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

More documents

Recommendations

Info

ω 1y = φ11 −1 d 1y + φ12 −1 d 2y = φ12 −1 d 2yω 2y = φ21 −1 d 1y + φ22 −1 d 2y = φ22 −1 d 2y = d 2ywhere φij−1 is the (i, j)th element of Φ −1 , andif i = j, φij−1 = 1. Then in an intermediate posex for α p , we have tanα p = d yd x. d y and d x aredelta values in x,y coordinates computed fromequation 3. With the above weight values, wehave:tanα p = d y= ω 1yφ(x − x 1 )+ω 2y φ(x − x 2 )d x ω 1x φ(x − x 1 )+ω 2x φ(x − x 2 )= φ 12 −1 d 2yφ(x − x 1 )+d 2y φ(x − x 2 )φ12 −1 d 2xφ(x − x 1 )+d 2x φ(x − x 2 )= d 2yd 2xWe can see this angle α p is a constant anddepends only on the value of delta in the secondpose d 2 =[d 2x ,d 2y ]. Then we take a lookat α Ip in the inverse case. We transform twoexamples to rest pose to obtain delta values:d r 1 =[d′ 1x ,d′ 1y ]=[0,0] and dr 2 =[d′ 2x ,d′ 2y ]. Sinceonly the second joint is rotating, we simplifySSD as a rotation transformation ignoring otherissues such as accumulating effects from the firstjoint:⎛SSD θ = ⎝cosθ sinθ 0−sinθ cosθ 00 0 1⎞⎠.where θ is the rotation angle of the second joint.Then for the vertex v r =[v 0x ,v 0y ], SSD θ (v r )transforms v from rest pose to [v SSDx ,v SSDy ]=[v 0x cosθ − v 0y sinθ,v 0x sinθ + v 0y cosθ]. In anintermediate pose x, we have its correspondingrest position as v r x =[v 0x + d x,v ′ 0y + d y], ′ andhere the [d x,d ′ y] ′ are interpolated result computingfrom equation 3. We just apply the simplifiedSSD to v r x to obtain v x Ip : v Inp x=(v 0x +d ′ x)cosθ − (v 0y + d ′ y)sinθ and v Inpy =(v 0x +d ′ x)sinθ +(v 0y + d y)cosθ. ′ Similarly, we computethe tangent of α Ip :tanα Ip = − v Inp y− v SSDyv Inpx − v SSDxwhere tanβ = d′ yd ′ x= − d′ x sinθ + d ′ y cosθ= −tan(β + θ)d x ′ cosθ − d y ′ sinθ= d′ 2y. Then we can see αd ′ Ip =2x−(θ + β), which is linearly proportional to thepose rotation θ.And now we take a look at a real cylindermodel with one vertex sculpted in the secondpose, shown in the Figure 4. Forward PSDand the corresponding inverse PSD in the sameposes (30, 45 and 60 degree of one rotated joint) are illustrated respectively in Figure 5. Wecan see that in forward case, the direction of deformedvertex always keeps the same with theexample cylinder (figure 4). For inverse PSDhowever, that direction is changed along withthe rotation of the joint. The case describedabove is quite common in practice when animatingshoulder, elbow, knee, hip-bone, neck,etc. All these parts would rotate from the restpose with some angle to other poses. On theother hand, as a matter of experience, PSD issupposed to be a method as “local” correction,which means pose space should not be extendedto a whole space that has to incorporate all influencedobjects. Otherwise, large amount of unnecessaryworks of building examples will be required,and the distance between different posesis also meaningless. For example how to measurethe distance between differing poses suchas “lying down” and “pitching”?Figure 4: A simple test case: two example poses with onevertex sculpted5 A Unified Framework forInverse Skinning ModelThe above discussions assume that the basicskinning algorithm is SSD, but in many circumstances,other deformation schemes will beadopted [9], [10], most of which have been implementedin most animation packages. Thereforewe propose a unified framework in whichno explicit inverse operation is necessitated.Given a basic skinning method supported byanimation packages we can deform the originalcharacter model from rest pose to another specificpose. In the more general case, we needto replace SSD with SKINNING in the inverseskinning algorithm as such for equation 4:v t i = v i + d i = SKINNING i (v r )+d iBut this time we don’t implement the inverseof SKINNING as in equation 5. To find delta d ′ i6
One advantage of Powell’s classic methodis that it does not need explicit computationof the function’s gradient [12]. Because weare treating the skinning operations as a “blackbox”, their gradient is not available, so Powell’smethod is suitable. Minimizing the functionf (d ′ ) in a particular direction is actually minimizationproblem of one variable, which is alsocalled line minimization. Powell’s idea is tryingto find each minimum of function f (d ′ ) indifferent direction until f (d ′ ) stops decreasing.How to choose the next direction is the mainconcern of Powell’s method, and it has beenproved that after repeated cycles of M line minimizationson conjugate directions, the optimizationwill in due course converge to the minimum[12]. M is the dimensionality of the function f ().Figure 5: Interpolated Poses: the rotated joint is at 30,45and 60 degree. left column: PSD; right column:inverse PSD.5.2 System Overviewin the rest pose:v i = SKINNING i (v r )+d i = SKINNING i (v r + d ′ i )we can setup a minimization problem to minimizethe function:f (d i ′ )=‖v i − SKINNING i (v r + d i ′ )‖2 (9)This function can be given to Powell’s methodto find d i ′ at the minimum of f (d ′ ). For eachexample pose P i ,wehavead i ′ , then we can applyradial basis function to d i ′ (i = 0...n − 1) inpose space to obtain ω i (i = 0...n − 1). In synthesisphase, a d x ′ in an intermediate pose x canbe computed by equation 3 based on its positionx in pose space d x ′ = ∑ n−1i=0 ω iφ(||x − x i ||). Thenwe have the final synthesis result:v x = SKINNING(v r + d ′ x) (10)5.1 Direction Set (Powell’s) Methods forMinimizationFor a minimization problem, there are manycandidate algorithms according to the form offunction, knowledge of the derivative, computingcapacity, and requirements for the rate ofconvergence, etc. In our situation, the functionform is not explicit, and the computing burdenincreases with the number of example poses increases.We will adopt Powell’s method as thesolution to this minimization problem.Figure 6: System Pipeline for Unified Inverse Skinning.We implement this unified approach as aMaya plug-in. In Maya, “tweaking” is a procedureadding delta values to original surfacevertices before any deformations. It is actu-7
Page 1 and 2: A Powell Optimization Approach for
Page 3 and 4: incorporate linear elements into RB
Page 5: it to example positions to obtain d

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

Create successful ePaper yourself

Delete template?

Save as template?