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