Canonical Skeletons for Shape Matching

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Canonical Skeletons for Shape Matching

• Matthijs van Eede
• University of Toronto
• August 22nd, 2006
• Joint work with Diego Macrini, Alex Telea,
  Cristian Sminchisescu, and Sven Dickinson

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Skeleton Definition
The skeleton of a shape yields a symmetry-based
parts decomposition (e.g., a shock graph) which
can support effective object indexing and
recognition, e.g., Siddiqi et al. (1999),
Sebastian et al. (2004). But, they suffer from
two forms of instability
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Skeletal Instabilities Type 1
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Skeletal Instabilities Type 1
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Skeletal Instabilities Type 2
Ligature segment
Ligature branch
Blum (1973)
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Goal

• Smooth these structural instabilities while
  retaining the objects salient shape structure.
• Two exemplar shapes drawn from the same category
  should therefore yield two graphs with the same
  structure.

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Approach

• Prune skeletal branches that dont contribute to
  the salient shape structure of the object.
• Simpler graphs with fewer unstable nodes lead to
  more efficient and more effective indexing and
  matching.
• But how do we measure branch saliency and when do
  we stop pruning?

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Skeletal Simplification as Optimization
Reconstruction error
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External Branch Pruning
Saliency favors elongated and thick parts
External branches rank-ordered by saliency
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4
8
2
5
1
3
6
9
12
10
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External Branch Pruning
The cost of external branch smoothing increased
reconstruction error
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Internal Branch Pruning

• Intuitively create similar topologies in the
  skeletons by pruning short (low
  saliency) ligature segments and branches

Ligature branch
Ligature segment
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Internal Branch Simplification
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Ligature Detection
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Internal Branch Pruning
Fit piecewise linear skeleton fragments subject
to endpoint and tangent constraints
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Internal Branch Smoothing
The cost of internal branch smoothing altering
the shapes appearance
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Cost Function

• Fact ? the medial axis transform of a shape is
  unique skeleton changes introduce reconstruction
  error
• Goal ? minimize a cost function that promotes
  simpler skeletons with low reconstruction error

branches
Reconstruction error
R(sp)
sp
p
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Final Algorithm

• Rank-order external branches by saliency
• Iteratively prune low-saliency external branches
  until cost function is minimized
• For internal branches, identify the ligature
  branches as candidates for pruning, and
  rank-order them by saliency
• Iteratively prune low-saliency candidate internal
  branches until cost function is minimized

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Canonical Skeleton
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Object Recognition Using Shock Graphs
Siddiqi, Shokoufandeh, Dickinson and Zucker, IJCV
1999
Type 1
Type 2
Type 3
Type 4
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Shock Graphs
Siddiqi et al. (1999)
LabelsType-ID
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Object Recognition Experiments

• Shock graphs are computed for 15 views of 8 three
  dimensional CAD models. A total of 120 shapes in
  the database.
• Each object view is removed from the database and
  used as a query
• Successful object recognition ? best ranked view
  belongs to the same object as query view
• Successful pose estimation ? neighbouring view of
  query is among top ranked views
• Noise is simulated by adding random bumps and
  notches to the query.

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Parameter Estimation on Database Without Noise
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Results of Experiments with Noise

• Object recognition performance increased up to
  16
• Pose estimation performance increased up to 20
• (r5) having a radius of 5 pixels

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Conclusions

• Skeletal descriptions of a shape offer a powerful
  shape representation for object recognition, yet
  their structural instability has long been an
  obstacle to their widespread use.
• Our structural simplification framework isolates
  this instability at both external and internal
  branches, and removes non-salient branches.
• The removal of internal branches requires a
  proper smoothing of neighboring branches so that
  the resulting skeleton is a MAT and
  reconstruction error is minimized.
• Results on a shock graph recognition experiment
  indicate a significant improvement in recognition
  and pose estimation performance when both query
  and database are structurally simplified prior to
  recognition.