Chapter 1: Motivation

1.1 Image analysis by computers
1.2 Humans, computers, and object recognition
1.3 Outline of the book
References

Chapter 2: Introduction to Object Recognition

2.1 Feature space

2.1.1 Metric spaces and norms
2.1.2 Equivalence and partition
2.1.3 Invariants
2.1.4 Covariants
2.1.5 Invariant-less approaches

2.2 Categories of the invariants

2.2.1 Simple shape features
2.2.2 Complete visual features
2.2.3 Transformation coefficient features
2.2.4 Textural features
2.2.5 Wavelet-based features
2.2.6 Differential invariants
2.2.7 Point set invariants
2.2.8 Moment invariants

2.3 Classifiers

2.3.1 Nearest-neighbor classifiers
2.3.2 Support vector machines
2.3.3 Neural network classifiers
2.3.4 Bayesian classifier
2.3.5 Decision trees
2.3.6 Unsupervised classification

2.4 Performance of the classifiers

2.4.1 Measuring the classifier performance
2.4.2 Fusing classifiers
2.4.3 Reduction of the feature space dimensionality

2.5 Conclusion
References

Chapter 3: 2D Moment Invariants to Translation, Rotation, and Scaling

3.1 Introduction

3.1.1 Mathematical preliminaries
3.1.2 Moments
3.1.3 Geometric moments in 2D
3.1.4 Other moments

3.2 TRS invariants from geometric moments

3.2.1 Invariants to translation
3.2.2 Invariants to uniform scaling
3.2.3 Invariants to non-uniform scaling

3.3 Rotation invariants using circular moments
3.4 Rotation invariants from complex moments

3.4.1 Complex moments
3.4.2 Construction of rotation invariants
3.4.3 Construction of the basis
3.4.4 Basis of the invariants of the 2nd and 3rd orders
3.4.5 Relationship to the Hu invariants

3.5 Pseudoinvariants
3.6 Combined invariants to TRS and contrast
3.7 Rotation invariants of symmetric objects

3.7.1 Logo recognition
3.7.2 Recognition of shapes with different fold numbers
3.7.3 Experiment with a baby toy

3.8 Rotation invariants via image normalization
3.9 Moment invariants of vector fields
3.10 Conclusion
References

Chapter 4: 3D Moment Invariants to Translation, Rotation, and Scaling

4.1 Introduction
4.2 Mathematical description of the 3D rotation
4.3 Translation and scaling invariance of 3D moments
4.4 3D rotation invariants by means of tensors

4.4.1 Tensors
4.4.2 Rotation invariants
4.4.3 Graph representation of the invariants
4.4.4 The number of the independent invariants
4.4.5 Possible dependencies among the invariants
4.4.6 Automatic generation of the invariants by the tensor method

4.5 Rotation invariants from 3D complex moments

4.5.1 Translation and scaling invariance of 3D complex moments
4.5.2 Invariants to rotation by means of the group representation theory
4.5.3 Construction of the rotation invariants
4.5.4 Automated generation of the invariants
4.5.5 Elimination of the reducible invariants
4.5.6 The irreducible invariants

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

4.6.1 Rotation normalization by geometric moments
4.6.2 Rotation normalization by complex moments

4.7 Invariants of symmetric objects

4.7.1 Rotation and reflection symmetry in 3D
4.7.2 The influence of symmetry on 3D complex moments
4.7.3 Dependencies among the invariants due to the symmetry

4.8 Invariants of 3D vector fields
4.9 Numerical Experiments

4.9.1 Implementation details
4.9.2 Experiment with archeological findings
4.9.3 Recognition of generic classes
4.9.4 Submarine recognition – robustness to noise test
4.9.5 Teddy bears – the experiment on real data
4.9.6 Artificial symmetric bodies
4.9.7 Symmetric objects from the Princeton Shape Benchmark

4.10 Conclusion
Appendix
References

Chapter 5: Affine Moment Invariants in 2D and 3D

5.1 Introduction

5.1.1 2D projective imaging of 3D world
5.1.2 Projective moment invariants
5.1.3 Affine transformation
5.1.4 2D Affine moment invariants – the history

5.2 AMIs derived from the Fundamental theorem
5.3 AMIs generated by graphs

5.3.1 The basic concept
5.3.2 Representing the AMIs by graphs
5.3.3 Automatic generation of the invariants by the graph method
5.3.4 Independence of the AMI’s
5.3.5 The AMIs and tensors

5.4 AMIs via image normalization

5.4.1 Decomposition of the affine transformation
5.4.2 Relation between the normalized moments and the AMIs
5.4.3 Violation of stability
5.4.4 Affine invariants via half normalization
5.4.5 Affine invariants from complex moments

5.5 The method of the transvectants
5.6 Derivation of the AMIs from the Cayley-Aronhold equation

5.6.1 Manual solution
5.6.2 Automatic solution

5.7 Numerical experiments

5.7.1 Invariance and robustness of the AMIs
5.7.2 Digit recognition
5.7.3 Recognition of symmetric patterns
5.7.4 The children’s mosaic
5.7.5 Scrabble tiles recognition

