ANSIG An Analytic Signature for ANSIG An Analytic Signature for Permutation Invariant 2D Shape...
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Transcript of ANSIG An Analytic Signature for ANSIG An Analytic Signature for Permutation Invariant 2D Shape...
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