Institute of Information Theory and AutomationPrague, Czech Republic
Flinders University of South AustraliaAdelaide, Australia
Object Recognition by Implicit Invariants
Jan Flusser Jaroslav Kautsky
Filip Šroubek
General motivationHow can we recognize deformed objects?
Curved surface deformation of the image
g = D(f)
D - unknown deformation operator
Problem formulation
What are explicit invariants?
Functionals defined on the image space L such that
• E(f) = E(D(f)) for all admissible D
• Fourier descriptors, moment invariants, ...
What are explicit invariants?
Functionals defined on the image space L such that
• E(f) = E(D(f)) for all admissible D
• For many deformations explicit invariants do not exist.
What are implicit invariants?
Functionals defined on L x L such that
• I(f,D(f)) = 0 for all admissible D
• Implicit invariants exist for much bigger set of deformations
Our assumption about D
Image deformation is a polynomial transform r(x) of order > 1 of the spatial coordinates
f’(r(x)) = f(x)
What are moments?
Moments are “projections” of the image function into a polynomial basis
How are the moments transformed?
• A depends on r and on the polynomial basis• A is not a square matrix• Transform r does not preserve the order of the
moments• Explicit moment invariants cannot exist.
If they existed, they would contain all moments.
m’ = A.m
Construction of implicit momentinvariants
• Eliminate the parameters of r from the system
• Each equation of the reduced system is an implicit invariant
m’ = A.m
Artificial example
Invariance property
Robustness to noise
Object recognitionAmsterdam Library of Object Images
http://staff.science.uva.nl/˜aloi/
ALOI database
99% recognition rate
The bottle
The bottle
The bottle again
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100% recognition rate
Implementation
How to avoid numerical problems with high
dynamic range of standard moments?
Implementation
How to avoid numerical problems with high
dynamic range of standard moments?
We used
orthogonal
Czebyshev
polynomials
Summary
• We proposed a new concept of implicit invariants
• We introduced implicit moment invariants to polynomial deformations of images
Thank you !
Any questions?
• Odtud dal uz to nebylo !
Common types of moments
Geometric moments
Special case
If an explicit invariant exist, then
I(f,g) = |E(f) – E(g)|
An example in 1D
Orthogonal moments
• Legendre
• Zernike
• Fourier-Mellin
• Czebyshev
• Krawtchuk, Hahn
Outlook for the futureand open problems
• Discriminability?
• Robustness?
• Other transforms?
How is it connected with image fusion?
Základní přístupy
• Brute force
• Normalized position inverse problem
• Description of the objects by invariants
Basic approaches
An example in 2D
Our assumption about D
Image degradation is a polynomial transform r(x) of the spatial coordinates of order > 1
Construction of implicit momentinvariants
• Eliminate the parameters of r from the system
• Each equation of the reduced system is an implicit invariant
How are the moments transformed?
• A depends on r and on the moment basis• A is not a square matrix• Transform r does not preserve the moment
orders• Explicit moment invariants cannot exist.
If they existed, they would contain all moments.
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