Dynamic Programming (DP), Shortest Paths (SP)

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Dynamic Programming (DP),Shortest Paths (SP)

• CS664 Lecture 22
• Thursday 11/11/04
• Some slides care of Yuri Boykov, Dan Huttenlocher

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Level sets
Donald Tanguay
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Level sets and curve evolution
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Shortest path problem
Lecture theme
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Dijkstra algorithm
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Shortest paths segmentation
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Shortest paths segmentation
Example find the shortest closed contour in a
given domain of a graph
Repeat for all points on the black line. Then
choose the optimal contour.
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DP (SP) for stereo
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Discrete snakes

• Represent the snake as a set of points
• Curve as spline, e.g. (particle method)
• Local update problem can be solved exactly
  (compute global min)
• Do this repeatedly
• Problems with collisions, change of topology

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Discrete snake energy
Best location of the last vertex vn depends only
the location of vn-1
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Discrete snakes example
control points
Fold data term into smoothness term
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Energy minimization by SP
sites
states
1
2

m
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Distance transform (DT)
Note can be generalized beyond 1P (DT of
arbitrary f)
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Computing distance transforms

• Depends on the distance measure (L1 or L2
  distance)
• Linear time algorithms based on dynamic
  programming
• Fast in practice
• Can think of this as smoothing in feature space

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Distance transform applications

• Primarily used in recognition
• Represent the model as a set of points
• Edges, or maybe corners
• Compare model to image
• Under some transformation of the model
• Chamfer matching L1 distance on distance
  transform
• Not robust at all

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Hausdorff distance

• Defined between two sets of points
• h(A,B)? if every point in A lies within ? of the
  nearest point in B
• ? is the smallest value for which this holds