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Mesh-Based Inverse Kinematics

Mesh-Based Inverse Kinematics. Robert W. Sumner Matthias Zwicker. Jovan Popovic MIT Craig Gotsman Harvard University. Demo. Demo. f 1. f 2. f 4. f 3. Compute Feature Vectors. f 5. w 2. w 3. w 4. w 5. w 1. Nonlinear Combination F= Σw i f i. w 6. w 10. f 6. w 7. w 9.

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Mesh-Based Inverse Kinematics

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  1. Mesh-Based Inverse Kinematics Robert W. Sumner Matthias Zwicker. Jovan Popovic MIT Craig Gotsman Harvard University

  2. Demo

  3. Demo

  4. f1 f2 f4 f3 Compute Feature Vectors f5 w2 w3 w4 w5 w1 Nonlinear Combination F= Σwifi w6 w10 f6 w7 w9 w8 f7 f8 f9 f10

  5. What is this paper about? • New framework – Feature Vectors • Traditional Point-based • Nonlinear mesh interpolation • Linear is good? • Efficient Optimization

  6. Feature Vectors • Feature Vector = a group of deformation gradients. • What is deformation gradient • affine mapping is • Φ(p) = T*p + d, T = rotation, scaling, skewing • dΦ(p) / dp = T is deformation gradient • So given two triangles, how to find T? ┌ │ │ └ ┐ │ │ ┘ ┌ │ │ └ ┐ │ │ ┘ ┌ │ │ └ ┐ │ │ ┘ X Y Z d T X +

  7. Deformed Reference

  8. Feature Vectors • Deformation gradient of affine mapping • Φ(p) = T*p+d, T = rotation, scaling, skewing • Illustration in 2-D curve Φ1 Φ2 Φ3 Φ4 Φ5 Φ6 F = [T1, T2, T3, T4, T5, T6 ]T

  9. In Mesh IK • Yet • Interpolation among feature vectors T1 T2 T3 T4 T5 T6 T d

  10. Output Output Output Output ? In 3D ? Target

  11. Output Output How it works? • Actually solving for position directly • x = argminx || f(x) - F || • f(x) = deformation gradient using x • x = argminx ||G’x – (F + c)||

  12. x = argmin x ||G’x – [(f1*w1 + f2*w2) + c] || f1 f2

  13. Outline • New framework • Non-linear mesh interpolation • How to interpolate linearly? • Polar decomposition • Exponential map • Efficient Optimization

  14. Linear Feature Space • We can find the desired mesh with feature vector fw = M*w, M is [f1,f2, … fn] • x*, w* = argmin || Gx – (Mw + c) || • If set Mw = davg + Σwidi • x*, w* = argmin || Gx – (Mw + c) || + k*||w|| ┌ │ │ │ │ └ ┐ │ │ │ │ ┘ *[1/3 2/3]T =

  15. NonLinear versus Linear

  16. Alternative: Nonlinear • Polar decomposition • rotation & scaling differently • Exponential map • different interpolation • “Matrix animation and Polar Decomposition” - Shoemake & Duff • “Linear Combination of Transformation” - Marc Alexa

  17. Polar Decomposition ┌ │ │ │ │ │ │ │ │ └ ┐ │ │ │ │ │ │ │ │ ┘ ┌ │ │ │ │ │ │ │ │ └ ┐ │ │ │ │ │ │ │ │ ┘ T1 T2 T3 . . . Tn R1S1 R2S2 R3S3 . . . RnSn = = R RS R must be orthogonal => SVD : UΣVT, QR : RL

  18. Exponential Map For T x M = Tb x Ta x M if a = b = 1/2 So ½ T x ½ T = T => ½ of T = T ½ AxB = BxA? (Commutative) T = => Half T = => For T ½ ⊙ T ½ ⊙= exp( Log(A) + Log(B) ) T ½⊙ T ½= exp( ½ Log(T) + ½ Log(T) ) = T

  19. Examples

  20. Nonlinear Feature Space • Polar decomposition Tj = Rj * Sj, • Exponential map • Tj(w) = exp(Σwi*log(Rij)) * Σwi*Sij • We can find the desired mesh with feature vector fw = M*w = [f1,f2, … fn] * [w1, w2, … wn]T • Defined as fw = [T1(w1), T2(w2), … Tn(Wn)] = M(w) ┌ │ │ │ │ └ ┐ │ │ │ │ ┘ *[1/3 2/3]T =

  21. Why use exponential map? • Easier to find derivatives, with respect to w • Tj(w) = R(w) * S(w) • dwkTj(w) = dR(w) * S(w) + R(w) * dS(w) • Then why do we need derivatives? • x*, w* = argmin || Gx – [M(w)+c] || • Gauss-Newton Algorithm • M(w +δ) = M(w) + dwM(w)* δ

  22. Gauss-Newton Method • For k-th iteration: • δk, xk+1 = argmin||Gx – dwM(wk)*δ–(M(wk)+c)|| • wk+1 = wk + δk • ATA * [x, δ]T = AT( M(wk) + c ) • A = • Take about a minute or longer to solve ┌ │ │ └ ┐ │ │ ┘ G | - J1 G | - J2 G | - J3 Ji = dwM(w)

  23. Outline • New framework • Non-linear mesh interpolation • Efficient Optimization • Specialized Cholesky-Factorization

  24. Optimized Solver ATA * [x, δ]T = AT( M(wk) + c ) • General Cholesky or QR factorization might not suffice • The structure of ATA in previous normal equation is well defined ┌ │ │ └ ┐ │ │ ┘ G | - J1 G | - J2 G | - J3 A = ┌ │ │ │ └ ┐ │ │ │ ┘ GTG - GTJ1 GTG - GTJ2 GTG - GTJ3 - J1TG - J2TG - J3TG ΣJiTJi ATA =

  25. Precomputation ATA * [x, δ]T = AT( M(wk) + c ) ┌ │ │ │ └ ┐ │ │ │ ┘ GTG - GTJ1 GTG - GTJ2 GTG - GTJ3 - J1TG - J2TG - J3TG ΣJiTJi ATA = Make U such that UTU = ATA ┌ │ │ │ └ ┐ │ │ │ ┘ ┌ │ │ └ ┐ │ │ ┘ R - R1 R - R2 R - R3 Rs G | - J1 G | - J2 G | - J3 U = A = where RTR = GTG, this R can be pre-computed

  26. Solving for ┌ │ │ │ └ ┐ │ │ │ ┘ GTG - GTJ1 GTG - GTJ2 GTG - GTJ3 - J1TG - J2TG - J3TG ΣJiTJi = ATA = ┌ │ │ │ └ ┐ │ │ │ ┘ RTR - RTR1 RTR - RTR2 RTR - RTR3 - R1TR - R2TR - R3TR ΣRiTRi + RsTRs UTU = 1. Solve Ri ,where RTRi = GTJi, 1<=i<=3 2. Solve Rs ,where RsTRs = ΣJiTJi - RiTRi 3. The bottleneck for MeshIK

  27. Numerical Result

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