Contents -- Authors' biographies -- Preface -- Acknowledgments -- 1 Introduction to moments -- 1.1 Motivation -- 1.2 What are invariants? -- 1.2.1 Categories of invariant -- 1.3 What are moments? -- 1.3.1 Geometric and complex moments -- 1.3.2 Orthogonal moments -- 1.4 Outline of the book -- References -- 2 Moment invariants to translation, rotation and scaling -- 2.1 Introduction -- 2.1.1 Invariants to translation -- 2.1.2 Invariants to uniform scaling -- 2.1.3 Traditional invariants to rotation -- 2.2 Rotation invariants from complex moments -- 2.2.1 Construction of rotation invariants -- 2.2.2 Construction of the basis -- 2.2.3 Basis of invariants of the second and third orders -- 2.2.4 Relationship to the Hu invariants -- 2.3 Pseudoinvariants -- 2.4 Combined invariants to TRS and contrast changes -- 2.5 Rotation invariants for recognition of symmetric objects -- 2.5.1 Logo recognition -- 2.5.2 Recognition of simple shapes -- 2.5.3 Experiment with a baby toy -- 2.6 Rotation invariants via image normalization -- 2.7 Invariants to nonuniform scaling -- 2.8 TRS invariants in 3D -- 2.9 Conclusion -- References -- 3 Affine moment invariants -- 3.1 Introduction -- 3.1.1 Projective imaging of a 3D world -- 3.1.2 Projective moment invariants -- 3.1.3 Affine transformation -- 3.1.4 AMIs -- 3.2 AMIs derived from the Fundamental theorem -- 3.3 AMIs generated by graphs -- 3.3.1 The basic concept -- 3.3.2 Representing the invariants by graphs -- 3.3.3 Independence of the AMIs -- 3.3.4 The AMIs and tensors -- 3.3.5 Robustness of the AMIs -- 3.4 AMIs via image normalization -- 3.4.1 Decomposition of the affine transform -- 3.4.2 Violation of stability -- 3.4.3 Relation between the normalized moments and the AMIs -- 3.4.4 Affine invariants via half normalization -- 3.4.5 Affine invariants from complex moments.

3.5 Derivation of the AMIs from the Cayley-Aronhold equation -- 3.5.1 Manual solution -- 3.5.2 Automatic solution -- 3.6 Numerical experiments -- 3.6.1 Digit recognition -- 3.6.2 Recognition of symmetric patterns -- 3.6.3 The children's mosaic -- 3.7 Affine invariants of color images -- 3.8 Generalization to three dimensions -- 3.8.1 Method of geometric primitives -- 3.8.2 Normalized moments in 3D -- 3.8.3 Half normalization in 3D -- 3.8.4 Direct solution of the Cayley-Aronhold equation -- 3.9 Conclusion -- Appendix -- References -- 4 Implicit invariants to elastic transformations -- 4.1 Introduction -- 4.2 General moments under a polynomial transform -- 4.3 Explicit and implicit invariants -- 4.4 Implicit invariants as a minimization task -- 4.5 Numerical experiments -- 4.5.1 Invariance and robustness test -- 4.5.2 ALOI classification experiment -- 4.5.3 Character recognition on a bottle -- 4.6 Conclusion -- References -- 5 Invariants to convolution -- 5.1 Introduction -- 5.2 Blur invariants for centrosymmetric PSFs -- 5.2.1 Template matching experiment -- 5.2.2 Invariants to linear motion blur -- 5.2.3 Extension to n dimensions -- 5.2.4 Possible applications and limitations -- 5.3 Blur invariants for N-fold symmetric PSFs -- 5.3.1 Blur invariants for circularly symmetric PSFs -- 5.3.2 Blur invariants for Gaussian PSFs -- 5.4 Combined invariants -- 5.4.1 Combined invariants to convolution and roation -- 5.4.2 Combined invariants to convolution and affine transform -- 5.5 Conclusion -- Appendix -- References -- 6 Orthogonal moments -- 6.1 Introduction -- 6.2 Moments orthogonal on a rectangle -- 6.2.1 Hypergeometric functions -- 6.2.2 Legendre moments -- 6.2.3 Chebyshev moments -- 6.2.4 Other moments orthogonal on a rectangle -- 6.2.5 OG moments of a discrete variable -- 6.3 Moments orthogonal on a disk -- 6.3.1 Zernike and Pseudo-Zernike moments.

6.3.2 Orthogonal Fourier-Mellin moments -- 6.3.3 Other moments orthogonal on a disk -- 6.4 Object recognition by ZMs -- 6.5 Image reconstruction from moments -- 6.5.1 Reconstruction by the direct calculation -- 6.5.2 Reconstruction in the Fourier domain -- 6.5.3 Reconstruction from OG moments -- 6.5.4 Reconstruction from noisy data -- 6.5.5 Numerical experiments with image reconstruction from OG moments -- 6.6 Three-dimensional OG moments -- 6.7 Conclusion -- References -- 7 Algorithms for moment computation -- 7.1 Introduction -- 7.2 Moments in a discrete domain -- 7.3 Geometric moments of binary images -- 7.3.1 Decomposition methods for binary images -- 7.3.2 Boundary-based methods for binary images -- 7.3.3 Other methods for binary images -- 7.4 Geometric moments of graylevel images -- 7.4.1 Intensity slicing -- 7.4.2 Approximation methods -- 7.5 Efficient methods for calculating OG moments -- 7.5.1 Methods using recurrent relations -- 7.5.2 Decomposition methods -- 7.5.3 Boundary-based methods -- 7.6 Generalization to n dimensions -- 7.7 Conclusion -- References -- 8 Applications -- 8.1 Introduction -- 8.2 Object representation and recognition -- 8.3 Image registration -- 8.3.1 Registration of satellite images -- 8.3.2 Image registration for image fusion -- 8.4 Robot navigation -- 8.4.1 Indoor robot navigation based on circular landmarks -- 8.4.2 Recognition of landmarks using fish-eye lens camera -- 8.5 Image retrieval -- 8.6 Watermarking -- 8.6.1 Watermarking based on the geometric moments -- 8.7 Medical imaging -- 8.7.1 Landmark recognition in the scoliosis study -- 8.8 Forensic applications -- 8.8.1 Detection of near-duplicated image regions -- 8.9 Miscellaneous applications -- 8.9.1 Noise-resistant optical flow estimation -- 8.9.2 Focus measure -- 8.9.3 Edge detection -- 8.9.4 Gas-liquid flow categorization -- 8.9.5 3D objects visualization.

8.10 Conclusion -- References -- 9 Conclusion -- Index.