Jan Flusser is a professor of Computer Science and a director of the Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic. His research interest covers moments and moment invariants, image registration, image fusion, multichannel blind deconvolution, and super-resolution imaging. He has authored and coauthored more than 200 research publications, including the monograph Moments and Moment Invariants in Pattern Recognition (Wiley, 2009), and has delivered 20 tutorials and invited/keynote talks at major conferences. His publications have received about 10,000 citations. Jan Flusser received several scientific awards and prizes, such as the Award of the Chairman of the Czech Science Foundation (2007), the Prize of the Czech Academy of Sciences (2007), the SCOPUS 1000 Award presented by Elsevier (2010), and the Felber Medal of the Czech Technical University for excellent contribution to research and education (2015).

Tomáš Suk received a Ph.D degree in computer science from the Czechoslovak Academy of Sciences in 1992. He is a senior research fellow with the Institute of Information Theory and Automation, Czech Academy of Sciences, Prague. His research interests include invariant features, moment and point-based invariants, color spaces, geometric transformations, and applications in botany, remote sensing, astronomy, medicine, and computer vision. He has authored and coauthored more than 30 journal papers and 50 conference papers in these areas, including tutorials on moment invariants held at the conferences ICIP'07 and SPPRA'09. He coauthored the monograph Moments and Moment Invariants in Pattern Recognition (Wiley, 2009). His publications have received about 1000 citations. In 2002 he received the Otto Wichterle Premium of the Czech Academy of Sciences for young scientists.

Barbara Zitová received her Ph.D degree in software systems from the Charles University, Prague, Czech Republic, in 2000. She is a head of Department of Image Processing at the Institute of Information Theory and Automation, Czech Academy of Sciences, Prague. She teaches courses on Digital Image Processing and Wavelets in Image Processing. Her research interests include geometric invariants, image enhancement, image registration, image fusion, medical image processing, and applications in cultural heritage. She has authored/coauthored more than 70 research publications in these areas, including the monograph Moments and Moment Invariants in Pattern Recognition (Wiley, 2009). In 2003 Barbara Zitová received the Josef Hlavka Student Prize, the Otto Wichterle Premium of the Czech Academy of Sciences for young scientists in 2006, and in 2010 she was awarded by the SCOPUS 1000 Award for receiving more than 1000 citations of a single paper.

