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

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Presentation on theme: "ANSIG An Analytic Signature for ANSIG  An Analytic Signature for Permutation Invariant 2D Shape Representation José Jerónimo Moreira Rodrigues."— Presentation transcript:

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

2 ANSIG Outline Motivation: shape representation Permutation invariance: ANSIG Dealing with geometric transformations Experiments Conclusion Real-life demonstration

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

4 ANSIG MotivationANSIG Geometric transformations ExperimentsConclusion Real-life demonstration Shape diversity

5 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

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

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

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

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

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

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

12 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

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

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

15 ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Geometrictransformations

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

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

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

19 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:

20 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

21 ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Experiments

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

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

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

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

26 ANSIG Motivation Geometric transformations ExperimentsConclusion Real-life demonstration ANSIG Conclusion

27 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 2008 - IEEE ICIP 2008 - Submitted to IEEE Transactions on PAMI

28 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

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

30 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

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