ANSIG An Analytic Signature for Permutation Invariant 2D Shape Representation
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Transcript of ANSIG An Analytic Signature for Permutation Invariant 2D Shape Representation
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ANSIG An Analytic Signature forPermutation Invariant 2D Shape Representation
José Jerónimo Moreira Rodrigues
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ANSIG
Outline
Motivation: shape representation
Permutation invariance: ANSIG
Dealing with geometric transformations
Experiments
Conclusion
Real-life demonstration
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ANSIGMotivation ANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
Motivation
The
Permutation
Problem
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ANSIGMotivation ANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
Shape diversity
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ANSIGMotivation ANSIG Geometric
transformations Experiments Conclusion 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
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ANSIGMotivation ANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
Without labels: the permutation problem
permutation matrix
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ANSIGMotivation ANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
Our approach:seek permutation invariant representations
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Motivation
ANSIGANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
ANSIG
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Motivation
ANSIGANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
The analytic signature (ANSIG) of a shape
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Motivation
ANSIGANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
Maximal invariance of ANSIG
same signature equal shapes
same signature equal shapes
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Motivation
ANSIGANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
Maximal invariance of ANSIG
Consider , such that
Since , their first nth order derivatives are equal:
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Motivation
ANSIGANSIG Geometric
transformations Experiments Conclusion 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
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Motivation
ANSIGANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
Storing ANSIGs
The ANSIG maps to an analytic function
How to store an ANSIG?
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Motivation
ANSIGANSIG Geometric
transformations Experiments Conclusion Real-life demonstration
Storing ANSIGs
2) Approximated by uniform sampling:
1) Cauchy representation formula:
512
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Geometrictransformations
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
(Maximal) Invariance to translation and scale
Remove mean and normalize scale:
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Sampling density
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transformations Experiments Conclusion Real-life demonstrationANSIG
Shape rotation: circular-shift of ANSIG
Rotation
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transformations Experiments Conclusion Real-life demonstrationANSIG
Efficient computation of rotation
Solution: maximum of correlation. Using FFTs,
“time” domain frequency domain
Optimization problem:
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Shape-based classificationSHAPE TOCLASSIFY
SHAPE 3
SHAPE 2
SHAPE 1
MÁX
Similarity
Similarity
Similarity
SHAPE
2
DAT
ABA
SE
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Experiments
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
MPEG7 database (216 shapes)
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Automatic trademark retrieval
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Robustness to model violation
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Object recognition
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Conclusion
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life 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
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life 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
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Real-lifedemonstration
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ANSIGMotivation Geometric
transformations Experiments Conclusion Real-life demonstrationANSIG
Shape-based image classfication
Shap
eda
taba
se
Pre-processing: morphological filter operations, segmentation, etc.
Image acquisition
system
Shape-based classification