Chapter 1: Introduction to Moments
1.1 Motivation
1.2 What are invariants?
1.2.1 Categories of the invariants
1.3 What are moments?
1.3.1 Geometric and complex moments
1.3.2 Orthogonal moments
1.4 The outline of the book
Bibliography

Chapter 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 2nd and 3rd 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 non-uniform scaling
2.8 TRS invariants in 3D
2.9 Conclusion
Bibliography

Chapter 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 Affine moment invariants
3.2 AMI's derived from the Fundamental theorem
3.3 AMI's generated by graphs
3.3.1 The basic concept
3.3.2 Representing the invariants by graphs
3.3.3 Independence of the AMI's
3.3.4 The AMI's and tensors
3.3.5 Robustness of the AMI's
3.4 Affine moment invariants 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 AMI's
3.4.4 Affine invariants via half normalization
3.4.5 Affine invariants from complex moments
3.5 Derivation of the AMI's 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
Bibliography

Chapter 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
Bibliography

Chapter 5: Invariants to Convolution
5.1 Introduction
5.2 Blur invariants for centrosymmetric PSF's
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 PSF's
5.3.1 Blur invariants for circularly symmetric PSF's
5.3.2 Blur invariants for Gaussian PSF's
5.4 Combined invariants
5.4.1 Combined invariants to convolution and rotation
5.4.2 Combined invariants to convolution and affine transform
5.5 Conclusion
Appendix
Bibliography

Chapter 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 Orthogonal 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 Zernike moments
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 orthogonal moments
6.5.4 Reconstruction from noisy data
6.5.5 Numerical experiments with image reconstruction from OG moments
6.6 Three-dimensional orthogonal moments
6.7 Conclusion
Bibliography

Chapter 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 orthogonal 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
Bibliography

Chapter 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 The 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
Bibliography

Chapter 9: Conclusion

Index