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CMPS 3120: Computational Geometry Spring 2013PowerPoint Presentation

CMPS 3120: Computational Geometry Spring 2013

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### CMPS 3120: Computational GeometrySpring 2013

A

B

d(B,A)

Shape Matching II

CMPS 3120 Computational Geometry

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

- 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 bB

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

- 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 distance

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

CMPS 3120 Computational Geometry

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