ANSIG An Analytic Signature for ANSIG An Analytic Signature for Permutation Invariant 2D Shape Representation José Jerónimo Moreira Rodrigues

ANSIG An Analytic Signature for

Permutation 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 ConclusionReal-life

demonstration

Motivation

The

Permutation

Problem




demonstration

Shape diversity




demonstration

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




demonstration

Without labels: the permutation problem

permutation matrix




demonstration

Our approach:seek permutation invariant representations

Motivation

ANSIGANSIG



demonstration

ANSIG

Motivation

ANSIGANSIG



demonstration

The analytic signature (ANSIG) of a shape

Motivation

ANSIGANSIG



demonstration

Maximal invariance of ANSIG

same signature equal shapes

same signature equal shapes

Motivation

ANSIGANSIG



demonstration


Consider , such that

Since , their first nth order derivatives are equal:

Motivation

ANSIGANSIG



demonstration


This set of equalities implies that - Newton’s identities

The derivatives are the moments of the zeros of the polynomials

Motivation

ANSIGANSIG



demonstration

Storing ANSIGs

The ANSIG maps to an analytic function

How to store an ANSIG?

Motivation

ANSIGANSIG



demonstration

Storing ANSIGs

2) Approximated by uniform sampling:

1) Cauchy representation formula:

512

ANSIGMotivation



demonstrationANSIG

Geometric

transformations

ANSIGMotivation



demonstrationANSIG

(Maximal) Invariance to translation and scale

Remove mean and normalize scale:

ANSIGMotivation



demonstrationANSIG

Sampling density

ANSIGMotivation



demonstrationANSIG

Shape rotation: circular-shift of ANSIG

Rotation

ANSIGMotivation



demonstrationANSIG

Efficient computation of rotation

Solution: maximum of correlation. Using FFTs,

“time” domain frequency domain

Optimization problem:

ANSIGMotivation



demonstrationANSIG

Shape-based classification

SHAPE TOCLASSIFY

SHAPE 3

SHAPE 2

SHAPE 1

MÁX

Similarity

Similarity

Similarity

SHAPE

2

DAT

AB

ASE

ANSIGMotivation



demonstrationANSIG

Experiments

Jeras

Experiments or results?

ANSIGMotivation



demonstrationANSIG

MPEG7 database (216 shapes)

ANSIGMotivation



demonstrationANSIG

Automatic trademark retrieval

ANSIGMotivation



demonstrationANSIG

Robustness to model violation

ANSIGMotivation



demonstrationANSIG

Object recognition

ANSIGMotivation



demonstrationANSIG

Conclusion

ANSIGMotivation



demonstrationANSIG

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

ANSIGMotivation



demonstrationANSIG

Future developments

Different sampling schemes

More than one ANSIG per shape class

Incomplete shapes, i.e., shape parts

Analytic functions for 3D shape representation

ANSIGMotivation



demonstrationANSIG

Real-life

demonstration

Jeras

Experiments or results?

ANSIGMotivation



demonstrationANSIG

Shape-based image classfication

Shap

eda

taba

se

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

Image acquisition

system

Shape-based classification

Documents

ANSIG An Analytic Signature for ANSIG An Analytic Signature for Permutation Invariant 2D Shape Representation José Jerónimo Moreira Rodrigues