Efficient belief propagation for early vision

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Efficient belief propagation for early vision

• Paper by P.F Felzenszwalb (university of
  chigago) and D.P. Huttenlocher (Cornell
  university)
• Presented by Diederik Roijers for the seminar on
  pattern recognition at Utrecht university, fall
  2009.

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Contents

• Task
• Theory
• Runtime
• Making it more efficient
• Making message calculation linear
• Using the bipartite property
• Multi-grids
• Summary

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Task

• Picture grid of pixels
• Input picture known
• Output labels for pixels
• Image restoration
• Stereo vision

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Theory

• No longer trees
• Standard sum-product does not apply
• Loopy belief propagation
• Iterative
• Local

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Loopy sum-product BP
Favors looking like the input
Favors smoothness
Most probable value for each pixel But we want
the most probable configuration
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Loopy max-product BP

• Again, logarithm holds the same maximum
• Minus makes it a minimizing the cost problem
• Then fq should minimize bq for the solution

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Loopy max-product BP

• Again, logarithm holds the same maximum
• Minus makes it a minimizing the cost problem
• O(nk2T)

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Contents

• Task
• Theory
• Runtime
• Making it more efficient
• Making message calculation linear
• Using the bipartite property
• Multi-grids
• Summary and critique

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More effiently

• Message calculation linear
• Number of messages per iteration in half
• Reducing the number of iterations to a small
  constant
• O(nkT)

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Linear messages

• Trick depending on scoring functions
• Example 1

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Linear messages

• There are also tricks for
• V(x-y) cx-y
• V(x-y) c(x-y)2
• And truncated versions of the both

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Cutting the number of messages

• Graph is bipartite
• Messages only depend on direct neighbors
• Checker board
• Black only depends on white and vice versa

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Problem

• Information on the one side of the picture
  propagates very slowly through the MRF
• Iterations should be at least
• About Sqrt(Width2height2) of the picture
• To reach every corner of the picture

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Solution

• Top level down.
• Instantiate the messages with the values of the
  higher levels.

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Contents

• Task
• Theory
• Runtime
• Making it more efficient
• Making message calculation linear
• Using the bipartite property
• Multi-grids
• Summary and critique

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Summary and critique

• Reduces the runtime of BP to O(nkT)?
• But is T a small constant? Dependent on n?
• Notation change when explaining the trick for
  V(x-y)cx-y