1. Notation of Tensor Indices and Components 32

2.4.2. Tensor Products 34

2.5. Tensor Distance Measures 35

2.5.1. Overview of Tensor Distances 37

2.5.1.1. Computation of Matrix Exponent and Logarithm Functions 38

2.5.2. Euclidean Image Distance and Standardizing Transform 42

2.6. Filtering of Tensor Fields 44

2.6.1. Order Statistic Filtering of Tensor Data 44

2.6.2. Anisotropic Diffusion Filtering 47

2.6.3. IMPLEMENTATION of Diffusion Processes 50

2.7. Looking into Images with the Structural Tensor 54

2.7.1. Structural Tensor in Two-Dimensional Image Space 56

2.7.2. Spatio-Temporal Structural Tensor 58

2.7.3. Multi-Channel and Scale-Space Structural Tensor 60

2.7.4. Extended Structural Tensor 61

2.7.4.1. IMPLEMENTATION of the Linear and Nonlinear Structural Tensor 63

2.8. Object Representation with Tensor of Inertia and Moments 67

2.8.1. IMPLEMENTATION of Moments and their Invariants 69

2.9. Eigen-Decomposition and Representation of Tensors 71

2.10. Tensor Invariants 74

2.11. Geometry of Multiple Views: The Multi-Focal Tensor 74

2.12. Multilinear Tensor Methods 76

2.12.1. Basic Concepts of the Multilinear Algebra 78

2.12.1.1. Tensor Flattening 78

2.12.1.2. The k-mode product of a tensor and a matrix 84

2.12.1.3. Ranks of a Tensor 88

2.12.1.4. IMPLEMENTATION of Basic Operations on Tensors 89

2.12.2. Higher-Order Singular Value Decomposition (HOSVD) 97

2.12.3. Computation of the HOSVD 99

2.12.3.1. Implementation of the HOSVD Decomposition 102

2.12.4. HOSVD Induced Bases 104

2.12.5. Tensor Best Rank-1 Approximation 106

2.12.6. Rank-1 Decomposition of Tensors 108

2.12.7. Best Rank-(R1, R2, …, RP) Approximations 113

2.12.8. Computation of the Best Rank-(R1, R2, …, RP) Approximations 116

2.12.8.1. IMPLEMENTATION - Rank Tensor Decompositions 118

2.12.8.2. CASE STUDY - Data Dimensionality Reduction 124

2.12.9. Subspace Data Representation 127

2.12.10. Nonnegative Matrix Factorization 129

2.12.11. Computation of the Nonnegative Matrix Factorization 131

2.12.12. Image Representation with NMF 135

2.12.13. Implementation of the Nonnegative Matrix Factorization 138

2.12.14. Nonnegative Tensor Factorization 143

2.12.15. Multilinear Methods of Object Recognition 146

2.13. Closure 150

2.13.1. Chapter Summary 150

2.13.2. Further Reading 151

2.13.3. Problems and Exercises 151

3. Classification Methods and Algorithms 153

3.1. Abstract 153

3.2. Classification Framework 153

3.2.1. IMPLEMENTATION Computer Representation of Features 155

3.3. Subspace Methods for Object Recognition 157

3.3.1. Principal Component Analysis 157

3.3.1.1. Computation of the PCA 160

Low dimensional features (L??N) 160

High dimensional features (L??N) 167

3.3.1.2. PCA for Multi-Channel Image Processing 169

3.3.1.3. PCA for Background Subtraction 172

3.3.2. Subspace Pattern Classification 174

3.4. Statistical Formulation of the Object Recognition 178

3.4.1. Parametric and Nonparametric Methods 178

3.4.2. Probabilistic Framework 178

3.4.3. Bayes Decision Rule 179

3.4.4. Maximum a posteriori classification scheme 179

3.4.5. Binary classification problem 181

3.5. Parametric Methods – Mixture of Gaussians 182

3.6. The Kalman Filter 186

3.7. Nonparametric Methods 189

3.7.1. Histogram Based Techniques 189

3.7.2. Comparing Histograms 191

3.7.3. IMPLEMENTATION - Multi-Dimensional Histograms 195

3.7.4. Parzen Method 197

3.7.4.1. Kernel Based Methods 198

3.7.4.2. Nearest-Neighbor Method 199

3.8. The Mean Shift Method 200

3.8.1. Introduction to the Mean Shift 201

3.8.2. Continuously Adaptive Mean Shift Method (CamShift) 205

3.8.3. Algorithmic Aspects of the Mean Shift Tracking 206

3.8.3.1. Tracking of Multiple Features 206

3.8.3.2. Tracking of Multiple Objects 207

3.8.3.3. Fuzzy Approach to the CamShift 207

3.8.3.4. Discrimination with Background Information 208

3.8.3.5. Adaptive update of the classifiers 209

3.8.4. IMPLEMENTATION of the CamShift Method 209

3.9. Neural Networks 211

3.9.1. Probabilistic Neural Network 212

3.9.2. IMPLEMENTATION - Probabilistic Neural Network 214

3.9.3. Hamming Neural Network 217

3.9.3.1. IMPLEMENTATION of the Hamming Neural Network 219

3.9.4. Morphological Neural Network 223

