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ANSIG  An Analytic Signature for Permutation Invariant 2D Shape Representation. José Jerónimo Moreira Rodrigues. Outline. Motivation: shape representation Permutation invariance : ANSIG Dealing with geometric transformations Experiments Conclusion Real-life demonstration.

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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


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