Praktikum zur Analyse von Formen - Abstandsma

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Praktikum zur Analyse von Formen- Abstandsmaße -
Helmut Alt Freie Universität Berlin
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Distance functions, Matching

• Distance functions d on patterns and shapes
  measuring their similarity
• Matching two shapes under a certain set of
  allowable transformations, e.g., translations,
  rigid motions, similarities, affine
  transformations finding the transformation t
  minimizing the distance between both
• d(t (A),B) min d(t(A),B)

0
0
t
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Hausdorff Distance

• Hausdorff distance for sets A,B

d(A,B) max ( max min a-b, max min
a-b) a ? A b ?
B b ? B a ? A
A
B
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Bad Example for Hausdorff distance
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Fréchet Distance
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Free Space Diagram

• dF(a,b) ? e iff there is a monotone ascending
• path in the free space from (0,0) to (1,1)
• Monotone path represents a reparametrization of b

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Finding a monotone path in O(mn) time
Regions on the boundaries of the cells that can
be reached by a monotone path from lower left
corner. Find these by traversing cells from
bottom to top and from left to right.
This algorithm solves the decision problem in
O(nm) time. For the computation problem Binary
search on e in O(nmk), where k of correct
bits Parametric search gives an O(nm log (nm))
algorithm