3.9.4.1. IMPLEMENTATION of the Morphological Neural Network 225

3.10. Kernels in Vision Pattern Recognition 230

3.10.1. Kernel Functions 233

3.10.2. IMPLEMENTATION - Kernels 237

3.11. Data Clustering 241

3.11.1. The k-Means Algorithm 242

3.11.2. Fuzzy c-Means 244

3.11.3. Kernel Fuzzy c-Means 246

3.11.4. Measures of Cluster Quality 247

3.11.5. IMPLEMENTATION issues 249

3.12. Support Vector Domain Description 256

3.12.1.1. Implementation of Support Vector Machines 261

3.12.2. Architecture of the Ensemble of One-Class Classifiers 261

3.13. Appendix – MATLAB and other Packages for Pattern Classification 263

3.14. Closure 263

3.14.1. Chapter Summary 263

3.14.2. Further Reading 263

3.14.3. Problems and Exercises 264

4. Object Detection and Tracking 265

4.1. Introduction 265

4.2. Direct Pixel Classification 265

4.2.1. Ground-Truth Data Collection 265

4.2.2. CASE STUDY – Human Skin Detection 266

4.2.3. CASE STUDY – Pixel Based Road Signs Detection 270

4.2.3.1. Fuzzy Approach 270

4.2.3.2. SVM Based Approach 273

4.2.4. Pixel Based Image Segmentation with Ensemble of Classifiers 276

4.3. Detection of Basic Shapes 279

4.3.1. Detection of Line Segments 280

4.3.2. Up-Write Detection of Convex Shapes 281

4.4. Figure Detection 284

4.4.1. Detection of Regular Shapes from Characteristic Points 284

4.4.2. Clustering of the Salient Points 287

4.4.3. Adaptive Window Growing Method 288

4.4.4. Figure Verification 289

4.4.5. CASE STUDY – Road Signs Detection System 290

4.5. CASE STUDY – Road Signs Tracking and Recognition 294

4.6. CASE STUDY – Framework for Object Tracking 297

4.7. Pedestrian Detection 303

4.8. Closure 307

4.8.1. Chapter Summary 307

4.8.2. Further Reading 307

4.8.3. Problems and Exercises 307

5. Object Recognition 309

5.1. Abstract 309

5.2. Recognition from Tensor Phase Histograms and Morphological Scale Space 309

5.2.1. Computation of the Tensor Phase Histograms in Morphological Scale 311

5.2.2. Matching of the Tensor Phase Histograms 312

5.2.3. CASE STUDY – Objects Recognition with Tensor  Phase Histograms in Morphological Scale-Space 313

5.3. Invariant Based Recognition 318

5.3.1. CASE STUDY – Pictogram Recognition with Affine Moment  Invariants 319

5.4. Template Based Recognition 321

5.4.1. Template Matching for Road Signs Recognition 322

5.4.2. Special Distances for Template Matching 324

5.4.3. Recognition with the Log-Polar and Scale Spaces 324

5.5. Recognition from Deformable Models 330

5.6. Ensembles of Classifiers 331

5.7. CASE STUDY – Ensemble of Classifiers for Road Signs Recognition from Deformed Prototypes 333

5.7.1. Architecture of the Road Signs Recognition System 334

5.7.2. Module for Recognition of Warning Signs 337

5.7.3. The arbitration unit 341

5.8. Recognition Based on Tensor Decompositions 342

5.8.1. Pattern Recognition in Sub-Spaces Spanned by the HOSVD Decomposition of Pattern Tensors 342

5.8.2. CASE STUDY - Road Signs Recognition System Based on Decomposition of Tensors with Deformable Pattern Prototypes 343

5.8.3. CASE STUDY - Handwritten Digit Recognition with Tensor Decomposition Method 349

5.8.4. IMPLEMENTATION of the Tensor Subspace Classifiers 352

5.9. Eye Recognition for Driver's State Monitoring 355

5.10. Object Category Recognition 361

5.10.1. Part-Based Object Recognition 361

5.10.2. Recognition with Bag-of-Visual-Words 362

5.11. Closure 364

5.11.1. Chapter Summary 364

5.11.2. Further Reading 365

5.11.3. Problems and Exercises 365

6. Appendix 366

6.1. Abstract 366

6.2. Morphological Scale-Space 366

6.3. Morphological Tensor Operators 368

6.4. Geometry of Quadratic Forms 368

6.5. Testing Classifiers 369

6.5.1. Implementation of the Confusion Matrix and Testing Object Detection in Images 372

6.6. Code Acceleration with OpenMP 375

6.6.1. Recipes for Object-Oriented Code Design with OpenMP 376

6.6.2. Hints on Using and Code Porting to OpenMP 380

6.6.3. Performance Analysis 383

6.1. Useful MATLAB Functions for Matrix and Tensor Processing 384

6.2. Short Guide to the Attached Software 389

6.3. Closure 389

6.3.1. Chapter Summary 389

6.3.2. Further Reading 390

6.3.3. Problems and Exercises 390

7. References 391

8. Index 409