Application of community identification methods for image segmentation

1. Application of community identification methods for image segmentation Francisco A. Rodrigues Luciano da F. Costa Gonzalo Travieso ~ Instituto de Fisica de Sao Carlos Dynamics on Complex Networks and Applications – February 2006

2. 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

3. 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).

4. 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 =S[eii – (Seij )2 ] • Q = 0 for random networks • Q > 0.3 network with community structure i i M. E. J. Newman, Eur. Phys. J. B, (2004).

5. 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

6. How to determine the precision of the methods • Computer generated networks are constructed by using two different probabilities pinand pout • n vertices are classified intoc 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

7. How to determine the precision of the methods • kin + kout = 16, n = 128, 4 communities kout = 3 kout = 1 kout = 7 kout = 5

8. 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).)

9. The faster method of Newman • Based on the maximization of the value of the modularity • Algorithm • Each node is in its own community • Compute the change of modularity when two communities are joined • Joining the communities with highest dQ • Repeat the second step until result in just one community It’s fast!! O(nlog2n) Fraction of correctly classified vertices A. Clauset, M. E. J. Newman and C. Moore, PRE (2004). kout/k

10. 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.

11. MOST PRECISE 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)

12. 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

13. 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

14. Computer vision Definition: Computer-based manipulation and interpretation of digital images. Complex networks approaches

15. Computer vision

16. Image segmentation To partition the image into its constituent parts (objects) Given any image, how to identify the objects? Autonomous segmentation: Can facilitate or disturb subsequent processes It is a very difficult problem!!!!

17. Complex networks X Image processing Each pixel in the image is associated to a node in a network mapping M: size of the image x, y: pixel position • Gray level • Texture • Color (RGB) • Position (movement) • … Each node has a vector of features associated: Edge weight: Euclidian distance between these feature vectors W(i,j) = || fi – fj || L. Da F. Costa, Complex Networks, Simple Vision, cond-mat/0403346

18. Complex networks X Computer vision characterization Inverse mapping Image acquisition Mapping into a network Recognition and classification segmentation Pre-processing Why is Computer Vision Difficult? We do not understand the recognition problem

19. 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

20. Community identification X Segmentation First approach: Weight = Gray level difference between every pair of pixel Objects with similar gray levels are identified together Inverse transformation Threshold (remove weak links) Community identification 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

21. 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

22. Future works Next approaches: • Weight : Gray level difference between pixel in a window • Weight : Euclidian distance between feature color vector of pixels We have a lot to do!! • 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

23. Suggestions are welcome!!! http://cyvision.if.sc.usp.br/~francisco/networks/ 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 Financial Support