2D and 3D Image Analysis by Moments

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 xviiAcknowledgements xxi1 Motivation 11.1 Image analysis by computers 11.2 Humans, computers, and object recognition 41.3 Outline of the book 5References 72 Introduction to Object Recognition 82.1 Feature space 82.1.1 Metric spaces and norms 92.1.2 Equivalence and partition 112.1.3 Invariants 122.1.4 Covariants 142.1.5 Invariant-less approaches 152.2 Categories of the invariants 152.2.1 Simple shape features 162.2.2 Complete visual features 182.2.3 Transformation coefficient features 202.2.4 Textural features 212.2.5 Wavelet-based features 232.2.6 Differential invariants 242.2.7 Point set invariants 252.2.8 Moment invariants 262.3 Classifiers 272.3.1 Nearest-neighbor classifiers 282.3.2 Support vector machines 312.3.3 Neural network classifiers 322.3.4 Bayesian classifier 342.3.5 Decision trees 352.3.6 Unsupervised classification 362.4 Performance of the classifiers 372.4.1 Measuring the classifier performance 372.4.2 Fusing classifiers 382.4.3 Reduction of the feature space dimensionality 382.5 Conclusion 40References 413 2D Moment Invariants to Translation, Rotation, and Scaling 453.1 Introduction 453.1.1 Mathematical preliminaries 453.1.2 Moments 473.1.3 Geometric moments in 2D 483.1.4 Other moments 493.2 TRS invariants from geometric moments 503.2.1 Invariants to translation 503.2.2 Invariants to uniform scaling 513.2.3 Invariants to non-uniform scaling 523.2.4 Traditional invariants to rotation 543.3 Rotation invariants using circular moments 563.4 Rotation invariants from complex moments 573.4.1 Complex moments 573.4.2 Construction of rotation invariants 583.4.3 Construction of the basis 593.4.4 Basis of the invariants of the second and third orders 623.4.5 Relationship to the Hu invariants 633.5 Pseudoinvariants 673.6 Combined invariants to TRS and contrast stretching 683.7 Rotation invariants for recognition of symmetric objects 693.7.1 Logo recognition 753.7.2 Recognition of shapes with different fold numbers 753.7.3 Experiment with a baby toy 773.8 Rotation invariants via image normalization 813.9 Moment invariants of vector fields 863.10 Conclusion 92References 924 3D Moment Invariants to Translation, Rotation, and Scaling 954.1 Introduction 954.2 Mathematical description of the 3D rotation 984.3 Translation and scaling invariance of 3D geometric moments 1004.4 3D rotation invariants by means of tensors 1014.4.1 Tensors 1014.4.2 Rotation invariants 1024.4.3 Graph representation of the invariants 1034.4.4 The number of the independent invariants 1044.4.5 Possible dependencies among the invariants 1054.4.6 Automatic generation of the invariants by the tensor method 1064.5 Rotation invariants from 3D complex moments 1084.5.1 Translation and scaling invariance of 3D complex moments 1124.5.2 Invariants to rotation by means of the group representation theory 1124.5.3 Construction of the rotation invariants 1154.5.4 Automated generation of the invariants 1174.5.5 Elimination of the reducible invariants 1184.5.6 The irreducible invariants 1184.6 3D translation, rotation, and scale invariants via normalization 1194.6.1 Rotation normalization by geometric moments 1204.6.2 Rotation normalization by complex moments 1234.7 Invariants of symmetric objects 1244.7.1 Rotation and reflection symmetry in 3D 1244.7.2 The influence of symmetry on 3D complex moments 1284.7.3 Dependencies among the invariants due to symmetry 1304.8 Invariants of 3D vector fields 1314.9 Numerical experiments 1314.9.1 Implementation details 1314.9.2 Experiment with archeological findings 1334.9.3 Recognition of generic classes 1354.9.4 Submarine recognition – robustness to noise test 1374.9.5 Teddy bears – the experiment on real data 1414.9.6 Artificial symmetric bodies 1424.9.7 Symmetric objects from the Princeton Shape Benchmark 1434.10 Conclusion 147Appendix 4.A 148Appendix 4.B 156Appendix 4.C 158References 1605 Affine Moment Invariants in 2D and 3D 1635.1 Introduction 1635.1.1 2D projective imaging of 3D world 1645.1.2 Projective moment invariants 1655.1.3 Affine transformation 1675.1.4 2D Affine moment invariants – the history 1685.2 AMIs derived from the Fundamental theorem 1705.3 AMIs generated by graphs 1715.3.1 The basic concept 1725.3.2 Representing the AMIs by graphs 1735.3.3 Automatic generation of the invariants by the graph method 1735.3.4 Independence of the AMIs 1745.3.5 The AMIs and tensors 1805.4 AMIs via image normalization 1815.4.1 Decomposition of the affine transformation 1825.4.2 Relation between the normalized moments and the AMIs 1855.4.3 Violation of stability 1865.4.4 Affine invariants via half normalization 1875.4.5 Affine invariants from complex moments 1875.5 The method of the transvectants 1905.6 Derivation of the AMIs from the Cayley-Aronhold equation 1955.6.1 Manual solution 1955.6.2 Automatic solution 1985.7 Numerical experiments 2015.7.1 Invariance and robustness of the AMIs 2015.7.2 Digit recognition 2015.7.3 Recognition of symmetric patterns 2045.7.4 The children’s mosaic 2085.7.5 Scrabble tiles recognition 2105.8 Affine invariants of color images 2145.8.1 Recognition of color pictures 2175.9 Affine invariants of 2D vector fields 2185.10 3D affine moment