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ANSIG  An Analytic Signature for Permutation Invariant 2D Shape Representation

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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  1. ANSIG  An Analytic Signature for Permutation Invariant 2D Shape Representation José JerónimoMoreiraRodrigues

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

  3. Motivation The Permutation Problem

  4. Shape diversity

  5. 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. Without labels: the permutation problem permutation matrix

  7. Our approach:seek permutation invariant representations

  8. ANSIG

  9. The analytic signature (ANSIG) of a shape

  10. Maximal invariance of ANSIG same signature equal shapes same signature equal shapes

  11. Maximal invariance of ANSIG Consider , such that Since , their first nth order derivatives are equal:

  12. Maximal invariance of ANSIG The derivatives are the moments of the zeros of the polynomials This set of equalities implies that - Newton’s identities

  13. StoringANSIGs The ANSIG maps to an analytic function How to store an ANSIG?

  14. StoringANSIGs 1) Cauchy representation formula: 2) Approximated by uniform sampling: 512

  15. Geometric transformations

  16. (Maximal) Invariance to translation and scale Remove mean and normalize scale:

  17. Sampling density

  18. Rotation Shape rotation: circular-shift of ANSIG

  19. Efficient computation of rotation Optimization problem: Solution: maximum of correlation. Using FFTs, “time” domain frequency domain

  20. 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

  21. Experiments

  22. MPEG7 database (216 shapes)

  23. Automatic trademark retrieval

  24. Robustness to model violation

  25. Object recognition

  26. Conclusion

  27. 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. 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. Real-life demonstration

  30. Pre-processing: morphological filter operations, segmentation, etc. Shape-based image classfication Image acquisition system Shape-based classification Shape database

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