Title: Praktikum zur Analyse von Formen - Abstandsma
1Praktikum zur Analyse von Formen- Abstandsmaße -
Helmut Alt Freie Universität Berlin
2Distance 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
3Hausdorff 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
4Bad Example for Hausdorff distance
5Fréchet Distance
6Free 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
7Finding 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