Image registration of satellite images

1
Object Recognition by Implicit Invariants
Jan Flusser Jaroslav
Kautsky Filip Šroubek
Institute of Information Theory and
AutomationPrague, Czech RepublicFlinders
University of South AustraliaAdelaide, Australia
2
General motivation
How can we recognize deformed objects?
3
(No Transcript)
4
(No Transcript)
5
Problem formulation

• Curved surface ? deformation of the image
• g D(f)
• D - unknown deformation operator

6
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, ...

7
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.

8
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

9
Our assumption about D

• Image deformation is a polynomial transform r(x)
  of order gt 1 of the spatial coordinates
• f(r(x)) f(x)

10
What are moments?

• Moments are projections of the image function
  into a polynomial basis

11
How are the moments transformed?
m A.m

• 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.

12
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
13
Artificial example
14
Invariance property
15
Robustness to noise
16
Object recognitionAmsterdam Library of Object
Images http//staff.science.uva.nl/aloi/
17
ALOI database
99 recognition rate
18
The bottle
19
The bottle
20
(No Transcript)
21
The bottle again
22
The bottle again
23
The bottle again
24
The bottle again
25
The bottle again
26
The bottle again
27
The bottle again
100 recognition rate
28
Implementation

• How to avoid numerical problems with high
• dynamic range of standard moments?

29
Implementation

• How to avoid numerical problems with high
• dynamic range of standard moments?
• We used
• orthogonal
• Czebyshev
• polynomials

30
Summary

• We proposed a new concept of implicit invariants
• We introduced implicit moment invariants to
  polynomial deformations of images

31

• Thank you !

Any questions?
32

• Odtud dal uz to nebylo !

33
Common types of moments

• Geometric moments

34
Special case

• If an explicit invariant exist, then
• I(f,g) E(f) E(g)

35
An example in 1D
36
Orthogonal moments

• Legendre
• Zernike
• Fourier-Mellin
• Czebyshev
• Krawtchuk, Hahn

37
(No Transcript)
38
(No Transcript)
39
Outlook for the futureand open problems

• Discriminability?
• Robustness?
• Other transforms?

40
How is it connected with image fusion?
41
Základní prístupy
Basic approaches

• Brute force
• Normalized position ? inverse problem
• Description of the objects by invariants

42
An example in 2D
43
Our assumption about D

• Image degradation is a polynomial transform r(x)
  of the spatial coordinates of order gt 1

44
Construction of implicit momentinvariants

• Eliminate the parameters of r from the system
• Each equation of the reduced system is an
  implicit invariant

45
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.

46
(No Transcript)