Probabilistic Procrustean Shape Analysis for Object Recognition

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Probabilistic Procrustean Shape Analysis for
Object Recognition

• Jared Glover
• Computer Science and Artificial Intelligence
  Laboratory
• Massachusetts Institute of Technology

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A Theory of Shape
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Normalizing Position
1
-1
1
-1
4
Normalizing Position
1
-1
1
-1
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Normalizing Position
1
-1
1
-1
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Normalizing Scale
1
-1
1
-1
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Normalizing Scale
1
-1
1
-1
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Normalizing Orientation
1
Match first edge?
-1
1
-1
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Normalizing Orientation
1
Match first edge?
-1
1
-1
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Normalizing Orientation
1
Match first edge?
-1
1
-1
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Normalizing Orientation
1
pre-shapes
-1
1
-1
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Normalizing Orientation
1
pre-shapes
-1
1
pre-shape space?
-1
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Normalizing Orientation
1
pre-shapes
-1
1
pre-shape space?
S2p-3
-1
hypersphere
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Normalizing Orientation

• An equivalence class of pre-shapes (over
  rotations)

great circle
S2p-3
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Shape Distance

• distance between shapes minimum distance
  between orbits

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Shape Distance

• distance between shapes minimum distance
  between orbits

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Shape Distance

• So, what is our minimization problem?

Procrustes distance
z rotated by -?
geodesic distance
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Shape Distance

• Now, minimize with respect to ?
• But cos-1 is monotonic decreasing,
• so minimize

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Shape Distance
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Shape Distance
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Shape Distance
Use this to find optimal ?
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Shape Distance

• Now, plug this
• into this
• to get

Procrustean Metric
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Shape Distance

• Q What does Procrustes distance correspond to
  in the real world?

• A The optimal ? in the Procrustes minimization
  problem will also minimize the sum of squared
  distances between corresponding landmarks!

Procrustes distance sum of squared errors
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A Good Approximation

• Tangent space

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A Good Approximation

• Tangent space
• at what point?

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A Good Approximation

• Tangent space
• at what point?
• Mean shape
• Given preshapes w1,,wn
• Let µ w1
• Repeat until µ converges
• Rotate w1,,wn to match µ
• Let

this will converge to
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A Good Approximation

• Tangent space coordinates
• Tangent Space Principle Components Analysis (PCA)
• eigenshape representation

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Reef Exploration

• Classify and track fish, coral
• eigenfish!
• (with Daniela Rus, Iuliu Vasilescu, Nicholas Roy)

AMOUR
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Reef Exploration

• Classify and track fish, coral
• eigenfish!
• (with Daniela Rus, Iuliu Vasilescu, Nicholas Roy)

AMOUR
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Getting Contours

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Finding Features
Local shape derivatives for feature detection
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Feature Correspondences
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Principle Components
Angelfish
Yellow Tang