Introduccin a las imgenes digitales

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Selected Chapters in Image Processing Prof.
Walter Kropatsch
183.151
Extracting Topological Information of 3D Digital
Images Dr. Rocio Gonzalez-Diaz rogodi_at_us.es
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183.151Selected Chapters in Image
ProcessingExtracting Topological Information of
3D Digital Images

• Web page http//www.prip.tuwien.ac.at/teaching/ss
  /akbv
• Speaker Rocío González-Díaz rogodi_at_us.esPerson
  al web-page http//www.personal.us.es/rogodi
• Organisation
• Registration TUWIS
• Termin Block von 22.4. bis 30.4. Ort
  Seminarraum 183/2

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183.151Selected Chapters in Image
ProcessingExtracting Topological Information of
3D Digital Images
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183.151Selected Chapters in Image
ProcessingExtracting Topological Information of
3D Digital Images
MOTIVATION

• Aim of the course to learn how to extract
  properties and features of digital images that
  correspond to topological properties or
  topological features of objects
• algorithms for thinning, border or surface
  tracing,
• counting of components or tunnels,
• or region-filling.

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183.151Selected Chapters in Image
ProcessingExtracting Topological Information of
3D Digital Images
APPLICATION DOMAINS

• Geometric Modeling

• Digital Image Processing

• Pattern Recognition

• Medical Imaging

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183.151Selected Chapters in Image
ProcessingExtracting Topological Information of
3D Digital Images
RELATED READING

• 3D Image Processing
• Lohmann G. Volumetric Image Analysis. Wiley
  Sons, 1998.
• Nikolaidis N., Pitas I. 3D Image Processing
  Algorithms. Wiley Sons, 2000.
• Girod, B., Greiner, G. Principles of 3D Image
  Analysis and Synthesis. Springer-Verlag, 2000

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183.151Selected Chapters in Image
ProcessingExtracting Topological Information of
3D Digital Images
RELATED READING

• Digital topology
• Kong, T.Y., Rosenfeld A. Topological Algorithms
  for Digital Image Processing. Elsevier, 1996.
• Klette, R., Rosenfeld A. Digital Geometry.
  Morgan Kaufmann Series in Computer Graphics and
  Geometric Modeling, 2004
• Topology and Algebraic Topology
• Munkres J. Topology. Prentice Hall, 1999.
• Hatcher A. Algebraic Topology. Cambridge
  University Press in 2002

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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

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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

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Selected Chapters in Image ProcessingExtracting
Topological Information of 3D Digital Images
What is Topology?
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Selected Chapters in Image ProcessingExtracting
Topological Information of 3D Digital Images
What is Topology?

• Topology
• Topological invariant
• Homotopy
• Metric spaces

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Selected Chapters in Image ProcessingExtracting
Topological Information of 3D Digital Images
What is Topology?

• Topology
• Topological invariant
• Homotopy
• Metric spaces

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What is topology?

Informally, topology is the study of properties
that remain unaffected by continuously
deforming the shape or size of a figure.
bending, squeezing, stretching, and compressing,
but not by breaking or tearing
The torus and a mug are topologically equivalent.
Informally, two objects are topologically
equivalent if one can be continuously deformed
into the other.
http//members.shaw.ca/jillbritton/qgoo/jbqgoo.htm
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What is topology?

What is the exact meaning of object and
continuity?
An object a topological space a set and a
topology on the set

• A topology on a set X is a collection T of
  subsets of X such that
• and X are in T.
• Union of elements of T is in T.
• Finite intersection of T is in T.

A subset U of X is open if U belongs to T . A
subset U of X is close if X-U is open.
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What is topology?

What is the exact meaning of object and
continuity?

• Examples of topologies
• The collection of all subsets of X (discrete
  topology).
• The collection consisting of X and
  (indiscrete topology).
• If X is the set of real numbers, the collection
  of all open intervals and unions of intervals
  (Euclidean topology).

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What is topology?

What is the exact meaning of object and
continuity?

• If is a set, a basis for a topology on
  is a collection of subsets of such
  that
• For each in , there exist one in
  containing .
• If belongs to and , then there
  exist in containing such that

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What is topology?

What is the exact meaning of continuity?
Let and be two topological spaces. A
function is continuous if for each open subset
of , the set is an open
subset of .
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What is topology?

MATHEMATICALLY

• Two topological spaces, X and Y, are
  topologically equivalent if there exists a
  function f X ? Y such that
• f X ? Y is a bijection,
• f X? Y is continuous,
• the inverse function f--1 Y ? X is continuous.

We say that f is a homeomorphism between X and
Y. We write X Y.
Topology is the study of those properties that do
not change under homeomorphism.
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What is topology?

Examples

• The open interval (-1, 1) and the real numbers
  R.
• The unit 2-disc D2 and the unit square in R2 .

• The function f  0, 2p) ? S1 defined by f(f)
  (cos(f), sin(f)) is bijective and continuous, but
  not a homeomorphism.

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What is topology?

Example classification of the letters of the
alphabet up to homeomorphism

Two letters in the same class are topologically
equivalent
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What is topology?

