A Blendshape Model that Incorporates Physical Interaction

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tems are attractive due to their low computationalcomplexity.Rest Length Animation. There are works thatsimulate muscle activation through controllingthe rest length of each spring in a mass-springsystem. Raibert and Hodgins [26] used restlength animation to simulate simple leg locomotion.Adjusting the rest length changes theforce at each spring so that it is able to initiate orterminate its motion. Tu and Terzopoulos [33]constructed a mass-spring system of a fish body,and assigned some springs to be muscle springsdriven by animated rest lengths. However, theseearlier techniques were not applied to shape interpolation.3 Blending Shapes with a MassSpring SystemThe proposed blendshape technique consists ofthe following two steps:1. Given a set of input blendshape targets,we construct a mass-spring system for eachblendshape target based on its structure.2. We interpolate rest lengths between two ormore aforementioned mass-spring systemsto generate a intermediate mass-spring system.The interpolated shape is the quasistaticstate of the intermediate system.3.1 Structure of the Mass Spring SystemIt is important to maintain the stability of themass-spring system during interpolation, andthis mostly depends on a good spring structure.Based on [10], we build the mass-spring systembased on input triangulated mesh as a combinationof structure springs and bending springs.The structure springs model the elastic propertiesof the mesh surface, connecting neighboringvertices. The bending springs define the bendingand flexural properties of the material, and connecta vertex to secondary neighboring vertices(i.e., at a distance of 2). These bending springshelp maintain the object’s resting shape and preservesurface curvature.Nevertheless, our experiments show that alarge surface which consists of many small trianglestends to be crumpled during interpolationG 0 G 1 G 2(0.70, 0.10, 0.20) (0.33, 0.33, 0.33) (0.12, 0.75, 0.12)Figure 3: Blending between multiple shapes.The top row shows three source shapesand the bottom row shows blendedresults with corresponding weightsshown as (w 0 ,w 1 ,w 2 ).due to insufficient numerical precision. To solvethis problem, we insert additional springs insidethe shape to help preserve volume. We applyconstrained Delaunay tetrahedralization [31] todetermine where to insert these internal springs.Similar to the mesh refinement step in [1], toprevent springs with long rest lengths we canalso perform mesh refinement by inserting newvertices. Those vertices have to be added to allthe input shapes correspondingly. Certain additionalvertices that are inside the input shapescan be assigned as an anchor as described inSection 3.4.A mass-spring system M = 〈V,S〉 is definedby a collection of vertices V = {v i |i = 1...n v }connected by springs S = {s q |q = 1...n s }. Eachspring s q ∈ S connects two vertices v eq0 andv eq1 , where e q0 = e(s q ,0) and e q1 = e(s q ,1) andfunction e returns the indices of spring s q ’s twovertices. We enforce 1 ≤ e q0 < e q1 ≤ n v . Inaddition, s q is characterized by a rest lengthr q = r(s q ) and spring constant k q = k(s q ), wherefunctions r and k return the rest length andspring constant of spring s q , respectively.3.2 Blending between Two ShapesWe first introduce our technique by explainingthe case of blending between two shapes G 0 and
G 1 . We build two consistent mass-spring systemsM 0 = 〈V 0 ,S 0 〉 and M 1 = 〈V 1 ,S 1 〉 fromG 0 and G 1 . The two mass-spring systems areinitially set to be in their own steady states, i.e.rq 0 =‖ v 0 e q0− v 0 e q1‖ and rq 1 =‖ v 1 e q0− v 1 e q1‖. Foreach α ∈ [0,1], the interpolated result from theinput shapes is the equilibrium state of a newmass-spring system ¯M = 〈 ¯V, S〉, ¯ where for eachspring,¯r q = (1 − α)rq 0 + αrq. 1 (1)In an equilibrium state, the force f (v i ) at eachvertex v i in M equals zero:f (v i ) = ∑ k q (‖v i − v j ‖ − r q ) v i − v j‖vj∈n(i)i − v j ‖ = 0,(2)where function n returns the indices of thosevertices that are adjacent to v i , and spring s qconnects v i and v j . Note that the velocity ofeach vertex can be ignored since our results arebased solely on quasi-static states. We assumethat each vertex has the same mass, thereforethe mass term can also be ignored. To formulateEq. (2) into a linear system, first we vectorize Vinto an one dimensional vector x as:x = [x(v 1 ),y(v 1 ),z(v 1 ),···,x(v nv ),y(v nv ),z(v nv )] T ,where x(v i ),y(v i ),z(v i ) are functions that returnthe X, Y and Z Cartesian coordinates of v i . Thesystem is then solved by using the Newton–Raphson method to determine the first order approximationof the optimal vertex configuration:f (x t+1 ) ≈ f (x t ) + J(x t )∆x t , (3)where x t+1 = x t + ∆x t and J(x t ) = ∂ f∂x (x t) is theglobal stiffness (Jacobian) matrix of f evaluatedat the current vertex positions x t . When thesystem is in its equilibrium state f (x t+1 ) = 0.Eq. (3) now becomes:J(x t )∆x t = − f (x t ). (4)The non-diagonal elements of the global stiffnessmatrix J i j are defined by:)J i j = J ji = k q(r q‖ d i j ‖ 2 I − d i j d T i j‖ d i j ‖ 3− I,where I is an identity matrix, d i j = v i − v j , andspring s q connects v i and v j ; or else a 3 × 3 matrixof zeros if no such a spring exists. The diagonalelements (i = j) are defined as:J ii = − ∑ J i j .j∈n(i)x 0 is initialized as the vertex positions of thesource shape V 0 . We then iteratively solveEq. (4) until ‖∆x t ‖ is smaller than a threshold( ¯M reaches its equilibrium). The final vertexpositions are then assigned to ¯V. Generally matrixJ is very sparse. There are both iterative(e.g. conjugate gradient) and direct methods forsolving this sparse linear system.3.3 Blending Multiple ShapesThe proposed technique can also be extended toblending multiple shapes by simply consideringthe interpolated rest length as a convex linearcombination of the spring rest lengths from theinput shapes:¯r q =n b∑i=0w i r i q,where ∑ n bi=0 w i = 1. This is illustrated in Figure3 where the interpolated shapes (shown inyellow) are the results of linearly-blended springrest lengths from three different input shapes(shown in blue). Notice that this shares exact thesame formulation for controlling blendshape targetsas as the traditional linear blendshape techniquedoes.3.4 Boundary ConditionsBoundary conditions must be specified in orderto solve the equilibrium of a mass-spring system.Without boundary conditions, the systemwill be under-constrained and the solution willnot be unique. Simply speaking, boundary conditionsare vertices which are fixed in a massspringsystem. In practice, this can be achievedby assigning Dirichlet boundary conditions tothe global stiffness matrix, i.e., by replacing theblock of the global stiffness matrix correspondingto boundary vertices with an identity matrix.The entries of the boundary vertices in fare replaced by zeros to enforce that the correspondingvertices do not move. A straightforwardmethod to assign the boundary conditionsis to find vertices which remain static betweenthe source and target shapes. However,
Page 1 and 2: A Blendshape Model that Incorporate
Page 3: 2 Related WorkOur technique relates
Page 7 and 8: the rest length:k q ∝ 1 r q.The s
Page 9 and 10: Figure 6: Results of shape interpol

A Blendshape Model that Incorporates Physical Interaction

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