Computational Geometry and Geometric Shape Matching

1
Computational Geometry and Geometric Shape
Matching
2
What is Computational Geometry?

• Algorithms for geometric objects

3
Convex Hull

• Given a set of pins on a pinboard
• And a rubber band around them
• How does the rubber band look when it snaps
  tight?

4
Convex Hull

• Given a set of pins on a pinboard
• And a rubber band around them
• How does the rubber band look when it snaps
  tight?

5
Voronoi Diagram

• Given all post offices in San Antonio
• Find a subdivision of San Antonio into cells
  such that points in a cell are all closest to
  one post office

6
Voronoi Diagram

• Given all post offices in San Antonio
• Find a subdivision of San Antonio into cells
  such that points in a cell are all closest to
  one post office

7
Security Art Gallery

• Given an art gallery
• How many guards do you need to guard the whole
  gallery? Where should they be located?

8
Data bases

• Given a set of points (data sets) in high
  dimensional space
• Preprocess them such that orthogonal range
  queries can be answered efficiently.

9
Geometric Shape Matching

• Consider geometric shapes to be composed of a
  number of basic objects

10
Geometric Shape Matching

• Consider geometric shapes to be composed of a
  number of basic objects such as

points
11
Geometric Shape Matching

• Consider geometric shapes to be composed of a
  number of basic objects such as

points
line segments
12
Geometric Shape Matching

• Consider geometric shapes to be composed of a
  number of basic objects such as

points
line segments
triangles
13
Geometric Shape Matching

• Consider geometric shapes to be composed of a
  number of basic objects such as

points
line segments
triangles

• How similar are two geometric shapes?

14
Geometric Shape Matching

• Consider geometric shapes to be composed of a
  number of basic objects such as

points
line segments
triangles

• How similar are two geometric shapes?

• Choice of distance measure
• Full or partial matching
• Exact or approximate matching
• Transformations (translations, rotations,
  scalings)

15
Computer-Aided Neurosurgery
FU Berlin, Functional Imaging Technologies GmbH
and the medical school Benjamin Franklin at FU
Berlin
16
Background

• Computer assisted neuro surgery (esp. brain
  tumor surgery)

17
Background

• Computer assisted neuro surgery (esp. brain
  tumor surgery)

• Before Surgery
• Functional MR scan of the brain
• 3D model of the brain

18
Background

• Computer assisted neuro surgery (esp. brain
  tumor surgery)

• Before Surgery
• Functional MR scan of the brain
• 3D model of the brain

During Surgery
19
Background

• Computer assisted neuro surgery (esp. brain
  tumor surgery)

• Before Surgery
• Functional MR scan of the brain
• 3D model of the brain

• During Surgery
• Electromagnetic pointing device
• Display positions in 3D model

20
Background

• Computer assisted neuro surgery (esp. brain
  tumor surgery)

• Before Surgery
• Functional MR scan of the brain
• 3D model of the brain

• During Surgery
• Electromagnetic pointing device
• Display positions in 3D model

Navigation aid mapping positions in the brain to
a prerecorded 3D MR image of the brain
21
Landmark Registration

• Set of markers attached to patients head

3D model
during surgery
image
world

• Small but very noisy point sets
• Find nearly rigid motion that maps image markers
  to world markers

22
Rigid Point Matching

• Pp1,p2,,pn Qq1,q2,,qm point sets in R3

P
Q
23
Rigid Point Matching

• Pp1,p2,,pn Qq1,q2,,qm point sets in R3

P
Q

• Rigid matching maps edges with same length onto
  each other

24
Rigid Point Matching

• Pp1,p2,,pn Qq1,q2,,qm point sets in R3

P
Q

• Rigid matching maps edges with same length onto
  each other

• Nearly rigid matching maps edges with similar
  lengths onto each other

25
Scoring Table
qu
pi
qv
pj

• Edges with similar lengths indicate
  a possible matching of and
  or vice versa

• For each pair of similar edges, increase the
  score of all pairs of involved endpoints

26
Scoring Table
qu
pi
qv
pj

• Edges with similar lengths indicate
  a possible matching of and
  or vice versa

• Maintain score for each pair
  indicating the quality of matching those two
  points

p1 pi pj pn q1 qu qv qm

• For each pair of similar edges, increase the
  score of all pairs of involved endpoints

27
Finding a Transformation

• Extract combinatorial matching
• from scoring table
• Least-Squares Approximation
• Find affine transformation A that minimizes the
  sum of the squared distances between
  corresponding points
• Test if A is nearly rigid (check determinant,
  unit vector images, etc.)

28
Computer-Aided Neurosurgery Summary

• Direct linear algebra approaches were
  numerically very unstable
• Geometric approach of splitting the problem into
  - finding the combinatorial matching and
  then - computing the nearly rigid
  transformation is very easy to implement and
  proved to be very robust.
• The algorithm is integrated into a commercial
  product and used in practice.

29
Protein Gel Matching
FU Berlin, UofA, German Heart Center Berlin
30
2D Gel Electrophoresis

• Two-dimensional Gel Electrophoresis (2DE) is
• an important method in proteome research
• a high resolution technique which is capable to
  separate thousands of proteins from a tissue
  sample

31
(No Transcript)
32
(No Transcript)
33
2D Gel Electrophoresis
34
2D Gel Electrophoresis

• Proteins are concentrated in so called spots
  of (axis- parallel) elliptic shape

35
2D Gel Electrophoresis

• Proteins are concentrated in so called spots
  of (axis- parallel) elliptic shape
• Protein analysis by mass spectrometry
  (expensive)

36
2D Gel Electrophoresis
37
2D Gel Electrophoresis
Gel Matching Protein identification by gel image
comparison is faster and not expensive
38
The Algorithmic Approach
Make use of ideas and methods from Computational
Geometry

• Spot detection
• Assign to each spot the coordinates of its
  center point and its intensity

• Point pattern matching
• Consider a gel as a point pattern. Then the
  problem reduces to a partial approximate point
  pattern matching.

39
GPS Curve Location
FU Berlin and UofA and UTSA
40
Finding a Curve in a Map

• Given
• A geometric graph G (embedded in R2 with line
  segments)
• A polygonal curve a
• Task
• Find a path p in G that
• is the most similar to a

41
Finding a Curve in a Map

• Given
• A geometric graph G (embedded in R2 with line
  segments)
• A polygonal curve a
• Task
• Find a path p in G that
• is the most similar to a

42
Application Map Construction

• Consider
• A given roadmap, and
• a sequence of GPS positions obtained from a
  person travelling on some of the roads while
  recording her positioning information using a GPS
  receiver polygonal curve
• Problem
• The noise of the GPS receiver distorts the
  polygonal curve inherently
• Task
• Find the roads in the roadmap that have been
  traveled