Matching 2D articulated shapes using

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Matching 2D articulated shapes using Generalized
Multidimensional Scaling
Michael M. Bronstein
Department of Computer Science Technion Israel
Institute of Technology

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Co-authors
Ron Kimmel
Alex Bronstein
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Main problems
Comparison of articulated shapes
Comparison of partially-missing articulated
shapes
Local differences between shapes
Correspondence between articulated shapes
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Ideal articulated shape
ISOMETRY

• Two-dimensional shape
• Geodesic distances induced by the
  boundary
• Consists of rigid parts and
  point joins
• Space of ideal articulated shapes
• Articulation an isometric deformation
  such that

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- articulated shape
-ISOMETRY

• Rigid parts and joints
  with
• Space of -articulated shapes
• Articulation an -isometry
  such that

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Articulation-invariant distance
A distance
between articulated shapes should satisfy

• Non-negativity
• Symmetry
• Triangle inequality
• Articulation invariance
  for all and all
• articulations of
• Dissimilarity if ,
  then there do not exist and
• two articulations of such that
  and
• Consistency to sampling if and are
  finite -coverings of
• and , then
• Efficiency can be efficiently computed

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Canonical forms distance (I)
Embed and into a common metric space
by minimum-distortion
embeddings and compare the images (canonical
forms)
A. Elad, R. Kimmel, CVPR 2001 H. Ling, D. Jacobs,
CVPR 2005
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Canonical forms distance (II)

• Approximately articulation invariant
• Approximately consistent to sampling
• Efficient computation using multidimensional
  scaling (MDS)

Given a sampling
the minimum-distortion embedding is found by
optimizing over the images
and not on itself
A. Elad, R. Kimmel, CVPR 2001
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Gromov-Hausdorff distance
Allow for an arbitrary embedding space

• A metric on the space
• Consistent to sampling if and are
  -coverings of and ,
• Computation untractable

M. Gromov, 1981
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Computing the Gromov-Hausdorff distance (I)
Equivalent definition in terms of metric
distortions
Where
Mémoli Sapiro (2005)

• Replace with a simpler
  expression
• Probabilistic bound on the error
• Combinatorial problem

F. Mémoli, G. Sapiro, Foundations Comp. Math,
2005
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Computing the Gromov-Hausdorff distance (II)
The Gromov-Hausdorff distance is essentially a
problem of finding minimum-distortion maps
between and , and can be computed in
an MDS-like spirit
Given the samplings
and , the
minimum-distortion embeddings are found by
optimizing over the images
and
B2K, PNAS 2006
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Generalized multidimensional scaling (GMDS)
G
MDS
MDS

• The distances have no analytic
  expression and must be
• approximated numerically
• Multiresolution scheme to prevent local
  convergence
• -norm can be used instead of for a
  more robust computation

B2K, PNAS 2006
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Example I comparison of shapes
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Example I comparison of shapes
Similarity patterns between different articulated
shapes
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Adding another axiom

• Partial matching If is a convex cut of
  , then

Convex cut guarantees
Partial matching is non-symmetric some
properties must be sacrificed
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Triangle inequality
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The danger of partial matching
does not necessarily
imply that
But if is an -covering of , then
Illustration Herluf Bidstrup
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Partial embedding distance (I)
Use the distortion as a measure of partial
similarity

• Non-symmetric
• Allows for partial matching
• Consistent to sampling
• In the discrete setting, posed as a
• GMDS problem
• Computationally efficient

B2K, PNAS 2006
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Example partial matching
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Local comparison
Use the contribution of a single point to the
distortion as a measure of local difference
between the shapes, or local distortion
B2K, PNAS 2006
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Example local differences
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Summary

• Isometric model of articulated shape
• Axiomatic approach to comparison of shapes
• Partial matching and correspondence
• GMDS - a generic tool for shape recognition and
  matching

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3D example
B2K, SIAM J. Sci. Comp, to appear
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3D example
Canonical forms distance (MDS, 500 points)
Gromov-Hausdorff distance (GMDS, 50 points)
B2K, SIAM J. Sci. Comp, to appear