5.8 Affine invariants of color images

5.8.1 Recognition of color pictures

5.9 Affine invariants of 2D vector fields
5.10 3D affine moment invariants

5.10.1 The method of geometric primitives
5.10.2 Normalized moments in 3D
5.10.3 Cayley-Aronhold equation in 3D

5.11 Beyond invariants

5.11.1 Invariant distance measure between images
5.11.2 Moment matching
5.11.3 Object recognition as a minimization problem
5.11.4 Numerical experiments

5.12 Conclusion
Appendix
References

Chapter 6: Invariants to Image Blurring

6.1 Introduction

6.1.1 Image blurring – the sources and modeling
6.1.2 The need for blur invariants
6.1.3 State of the art of blur invariants
6.1.4 The Chapter outline

6.2 An intuitive approach to blur invariants
6.3 Projection operators in Fourier domain
6.4 Blur invariants from image moments
6.5 Invariants to centrosymmetric blur
6.6 Invariants to circular blur
6.7 Invariants to N-FRS blur
6.8 Invariants to dihedral blur
6.9 Invariants to directional blur
6.10 Invariants to Gaussian blur

6.10.1 1D Gaussian blur invariants
6.10.2 Multidimensional Gaussian blur invariants
6.10.3 2D Gaussian blur invariants from complex moments

6.11 Invariants to other blurs
6.12 Combined invariants to blur and spatial tr

6.12.1 Invariants to blur and rotation
6.12.2 Invariants to convolution and affine transform

6.13 Computational issues
6.14 Experiments with blur invariants

6.14.1 A simple test of blur invariance property
6.14.2 Template matching in satellite images
6.14.3 Template matching in outdoor images
6.14.4 Template matching in astronomical images
6.14.5 Face recognition on blurred and noisy photographs
6.14.6 Traffic sign recognition

6.15 Conclusion
Appendix
References

Chapter 7: Orthogonal Moments

7.1 Introduction
7.2 2D moments orthogonal on a square

7.2.1 Hypergeometric functions
7.2.2 Legendre moments
7.2.3 Chebyshev moments
7.2.4 Hermite moments
7.2.5 Other moments orthogonal on a rectangle
7.2.6 Orthogonal moments of a discrete variable
7.2.7 Rotation invariants from moments orthogonal on a square

7.3 2D moments orthogonal on a disk

7.3.1 Zernike and Pseudo-Zernike moments
7.3.2 Fourier-Mellin moments
7.3.3 Other moments orthogonal on a disk

7.4 Object recognition by Zernike moments
7.5 Image reconstruction from moments

7.5.1 Reconstruction by direct calculation
7.5.2 Reconstruction in the Fourier domain
7.5.3 Reconstruction from orthogonal moments
7.5.4 Reconstruction from noisy data
7.5.5 Numerical experiments with a reconstruction from OG moments

7.6 3D orthogonal moments

7.6.1 3D moments orthogonal on a cube
7.6.2 3D moments orthogonal on a sphere
7.6.3 3D moments orthogonal on a cylinder
7.6.4 Object recognition of 3D objects by orthogonal moments
7.6.5 Object reconstruction from 3D moments

7.7 Conclusion
References

Chapter 8: Algorithms for Moment Computation

8.1 Introduction
8.2 Digital image and its moments

8.2.1 Digital image
8.2.2 Discrete moments

8.3 Moments of binary images

8.3.1 Moments of a rectangle
8.3.2 Moments of a general-shaped binary object

8.4 Boundary-based methods for binary images

8.4.1 The methods based on the Green’s theorem
8.4.2 The methods based on boundary approximations
8.4.3 Boundary-based methods for 3D objects

8.5 Decomposition methods for binary images

8.5.1 The “delta” method
8.5.3 Morphological decomposition
8.5.4 Graph-based decomposition
8.5.5 Computing binary OG moments by means of decomposition methods
8.5.6 Experimental comparison of decomposition methods
8.5.7 3D decomposition methods

8.6 Geometric moments of graylevel images

8.6.1 Intensity slicing
8.6.2 Bit slicing
8.6.3 Approximation methods

8.7 Orthogonal moments of graylevel images

8.7.1 Recurrent relations for moments orthogonal on a square
8.7.2 Recurrent relations for moments orthogonal on a disk
8.7.3 Other methods

8.8 Conclusion

8.8.1 Filling the holes of the triangulation

References

Chapter 9: Applications

9.1 Introduction
9.2 Image understanding

9.2.1 Recognition of animals
9.2.2 Face and other human parts recognition
9.2.3 Character and logo recognition
9.2.4 Recognition of vegetation and of microscopic structures
9.2.5 Traffic-related recognition
9.2.6 Industrial recognition
9.2.7 Miscellaneous applications

9.3 Image registration

9.3.1 Landmark-based registration
9.3.2 Landmark-free registration methods

9.4 Robot & autonomous vehicle navigation
9.5 Focus and image quality measure
9.6 Image retrieval
9.7 Watermarking
9.8 Medical imaging
9.9 Forensic applications
9.10 Miscellaneous applications

9.10.1 Noise resistant optical flow estimation
9.10.2 Edge detection
9.10.3 Description of solar flares
9.10.4 Gas-liquid flow categorization
9.10.5 3D object visualization
9.10.6 Object tracking

9.11 Conclusion
References

Chapter 10: Conclusion

10.1 Summary of the book
10.2 Pros and cons of moment invariants
10.3 Outlook to the future

Index