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J. Flusser, T. Suk, and B. Zitová
Moments and Moment Invariants
in Pattern Recognition
http://zoi.utia.cas.cz/moment_invariants
The slides accompanying
the book
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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
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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
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Chapter 5
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Invariants to convolution
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Motivation – character recognition
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Motivation – template matching
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Image degradation models
Space-variant
Space-invariant convolution
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Two approaches
Traditional approach: Image restoration (blind deconvolution) and traditional invariants
Proposed approach: Invariants to convolution
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Invariants to convolution
for any admissible h
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The moments under convolution
Geometric/central
Complex
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PSF is centrosymmetric with respect to its centroid and has a unit integral
Assumptions on the PSF
supp(h)
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Common PSF’s are centrosymmetric
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Invariants to centrosymmetric convolution
where (p + q) is odd
What is the intuitive meaning of the invariants?
“Measure of anti-symmetry”
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Invariants to centrosymmetric convolution
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Invariants to centrosymmetric convolution
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Convolution invariants in FT domain
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Convolution invariants in FT domain
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Relationship between FT and moment invariants
where M(k,j) is the same as C(k,j) but with geometric moments
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Template matching
sharp image
with the templates
the templates (close-up)
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Template matching
a frame where
the templates were
located successfully
the templates (close-up)
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Template matching performance
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Valid region
Boundary effect in template matching
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Invalid region
- invariance is
violated
Boundary effect in template matching
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Boundary effect in template matching
zero-padding
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Extension to n dimensions
where |p| is odd
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• N-fold rotation symmetry, N > 2
• Circular symmetry
• Gaussian PSF
• Motion blur PSF
Other types of the blurring PSF
The more we know about the PSF, the more invariants and the higher discriminability we get
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Invariants to N-fold PSF
If (p-q)/N is integer the invariant is zero
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Invariants to circularly symmetric PSF
If p = q the invariant is zero (except p = q = 0)
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Invariants to Gaussian PSF
If p = q = 1 the invariant is zero
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Discrimination power
The null-space of the blur invariants = a set of all images having the same symmetry as the PSF.
One cannot distinguish among symmetric objects.
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Recommendation for practice
It is important to learn as much as possible about the PSF and to use proper invariants for object recognition
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Combined invariants
Convolution and rotation
For any N
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Satellite image registration by combined invariants
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( v11, v21, v31, …
)
( v12, v22, v32, …
)
min distance(( v1k, v2k, v3k, … ) , ( v1m, v2m, v3m, … ))k,m
Control point detection
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Control point matching
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Transformation and resampling
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Combined blur-affine invariants
• Let I(μ00,…, μPQ) be an affine moment invariant. Then I(C(0,0),…,C(P,Q)), where C(p,q) are blur invariants, is a combined blur-affine invariant (CBAI).
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Digit recognition by CBAI
CBAI AMI