Hierarchical Mesh Decomposition using Fuzzy Clustering and Cuts

Hierarchical MeshDecomposition using FuzzyClustering and Cuts報 告 者 : R95922120 黃 維 嘉

Outline• Introduction• Algorithm – binary case• Algorithm – k way case• Result• Conclusion

Introduction• Fuzzy decomposition

Binary case (1/4)• Assign ProbabilityChoose two faces REP A and REP B with largest distanceas initial representative of the two clusterAssign probabilities to every faces according to thedistances to the initial patches:P ( fBiDist(fi,REPA)) PA( fi)1PB( fi)Dist(f , REP ) Dist(f , REP )iADist(f i ,f j ) is the shortest path from f i to f j in mesh’s dualgraph withiGeod ( fi,fj) Ang _ Dist(ij)Weight ( dual(fi), dual(fj)) (1 ) avg(Geod ) avg(Ang _ Dist)Ang _ Dist( ) (1cos )ijijB

Binary case (2/4)• Generating fuzzy decompositionGoal: cluster faces by minimize the functionAlgorithmF probability ( f patch ( p)) Dist(f , p)1. Compute the probabilities of faces to belong to each patch2. Re-compute the set of representatives to minimize F byREPA minf (1 PB( fi)) Dist(f , fi)REPB minf PB( fi) Dist(f , fi)fi3. If the representative set is changed, go back to 14. If no change, partition faces aspffiA { fB { fiiC { fi| PB( fi| P ( fBi) 0.5 }) 0.5 }| 0.5 PB( fi) 0.5 }

Binary case (3/4)• Generate final decompositionG(V,E) : dual graph of mesh.VA,VB : the set of dual vertices of patches A and B respectivelyGoal: partition V into VA’, VB’ such thatVA VA, Vweight ( Cut(V'B V)) (u,v)is minimalConstruct a flow network graph G(V’,E’) such thatVE''VC ECVCAVCB{(S,v),vVA'B', V{S,T}CAB'With capacity Cap(I,j) and use max flow algorithm to find min cutuVA', vVB'} {(T,v),vVCB} {eij E | i VC, j {VCAVCB}} 1Ang _ Dist( ) 1ijCap(i,j)avg(Ang _ Dist)if {i, j S,T}else

Binary case (4/4)• Hierarchy decompositionEvery patches can be recursively decomposedso that we can get a hierarchy structure• Stop conditionThe hierarchy decomposition is stopped when(a)(b)(c)Distance between representative is smaller than a thresholdDifference between max and min dihedral angle is smaller thanthresholdThe ratio between the average distance in the patch and that ofthe overall object does not exceed a threshold

K way case(1/3)• A generalization of binary case• Representatives are choose iteratively 1st representative: Choose the face havingthe minimal sum of distances from all otherfaces (body) Other representative: added in turn so as tomaximize their minimal distance from previousassigned representatives

K way case(2/3)• The number of representatives is chosen to minimizethe first derivative ofG k) min ( Dist(REP , REP ))(ikk i

K way case(3/3)• The probability that face f i belonging to patch P j is1defined as• To extract the fuzzy area, each pair of neighboringcomponents are proceeded similarly to the binarycase• Complexity: O(V 2 logV+IV 2 )P j( fi) 1Dist(f , REP(P ))V is number of vertex, I is number of iterationV 2 logV for all pair shortest path and minimum cut algorithmIV 2 for assigning faces to patcheslDist(f , REP(P ))iijl

Result

Conclusion• The hierarchically decomposition algorithmavoid jaggy boundaries as well as oversegmentation• Different distance function and capacityfunction can be experimented with• Non-geometric features such as color andtexture can be embedded in the algorithm

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