Praktikum zur Analyse von Formen - Abstandsmaße - Helmut Alt Freie Universität Berlin

Distance functions, Matching  Distance functions  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:   (t (A),B) = min  (t(A),B) t 0 0

Hausdorff Distance  (  ) = max ( max min ||a-b||, max min ||a-b||) a  A b  B b  B a  A –Hausdorff distance for sets A,B: A B

Bad Example for Hausdorff distance

Fréchet Distance    F ( ,  ) = inf max ||  (f(t))-  (g(t))|| f,g : [0,1 ] [0,1] t  [0,1] where f and g range over continuous non-decreasing reparametrizations.  Fréchet distance for parametrized curves ,  :

Free Space Diagram       F ( ,  )   iff there is a monotone ascending path in the free space from (0,0) to (1,1)  Monotone path represents a reparametrization of 

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  in O(nmk), where k = # of correct bits Parametric search gives an O(nm log (nm)) algorithm