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

The bottle again

The bottle again

The bottle again

The bottle again

The bottle again

The bottle again

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.