A Powell Optimization Approach for Example-Based ... - JP Lewis

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simple illustration of SSD with only two jointframes. Rectangles represent animated sectionsin each of two frames and the curve shows theblended result of both frames. Since vertextransformations can be easily implemented inthe graphic card, SSD is very popular in circumstancesthat require animating a number of charactersin real time. Some opportunities for controlare provided to the animators. When a charactergoes wrong in some pose, animators canadjust joint influence weights. But the domainof adjusting one vertex in this way is strictly limitedto the linear subspace formed by the vertexas transformed by joints influencing this vertex.3.1 Pose Space DeformationThe famous SSD problem of “collapsed elbow”is recognized in [1] as being due to the fact thatdeforming is limited to a linear subspace. Becauseof this limitation SSD cannot synthesizemany parts of a character skin involving complicatedjoint structures. Built on the SSD scheme,the Pose Space Deformation (PSD) is proposedby [1] as a combination of SSD and shape blendingproviding nice solution to above mentionedproblems.PSD can be performed in the following steps:• example pose setup: move the character toproblematic poses, sculpt and store pose information(joint configuration x) includingthe amount of movement of each sculptedvertex (delta values d).• In the inverse PSD approach in this paper,we transform example models (d) totherest pose (d r ): This step is trivial if the basicskinning, say SSD, is explicit. In ourproposed framework, this step will be replacedby an optimization routine.• solving a linear system: we setup an interpolationscheme for delta Φ(x)ω = d andsolve it to obtain weights ω for all exampleposes.• realtime synthesis: for an intermediatepose, we obtain the delta in the rest poseby interpolating example poses at runtime.We add this delta to the original charactersurface and then let SSD or any other skinningscheme finish the final transformation.For a vertex v, if sculpted in N example poses,then there are N delta d i ,i = 0,...,N − 1 correspondingto each pose x i ,i = 0,...,N −1. Theseare converted to rest pose displacements usingdi r = SSD −1 (d i ). We adopt Gaussian Radial Basisfunctions to interpolate di r.First a N ∗ Nmatrix Φ is built with the (i, j)th element asφ(‖ x i − x j ‖), where ‖ x i − x j ‖ means the Euclideandistance between pose x i and pose x j ,then we have a linear system:W = Φ −1 D r (2)Here W and D r are column vectors with ithelement ω i and d r i respectively. In the synthesisphase, for an intermediate pose x, we can obtainthe delta d for this vertex by:N−1d =∑i=0ω i φ(‖ x i − x ‖) (3)For the Gaussian function φ(x) =e − x2σ 2 , σ isused to control the “fall-off”. In Figure 1 (b).,we use Gaussian Radial Basis functions to interpolate3 points. The blue and green curve representσ = 1.0 and σ = 2.0 respectively. Otherbasis functions also can be candidates.Although PSD and improved example-basedschemes have been discussed in many publications[2], [3], [4], the reason why the inverseshould be performed is still ambiguous. Inthe next section we will analyze this issue anddemonstrate why inverting the SSD (and otherdeformations) in order to interpolate the examplesin the rest pose is a the right choice.4 Inverse OperationThis section will describe the implementation ofour inverse algorithm and why it is an improvement.We still study SSD as the underlying skinning,since an explicit form of basic skinningcan help to simplify our task of explanation. Wecall the PSD scheme without the inverse operationas “forward PSD”, and comparison to it willbe used to demonstrate the superiority of the inversemethod.4.1 ImplementationIn this sub-section, we will explain how inversePSD works. For N examples, a vertex v isfirst transformed from rest pose by SSD to positionsv i ,i = 0,...,N − 1, then animators move4
it to example positions to obtain delta valuesd i ,i = 0,...,N − 1. The final positions of v inexample poses are v i + d i ,i = 0,...,N − 1, andwe call them target positions v t i . We summarizeabove operations as:v t i = v i + d i = SSD i (v r )+d i (4)where v r means the rest position of v andSSD i (∗) represents the equation 1. The “forwardPSD” approach then concludes by interpolatingd i as a function of pose.In the inverse approach we instead apply theinverse of SSD i (∗) to v t i to obtain a modified restpose vertex v r i . The difference of vr i and v r producesnew delta value d r i , which will be the inputof linear system (equation 2) introduced inthe previous section.d r i = SSD −1i (v t i ) − vr (5)In this step we need implement the inverse skinningoperator SSD −1 . Since SSD is a 3D transformation,SSD −1 simply is the inverse transformationmatrix generated by SSD. For the situationwhere other unknown skinning operationsare adopted, we propose a unified frameworkwhich will be discussed in the following section.Next we build a new delta vector D r with ith elementas d r i , and replace D in equation 2 withd r i to get a new weight vector W r .W r = Φ −1 D r (6)In the synthesis phase, for an intermediate posex we have:d r N−1x = ∑ ω r i φ(‖ x i − x ‖) (7)i=0then we add this d r x to v r and let SSD finish therest of the job:v x = SSD x (v r + d r x) (8)v x represents the final position of vertex v inpose x.4.2 Deformation Direction AnalysisIn this section we will analyze and compare thedeforming effects of both forward PSD and inversePSD, explaining why the inverse methodis superior..Given two examples as shown in Figure 2 (a)and (b) respectively, vertex v with the position v rin the rest pose(0degrees ) is sculpted to a “targetposition” v t i in an example pose (90 degrees).The delta value in the first pose is zero. ThenFigure 2: (a). rest pose; (b). example pose with rotationof 90 degreesFigure 3: deformation directionwe apply forward and inverse PSD respectivelyto interpolate these two poses. For an intermediatepose x, we have two distinct deformingvertices resulting from two algorithms, as illustratedin Figure 3, v x ssd ,vx p,v x Ip are the deformedpositions from SSD, forward and inverse PSDin an intermediate pose x. We use two anglesα p and α Ip to analyze how directions of a deformedvertex change with the pose. In the forwardcase, α p is formed by the vector (v x ssd ,vx p)and the line y = Y v xssd, where Y v xssdis the y coordinateof v x ssd. For two examples shown in Figure2 (a) and (b), we have delta values d 1 =[d 1x ,d 1y ]and d 2 =[d 2x ,d 2y ]. For equation 2:∣ ω ∣ 1x ω 1y ∣∣∣ =ω 2x ω 2y∣ φ ∣11 φ 12 ∣∣∣ −1 ∣ ∣ ∣∣ d 1x d 1y ∣∣∣φ 21 φ 22 d 2x d 2yBecause in the rest pose, we didn’t generateany movement for v , then d 1 =[d 1x ,d 1y ]=[0,0].Taking the model in rest pose as an example isa common practice when applying shape interpolation,since interpolating effects from otherexamples should not change the original modelin rest pose. Therefore, by solving above equationwe have:ω 1x = φ11 −1 1x + φ12 −1 2x = φ12 −1 2xω 2x = φ21 −1 1x + φ22 −1 2x = φ22 −1 2x = d 2x5
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