ANSIG An Analytic Signature for Permutation Invariant 2D Shape Representation

ANSIG An Analytic Signature forPermutation Invariant 2D Shape Representation

José Jerónimo Moreira Rodrigues

ANSIG

Outline

Motivation: shape representation

Permutation invariance: ANSIG

Dealing with geometric transformations

Experiments

Conclusion

Real-life demonstration

ANSIGMotivation ANSIG 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

Motivation

ANSIGANSIG Geometric


ANSIG

Motivation



The analytic signature (ANSIG) of a shape

Motivation



Maximal invariance of ANSIG

same signature equal shapes

same signature equal shapes

Motivation




Consider , such that

Since , their first nth order derivatives are equal:

Motivation




This set of equalities implies that - Newton’s identities

The derivatives are the moments of the zeros of the polynomials

Motivation



Storing ANSIGs

The ANSIG maps to an analytic function

How to store an ANSIG?

Motivation



Storing ANSIGs

2) Approximated by uniform sampling:

1) Cauchy representation formula:

512

ANSIGMotivation Geometric

transformations Experiments Conclusion Real-life demonstrationANSIG

Geometrictransformations



(Maximal) Invariance to translation and scale

Remove mean and normalize scale:



Sampling density



Shape rotation: circular-shift of ANSIG

Rotation



Efficient computation of rotation

Solution: maximum of correlation. Using FFTs,

“time” domain frequency domain

Optimization problem:



Shape-based classificationSHAPE TOCLASSIFY

SHAPE 3

SHAPE 2

SHAPE 1

MÁX

Similarity

Similarity

Similarity

SHAPE

2

DAT

ABA

SE



Experiments

Jeras

Experiments or results?



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-lifedemonstration

Jeras

Experiments or results?



Shape-based image classfication

Shap

eda

taba

se

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

Image acquisition

system

Shape-based classification

ANSIG An Analytic Signature for Permutation Invariant 2D Shape Representation

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