Inverse Kinematics for Reduced Deformable Models Keven G. Der, Stanford Robert W. Sumner, ETH Jovan Popovic, MIT to appear in proceeding of SIGGRAPH 2006

Infer Meaningful Control Reduce Complexity

Deformation Gradient Skinning Mesh Animations IK for Reduced Deformable Model Mesh Based Inverse Kinematics Deformation Transfer Trilogy

Deformation Gradient Skinning Mesh Animations IK for Reduced Deformable Model Mesh Based Inverse Kinematics Deformation Transfer Road Map

Deformation Gradient Concept of deformation gradient: –Global manipulation –Local descriptor –What is it? Ex. Affine transform Φ(p) F XYZXYZ d X + => F dΦ(p) / dp Rotation x Sheer x Scale

Deformation Gradient Skinning Mesh Animations IK for Reduced Deformable Model Mesh Based Inverse Kinematics Deformation Transfer Road Map

Deformation Transfer Output ? Target Output ?

Deformation Transfer Mathematically –f(X) = Deformation Gradient –X* = argmin x || f(X) - Deformation Gradient || 2 –f( ) is linear in X, f(X) = G * X G x = f ? ?

Deformation Gradient Skinning Mesh Animations IK for Reduced Deformable Model Mesh Based Inverse Kinematics Deformation Transfer Road Map Unknown Deformation Known Deformation

MeshIK = *[1/3 2/3] T ┌││││└┌││││└ ┐││││┘┐││││┘ =Non-linear function “m” of ( and [1/3 2/3] T ) G x

MeshIK Mathematically –X*, β* = argmin x, β || G*X – m(β) || 2 G x= m(, β ) ┌││││└┌││││└ ┐││││┘┐││││┘ ? ?

=+ Deformation Gradient Skinning Mesh Animations IK for Reduced Deformable Model Mesh Based Inverse Kinematics Deformation Transfer Road Map

SMA F 1, d 1 F 2, d 2 α 1 (F 2, d 2 ) + α 2 (F 2, d 2 ) + …

=+ Deformation Gradient Skinning Mesh Animations IK for Reduced Deformable Model Mesh Based Inverse Kinematics Deformation Transfer Road Map

Reduced Deformable Model V = Σ i α i (X) * (F i *X + d i ) dV = Σ i [ dα i (F i *X + d i ) + α i (F i ) ] = Σ i [ F i (dα i X + α i I (3x3) ) + d i d α i ] For K th vertex, dV K is formulated as ….dα i X + α i …… | …dα i …… x... … F i … _ … d i … dVKdVK =

In Matrix Form ….dα i + α i …… | …dα i ……... … F … _ … d … x G x t ….dα j + α j …… | …dα j …… ….dα K + α K …… | …dα K ……... … dV i dV j dV k … = = dVdV m( β ) =

m(,β ) = G x Conceptually F i, d i F j, d j Deformation Gradient of G x t =

Complexity G x t # of {F i, d i }, 20~30 # of vertex, thousands dVdV = t*, β* = argmin t, β || G*t – m(β) || 2 m( β )

Using Proxy Vertices dVdV

Complexity G x t # of {F i, d i }, 20~30 # of vertex, hundreds dVdV = t*, β* = argmin t, β || G*t – m(β) || 2 m( β )

Pose Editing What if users impose constraints? In addition to t* β* = argmin t || G*t – m(β) || 2 C*t = b Σ i α i (X) * (F i *X + d i ) = V

Constrained Solution Under-determined equation C*t = b t = C + *b + N*t 1 Constrained t*, β* = argmin t, β || G*Nt 1 – (m(β)-GC + b) || 2 Undetermined

Solution to the minimization Identical to MeshIK Yet a magnitude faster (8ms v.s. 2s) –Time does not scale with geometry-complexity Complexity is O(p*c*e + e 3 ) –In MeshIK p ~= thousands c ~= tens of thousnad e ~= 10 –In Reduced, p~= 300, c~= 40, e ~= 10

If you already know MeshIK G x t dVdV = X

Conclusion Interaction is crucial for all kinds of computer graphics application Geometry Independent Deformation Complexity Dependent

Limitation Title for this paper states clearly –Do not apply to all animation Two APPROXIMATION to make G small Need examples, and IDENTICAL connectivity