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Application of communityidentification methods for image segmentation

Francisco A. Rodrigues

Instituto de Fisica de Sao Carlos

Luciano da F. CostaGonzalo Travieso

Dynamics on Complex Networks and Applications – February 2006

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Outline Community definition Methods for community identification

How to determine the precision of the method Edge Betweenness centrality based method The faster method of Newman The most precise Local methods Hybrid methods

Computer vision Complex Networks X Computer vision Community identification X image segmentation Future works

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Community definition Definition of Community in a Strong Sense

The subgraph V is a community in a strong sense if

k(i)in(V) > k(i)out(V), for all in V

(Radicchi et al. PNAS, 2004).

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Modularity For a network divided in g groups:

Define a matrix g X g eij : fraction of vertices that connects the group i to j

Q = [eii – (eij )2 ] Q = 0 for random networks Q > 0.3 network with community structure

i

M. E. J. Newman, Eur. Phys. J. B, (2004).

i

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Outline Community definition Methods for community identification

How to determine the precision of the method Edge Betweenness centrality based method The fastest method The most precise Local methods Hybrid methods

Image processing Complex Networks X Image processing Community identification X image segmentation Conclusions and future works

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Computer generated networks are constructed by using two different probabilities pin and pout

n vertices are classified into c communities. At each

subsequent step, two vertices are selected and linked

with probability pin if they are in the same community,

or pout in case they are belonging to different

communities. In general pin > pout

How to determine the precision of the methods

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How to determine the precision of the methods

kin + kout = 16, n = 128, 4 communities

kout = 1 kout = 3

kout = 5 kout = 7

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Edges Betweenness centrality method Edge betweenness centrality is given by the number of shortest paths between pairs of vertices that run along the edge

Algorithm of Girvan and Newman:

1 - Calculate the betweenness score for each of the edges.

2 - Remove the edge with the highest score.

3 - Compute the modularity for the network.

4 - Go back to step 1 until all edges of the networks are removed

Limitation: time of processing O(n3)

(M. Girvan and M. E. J. Newman,PNAS (2002).)

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The faster method of Newman

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kout/k

Based on the maximization of the value of the modularity

Algorithm

1. Each node is in its own community

2. Compute the change of modularity when two communities are joined

3. Joining the communities with highest dQ

4. Repeat the second step until result in just one community

It’s fast!! O(nlog2n)

A. Clauset, M. E. J. Newman and C. Moore, PRE (2004).

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The most precise method

Construct the modularity matrix and find its most positive eigenvalue and eigenvector

Divide the network in two part according to the signs of the elements of this vector

Repeat the process for each part

When the proposed split makes a zero or negative contribution to the modularity, we leave the subgraph undivided

Stop when the entire network has been decomposed into indivisible subgraphs.

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The most precise method

It’s not very fast!! O(n2log2n)But it is the most precise!

M. E. J. Newman, physics/0602124 (Feb. 2006)J. Duch and A. Arenas, PRE (2005)

MOST PRECISE

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Local methods and Hybrid methods Local: Based on growing network for a single vertex

J.P. Bagrow, E. M. Bollt, A Local Method for Detecting Communities, cond-mat/0412482, Phys. Rev. E, 72 046108

Aaron Clauset, Finding local community structure in networks, Phys. Rev. E 72, 026132 (2005)

• Hybrid: Based on local and global network information

• L. da F. Costa, Hub-Based Community Finding, cond-mat/0405022

• F. A. Rodrigues, G. Travieso and L. da F. Costa , Fast Community Identification by Hierarchical Growth, physics/0602144

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Outline Community definition Methods for community identification

How to determine the precision of the method Edge Betweenness centrality based method The faster method of Newman The most precise Local methods Hybrid methods

Computer vision Complex Networks X Computer vision Community identification X image segmentation Conclusions and future works

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Computer vision

Definition: Computer-based manipulation and interpretation of digital images.

Complex networks approaches

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Computer vision

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Image segmentation

Given any image, how to identify the objects?

It is a very difficult problem!!!!

To partition the image into its constituent parts (objects)

Autonomous segmentation: Can facilitate or disturb subsequent processes

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Complex networks X Image processing

Each pixel in the image is associatedto a node in a network

mapping

M: size of the imagex, y: pixel position

Each node has a vector of features associated:

• Gray level• Texture• Color (RGB)• Position (movement)• …

W(i,j) = || fi – fj ||

Edge weight: Euclidian distance between these feature vectors

L. Da F. Costa, Complex Networks, Simple Vision, cond-mat/0403346

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Complex networks X Computer vision

Image acquisition

Pre-processing

Mapping into a network

segmentation

characterization

Recognition andclassification

Inversemapping

Why is Computer Vision Difficult? We do not understand the recognition problem

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Outline Community definition Methods for community identification

How to determine the precision of the method Edge Betweenness centrality based method The faster method of Newman The most precise Local methods Hybrid methods

Image processing Complex Networks X Computer vision Community identification X image segmentation Future works

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Community identification X Segmentation

Objects with similar gray levels are identified

together

Threshold(remove weak links)

Communityidentification

First approach: Weight = Gray level difference between every pair of pixel

Picture extracted from: http://gallery.hd.org/_c/money/_more2000/_more04/Sweden-coins-10-5-Kronor-50-oere-silver-copper-gold-colours-SEK-1-DHD.jpg.html

Inversetransformation

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Outline Community definition Methods for community identification

How to determine the precision of the method Edge Betweenness centrality based method The faster method of Newman The most precise Local methods Hybrid methods

Image processing Complex Networks X Computer vision Community identification X image segmentationFuture works

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Future worksNext approaches:

• Weight : Gray level difference between pixel in a window

• Weight : Euclidian distance between feature color vector of pixels

Compare the complex network approach with popular methods of image segmentation in terms of precision and time of processing

Develop new approaches to classify and recognize images in terms of complex networks

…Picture extracted from: http://gallery.hd.org/_c/money/_more2000/_more04/Sweden-coins-10-5-Kronor-50-oere-silver-copper-gold-colours-SEK-1-DHD.jpg.html

We have a lot to do!!

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COMPLEX NETWORKS

• Application of complex networks to image analysis and computer vision

• Development of new methods for community identification

• Investigations about network of word associations (next talk)

• Biological networks

• Characterization of complex networks: A survey of measurements (cond-mat/0505185 )

• Development of Hierarchical measurements

http://cyvision.if.sc.usp.br/~francisco/networks/

Suggestions are welcome!!!

Financial Support