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Praktikum zur Analyse von Formen - Abstandsmaße -. Helmut Alt Freie Universität Berlin. Distance functions, Matching. Distance functions d on patterns and shapes measuring their similarity

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Praktikum zur analyse von formen abstandsma e

Praktikum zur Analyse von Formen- Abstandsmaße -

Helmut Alt

Freie Universität Berlin


Distance functions matching
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


Hausdorff distance
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



Fr chet distance

dF(a,b) = inf max ||a(f(t))-b(g(t))||

f,g : [0,1 ] [0,1] t [0,1]

where f and g range over continuous non-decreasing

reparametrizations.

a

b

Fréchet Distance


Free space diagram

b

a

b

a

Free Space Diagram

  • dF(a,b) eiff there is a monotone ascending

    path in the free space from (0,0) to (1,1)

  • Monotone path represents a reparametrization of b


Finding a monotone path in o mn time

b

a

b

a

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


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