Introduccin a las imgenes digitales

1
183.151Selected Chapters in Image
ProcessingExtracting Topological Information of
3D Digital Images
Rocío González-Díaz rogodi_at_us.es

• Content
• What is topology? Bases for topological spaces
• 3D Digital images.
• Digital topology and continuous analogues
• Cellular complexes, Betti numbers and homology
• Algebraic-topological invariants of 3D digital
  images
• Examples

2
Cellular complexes, Betti numbers and homology
groups
3
Cellular complexes, Betti numbers and homology
groups
4
Cellular complexes, Betti numbers and homology
groups
CELL COMPLEX-gtCHAIN COMPLEX-gtHOMOLOGY
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Cellular complexes, Betti numbers and homology
groups
CELL COMPLEX-gtCHAIN COMPLEX-gtHOMOLOGY
Examples
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Cellular complexes, Betti numbers and homology
groups
Question 1 how can we compute homology groups?
7
Cellular complexes, Betti numbers and homology
groups
Question 2 How can we compute finer invariants
than homology?
DIFERENT EMBEDDINGS OF TWO DONUTS

• 1 connected component
• 2 tunnels
• 0 cavities

8
Cellular complexes, Betti numbers and homology
groups
Question 2 How can we compute finer invariants
than homology?
DIFFERENT CONFIGURATIONS OF THREE TORUS

• 1 connected component
• 4 tunnels
• 3 cavities

9
183.151Selected Chapters in Image
ProcessingExtracting Topological Information of
3D Digital Images
Rocío González-Díaz rogodi_at_us.es

• Content
• What is topology? Bases for topological spaces
• 3D Digital images.
• Digital topology and continuous analogues
• Cellular complexes, Betti numbers and homology
• Algebraic-topological invariants of 3D digital
  images
• Examples

10
Algebraic-topological invariants
A cellular complex
homology groups and generators cohomology ring
(finer invariant than homology)
11
Algebraic-topological invariants
Question 1 how can we compute homology?
INCREMENTA ALGORITHM (Edelsbrunner et al 2002)
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Algebraic-topological invariants
Question 1 how can we compute homology?
INCREMENTA ALGORITHM (Edelsbrunner et al 2002)
13
Algebraic-topological invariants
Question 1 how can we compute homology?
INCREMENTA ALGORITHM (Edelsbrunner et al 2002)
14
Algebraic-topological invariants
Question 1 how can we compute homology?
INCREMENTA ALGORITHM (Edelsbrunner et al 2002)
15
Algebraic-topological invariants
Question 1 how can we compute homology?
ALGEBRAIC VERSION OF INCREMENTAL ALGORITHM (G-D,
Real 2003)
EXAMPLE
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Algebraic-topological invariants
Question 1 how can we compute homology?
ALGEBRAIC VERSION OF INCREMENTAL ALGORITHM (G-D,
Real 2003)
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Algebraic-topological invariants
Question 1 how can we compute homology?
ALGEBRAIC VERSION OF INCREMENTAL ALGORITHM (G-D,
Real 2003)
18
Algebraic-topological invariants
Question 1 how can we compute homology?
ALGEBRAIC VERSION OF INCREMENTAL ALGORITHM (G-D,
Real 2003)
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Algebraic-topological invariants
Question 1 how can we compute homology?
ALGEBRAIC VERSION OF INCREMENTAL ALGORITHM (G-D,
Real 2003)
EXAMPLE
H0 Z2(0) H1 Z2(1,2)
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Algebraic-topological invariants
Question 1 how can we compute homology?
ALGEBRAIC VERSION OF INCREMENTAL ALGORITHM (G-D,
Real 2003)
Why does it work?
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Algebraic-topological invariants
Question 1 how can we compute homology?
ALGEBRAIC VERSION OF INCREMENTAL ALGORITHM (G-D,
Real 2003)
Why does it work?
22
Algebraic-topological invariants
Question 1 how can we compute homology?
ALGEBRAIC VERSION OF INCREMENTAL ALGORITHM (G-D,
Real 2003)
Some properties
23
Algebraic-topological invariants
24
Algebraic-topological invariants
Question 2 How can we compute finer invariants
than homology?
DIFFERENT CONFIGURATIONS OF THREE TORUS
25
Algebraic-topological invariants
COHOMOLOGY
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Cohomology
COHOMOLOGY GROUPS
H0 Z2(0) H0 Z2(0) H1 Z2(1,2)
H1 Z2(1,2)
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Cohomology
Z2-COHOMOLOGY RING
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Cohomology
Z2-COHOMOLOGY RING
H0 Z2(0) H0 Z2(0) H1 Z2(1,2)
H1 Z2(1,2)
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Cohomology
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Cohomology
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Cohomology
Example 1 Z2-Cohomology ring of a sphere with
two lops
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Cohomology
Example 1 Z2-Cohomology ring of a sphere with
two lops
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Cohomology
Example 1 Z2-Cohomology ring of a sphere with
two lops
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Cohomology
Example 2 Z2-Cohomology ring of the torus
35
Cohomology
Example 2 Z2-Cohomology ring of the torus
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Cohomology
HB11
HB10
They are not homotopic nor homeomorphic
37
Algebraic-topological invariants
Bibliography

• H. Edelsbrunner, D. Letscher, and A. Zomorodian,
  Topological Persistence and Simplification,
  Discrete Comput. Geom., 28 (2002), 511--533.
• GonzálezDíaz R., Real P. Towards Digital
  Cohomology. DGCI 2003, LNCS, Springer 2886 (2003)
  92101
• GonzálezDíaz R., Real P. On the Cohomology of
  3D Digital Images. Discrete Applied Math 147
  (2005) 245263

38
Algebraic-topological invariants
A cellular complex
homology groups and generators cohomology ring
(finer invariant than homology)