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Large Mesh Deformation Using the Volumetric Graph Laplacian Kun Zhou Jin Huang* John Snyder^ Xinguo Liu Hujun Bao* Baining Guo Heung-Yeung Shum Microsoft Research Asia *Zhejiang University ^Microsoft Research

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Mesh Deformation Smooth geometry – Freeform deformation [Barr84, Singh98, Bendels03] – Energy minimization [Welch94, Taubin95, Botsch04] Detailed geometry – Multi-resolution editing [Zorin97, Kobbelt98, Guskov99] – Differential domain methods: Poisson mesh editing [Yu04] Laplacian surface editing [Sorkine04, Lipman05]

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Large Mesh Deformation Challenge to existing techniques – Local self-intersection, unnatural volume change BendingTwisting

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Large Mesh Deformation Challenge to existing techniques – Local self-intersection, unnatural volume change Poisson Mesh EditingVGL

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Large Mesh Deformation Challenge to existing techniques – Local self-intersection, unnatural volume change Poisson Mesh EditingVGL

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Large Deformation: Why Difficult? Differential domain methods [Yu04, Sorkine04] Uniform error distribution using least-squares optimization Only surface details, volume ignored Displacement volumes [Botsch03] Volumetric constraints Iterative relaxation produces artifacts Solution: Volumetric Constraints & Least-Squares Optimization

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Poisson Mesh Editing Step 1: Specify control curveStep 2: Edit control curve F F

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Poisson Mesh Editing Step 3: Propagate local frame transformations Step 4: Solve Poisson equation

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Surface Details & Laplacian

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Volumetric Details & Laplacian

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Quadric Energy Function Surface DetailsVolumetric DetailsPosition Constraints

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1. Construct the volumetric graph 2. Compute Laplacian coordinates 3. Compute and apply local transformation 4. Solve the sparse linear system Deformation Using VGL

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Volumetric Graph Construction

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Construct an inner shell

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Volumetric Graph Construction Embed both the mesh and shell in a lattice

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Volumetric Graph Construction Build edge connections

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Volumetric Graph Construction Simplify and smooth the graph Not Tetrahedral Mesh

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Deformation Comparison Laplacian surface [Sorkine04] Poisson mesh [Yu04] VGL

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Poisson mesh [Yu04] Laplacian surface [Sorkine04] Deformation Comparison VGL

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Deformation Comparison Original modelPoisson meshVGL

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Deformation Interface 3D space manipulation [Yu04] – Tedious and require artistic skill 2D sketch-based interface – Modeling: Teddy [Igarashi99] – Editing: [Zelinka04, Kho05, Nealen05] Teddy-like deformation: intuitive and easy to use

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2D Sketch-based Deformation

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Deformation Retargeting

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Results

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Conclusion Volumetric graph Laplacian (VGL) – Volumetric constraints – Least squares minimization – No tetrahedral mesh construction 2D sketch-based deformation system – Teddy-like deformation system – Cartoon deformation retargeting

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Future Work Anchor-based deformation Dynamic connectivity Automatic contour tracking

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Acknowledgement Cartoons from Disney Feature Animation and Dongyu Cao 3D models from Stanford, MIT, Cyberware Xin Sun, Jianwei Han Steve Lin, Bo Zhang NSFC and 973 Program of China

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Thank You !

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Poisson Mesh Editing Mesh GeometryGuidance FieldBoundary Condition

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Quadric Energy Function Surface DetailsVolumetric DetailsPosition Constraints : unknown positions : control points : mesh Laplacian : new mesh Laplacian coords : volumetric graph Laplacian : new graph Laplacian coords : control parameters

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Geodesic distance fields-based blending – Compute the local transform for each control point – Blend the transforms of all control points – Blend the transform with the identity Compute Local Transformation

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