j. flusser, t. suk, and b. zitová moments and moment invariants in pattern recognition
DESCRIPTION
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. © Jan Flusser, Tomas Suk, and Barbara Zitová, 2008TRANSCRIPT
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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
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Introduction to Moments
Slides to Chapter 1
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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.
© Jan Flusser, Tomas Suk, and Barbara Zitová,
2008
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General motivationHow can we recognize objects on non-ideal images?
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Traffic surveillance - can we recognize the license plates?
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Recognition (classification) = assigning a pattern/object to one of pre-defined classes
The object is described by its features
Features – measurable quantities, usually form an n-D vector in a metric space
Object recognition
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Non-ideal imaging conditions degradation of the image
g = D(f)
D - unknown degradation operator
Problem formulation
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Základní přístupy
• Brute force
• Normalized position inverse problem
• Description of the objects by invariants
Basic approaches
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What are invariants?
Invariants are functionals defined on the image space such that
• I(f) = I(D(f)) for all admissible D
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Example: TRS
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What are invariants?
Invariants are functionals defined on the image space such that
• I(f) = I(D(f)) for all admissible D
• I(f1), I(f2) “different enough“ for different f1, f2
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Discrimination power
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Major categories of invariants
Simple shape descriptors- compactness, convexity, elongation, ...Transform coefficient invariants- Fourier descriptors, wavelet features, ...Point set invariants- positions of dominant points Differential invariants- derivatives of the boundaryMoment invariants
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What are moment invariants?Functions of image moments, invariant to certain class of image degradations
• Rotation, translation, scaling• Affine transform• Elastic deformations• Convolution/blurring• Combined invariants
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What are moments?
Moments are “projections” of the image function into a polynomial basis
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The most common moments
Geometric moments
(p + q) - the order of the moment
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Geometric moments – the meaning
0th order - area1st order - center of gravity
2nd order - moments of inertia 3rd order - skewness
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Uniqueness theorem
Geometric moments
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Complex moments
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Basic relations between the moments
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Invariants to translationCentral moments
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Invariants to translationCentral moments
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Invariants to translation and scaling
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Invariants to translation and scaling
Normalized central moments
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Invariants to translation and scaling
Normalized central moments
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Invariants to translation and scaling
Normalized central moments