Graph Cuts and Normalized Cuts
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1 Definition: University of Alicante (Spain) Matrix Computing (subject 3168 Degree in Maths) 30 hours (theory)) + 15 hours (practical assignment)
2 Contents 1. Graph Partitiond, Cuts and Normalization 1. Graph partition 2. Cut and Normalized Cut. What is the optimal partition? 3. The purpose of normalization 4. Properties of the Optimal Partition 2. Problem Formulation (binary) 1. NCut and Laplacian 2. Relaxation to real values. Friedler vector analysis 3. Extenstion: recursive sub-division 3. Application to point and image clustering 1. Point clustering 2. Clustering in images
3 Graph Partitions, Cuts and Normalization Objective,cuts and normalized cuts: Find two disjoint partitions A and B of the vertices V of a graph, so that A B = V and A B =. What is a good partition? Define a precise criterion Given a similaritymeasure w(i,j) between two vertices (e.g. identity when they are connected) a cutvalue(anditsnormalizedversion) is defined as: Optimal bi-partition: Is the oneminimizing the cut value. There are an exponential number of partitions. Can we solve the problem efficiently? Use spectral graph theory!
4 Graph Partitions, Cuts and Normalization Example #7: A1=(1,2,,9), B1 =(10,11,12) cut(a1,b1) = 1 (link between 2 and 10) A2=(1,2,4,,12), B2 =(3) cut(a1,b1) = 1 (link between 3 and 1) asso(a1,v) = 9, asso(b1,v) = 4 Ncut(A1,B1)=1/9 +1/4 =13/36=0.36 asso(a2,v) = 12, asso(b2,v) = 1 Ncut (A2,B2)= 1/12 +1/1 =13/12=1.08 NCut considers the total edge weight connecting a partition with the rest of vertices in the graph, and thus isolated vertices as partitions are avoided!
5 Graph Partitions, Cuts and Normalization Example #2: A1=(1,2,,9), B1 =(10,11,12) cut(a1,b1) = 1 (link between 2 and 10) asso(a1,v) = 9, asso(b1,v) = 4 asso(a1,a1) = 8, asso(b1,b1)=3 Nasso(A1,B1)=8/9+ 3/4 = = 1.63 A2=(1,2,4,,12), B2 =(3) cut(a2,b2) = 1 (link between 3 and 1) asso(a2,v) = 12, asso(b2,v) = 1 asso(a2,a2) = 11, asso(b2,b2) = 0 Nasso(A2,B2) = 11/12 + 0/1 =0.91
6 Graph Partitions, Cuts and Normalization Properties of the optimal partition: Considering NCut we seek theminimization of disassociation between the groups A and B and themaximization of the association within each group: Exercise #5 (proof) Nasso is encoding the association within the group, that is, the ratio between how many weight remains inside and goes outside for both groups.
7 Problem formulation (binary) Using dimensional indicators: [Shi & Malik,00] Given a partition of V in sets A and B. letxa N= V indicatorvector so that x i =1 if node i is in A and x i =-1 if node i is in B. Let Then
8 Problem formulation (binary) Going towards the Laplacian: [Shi & Malik,00] BeingDthe degree matrix andwthe attribute matrix: Definingy We have
9 Problem formulation (binary) Solving the eigensystem: [Shi & Malik,00] If y is relaxed to take real values the latter minimization by solving the following generalized eigenvalue system: But we have to consider the constrainty T D1=0 and relaxing +1-1 to real values: Therefore, the Friedler vector (second vector of the eigensystem) is the solution (close to 1 in cluster A, close to -1 in cluster B) What happens with close to zero values
10 Applications: Graphs W=I Example #2:
11 Applications: Segmentation Example #7: X(red)=(1,0,0) X(blue)=(0,0,1) Waa = Wgg=Wbb=e -0 =1; Wxy x~=y Wxy = e -sqrt(2) =
12 Applications: Segmentation Example #8: X(red)=(1,0,0) X(blue)=(0,0,1) X(black)=(0,0,0) Waa = Wgg=Wbb=e -0 =1; Wrb=Wbr= e -sqrt(2) = Wrblack = Wblackr = e -1 = Wbblack = Wblackb = e-1 =
13 Applications: Segmentation Example #9: X(red)=(1,0,0) X(green)=(0,1,0) X(blue)=(0,0,1) Waa = Wgg=Wbb=e -0 =1; Wxy x~=y Wxy = e -sqrt(2) =
14 Applications: Clustering Example #10: W clustering result
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