ANSIG An Analytic Signature for Permutation Invariant 2D Shape Representation

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ANSIG An Analytic Signature for Permutation Invariant 2D Shape Representation

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ANSIG An Analytic Signature for Permutation Invariant 2D Shape Representation

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ANSIG An Analytic Signature for

Permutation Invariant 2D Shape Representation

José JerónimoMoreiraRodrigues

Outline

Motivation: shape representation

Permutation invariance: ANSIG

Dealing with geometric transformations

Experiments

Conclusion

Real-life demonstration

Motivation

The Permutation Problem

Shape diversity

When the labels are known: Kendall’s shape

‘Shape’ is the geometrical information that remains when location/scale/rotation effects are removed.

Limitation:points must have labels, i.e.,vectors must be ordered, i.e.,correspondences must be known

Without labels: the permutation problem

permutation matrix

Our approach:seek permutation invariant representations

ANSIG

The analytic signature (ANSIG) of a shape

Maximal invariance of ANSIG

same signature equal shapes

same signature equal shapes

Maximal invariance of ANSIG

Consider , such that

Since , their first nth order derivatives are equal:

Maximal invariance of ANSIG

The derivatives are the moments of the zeros of the polynomials

This set of equalities implies that - Newton’s identities

StoringANSIGs

The ANSIG maps to an analytic function

How to store an ANSIG?

StoringANSIGs

1) Cauchy representation formula:

2) Approximated by uniform sampling:

512

Geometric

transformations

(Maximal) Invariance to translation and scale

Remove mean and normalize scale:

Sampling density

Rotation

Shape rotation: circular-shift of ANSIG

Efficient computation of rotation

Optimization problem:

Solution: maximum of correlation. Using FFTs,

“time” domain frequency domain

Shape-based classification

SHAPE TO

CLASSIFY

SHAPE 1

DATABASE

Similarity

S

H

A

P

E

2

M

Á

X

SHAPE 2

Similarity

SHAPE 3

Similarity

Experiments

MPEG7 database (216 shapes)

Automatic trademark retrieval

Robustness to model violation

Object recognition

Conclusion

Summary and conclusion

- ANSIG: novel 2D-shape representation
- - Maximally invariant to permutation (and scale, translation)
- - Deals with rotations and very different number of points
- - Robust to noise and model violations

- Relevant for several applications
- Development of software packages for demonstration
- Publications:
- - IEEE CVPR 2008
- - IEEE ICIP 2008
- - Submitted to IEEE Transactions on PAMI

Future developments

Different sampling schemes

More than one ANSIG per shape class

Incomplete shapes, i.e., shape parts

Analytic functions for 3D shape representation

Real-life

demonstration

Pre-processing: morphological filter operations, segmentation, etc.

Shape-based image classfication

Image acquisition system

Shape-based classification

Shape

database