Preface xvii

Acknowledgements xxi

1 Motivation 1

1.1 Image analysis by computers 1

1.2 Humans, computers, and object recognition 4

1.3 Outline of the book 5

References 7

2 Introduction to Object Recognition 8

2.1 Feature space 8

2.1.1 Metric spaces and norms 9

2.1.2 Equivalence and partition 11

2.1.3 Invariants 12

2.1.4 Covariants 14

2.1.5 Invariant-less approaches 15

2.2 Categories of the invariants 15

2.2.1 Simple shape features 16

2.2.2 Complete visual features 18

2.2.3 Transformation coefficient features 20

2.2.4 Textural features 21

2.2.5 Wavelet-based features 23

2.2.6 Differential invariants 24

2.2.7 Point set invariants 25

2.2.8 Moment invariants 26

2.3 Classifiers 27

2.3.1 Nearest-neighbor classifiers 28

2.3.2 Support vector machines 31

2.3.3 Neural network classifiers 32

2.3.4 Bayesian classifier 34

2.3.5 Decision trees 35

2.3.6 Unsupervised classification 36

2.4 Performance of the classifiers 37

2.4.1 Measuring the classifier performance 37

2.4.2 Fusing classifiers 38

2.4.3 Reduction of the feature space dimensionality 38

2.5 Conclusion 40

References 41

3 2D Moment Invariants to Translation, Rotation, and Scaling 45

3.1 Introduction 45

3.1.1 Mathematical preliminaries 45

3.1.2 Moments 47

3.1.3 Geometric moments in 2D 48

3.1.4 Other moments 49

3.2 TRS invariants from geometric moments 50

3.2.1 Invariants to translation 50

3.2.2 Invariants to uniform scaling 51

3.2.3 Invariants to non-uniform scaling 52

3.2.4 Traditional invariants to rotation 54

3.3 Rotation invariants using circular moments 56

3.4 Rotation invariants from complex moments 57

3.4.1 Complex moments 57

3.4.2 Construction of rotation invariants 58

3.4.3 Construction of the basis 59

3.4.4 Basis of the invariants of the second and third orders 62

3.4.5 Relationship to the Hu invariants 63

3.5 Pseudoinvariants 67

3.6 Combined invariants to TRS and contrast stretching 68

3.7 Rotation invariants for recognition of symmetric objects 69

3.7.1 Logo recognition 75

3.7.2 Recognition of shapes with different fold numbers 75

3.7.3 Experiment with a baby toy 77

3.8 Rotation invariants via image normalization 81

3.9 Moment invariants of vector fields 86

3.10 Conclusion 92

References 92

4 3D Moment Invariants to Translation, Rotation, and Scaling 95

4.1 Introduction 95

4.2 Mathematical description of the 3D rotation 98

4.3 Translation and scaling invariance of 3D geometric moments 100

4.4 3D rotation invariants by means of tensors 101

4.4.1 Tensors 101

4.4.2 Rotation invariants 102

4.4.3 Graph representation of the invariants 103

4.4.4 The number of the independent invariants 104

4.4.5 Possible dependencies among the invariants 105

4.4.6 Automatic generation of the invariants by the tensor method 106

4.5 Rotation invariants from 3D complex moments 108

4.5.1 Translation and scaling invariance of 3D complex moments 112

4.5.2 Invariants to rotation by means of the group representation theory 112

4.5.3 Construction of the rotation invariants 115

4.5.4 Automated generation of the invariants 117

4.5.5 Elimination of the reducible invariants 118

4.5.6 The irreducible invariants 118

4.6 3D translation, rotation, and scale invariants via normalization 119

4.6.1 Rotation normalization by geometric moments 120

4.6.2 Rotation normalization by complex moments 123

4.7 Invariants of symmetric objects 124

4.7.1 Rotation and reflection symmetry in 3D 124

4.7.2 The influence of symmetry on 3D complex moments 128

4.7.3 Dependencies among the invariants due to symmetry 130

4.8 Invariants of 3D vector fields 131

4.9 Numerical experiments 131

4.9.1 Implementation details 131

4.9.2 Experiment with archeological findings 133

4.9.3 Recognition of generic classes 135

4.9.4 Submarine recognition – robustness to noise test 137

4.9.5 Teddy bears – the experiment on real data 141

4.9.6 Artificial symmetric bodies 142

4.9.7 Symmetric objects from the Princeton Shape Benchmark 143

4.10 Conclusion 147

Appendix 4.A 148

Appendix 4.B 156

Appendix 4.C 158

References 160

5 Affine Moment Invariants in 2D and 3D 163

5.1 Introduction 163

5.1.1 2D projective imaging of 3D world 164

5.1.2 Projective moment invariants 165

5.1.3 Affine transformation 167

5.1.4 2D Affine moment invariants – the history 168

5.2 AMIs derived from the Fundamental theorem 170

5.3 AMIs generated by graphs 171

5.3.1 The basic concept 172

5.3.2 Representing the AMIs by graphs 173

5.3.3 Automatic generation of the invariants by the graph method 173

5.3.4 Independence of the AMIs 174

5.3.5 The AMIs and tensors 180

5.4 AMIs via image normalization 181

5.4.1 Decomposition of the affine transformation 182

5.4.2 Relation between the normalized moments and the AMIs 185

5.4.3 Violation of stability 186

5.4.4 Affine invariants via half normalization 187

5.4.5 Affine invariants from complex moments 187

5.5 The method of the transvectants 190

5.6 Derivation of the AMIs from the Cayley-Aronhold equation 195

5.6.1 Manual solution 195

5.6.2 Automatic solution 198

5.7 Numerical experiments 201

5.7.1 Invariance and robustness of the AMIs 201

5.7.2 Digit recognition 201

5.7.3 Recognition of symmetric patterns 204

5.7.4 The children’s mosaic 208

5.7.5 Scrabble tiles recognition 210

5.8 Affine invariants of color images 214

5.8.1 Recognition of color pictures 217

5.9 Affine invariants of 2D vector fields 218

5.10 3D affine moment invariants 221

5.10.1 The method of geometric primitives 222

