Cmps 3120 computational geometry spring 2013
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CMPS 3120: Computational Geometry Spring 2013. A. B. d  (B,A). Shape Matching II. Geometric Shape Matching. Given:. Two geometric shapes , each composed of a number of basic objects such as. points. line segments. triangles.

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Cmps 3120 computational geometry spring 2013

CMPS 3120: Computational GeometrySpring 2013

A

B

d(B,A)

Shape Matching II

CMPS 3120 Computational Geometry


Geometric Shape Matching

Given:

Twogeometric shapes, each composed of a number of basic objects such as

points

line segments

triangles

An application dependantdistance measured, e.g., the Hausdorff distance

A set of transformations T, e.g., translations, rigid motions, or none

min d(T(A),B)

Matching Task: Compute

TÎT

CMPS 3120 Computational Geometry


Hausdorff distance
Hausdorff distance

  • Directed Hausdorff distance

    d(A,B) = max min || a-b ||

  • Undirected Hausdorff-distance

    d(A,B) = max (d(A,B) , d(B,A) )

A

B

a A bB

d(B,A)

d(A,B)

  • If A, B are discrete point sets, |A|=m, |A|=n

    • In d dimensions: Compute in O(mn) time brute-force

    • In 2 dimensions: Compute in O((m+n) log (m+n) time

      • Compute d(A,B) by computing VD(B) and performing nearest neighbor search for every point in A

CMPS 3120 Computational Geometry


Shape matching applications
Shape Matching - Applications

  • Character Recognition

  • Fingerprint Identification

  • Molecule Docking, Drug Design

  • Image Interpretation and Segmentation

  • Quality Control of Workpieces

  • Robotics

  • Pose Determination of Satellites

  • Puzzling

  • . . .

CMPS 3120 Computational Geometry


Hausdorff distance1
Hausdorff distance

  • W(m2n2) lower bound construction for directed Hausdorff distance of line segments under translations

CMPS 3120 Computational Geometry


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