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
3D Digital Images
Aim To describe a 3D digital image in terms of a
set of features.

• Process
• Acquisition.
• Segmentation.
• Extracting topological features.

3
3D Digital Images
Adjacency relation between voxels.
CUBIC GRID
18-adjacency
26-adjacency
6-adjacency
BCC GRID
14-adjacency
4
3D Digital Images
Let be a picture.
A black (resp. white) path in is a sequence
of black
(white) points in in which each is
-adjacent to The length of the path is
Two black (white) points are
connected if there exist a path in from
to is
the length of the shortest path from to in

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3D Digital Images
Let be a picture.
The open ball of radius r (gt0) about in
is the set
Examples
Open balls of radius 1 considering
6,18,26-adjacency, respectively.
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3D Digital Image
A regular DPS satisfy the Digital Surface Theorem.
Example
7
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

8
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

9
Digital Topology
Digital topology deals with properties and
features of a binary image array that correspond
to topological properties and features

• Questions that are typically attacked by digital
  topology
• Definition of connectedness
• Topological classification of voxels
• Formulation of Jordans Curve Theorem
• Computation of Euler number
• Definition of continuity or homotopy

10
Digital Topology

• Several approaches to Digital Topology
• Khalimskys approach.
• Rosenfelds graph-based approach.
• Kovalevskys approach.

11
Digital Topology

• Several approaches to Digital Topology
• Khalimskys approach.
• Certain subsets of the underlying digital
  structure are declared to be open sets and to
  fulfill certain axioms chosen in such a way that
  digital structures gets properties which are as
  close as possible to the properties of usual
  topology.

12
Digital Topology

• Several approaches to Digital Topology
• Khalimskys approach.
• Comments Very elegant (mathematically speaking)
  but it does not directly provide the language
  wanted in applications.

13
Digital Topology

• Several approaches to Digital Topology
• Rosenfelds graph-based approach.
• A graph is obtained when a neighborhood relation
  is introduced into the digital set. Such a
  structure allows to investigate connectivity.

14
Digital Topology

• Several approaches to Digital Topology
• Rosenfelds graph-based approach.
• Comments It yields directly connectedness but it
  becomes very difficult to handle more complicated
  concepts such as homotopy.

15
Digital Topology

• Several approaches to Digital Topology
• Kovalevskys approach.
• The discrete structure is embedded into a known
  continuous one (for example, Euclidean space).
  Topological properties of discrete objects are
  then defined by means of their continuous
  analogues.

16
Digital Topology

• Several approaches to Digital Topology
• Kovalevskys approach.
• Comments it is adequate for structures which can
  be related to an Euclidean space. The problem is
  that ones has to find for each question an
  appropriate embedding.

17
Digital Topology

• Several approaches to Digital Topology
• Khalimskys approach.
• Rosenfelds graph-based approach.
• Kovalevskys approach.

18
Digital Topology

• Several approaches to Digital Topology
• Khalimskys approach.
• Rosenfelds graph-based approach.
• Kovalevskys approach.

19
Digital Topology

• Several approaches to Digital Topology
• Khalimskys approach.
• Certain subsets of the underlying digital
  structure are declared to be open sets and to
  fulfill certain axioms chosen in such a way that
  digital structures gests properties which are as
  close as possible to the properties of usual
  topology.

20
Digital Topology
Khalimskys approach
Example 1 On the cubic grid, a topological basis
B is p p in Z3 where
The topology T consists in the set of all the
possible unions of elements of B.

• Drawback
• The only connected sets are reduced to single
  points

21
Digital Topology
Khalimskys approach It consists in defining a
topological basis of the grid.
Example On the cubic grid, a topological basis
is U(p) p(x,y,z) in Z3 where
N6(p) is the set of voxels that are 6-adjacent to
p.

• Drawback
• The topology is not invariant under translation.

22
Digital Topology

• Several approaches to Digital Topology
• Khalimskys approach.
• Rosenfelds graph-based approach.
• Kovalevskys approach.

23
Digital Topology

• Several approaches to Digital Topology
• Rosenfelds graph-based approach.
• A graph is obtained when a neighborhood relation
  is introduced into the digital set. Such a
  structure allows to investigate connectivity.

24
Digital Topology

• Rosenfelds graph-based approach
• Consider a digital image as a graph
• The vertices of the graph are the grid points.
• The edges are the adjacency relations between
  points.

25
Digital Topology
Examples of adjacency graphs
6-adjacency
18-adjacency
26-adjacency
14-adjacency
Remember the DPS considered are
14-adjacency in the BCC grid
26
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Counting connected components (I)

• Consider only black voxels.
• Consider the 26-adjacency graph.
• Construct a cover forest of the graph using DPS
  or BPS.
• Assign a label to each tree.

Drawback the stack which implements
last-in-first-out strategy or the queue which
implements first-in-first-out strategy can be
very large
27
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Counting connected components (II)

• The input image is scanned slice by slice and
  each slice, row by row.
• If a black voxel is encountered, check if it is
  26-adjacent to an already labelled voxel.

• If so, it receives the same label as its
  neighbour.
• If it has different-labelled voxels in his
  neighborhoud, record that this labels are
  equivalent using a tree structure with the
  smallest label at its root.

• If not, a new label is generated and attached to
  this voxel.

28
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Counting connected components (II)
After the entire image is processed, each voxel
is associated with a tree.

• Now, scan the image slice by slice and each
  slice, row by row in inverse order.
• Each voxel is re-labelled with a label
  corresponding to the tree it belongs to.

29
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Counting cavities
A cavity is a totally enclosed connected
component of the background. So

• Invert the image so black voxels are white and
  viceversa.

