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

Copyright notice

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© Jan Flusser, Tomas Suk, and Barbara Zitová,

2009

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

Chapter 5

Invariants to convolution

Motivation – character recognition

Motivation – template matching

Image degradation models

Space-variant

Space-invariant convolution

Two approaches

Traditional approach: Image restoration (blind deconvolution) and traditional invariants

Proposed approach: Invariants to convolution

Invariants to convolution

for any admissible h

The moments under convolution

Geometric/central

Complex

PSF is centrosymmetric with respect to its centroid and has a unit integral

Assumptions on the PSF

supp(h)

Common PSF’s are centrosymmetric

Invariants to centrosymmetric convolution

where (p + q) is odd

What is the intuitive meaning of the invariants?

“Measure of anti-symmetry”

Invariants to centrosymmetric convolution

Invariants to centrosymmetric convolution

Convolution invariants in FT domain

Convolution invariants in FT domain

Relationship between FT and moment invariants

where M(k,j) is the same as C(k,j) but with geometric moments

Template matching

sharp image

with the templates

the templates (close-up)

Template matching

a frame where

the templates were

located successfully

the templates (close-up)

Template matching performance

Valid region

Boundary effect in template matching

Invalid region

- invariance is

violated

Boundary effect in template matching

Boundary effect in template matching

zero-padding

Extension to n dimensions

where |p| is odd

• 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

Invariants to N-fold PSF

If (p-q)/N is integer the invariant is zero

Invariants to circularly symmetric PSF

If p = q the invariant is zero (except p = q = 0)

Invariants to Gaussian PSF

If p = q = 1 the invariant is zero

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.

Recommendation for practice

It is important to learn as much as possible about the PSF and to use proper invariants for object recognition

Combined invariants

Convolution and rotation

For any N

Satellite image registration by combined invariants

( v11, v21, v31, …

)

( v12, v22, v32, …

)

min distance(( v1k, v2k, v3k, … ) , ( v1m, v2m, v3m, … ))k,m

Control point detection

Control point matching

Transformation and resampling

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).

Digit recognition by CBAI

CBAI AMI