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Segmentation via Graph Cuts

Segmentation via Graph Cuts. Balakrishna Thiagarajan Computer Science and Engineering State University of New York at Buffalo. Introduction. Gestalt approach ( From) Normalized Cuts and Image Segmentation. Image Segmentation.

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Segmentation via Graph Cuts

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  1. Segmentation via Graph Cuts Balakrishna Thiagarajan Computer Science and Engineering State University of New York at Buffalo

  2. Introduction Gestalt approach (From) Normalized Cuts and Image Segmentation

  3. Image Segmentation • Segmentation refers to the process of partitioning a digital image into multiple segments. • The goal of segmentation is to simplify and/or change the representation of an image into something that is more meaningful and easier to analyze. • Similarity (gray level difference), Proximity and continuity are important factors in segmentation.

  4. Graph representation of Image • Set of points of the feature space represented as a weighted, undirected graph, G = (V, E). • The points of the feature space are the nodes of the graph. • Edge between every pair of nodes.

  5. Graph representation of Image • Weight on each edge, w(i, j), is a function of the similarity between the nodes i and j. • Partition the set of vertices into disjoint sets where similarity within the sets is high and across the sets is low. • Therefore, Segmentation is equivalent to Graph partition.

  6. Graph representation of Image Cut: • A cut in a graph is a set of edges whose removal disconnects the graph. • A minimum cut is a cut with a minimum number of edges. It is denoted by S. In the weighted graph, cut(A,B) = ∑ w(u,v) u E A,v E B where A & B are two partitioned disjoint sets

  7. Segmentation by Min Cut method • By Min Cut method, the graph is partitioned into clusters. • Each cluster is considered as an image segment. • Min Cut method uses the HCS (Highly Connected Subgraphs) Algorithm to find the clusters.

  8. Segmentation by Min Cut method • Edge Connectivity is the minimum number of edges whose removal results in a disconnected graph. It is denoted by k(G). • For a graph with vertices n > 1 to be highly connected if its edge-connectivity k(G) > n/2. • A highly connected subgraph (HCS) is an induced subgraph H in G such that H is highly connected.

  9. Segmentation by Min Cut method Example: No. of nodes = 5 Edge Connectivity = 1 A B Not HCS! D C E

  10. Segmentation by Min Cut method Example continued: No. of nodes = 4 Edge Connectivity = 3 A B HCS! D C

  11. HCS Algorithm - Min Cut method HCS(G(V,E)) begin (H, H’,C)  MINCUT(G) if G is highly connected then return (G) else HCS(H) HCS(H’) end if end

  12. HCS Algorithm - Min Cut method • The procedure MINCUT(G) returns H, H’ and C where C is the minimum cut which separates G into the subgraphs H and H’. • Procedure HCS returns a graph in case it identifies it as a cluster. • Single vertices are not considered clusters and are grouped into singletons set S.

  13. HCS Algorithm - Min Cut method Example

  14. HCS Algorithm - Min Cut method Example Continued

  15. HCS Algorithm - Min Cut method Example Continued Cluster 2 Cluster 1 Cluster 3

  16. HCS Algorithm - Min Cut method • The HCS Algorithm returns clusters which are nothing but image segments. • The problem with HCS Algorithm – Min Cut method is that there is a chance that many singletons may be separated. • To overcome this we use normalized-cut for image segmentation.

  17. HCS Algorithm – Main Problem Example

  18. HCS Algorithm – Main Problem Example – Another possible cut

  19. Max Flow - Min Cut Method • Theorem In graph G, the maximum source-to-sink flow possible is equal to the capacity of the minimum cut in G. (L. R. Foulds, Graph Theory Applications, 1992 Springer-Verlag New York Inc., 247-248)

  20. Max Flow - Min Cut Method • Min Cut can be made by using Ford-Fulkerson Algorithm to find the max-flow in the graph. • In the graph temporarily add Source ‘s’ and Sink ‘t’. • Consider the weight associated with each each node as a flow capacity. • Starting from zero flow, increase the flow gradually by finding a path from s to t along which more flow can be sent, until a max-flow is achieved.

  21. Max Flow - Min Cut Method

  22. Max Flow - Min Cut Method • After the max-flow is found, the minimum cut is by S = {All vertices reachable from s} T = G \ S

  23. Max Flow - Min Cut Method - Illustration (From) Image Segmentation Using Minimal Graph Cuts by Anders P. Eriksson, Olof Barr and KalleAstrom

  24. Normalized Cut • Normalized Cut (Ncut) : Ncut(A,B) = cut(A,B) + cut(A,B) assoc(A,V) assoc(B,V) where, assoc(A,V) = ∑ w(u,t) u E A, t E V • The cut is made such that the normalized cut value between the two sets A & B is minimum.

  25. Normalized Cut • Minimum cut criteria favors cutting small sets of isolated nodes in the graph • (From) Normalized Cuts and Image Segmentation – • Shi & Malik

  26. Normalized Cut • Exact solution to minimizing normalized cut is an NP-complete problem. • However, approximate discrete solutions can be found efficiently. • Normalized cut criterion can be computed efficiently by solving a generalized eigenvalue problem.

  27. Normalized Cut – Physical Interpretation • Think of the weighted graph as a spring mass system. • Graph nodes  physical masses • Graph edges  springs • Graph edge weight  spring stiffness • Total incoming edge weights  mass of the node

  28. Normalized Cut – Physical Interpretation • Imagine giving a hard shake to this spring-mass system, forcing the nodes to oscillate in the direction perpendicular to the image plane. • Nodes that have stronger spring connections among them will likely oscillate together. • Eventually, the group will “pop” off from the image plane.

  29. Normalized Cut (From) Normalized Cuts and Image Segmentation - Shi & Malik

  30. Average Cut • Average Cut: Averagecut(A,B) = cut(A,B) + cut(A,B) |A| |B| • The average cut measure is proportional to the total link weight cut, normalized by the sizes of the partitions. • The problem with Average Cut is that one cannot ensure the two partitions computed will have tight within-group similarity.

  31. Comparison • Different types of graph cuts - MinCut, Average Cut, and Normalized Cut were analysed by Padmanabhan Soundararajan and Sudeep Sarkar. • Totally 50 images were used. 5 different image classes with 10 images in each class.

  32. Comparison • The 5 classes of images were • natural objects in indoor backgrounds • natural objects in outdoor backgrounds • man-made objects in indoor backgrounds • man-made objects in outdoor backgrounds • aerial images that contained man-made objects in natural surroundings.

  33. Comparison • The best possible performances on each image for the three different cut measures were not significantly different from each other. • The performance with the three measures is worst for man made objects in indoor surroundings and for aerial images. • The normalized cut based partition took significantly larger time to compute than the average and the mincut.

  34. References • J. Shi and J. Malik, “Normalized Cuts and Image Segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 8, pp. 888-905, Aug. 2000. • Erez Hartuv and Ron Shamir, “A Clustering Algorithm based on Graph Connectivity”, Mar. 1999. • Padmanabhan Soundararajan and Sudeep Sark, “Analysis of MinCut, Average Cut, and Normalized Cut Measures”, May 2000. • Anders P. Eriksson, Olof Barr and Kalle Astrom,”Image Segmentation Using Minimal Graph Cuts”.

  35. Thank You!!

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