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

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Presentation transcript:

ANSIG An Analytic Signature for 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

ANSIG MotivationANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Motivation The Permutation Problem

ANSIG MotivationANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Shape diversity

ANSIG MotivationANSIG Geometric transformations ExperimentsConclusion Real-life 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

ANSIG MotivationANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Without labels: the permutation problem permutation matrix

ANSIG MotivationANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Our approach: seek permutation invariant representations

Motivation ANSIG ANSIG Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG

Motivation ANSIG ANSIG Geometric transformations ExperimentsConclusion Real-life demonstration The analytic signature (ANSIG) of a shape

Motivation ANSIG ANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Maximal invariance of ANSIG same signature equal shapes

Motivation ANSIG ANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Maximal invariance of ANSIG Consider, such that Since, their first nth order derivatives are equal:

Motivation ANSIG ANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Maximal invariance of ANSIG This set of equalities implies that - Newton’s identities The derivatives are the moments of the zeros of the polynomials

Motivation ANSIG ANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Storing ANSIGs The ANSIG maps to an analytic function How to store an ANSIG?

Motivation ANSIG ANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Storing ANSIGs 2) Approximated by uniform sampling: 1) Cauchy representation formula: 512

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Geometrictransformations

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG (Maximal) Invariance to translation and scale Remove mean and normalize scale:

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Sampling density

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Shape rotation: circular-shift of ANSIG Rotation

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Efficient computation of rotation Solution: maximum of correlation. Using FFTs, “time” domain frequency domain Optimization problem:

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Shape-based classification SHAPE TO CLASSIFY SHAPE 3 SHAPE 2 SHAPE 1 MÁXMÁX Similarity SHAPE2SHAPE2 DATABASE

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Experiments

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG MPEG7 database (216 shapes)

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Automatic trademark retrieval

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Robustness to model violation

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Object recognition

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Conclusion

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG 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 IEEE ICIP Submitted to IEEE Transactions on PAMI

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Future developments Different sampling schemes More than one ANSIG per shape class Incomplete shapes, i.e., shape parts Analytic functions for 3D shape representation

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Real-lifedemonstration

ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Shape-based image classfication Shape database Pre-processing: morphological filter operations, segmentation, etc. Image acquisition system Shape-based classification