• Use one of the algorithm for computing connected
  components (using the 6-adjacency for black
  voxels).
• Subtract the infinity component.

30
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Classification of voxels according to their
topological type

• Interior voxel
• Isolated voxel
• Simple voxel
• Curve voxel
• Curve(s) junction
• Surface voxel
• Surface/curve(s) junction
• Surfaces junction
• Surfaces/curve(s) junction

31
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Classification of voxels according to their
topological type
32
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Classification of voxels according to their
topological type
Nk(x) the k-neighbourhood of x including
x. Nk(x) Nk(x) \ x. C number of
26-connected components of N26(x)n X that are
26-adjacent to x. Cnumber of 6-connected
components of N18(x)n X that are 6-adjacent to
x.
33
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Classification of voxels according to their
topological type
Example
C 1
34
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Classification of voxels according to their
topological type
35
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Classification of voxels according to their
topological type
Exercises Classify the red voxels
This problem can be rectified, reclassifying
every point as a curve junction if it has more
than two curves points in its 26-neighbourhood
36
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Classification of voxels according to their
topological type
There are also problems in classifying surface
points
To overcome this difficulty, the basic idea is to
detect the simple surfaces defining a missed
junction instead of the junction itself.
Let S(x) be the set of the surface points
26-adjacency to x. A point x is a junction point
if S(x) is not included in a single simple
surface. Otherwise it is a surface point.
37
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
38
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Skeleton
Digital
Digital thinning
DIGITAL TOPOLOGY

• We want to reduce the image content to its
  essential. Idea
• Eliminate border voxels until only a skeleton of
  the original image remains.

A black voxel is a border if it has at least one
white 6-neighbour (a white voxel is a border if
it has at least one black 26-neighbour).
39
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Thinning algorithm
Basic idea delete as many voxels as possible,
using local criteria, from the binary image
without destroying its topology, preserving its
basic shape.

• In the process of deleting voxels we cannot
• disconnect formerly connected components or
  connect unconnected ones,
• close cavities or produces new ones,
• close handles or produces new ones.

40
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Thinning algorithm
Solution delete simple points of X.
41
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Parallel thinning algorithm
Let I be a 3D binary image, let x be a black
voxel in I. The voxel x is a directed simple
voxel of direction N,E,W,S,T,B if the voxel
adjacent to p in direction N,E,W,S,T,B is white.
A simple voxel x in I is a final point if the
number of black voxels in its 26-neighborhood is
less than 2.
42
Digital Topology
Consider Rosenfeld graph-based approach,
26-adjacency for the foreground and 6-adjacency
for the background.
Parallel thinning algorithm
Algorithm

• Until no further deletion is possible.
• Alternatively, for each type of directed simple
  points
• All deletable voxels in one direction are marked
  for deletion during a first scan through the
  image
• They are removed in parallel during a second
  scan.

43
Digital Topology

• Several approaches to Digital Topology
• Khalimskys approach.
• Rosenfelds graph-based approach.
• Kovalevskys approach.

44
Digital Topology

• Several approaches to Digital Topology
• Kovalevskys approach.
• The discrete structure is embedded into a known
  continuous structure (for example, Euclidean
  space). Topological properties of discrete
  objects are then defined by means of their
  continuous analogues.

45
Digital topology
Kovalevskys approach.
In order to prove that a digital topology
operation correctly reflects the topology of a
digital picture, the main idea is to associate a
continuous analogue C(I) with the digital picture
I. C(I) fills the gap between black points of I
in a way that strongly depends on the grid and
adjacency relations chosen for the digital
picture I.
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Continuos analogues
Following Kovalevskys idea
47
Continuos analogues
SIMPLICIAL COMPLEXES
48
Continuos analogues
SIMPLICIAL COMPLEXES
49
Continuos analogues
SIMPLICIAL COMPLEXES
50
Continuos analogues
SIMPLICIAL COMPLEXES
51
Digital topology
In order to prove that a digital topology
operation correctly reflects the topology of a
digital picture, the main idea is to associate a
continuous analogue C(I) with the digital picture
I. C(I) fills the gap between black points of I
in a way that strongly depends on the grid and
adjacency relations chosen for the digital
picture I.
Consider the BCC grid and the (14,14)-adjacency
52
Continuos analogues
53
Continuos analogues

Example
14-adjacency
14-adjacency
54
Continuos analogues
Computing Euler characteristic of digital
images Compute the Euler characteristic, E, of
its continuous analogue
Counting tunnels Compute the connected components
and cavities. Then tunnels connected
components-E cavities
55
Digital Topology
Bibliography

• E.D. Khalimsky, R.D. Kopperman, P.R. Meyer,
  Computer graphics and connected topologies on
  finite ordered sets, Topology Appl. 36 (1990)
  1-17.
• T.Y. Kong, A.W. Roscoe, A. Rosenfeld, Concepts
  of digital topology, Topology Appl. 46 (1992)
  219-262
• T.Y.Kong, A.Rosenfeld, Digital Topology
  Introduction and Survey, Computer Vision,
  Graphics and Image Processing, 48, 1989, 357-393.
  
• G. Bertrand, G. Malandain. A new
  characterization of 3D simple points. Pattern
  Recognition Letters, 15, 169-175 (1994)
• V.A. Kovalevsky, Finite topology as applied to
  image analysis, Computer vision, graphics,and
  image processing 46 (1989),141161. R.
  Gonzalez-Diaz, P. Real. On the Cohomology of 3D
  Digital Images. Discrete and Applied
  Math, 147 (2005) 245 - 263 

56
Digital Topology

• Several approaches to Digital Topology
• Khalimskys approach based in a finite
  topological space.
• Rosenfelds graph-based approach.
• Kovalevskys approach based on abstract cell
  complexes.