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Recent Work on Laplacian Mesh Deformation Speaker: Qianqian Hu Date: Nov. 8, 2006
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Mesh Deformation Producing visually pleasing results Preserving surface details
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Approaches Freeform deformation (FFD) Multi-resolution Gradient domain techniques
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FFD FFD is defined by uniformly spaced feature points in a parallelepiped lattice. Lattice-based (Sederberg et al, 1986) Curve-based (Singh et al, 1998) Point-based (Hsu et al, 1992)
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Multi-resolution
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Gradient domain Techniques Surface details: local differences or derivatives An energy minimization problem Least squares method (Linear) Alexa 03; Lipman 04; Yu 04; Sorkine 04; Zhou 05; Lipman 05; Nealen 05. Iteration (Nonlinear) Huang 06.
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References Zhou, K, Huang, J., Snyder, J., Liu, X., Bao, H., and Shum, H.Y. 2005. Large Mesh Deformation Using the Volumetric Graph Laplacian. ACM Trans. Graph. 24, 3, 496-503. Huang, J., Shi, X., Liu, X., Zhou, K., Wei, L., Teng, S.H., Bao, H., G, B., Shum, H.Y. 2006. Subspace Gradient Domain Mesh Deformation. In Siggraph ’ 06 Sorkine, O., Lipman, Y., Cohen-or,D., Alexa, M., Rossl, C., Seidel, H.P. 2004. Laplacian surface editing. In Symposium on Geometry Processing, ACM SIGGRAPH/Eurographics, 179-188.
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Differential Coordinates Invariant only under translation!
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Geometric meaning Approximating the local shape characteristics The normal direction The mean curvature
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Laplacian Matrix The transformation from absolute Cartesian coordinates to differential coordinates A sparse matrix
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Energy function The energy function with position constraints The least squares method
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Characters Advantages Detail preservation Linear system Sparse matrix Disadvantages No rotation and scale invariants
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Example
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OriginalEdited 1) Isotropic scale 2) Rotation
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Definition of T i A linear approximation to where is such that γ=0, i.e.,
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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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Comparison
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Contribution Be fit for large deformation No local self-intersection Visually-pleasing deformation results
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Outline Construct VG (Volumetric Graph) G in (avoid large volume changes) G out (avoid local self-intersection) Deform VG based on volumetric graph laplacian Deform from 2D curves
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Volumetric Graph Step 1: Construct an inner shell Min for the mesh by offsetting each vertex a distance opposite its normal. An iterative method based on simplification envelopes
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Volumetric Graph Step 2: Embed Min and M in a body-centered cubic lattice. Remove lattice nodes outside Min.
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Volumetric Graph Step 3:Build edge connections among M, Min, and lattice nodes.
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Edge connection
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Volumetric Graph Step 4: Simplify the graph using edge collapse and smooth the graph. Simplification: Smoothing:
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VG Example Left: G in (Red); Right: G out (Green); Original Mesh (Blue)
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Laplacian Approximation The quadratic minimization problem The deformed laplacian coordinates T i : a rotation and isotropic scale.
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Volumetric Graph LA The energy function is Preserving surface details Enforcing the user- specified deformation locations Preserving volumetric details
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Weighting Scheme For mesh laplacian, For graph laplacian, i j-1 j+1 j β ij α ij pipi p1p1 p2p2 P j-1 pjpj P j+1
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Local Transforms Propagating the local transforms over the whole mesh.
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Deformed neighbor points C(u)C(u) p upup t(u)t(u) C’(u)C’(u) P ’ UpUp t’ (u)t’ (u)
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Local Transformation For each point on the control curve Rotation: normal: linear combination of face normals tangent vector Scale: s(u p )
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Propagation Scheme The transform is propagated to all graph points via a deformation strength field f(p) Constant Linear Gaussian The shortest edge path
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Propagation Scheme A smoother result: computing a weighted average over all the vertices on the control curve. Weight: The reciprocal of distance: A Gaussian function: Transform matrix:
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Solution By least square method A sparse linear system: Ax=b Precomputing A -1 using LU decomposition
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Example
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Deformation from 2D curves 2D Projection Back projection 3D Deformation 2D Deformation
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Curve Editing C Least square fitting 3 bspline curve CbCb CdCd Editing C ’ b C ’ d A rotation and scale mapping T i discrete C ’ Laplacian deformation
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Example Demo
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Subspace Gradient Domain Mesh Deformation Jin Huang, Xiaohan Shi, Xinguo Liu, Kun Zhou, Liyi Wei, Shang-Hua Teng, Hujun Bao, Baining Guo, Heung- Yeung Shum Microsoft Research Asia, Zhejiang University, Boston University
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Contributions Linear and nonlinear constraints Volume constraint Skeleton constraint Projection constraint Fit for non-manifold surface or objects with multiple disjoint components
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Example Deformation with nonlinear constraints
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Example Deformation of multi-component mesh
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Laplacian Deformation The unconstrained energy minimization problem where are various deformation constraints
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Constraint Classification Soft constraints a nonlinear constraint which is quasi-linear. AX=b(X) A: a constant matrix, b(X): a vector function, ||J b ||<<||A|| Hard constraints those with low-dimensional restriction and nonlinear constraints that are not quasi-linear
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Deformation with constraints The energy minimization problem where L is a constant matrix and g(X) = 0 represents all hard constraints. Soft constraints: laplacian, skeleton, position constraints Hard constraints: volume, projection constraints
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Subspace Deformation Build a coarse control mesh Control mesh is related to original mesh X=WP using mean value interpolation The energy minimization problem
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Example
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Constraints Laplacian constraint Skeleton constraint Volume constraint Projection constraint
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Laplacian constraint a) the Laplacian is a discrete approximation of the curvature normal b) the cotangent form Laplacian lies exactly in the linear space spanned by the normals of the incident triangles xixi X i,j-1 X i,j X i,j+1
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Laplacian coordinate For the original mesh, In matrix form, δ i = A i μ i, then μ i = A i + δ i For deformed mesh The differential coordinate
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Skeleton constraint For deforming articulated figures, some parts require unbendable constraint. Eg, human ’ s arm, leg.
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Skeleton specificaation A closed mesh: two virtual vertices(c1,c2), the centroids of the boundary curve of the open ends: Line segment ab: approximating the middle of the front and back intersections(blue)
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Skeleton constraint Preserving both the straightness and the length In matrix form, a b sisi S i+1
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Volume constraint The total signed volume: The volume constraint is the total volume of the original mesh
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Example Notice: volume constraint can also be applied to local body parts
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Projection constraint Let p=Q p X, the projection constraint p (ω x,ω y ) Object spaceEye spaceProjection plane
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Projection constraint The projection of p(=Q p X) In matrix form, i.e.,
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Example
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Constrained Nonlinear Least Squares The energy minimization problem
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Iterative algorithm Following the Gauss-Newton method, f(X) = LX-b(X) is linearized as
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Iterative algorithm At each iteration, then When X k =X k-1, stop
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Stability Comparison
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Example(Skeleton)
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Example(Volume)
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Example(non-manifold) Demo
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Thanks a lot!
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