ICVPx

molassesitalianΤεχνίτη Νοημοσύνη και Ρομποτική

6 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

75 εμφανίσεις


Goal: a graph representation of the topology of a
gray scale image.


The graph represents the hierarchy of the lower
and upper level sets of the gray level function.


This graph contains the inclusion trees, but it is
not a tree.


The topological tools:


cell decomposition: the image is represented as a
combination of pixels as well as edges and vertices.


cycles: both upper and lower level sets are captured by
circular sequences of edges.


Segmentation: capturing upper and lower level sets of
the gray level function of the image.


Rationale: the connected components of these sets are
building blocks of real items depicted in the image.


The connected components of the
lower level sets have a clear
hierarchy based on inclusion.
This hierarchy provides a graph
representation of the topology of
the image.




.

The inclusion trees for upper and lower level
sets, if considered separately, do not help in
finding out which object has which hole.





Therefore, in order to capture the topology of
the image, the two trees have to be combined
in some way.


P.
Monasse

and F.
Guichard
,
Fast computation
of a contrast invariant image representation
.
IEEE Transactions on Image Processing, 9(5),
pp. 860

872, 2000.


Jordan Theorem: A component of a level set
encircles or is encircled by components of
other level sets.








+




=





The lower level sets are mixed with the upper level
sets.


The gray levels are also mixed.








+



=





The
topology graph
of the image



The lower and upper inclusion trees remain intact
within the graph.


The graph breaks into layers that coincide with the
topology graphs of the corresponding binary images.


The topology graph is not a tree in general.



Our goal is to capture the
topological
features

present in the image: connected
components and their holes.


We think of black objects as connected
components and white objects as holes in the
dark objects.

A binary image is a rectangle covered by black
and white pixels arranged in a grid.

A pixel is a square, or a tile: [n, n + 1]
×

[m, m
+ 1].



a vertex {n}
×
{m} is a
0
-
cell
,


an edge {n}
×
(m, m + 1) is a
1
-
cell
, and


a face (n, n + 1)
×
(m, m + 1) is a
2
-
cell
.




Two adjacent edges
are 1
-
cells and they
share a vertex, a 0
-
cell;


Two adjacent faces
are 2
-
cells and they
share an edge, a 1
-
cell.


Cycles are used as a tool of image segmentation.

Both connected components and holes are captured
by cycles:


a
0
-
cycle

as a sequence of vertices that follows the
outer boundary of a connected component;


a
1
-
cycle

as a sequence of edges that follows the
outer boundary of a hole.




The nodes of the topology graph are the cycles in the
image and there is an arrow from node A to node B if:


0
-
cycle B has 0
-
cycle A inside, provided A and B
correspond to consecutive gray levels.


0
-
cycle B has 1
-
cycle A inside, provided A and B
correspond to the same gray level.


And vice versa.


All pixels in the image are ordered in such a way
that all black pixels come before white ones.


Following this order, each pixel is processed:


add its vertices, unless those are already present as parts of
other pixels;


add its edges, unless those are already present as parts of
other pixels;


add the face of the pixel.


At every step, the graph is given a new node and
arrows that connect the nodes in order to represent
the merging and the splitting of the cycles:


adding a new vertex creates a new component;


adding a new edge may connect two components, or
create, or split a hole;


adding the face to the hole eliminates the hole.


Suppose N is the number of pixels in the
image. Then


The memory usage is O(N).


The complexity of the algorithm is O(N
2
).

Intel Core 2
Dual CPU
T7500
2.2GHz

If a 0
-
cycle is an ancestor of another, only one
of them is taken into account.

If a 0
-
cycle

is an ancestor of
another, only one
of them is taken into account.


The approach and the method are justified by
appealing to classical mathematics.


The new representation of the topology of a gray
scale image is a graph that isn’t a tree in general.


This data structure allows components and holes
to be treated simultaneously but kept separate.


The algorithm and its interpretation are intuitive.


The algorithm is fast enough to be practical.


The analysis produces meaningful results for
various gray scale images.





Thank you