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Pattern-Based Quadrangulation for N-sided Patches Kenshi Takayama Daniele Panozzo Olga Sorkine-Hornung ETH Zurich Symposium on Geometry Processing 2014

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Quad meshing is important 210 July 2014Kenshi Takayama Subdivision surfacesFinite element analysis

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Sketch-based approach [Takayama13] 310 July 2014Kenshi Takayama User: Sketches curves N-sided patches

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Sketch-based approach [Takayama13] User: Sketches curves N-sided patches Specifies num of edge subdiv System: Quadrangulates each patch 410 July 2014Kenshi Takayama We propose a new algorithm

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Sketch-based approach [Takayama13] 510 July 2014Kenshi Takayama x4 speed

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Problem definition 610 July 2014Kenshi Takayama Input: Num of edge subdiv at each side Output: Quadrangulation satisfying the input l 0 =5 l 1 =2l 2 =1 ∑ i l i must be even 4 #F = 2 #E interior + #E boundary Singularities at boundary Valence counted accordingly Topology only Geometry by Laplacian smoothing Note: valence-5 singularity valence-3 singularity

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Problem examples 710 July 2014Kenshi Takayama

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Previous work 810 July 2014Kenshi Takayama Ours: Guaranteed to work for arbitrary cases Don’t work when num of edge subdiv is extremely small Advancing front [Schaefer04] Greedy heuristic [Peng14] Pattern-based [Nasri09,Yasseen13] Complicated formulation Smaller set of patterns Our SIGGRAPH paper [Takayama13] main focus on the UI

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Algorithm 910 July 2014Kenshi Takayama

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Infinitely many cases to consider ● Our idea: Reduce the problem into a smaller subproblem 1010 July 2014Kenshi Takayama

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Reducing the problem by trimming 1110 July 2014Kenshi Takayama x4 grid 3x1 grid Reduced problem can be quadrangulated Original problem can be quadrangulated (Way of trimming is not unique) Maximally reduced

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Reducing the problem by trimming 1210 July 2014Kenshi Takayama

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Reducing the problem by trimming 1310 July 2014Kenshi Takayama Common form of reduced problem: How can we quadrangulate it? 11 α

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Topological pattern ● Minimal quad mesh ● Parameter x: num of edge loops inserted 1410 July 2014Kenshi Takayama x 4 x=0 8 x=2 4+2x6 x=1...

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Using topological pattern ● α must be even ● Guaranteed to work for all the cases! 1510 July 2014Kenshi Takayama x 11 α Case I: α =2Case II: α =4+2x already a quad

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Complete set of patterns 1610 July 2014Kenshi Takayama

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Reduced input topological pattern See the paper for more details 1710 July 2014Kenshi Takayama Input Condition ( α, 1) α =3+2x (2, 2) -- Input Condition α =2 α =4+2x Input Condition (1, 1, 1, 1) -- ( α, 1, 1, 1) α =3+2x α = β =2+x α =4+2x+y, β =2+y ( α, β, 1, 1) Pattern ( α, 1, 1)

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Reduced input topological pattern See the paper for more details 1810 July 2014Kenshi Takayama Input Condition α =2 α =4+2x α =2+x, β =1+x α =5+2x+y, β =2+y ( α, β, 1, 1, 1) Input Condition (1, 1, 1, 1, 1, 1) -- ( α, 1, 1, 1, 1, 1) α =3+2x α = β =2+x α =4+2x+y, β =2+y ( α, β, 1, 1, 1, 1) α = β =1+x α =4+2x+y, β =2+y ( α, 1, 1, β, 1, 1) Pattern ( α, 1, 1, 1, 1) Enumerating reduced input is easy, but finding patterns is hard (done by hand)

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How to trim the patch optimally? 1910 July 2014Kenshi Takayama x x x2 1x1

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Optimal trimming 2010 July 2014Kenshi Takayama x p0p0 p i : padding parameter ( trimming)

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Optimal trimming 2110 July 2014Kenshi Takayama x p0p0 p1p1 p i : padding parameter ( trimming)

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Optimal trimming 2210 July 2014Kenshi Takayama l0l0 l1l1 l2l2 x p0p0 p1p1 p2p2 p i : padding parameter ( trimming)

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Optimal trimming 2310 July 2014Kenshi Takayama l 0 =11 l 1 =5l 2 =4 x p0p0 p1p1 p2p2 p 0 =1 p 1 =2 p 2 =3 x=1 l0l0 l1l1 l2l2 x p0p0 p1p1 p2p2 p i : padding parameter ( trimming)

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Optimal trimming 2410 July 2014Kenshi Takayama l0l0 l1l1 l2l p0p1p2xp0p1p2x l0l1l2l0l1l2 = l 0 =11 l 1 =5l 2 =4 p0p0 p1p1 p2p2 x p 0 =1 p 1 =2 p 2 =3 x=1

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Optimal trimming 2510 July 2014Kenshi Takayama p1p1 x l0l0 l1l1 l2l2 p0p0 p1p1 p2p2 l0l0 l1l1 l2l2 l 0 =4 l 1 =7l 2 = p0p1p2xp0p1p2x l0l1l2l0l1l2 = p0p1p2p0p1p2 l0l1l2l0l1l2 = p0p0 p2p2

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Optimal trimming 2610 July 2014Kenshi Takayama p1p1 x l0l0 l1l1 l2l2 p0p0 p1p1 p2p2 l0l0 l1l1 l2l2 p0p0 p2p p0p1p2p0p1p2 = = p0p1p2xp0p1p2x Non-negative integer solution? Integer Linear Programming f maximize: subject to: ∑ i p i A y = b y M y ≥ 0

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Underconstrained system ● Multiple feasible solutions ● UI for exploring the solution space Increment/decrement selected parameter Different constraint & objective in ILP 2710 July 2014Kenshi Takayama A y b = Decrement xIncrement x x=0x=1x=2

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Demo 2810 July 2014Kenshi Takayama

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Integrated into sketch-retopo system 2910 July 2014Kenshi Takayama [Takayama13]

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Future work 3010 July 2014Kenshi Takayama Better geometry optimizationMore complex patterns Generalization to higher N-gonsGeneralization to hex meshing

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Thank you! ● C++ code available at: ● Acknowledgements: Alexander Sorkine-Hornung Maurizio Nitti ERC grant iModel (StG ) JSPS Postdoc Fellowships for Research Abroad 10 July 2014Kenshi Takayama31

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Equations for other patterns 3210 July 2014Kenshi Takayama p0p1p2p3xyp0p1p2p3xy l0l1l2l3l0l1l2l3 = x y p0p0 p1p1 p3p3 p2p2 l0l0 l2l2 l1l1 l3l3 l0l0 l1l1 l2l2 l3l3 l4l4 x y p0p0 p1p1 p4p4 p3p3 p2p p0p1p2p3p4xyp0p1p2p3p4xy l0l1l2l3l4l0l1l2l3l4 =

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Comparison 3310 July 2014Kenshi Takayama [Yassen13] Ours

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