Example classification of the letters of the
alphabet up to homeomorphism

Two letters in the same class are topologically
equivalent
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What is topology?
So, how to know if two objects are homeomorphic
(topologically equivalent)?
A trefoil
A donut
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Selected Chapters in Image ProcessingExtracting
Topological Information of 3D Digital Images
What is Topology?

• Topology
• Topological invariant
• Homotopy
• Metric spaces

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What is topology?
A topological invariant is a property which
remain unaffected by homeomorphism.
Two objects are homeomorphic
They have the same topological invariants
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What is topology?

Connectedness is a topological invariant.
The existence of a route by metro between two
places in Wien would not depend on the lengths of
the metro lines, but only on connectivity
properties.
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What is topology?

Connectedness is a topological invariant.
A topological space X is said to be connected if
it cannot be contained in two disjoint nonempty
open sets.
The topological space X is path-connected space
if any two points x and y can be joined by a path
(a continuous function f 0,1 X with f(0)
x and f(1) y ).
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What is topology?
Connectedness is a topological invariant.
Path-connectedness connectedness
sin(1/x)U0 is connected but not path-connected
In finite topological spaces, path-connectedness
connectedness
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What is topology?

• Some topological invariants
• Number of connected components
• Number of independent loops
• Number of cavities
• Euler number

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What is topology?

So, how to know if two objects are homeomorphic?
Answer 1 seeking for a homeomorphism between the
objects.
Answer 2 seeking for topological invariants in
both objects.
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What is topology?

How to seek for topological invariants of an
object?
Idea Thinning the object and computing
topological invariants in the thinned object.
THINNING f
THINNED OBJECT A
Homotopy equivalence
GENERAL TOPOLOGY
Skeleton
Digital
Digital thinning
DIGITAL TOPOLOGY
Combinatorial
Homology
Chain contraction
ALGEBRAIC TOPOLOGY
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Selected Chapters in Image ProcessingExtracting
Topological Information of 3D Digital Images
What is Topology?

• Topology
• Topological invariant
• Homotopy
• Metric spaces

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What is topology?

HOMOTOPY
OBJECT X
THINNING f
THINNED OBJECT A
Homotopy equivalence
A homotopy between two continuous functions f and
g from a topological space X to a topological
space Y is a continuous function H X 0,1 ? Y
such that, for all points x in X, H(x,0)f(x) and
H(x,1)g(x).
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What is topology?

HOMOTOPY EQUIVALENCE
A homotopy equivalence between two topological
spaces and is a continuous map
satisfying that there exists
another continuous map
such that g o f is homotopic to the indentity
map of and f o g is homotopic to identity
map of
We say that X and Y are homotopy equivalent or
have the same homotopy type
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What is topology?
HOMOTOPY EQUIVALENCE
Example

They have the same homotopy type but they are not
homeomorphic
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What is topology?

Remarks
If two objects are topologically equivalent, then
they have the same homotopy type
Homotopy class
Homeomorphism class
Number of connected components, independent loops
and cavities and Euler number are also homotopy
invariants.
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What is topology?

Example classification of the letters of the
alphabet up to homotopy

Two letters in the same class are homotopy
equivalent
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What is topology?

Example classification of the letters of the
alphabet up to homotopy

Two letters in the same class are homotopy
equivalent
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Selected Chapters in Image ProcessingExtracting
Topological Information of 3D Digital Images
What is Topology?

• Topology
• Topological invariant
• Strong deformation retraction
• Homotopy
• Metric spaces

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What is topology?

Intuitively, a continuous function maps points
that are "close together" to points that are
close together.
How can we express mathematically the idea of
close together?

Metric spaces A metric on a set X is a function
(called distance) d X X ? R such that for all
x, y, z in X d(x, y) 0 (non-negativity)
d(x, y) 0 if and only if x y
(identity of indiscernibles) d(x, y) d(y, x)
(symmetry) d(x, z) d(x, y) d(y, z)
(triangle inequality).
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What is topology?

Distances in R3 Euclidean distance Manhatta
n distance Chebyshev distance

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What is topology?

Let (X,d) be a metric space.
The open ball of radius r (gt0) about x in X is
the set B(x r) y in M d(x,y) lt r.
Example of open balls
Euclidean distance
Manhattan distance
Chebyshev distance
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What is topology?

• Jordan Surface Theorem Let X be a continuous,
  injective mapping of the sphere into R3. Then,
  the complement of the image of X consists of two
  distinct connected components
• One is bounded (the interior)
• The other is unbounded (the exterior).
• The image of X is their common boundary.

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What is topology?

The problem of the embedding
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What is topology?

Isotopy
Let f and g be two homeomorphisms from the
topological space X to the topological space Y.
An isotopy is a homotopy, H, such that for each
fixed t, H(x,t) gives a homeomorphism.
Example when should two knots K and K in 3D
space be considered the same? When there exists
an isotopy starting with the identity
homeomorphism of 3D space, and ending at a
homeomorphism, h, such that h moves K to K.
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What is topology?

Isotopy
These knots are homeomorphic but not isotopic
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Selected Chapters in Image ProcessingExtracting
Topological Information of 3D Digital Images
What is Topology?

• Topology
• Topological invariant
• Homotopy
• Metric spaces

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