5.10.2 Normalized moments in 3D 224

5.10.3 Cayley-Aronhold equation in 3D 225

5.11 Beyond invariants 225

5.11.1 Invariant distance measure between images 225

5.11.2 Moment matching 227

5.11.3 Object recognition as a minimization problem 229

5.11.4 Numerical experiments 229

5.12 Conclusion 231

Appendix 5.A 232

Appendix 5.B 233

References 234

6 Invariants to Image Blurring 237

6.1 Introduction 237

6.1.1 Image blurring – the sources and modeling 237

6.1.2 The need for blur invariants 239

6.1.3 State of the art of blur invariants 239

6.1.4 The chapter outline 246

6.2 An intuitive approach to blur invariants 247

6.3 Projection operators and blur invariants in Fourier domain 249

6.4 Blur invariants from image moments 252

6.5 Invariants to centrosymmetric blur 254

6.6 Invariants to circular blur 256

6.7 Invariants to N-FRS blur 259

6.8 Invariants to dihedral blur 265

6.9 Invariants to directional blur 269

6.10 Invariants to Gaussian blur 272

6.10.1 1D Gaussian blur invariants 274

6.10.2 Multidimensional Gaussian blur invariants 278

6.10.3 2D Gaussian blur invariants from complex moments 279

6.11 Invariants to other blurs 280

6.12 Combined invariants to blur and spatial transformations 282

6.12.1 Invariants to blur and rotation 282

6.12.2 Invariants to blur and affine transformation 283

6.13 Computational issues 284

6.14 Experiments with blur invariants 285

6.14.1 A simple test of blur invariance property 285

6.14.2 Template matching in satellite images 286

6.14.3 Template matching in outdoor images 291

6.14.4 Template matching in astronomical images 291

6.14.5 Face recognition on blurred and noisy photographs 292

6.14.6 Traffic sign recognition 294

6.15 Conclusion 302

Appendix 6.A 303

Appendix 6.B 304

Appendix 6.C 306

Appendix 6.D 308

Appendix 6.E 310

Appendix 6.F 310

Appendix 6.G 311

References 315

7 2D and 3D Orthogonal Moments 320

7.1 Introduction 320

7.2 2D moments orthogonal on a square 322

7.2.1 Hypergeometric functions 323

7.2.2 Legendre moments 324

7.2.3 Chebyshev moments 327

7.2.4 Gaussian-Hermite moments 331

7.2.5 Other moments orthogonal on a square 334

7.2.6 Orthogonal moments of a discrete variable 338

7.2.7 Rotation invariants from moments orthogonal on a square 348

7.3 2D moments orthogonal on a disk 351

7.3.1 Zernike and Pseudo-Zernike moments 352

7.3.2 Fourier-Mellin moments 358

7.3.3 Other moments orthogonal on a disk 361

7.4 Object recognition by Zernike moments 363

7.5 Image reconstruction from moments 365

7.5.1 Reconstruction by direct calculation 367

7.5.2 Reconstruction in the Fourier domain 369

7.5.3 Reconstruction from orthogonal moments 370

7.5.4 Reconstruction from noisy data 373

7.5.5 Numerical experiments with a reconstruction from OG moments 373

7.6 3D orthogonal moments 377

7.6.1 3D moments orthogonal on a cube 380

7.6.2 3D moments orthogonal on a sphere 381

7.6.3 3D moments orthogonal on a cylinder 383

7.6.4 Object recognition of 3D objects by orthogonal moments 383

7.6.5 Object reconstruction from 3D moments 387

7.7 Conclusion 389

References 389

8 Algorithms for Moment Computation 398

8.1 Introduction 398

8.2 Digital image and its moments 399

8.2.1 Digital image 399

8.2.2 Discrete moments 400

8.3 Moments of binary images 402

8.3.1 Moments of a rectangle 402

8.3.2 Moments of a general-shaped binary object 403

8.4 Boundary-based methods for binary images 404

8.4.1 The methods based on Green’s theorem 404

8.4.2 The methods based on boundary approximations 406

8.4.3 Boundary-based methods for 3D objects 407

8.5 Decomposition methods for binary images 410

8.5.1 The "delta" method 412

8.5.2 Quadtree decomposition 413

8.5.3 Morphological decomposition 415

8.5.4 Graph-based decomposition 416

8.5.5 Computing binary OG moments by means of decomposition methods 420

8.5.6 Experimental comparison of decomposition methods 422

8.5.7 3D decomposition methods 423

8.6 Geometric moments of graylevel images 428

8.6.1 Intensity slicing 429

8.6.2 Bit slicing 430

8.6.3 Approximation methods 433

8.7 Orthogonal moments of graylevel images 435

8.7.1 Recurrent relations for moments orthogonal on a square 435

8.7.2 Recurrent relations for moments orthogonal on a disk 436

8.7.3 Other methods 438

8.8 Conclusion 440

Appendix 8.A 441

References 443

9 Applications 448

9.1 Introduction 448

9.2 Image understanding 448

9.2.1 Recognition of animals 449

9.2.2 Face and other human parts recognition 450

9.2.3 Character and logo recognition 453

9.2.4 Recognition of vegetation and of microscopic natural structures 454

9.2.5 Traffic-related recognition 455

9.2.6 Industrial recognition 456

9.2.7 Miscellaneous applications 457

9.3 Image registration 459

9.3.1 Landmark-based registration 460

9.3.2 Landmark-free registration methods 467

9.4 Robot and autonomous vehicle navigation and visual servoing 470

9.5 Focus and image quality measure 474

9.6 Image retrieval 476

9.7 Watermarking 481

9.8 Medical imaging 486

9.9 Forensic applications 489

9.10 Miscellaneous applications 496

9.10.1 Noise resistant optical flow estimation 496

9.10.2 Edge detection 497

9.10.3 Description of solar flares 498

9.10.4 Gas-liquid flow categorization 499

9.10.5 3D object visualization 500

9.10.6 Object tracking 500

9.11 Conclusion 501

References 501

10 Conclusion 518

10.1 Summary of the book 518

10.2 Pros and cons of moment invariants 519

10.3 Outlook to the future 520

Index 521

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