R' Marfil, L' MolinaTanco, A' Bandera and F' Sandoval

1
The construction of Bounded Irregular Pyramids
with a union-find decimation process

• R. Marfil, L. Molina-Tanco, A. Bandera and F.
  Sandoval

6TH IAPR-TC15 Worshop on Graph-based
Representations Alicante (Spain)
2
Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

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Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

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• Regular pyramids
• Irregular pyramids

• Regular pyramids
• Irregular pyramids

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• Regular pyramids
• Irregular pyramids

They can be represented as a hierarchy of images
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• Regular pyramids
• Irregular pyramids

Variable data structure and decimation processes
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The key idea behind the BIP is to use a regular
approach in the homogenous regions of the input
image and an irregular approach in the rest of
regions.
The BIP is a combination of a 4 to 1 regular
structure with a simple graph irregular one.
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Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

9
Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

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The BIP is a combination of a 4 to 1 regular
structure with a simple graph irregular one.
There are two types of nodes nodes belonging to
the regular structure (regular nodes) and nodes
belonging to the irregular structure (virtual
nodes).
The BIP can be seen as a graph hierarchy. G0 is a
8-connected graph where all the nodes are regular
nodes. Each node of G0 is a pixel of the original
image.

• The process to build a graph Gl1 from the graph
  below Gl has three main steps
• Regular decimation
• Union-find
• Intra-level edge generation

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• Regular decimation
• Union-find
• Intra-level edge

LEVEL 1
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• Regular decimation
• Union-find
• Parent search and intra-level twining
• Virtual parent search and virtual node linking
• Intra-level edge

LEVEL 1
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• Regular decimation
• Union-find
• Parent search and intra-level twining
• Virtual parent search and virtual node linking
• Intra-level edge

LEVEL 1
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• Regular decimation

LEVEL 2
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• Union-find
• Parent search and intra-level twining

LEVEL 2
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• Union-find
• Virtual parent search and
• virtual node linking

LEVEL 2
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• Intra-level edge

LEVEL 2
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• Union-find
• Virtual parent search and
• virtual node linking

LEVEL 3
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Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

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Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

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A receptive field is always generated (or
extended) by the linking of neighbour nodes.
Proof
Theorem 1 If two nodes ni and nj are neighbours
in Gl, their receptive fields ri and rj are
neighbours in G0
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Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

23
Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

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Theorem 2 If all the nodes of an elongated
object of one-pixel width have similar colour in
G0, then there exists a virtual node whose
receptive field is formed by them.
Proof The nodes of the elongated object will be
gruped in subset of nodes (reduction windows)
generating a set of parents in the upper level.
These parents are connected because their
reduction windows are connected. Therefore, they
also generate a set of parents in the upper level
which are connected. This process stops when a
level with only a node is reached. The receptive
field of this node is the elongated object.
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Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

26
Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

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Q function
I - Segmented input image. NxM Size of the
segmented image. R Number of segmented
regions. ei Average colour error of region i
compared with the original image. Ai Area of
segmented region i. R(Ai) Number of segmented
regions with area equal to Ai.

• The obtained regions must be uniform and
  homogeneous.
• The interior of the regions must be simple,
  without small holes.
• Adjacent regions must be significantly
  different.
• The existence of small regions is penalized.

M. Borsotti, P. Campadelli, and R. Schettini.
Quantitative evaluation of color image
segmentation results. Pattern Recognition
Letters, 19(8)741747, June 1998.
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Comparative study

• Irregular pyramids
• Classical RAG hierarchy (ClIP) (Bertolino and
  Montanvert, 1996)
• Localized pyramid (LIP) (Huart and Bertolino,
  2005)
• Classical RAG hierarchy with region growing
  stopping (MIP) (Lallich et al., 2003)
• Hierarchy of image partitions (HIP) (Haxhimusa
  and Kropatsch, 2004)
• Combinatorial pyramid (CoIP) (Brun and
  Kropatsch, 2003)

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Marfil, R., Molina-Tanco, L., Bandera, A.,
Rodriguez, J., Sandoval, F. Pyramid segmentation
algorithms revisited. Pattern Recognition 39(8)
(2006) 14301451
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Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

32
Index

• Introduction
• Construction of the BIP
• Preservation of connectivity
• Representation of elongated objects
• Results
• Conclusions and future work

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• This new implementation of the Bounded Irregular
  Pyramid is a mixture between a regular structure
  and an irregular one, which combines the
  advantages of both types of structures low
  computational cost and accurate results.
• The regular part of the BIP is a 4-to-1 data
  structure and the irregular part consists in a
  simple graph data structure combined with a
  union-find decimation strategy.
• The obtained segmentation results are similar to
  the obtained with the main irregular structures
  but with lower computational cost.
• Future work will be focused on increasing the
  degree of mixture between the regular and
  irregular parts of the BIP.

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The construction of Bounded Irregular Pyramids
with a union-find decimation process

• Thanks for your attention!!
• Any questions/advise?

• R. Marfil, L. Molina-Tanco, A. Bandera and F.
  Sandoval

6TH IAPR-TC15 Worshop on Graph-based
Representations Alicante (Spain)