Recent Work on Laplacian Mesh Deformation Speaker: Qianqian Hu Date: Nov. 8, 2006

Mesh Deformation Producing visually pleasing results Preserving surface details

Approaches Freeform deformation (FFD) Multi-resolution Gradient domain techniques

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)

Multi-resolution

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.

References Zhou, K, Huang, J., Snyder, J., Liu, X., Bao, H., and Shum, H.Y Large Mesh Deformation Using the Volumetric Graph Laplacian. ACM Trans. Graph. 24, 3, Huang, J., Shi, X., Liu, X., Zhou, K., Wei, L., Teng, S.H., Bao, H., G, B., Shum, H.Y Subspace Gradient Domain Mesh Deformation. In Siggraph ’ 06 Sorkine, O., Lipman, Y., Cohen-or,D., Alexa, M., Rossl, C., Seidel, H.P Laplacian surface editing. In Symposium on Geometry Processing, ACM SIGGRAPH/Eurographics,

Differential Coordinates Invariant only under translation!

Geometric meaning Approximating the local shape characteristics The normal direction The mean curvature

Laplacian Matrix The transformation from absolute Cartesian coordinates to differential coordinates A sparse matrix

Energy function The energy function with position constraints The least squares method

Characters Advantages Detail preservation Linear system Sparse matrix Disadvantages No rotation and scale invariants

Example

OriginalEdited 1) Isotropic scale 2) Rotation

Definition of T i A linear approximation to where is such that γ=0, i.e.,

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

Comparison

Contribution Be fit for large deformation No local self-intersection Visually-pleasing deformation results

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

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

Volumetric Graph Step 2: Embed Min and M in a body-centered cubic lattice. Remove lattice nodes outside Min.

Volumetric Graph Step 3:Build edge connections among M, Min, and lattice nodes.

Edge connection

Volumetric Graph Step 4: Simplify the graph using edge collapse and smooth the graph. Simplification: Smoothing:

VG Example Left: G in (Red); Right: G out (Green); Original Mesh (Blue)

Laplacian Approximation The quadratic minimization problem The deformed laplacian coordinates T i : a rotation and isotropic scale.

Volumetric Graph LA The energy function is Preserving surface details Enforcing the user- specified deformation locations Preserving volumetric details

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

Local Transforms Propagating the local transforms over the whole mesh.

Deformed neighbor points C(u)C(u) p upup t(u)t(u) C’(u)C’(u) P ’ UpUp t’ (u)t’ (u)

Local Transformation For each point on the control curve Rotation: normal: linear combination of face normals tangent vector Scale: s(u p )

Propagation Scheme The transform is propagated to all graph points via a deformation strength field f(p) Constant Linear Gaussian The shortest edge path

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:

Solution By least square method A sparse linear system: Ax=b Precomputing A -1 using LU decomposition

Example

Deformation from 2D curves 2D Projection Back projection 3D Deformation 2D Deformation

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

Example Demo

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

Contributions Linear and nonlinear constraints Volume constraint Skeleton constraint Projection constraint Fit for non-manifold surface or objects with multiple disjoint components

Example Deformation with nonlinear constraints

Example Deformation of multi-component mesh

Laplacian Deformation The unconstrained energy minimization problem where are various deformation constraints

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

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

Subspace Deformation Build a coarse control mesh Control mesh is related to original mesh X=WP using mean value interpolation The energy minimization problem

Example

Constraints Laplacian constraint Skeleton constraint Volume constraint Projection constraint

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

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

Skeleton constraint For deforming articulated figures, some parts require unbendable constraint. Eg, human ’ s arm, leg.

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)

Skeleton constraint Preserving both the straightness and the length In matrix form, a b sisi S i+1

Volume constraint The total signed volume: The volume constraint is the total volume of the original mesh

Example Notice: volume constraint can also be applied to local body parts

Projection constraint Let p=Q p X, the projection constraint p (ω x,ω y ) Object spaceEye spaceProjection plane

Projection constraint The projection of p(=Q p X) In matrix form, i.e.,

Example

Constrained Nonlinear Least Squares The energy minimization problem

Iterative algorithm Following the Gauss-Newton method, f(X) = LX-b(X) is linearized as

Iterative algorithm At each iteration, then When X k =X k-1, stop

Stability Comparison

Example(Skeleton)

Example(Volume)

Example(non-manifold) Demo

Thanks a lot!