invariants 2215.10.1 The method of geometric primitives 2225.10.2 Normalized moments in 3D 2245.10.3 Cayley-Aronhold equation in 3D 2255.11 Beyond invariants 2255.11.1 Invariant distance measure between images 2255.11.2 Moment matching 2275.11.3 Object recognition as a minimization problem 2295.11.4 Numerical experiments 2295.12 Conclusion 231Appendix 5.A 232Appendix 5.B 233References 2346 Invariants to Image Blurring 2376.1 Introduction 2376.1.1 Image blurring – the sources and modeling 2376.1.2 The need for blur invariants 2396.1.3 State of the art of blur invariants 2396.1.4 The chapter outline 2466.2 An intuitive approach to blur invariants 2476.3 Projection operators and blur invariants in Fourier domain 2496.4 Blur invariants from image moments 2526.5 Invariants to centrosymmetric blur 2546.6 Invariants to circular blur 2566.7 Invariants to N-FRS blur 2596.8 Invariants to dihedral blur 2656.9 Invariants to directional blur 2696.10 Invariants to Gaussian blur 2726.10.1 1D Gaussian blur invariants 2746.10.2 Multidimensional Gaussian blur invariants 2786.10.3 2D Gaussian blur invariants from complex moments 2796.11 Invariants to other blurs 2806.12 Combined invariants to blur and spatial transformations 2826.12.1 Invariants to blur and rotation 2826.12.2 Invariants to blur and affine transformation 2836.13 Computational issues 2846.14 Experiments with blur invariants 2856.14.1 A simple test of blur invariance property 2856.14.2 Template matching in satellite images 2866.14.3 Template matching in outdoor images 2916.14.4 Template matching in astronomical images 2916.14.5 Face recognition on blurred and noisy photographs 2926.14.6 Traffic sign recognition 2946.15 Conclusion 302Appendix 6.A 303Appendix 6.B 304Appendix 6.C 306Appendix 6.D 308Appendix 6.E 310Appendix 6.F 310Appendix 6.G 311References 3157 2D and 3D Orthogonal Moments 3207.1 Introduction 3207.2 2D moments orthogonal on a square 3227.2.1 Hypergeometric functions 3237.2.2 Legendre moments 3247.2.3 Chebyshev moments 3277.2.4 Gaussian-Hermite moments 3317.2.5 Other moments orthogonal on a square 3347.2.6 Orthogonal moments of a discrete variable 3387.2.7 Rotation invariants from moments orthogonal on a square 3487.3 2D moments orthogonal on a disk 3517.3.1 Zernike and Pseudo-Zernike moments 3527.3.2 Fourier-Mellin moments 3587.3.3 Other moments orthogonal on a disk 3617.4 Object recognition by Zernike moments 3637.5 Image reconstruction from moments 3657.5.1 Reconstruction by direct calculation 3677.5.2 Reconstruction in the Fourier domain 3697.5.3 Reconstruction from orthogonal moments 3707.5.4 Reconstruction from noisy data 3737.5.5 Numerical experiments with a reconstruction from OG moments 3737.6 3D orthogonal moments 3777.6.1 3D moments orthogonal on a cube 3807.6.2 3D moments orthogonal on a sphere 3817.6.3 3D moments orthogonal on a cylinder 3837.6.4 Object recognition of 3D objects by orthogonal moments 3837.6.5 Object reconstruction from 3D moments 3877.7 Conclusion 389References 3898 Algorithms for Moment Computation 3988.1 Introduction 3988.2 Digital image and its moments 3998.2.1 Digital image 3998.2.2 Discrete moments 4008.3 Moments of binary images 4028.3.1 Moments of a rectangle 4028.3.2 Moments of a general-shaped binary object 4038.4 Boundary-based methods for binary images 4048.4.1 The methods based on Green’s theorem 4048.4.2 The methods based on boundary approximations 4068.4.3 Boundary-based methods for 3D objects 4078.5 Decomposition methods for binary images 4108.5.1 The "delta" method 4128.5.2 Quadtree decomposition 4138.5.3 Morphological decomposition 4158.5.4 Graph-based decomposition 4168.5.5 Computing binary OG moments by means of decomposition methods 4208.5.6 Experimental comparison of decomposition methods 4228.5.7 3D decomposition methods 4238.6 Geometric moments of graylevel images 4288.6.1 Intensity slicing 4298.6.2 Bit slicing 4308.6.3 Approximation methods 4338.7 Orthogonal moments of graylevel images 4358.7.1 Recurrent relations for moments orthogonal on a square 4358.7.2 Recurrent relations for moments orthogonal on a disk 4368.7.3 Other methods 4388.8 Conclusion 440Appendix 8.A 441References 4439 Applications 4489.1 Introduction 4489.2 Image understanding 4489.2.1 Recognition of animals 4499.2.2 Face and other human parts recognition 4509.2.3 Character and logo recognition 4539.2.4 Recognition of vegetation and of microscopic natural structures 4549.2.5 Traffic-related recognition 4559.2.6 Industrial recognition 4569.2.7 Miscellaneous applications 4579.3 Image registration 4599.3.1 Landmark-based registration 4609.3.2 Landmark-free registration methods 4679.4 Robot and autonomous vehicle navigation and visual servoing 4709.5 Focus and image quality measure 4749.6 Image retrieval 4769.7 Watermarking 4819.8 Medical imaging 4869.9 Forensic applications 4899.10 Miscellaneous applications 4969.10.1 Noise resistant optical flow estimation 4969.10.2 Edge detection 4979.10.3 Description of solar flares 4989.10.4 Gas-liquid flow categorization 4999.10.5 3D object visualization 5009.10.6 Object tracking 5009.11 Conclusion 501References 50110 Conclusion 51810.1 Summary of the book 51810.2 Pros and cons of moment invariants 51910.3 Outlook to the future 520Index 521