http://zoi.utia.cas.cz/moment_invariants
The slides accompanying
the book
J. Flusser, T. Suk, and B. Zitová
Moments and Moment Invariants
in Pattern Recognition
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 6
Orthogonal moments
- set of orthogonal polynomials
Motivation for using OG moments
• Stable calculation by recurrent relations
• Easier and stable image reconstruction
Numerical stability
How to avoid numerical problems with high
dynamic range of geometric moments?
Standard powers
Orthogonal polynomials
Calculation using recurrent relations
Two kinds of orthogonality
• Moments (polynomials) orthogonal on a unit
square
• Moments (polynomials) orthogonal on a unit
disk
Moments orthogonal on a square
is a system of 1D orthogonal polynomials
Common 1D orthogonal polynomials
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•
•
•
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Legendre
Chebyshev
Gegenbauer
Jacobi
(generalized) Laguerre
Hermite
<-1,1>
<-1,1>
<-1,1>
<-1,1> or <0,1>
<0,∞)
(-∞,∞)
Legendre polynomials
Definition
Orthogonality
Legendre polynomials explicitly
Legendre polynomials in 1D
Legendre polynomials in 2D
Legendre polynomials
Explicit formula
Recurrence relation
Legendre moments
Legendre and geometric moments
Chebyshev polynomials
First kind
Second kind
Chebyshev polynomials on <-1,1>
First kind
Second kind
Chebyshev polynomials on <-2,2>
First kind
Second kind
Chebyshev polynomials in 2D
Chebyshev polynomials – orthogonality
Continuous
They can be normalized to discrete
orthogonality
Chebyshev moments
Gegenbauer polynomials
Generalization of both Legendre and
Chebyshev polynomials – parameter λ
- special initial values:
Gegenbauer polynomials
Jacobi polynomials
Further generalization, parameters α and β
Laguerre and Hermite polynomials
• Infinite interval of orthogonality
• Suitable for particular applications only
Discrete OG polynomials on a
square
Discrete variable
Discrete orthogonality
Discrete 1D OG polynomials
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•
•
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Discrete Chebyshev
Discrete Laguerre
Krawtchouk
Hahn
Dual-Hahn
Racah
Discrete Chebyshev polynomials
Krawtchouk polynomials
Krawtchouk polynomials
Weighted Krawtchouk polynomials
p=0.5
p=0.2
Dual-Hahn and Racah polynomials
Nonuniform lattice
They were adapted such that s is a traditional
coordinate in a discrete image.
Moments orthogonal on a disk
Radial part
Angular part
Moments orthogonal on a disk
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•
•
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Zernike
Pseudo-Zernike
Orthogonal Fourier-Mellin
Jacobi-Fourier
Chebyshev-Fourier
Radial harmonic Fourier
Zernike polynomials
Definition
Orthogonality
Zernike polynomials – radial part in 1D
Zernike polynomials – radial part in 2D
Zernike polynomials
Zernike moments
Mapping of Cartesian coordinates x,y to
polar coordinates r,φ:
• Whole image is mapped inside the unit disk
• Translation and scaling invariance
Rotation property of Zernike moments
The magnitude is preserved, the phase is shifted by ℓθ.
Invariants are constructed by phase cancellation
Zernike rotation invariants
Phase cancellation by multiplication
Normalization to rotation
Pseudo - Zernike polynomials
need not be even → redundancy
Orthogonal Fourier-Mellin moments
Orthogonal FM moments – radial part
Orthogonal FM moments – the basis functions
Jacobi-Fourier and Chebyshev-Fourier
moments
Combination:
radial part – 1D orthogonal polynomials
angular part – traditional harmonic function
1D OG polynomials – shifted version:
Orthogonality on <0,1>
Radial harmonic Fourier moments
Tn(r) is not polynomial → they are not moments
in a strict sense
Recognition by Zernike rotation invariants
Insufficient separability
Sufficient separability
Image reconstruction
• Direct reconstruction from geometric moments
• Solution of a system of equations
• Works for very small images only
• For larger images the system is ill-conditioned
Image reconstruction by direct
computation
12 x 12
13 x 13
Image reconstruction
• Reconstruction from geometric moments via FT
Image reconstruction via Fourier transform
Image reconstruction
• Image reconstruction from OG moments on
a square
• Image reconstruction from OG moments on
a disk (Zernike)
Image reconstruction from Legendre
moments
Continuous orthogonality
Image reconstruction from Zernike
moments
Better for polar raster
Image reconstruction from discrete
Chebyshev moments
– precise reconstruction
Illustrates discrimination power of the moments
Reconstruction of large images
Discrete Chebyshev moments – precise up to 1075th order
Limitation is imposed by floating-point underflow
Orthogonal moments in 3D
Ω - Rectangular cuboid
Ω - Cylinder
Ω - Sphere
- spherical harmonics
Example: 3D Zernike moments
Summary of OG moments
• OG moments are used because of their favorable
numerical properties, not because of theoretical
contribution
• OG moments should be never used outside the area
of orthogonality
• OG moments should be always calculated by
recurrent relations, not by expanding into powers
• Preferably discrete OG polynomials should be used
• Moments OG on a square do not provide easy
rotation invariance
• Even if the reconstruction from OG moments is
seemingly simple, moments are not a good tool
for image compression