J. Flusser, T. Suk, and B. Zitová Moments and Moment Invariants in Pattern Recognition The slides accompanying the book

Copyright notice The slides can be used freely for non-profit education provided that the source is appropriately cited. Please report any usage on a regular basis (namely in university courses) to the authors. For commercial usage ask the authors for permission. The slides containing animations are not appropriate to print. © Jan Flusser, Tomas Suk, and Barbara Zitová, 2009

Contents 1. Introduction to moments 2. Invariants to translation, rotation and scaling 3. Affine moment invariants 4. Implicit invariants to elastic transformations 5. Invariants to convolution 6. Orthogonal moments 7. Algorithms for moment computation 8. Applications 9. Conclusion

Chapter 3

Invariants to affine transform What is affine transform ?

Invariants to affine transform What is affine transform ?

Why is affine transform important? Affine transform is a good approximation of projective transform Projective transform describes a perspective projection of 3-D objects onto 2-D plane by a central camera

Projective deformation

Why not projective moment invariants? Do not exist when using any finite set of moments Do not exist when using infinite set of (all) moments Exist formally as infinite series of moments of both positive and negative indexes

Theory of algebraic invariants (Fundamental theorem) Graph method Image normalization Cayley-Aronhold equation Hybrid approaches Affine moment invariants All methods lead to equivalent invariants … Many ways how to derive them

Two simplest AMI’s, frequently cited … such as

AMI’s by means of the Fundamental theorem Binary algebraic form Algebraic invariant of weight w

AMI’s by means of the Fundamental theorem

AMI’s by means of the graph method - arbitrary points r points, n kj – non-negative integers

Affine Moment Invariants AMI’s by means of the graph method where

Simple examples of the AMI’s 1),

Simple examples of the AMI’s 2),

Graph representation of the AMI’s

Dependence among invariants Trivial invariants (always zero or identical)

Dependence among invariants Trivial invariants, identical invariants Reducible invariants (products, linear combinations) Irreducible invariants (polynomials, polynomials of products) Independent invariants

Removing dependence For w ≤ 12 : invariants (graphs) altogether zero invariants identical invariants linear combinations products irreducible invariants 80 independent invariants

Removing dependence The most difficult step: How to proceed from irreducible to independent invariants? Exhaustive search of all possible polynomial dependences The dependences themselves may be dependent ! (2 nd -order dependencies)

Higher-order dependencies The number of independent invariants:

Numerical experiments with the AMI’s

Robustness of the AMI’s to distortions

Affine invariants via normalization Many possibilities how to define normalization constraints Several possible decompositions of the affine transform

Decomposition of the affine transform Horizontal and vertical translations Uniform scaling First rotation Stretching Second rotation Mirror reflection

Normalization to partial transforms Horizontal and vertical translation -- m 01 = m 10 = 0 Scaling -- c 00 = 1 First rotation -- c 20 real and positive Stretching -- c 20 = 0 (μ 20 =μ 02 ) Second rotation -- c 21 real and positive

Moment values after the normalization Translation, uniform scaling and the first rotation Stretching

Moment values after the normalization Second rotation

Possible volatility of the normalization

Affine invariants via half-normalization “Hybrid” approach. The image is normalized to translation, scaling, first rotation and stretching. Then, rotation invariants are used to handle the second rotation. More stable in some cases.

Affine invariants from complex moments

Affine invariants from Cayley- Aronhold equation Skewing parameter t

Digit recognition by the AMI’s

Recognition of symmetric patterns

Recognition of children’s mosaic

Affine invariants of color images Color moments Algebraic invariants of more than one binary forms

Affine invariants of color images Common centroid of color channels Additional invariants

Affine invariants in 3D 3D affine transform Analogy with the graph method

Affine invariants in 3D An example Corresponding hypergraph

Affine invariants in 3D Corresponding hypergraph

Affine normalization in 3D Theory based on spherical harmonics (analogy to complex moments)

Cayley-Aronhold equation in 3D Analogy to 2D