Mesh-Based Inverse Kinematics R. W. Sumner, M. Zwicker, C. Gotsman, - - PowerPoint PPT Presentation

Mesh-Based Inverse Kinematics

• R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´

c

CS468, Wed Nov 9th 2005

SIGGRAPH 2005

The problem

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

General approach

2 Learn from experience...

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As-rigid-as-possible shape interpolation

[ACL00]

Outline

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

• Related work
• Overview of the method
• The method step by step
• Numerics
• Experiments

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

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Related work

Subdivision

[ZSS97, KCVS98, KBS00, GSS99]

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

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Skeleton-based kinematics

Related work

[JT05, ASKTR05]

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Relation to mesh editing

Inverse kinematics coating transfer mesh transplanting

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Intrinsic mesh editing methods

Related work

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Laplacian (or differential) coordinates

[SLCARS04]

Overview of the method

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Let M0, · · · , Mk be the input meshes (M0 is the reference mesh).

• 1. map each transformation T i (i ≥ 1) to a feature vector fi ∈ R9m,

where m is the number of triangles of the base mesh

• 2. define the space Σ of admissible transformations as the span of

the {fi}i≥1

• 3. given a set of specified vertex positions v1, · · · , vl, find a feature

vector f close to Σ that maps each constrained vertex ¯ vj to a point close to its specified position vj

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Feature vectors

Given i ≥ 1, fi encodes the deformation gradients of the piecewise affine map from M0 to Mi

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

For the jth triangle of M0, let φj be the affine transformation that maps the triangle onto its image in Mi: ∀p ∈ R3, φj := Tj p + tj

linear (3x3 matrix) translation vector

• only the gradient of φj is encoded: Dφj(p) = Tj
• to make Tj unique, a 4th vertex is added to the jth triangle, along

the normal direction

¯ v1 ¯ v2 ¯ v3 ¯ v4 v1 v2 v3 v4

φj

from [SP04]

Feature vectors

Given i ≥ 1, fi encodes the deformation gradients of the piecewise affine map from M0 to Mi

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

For the jth triangle of M0, let φj be the affine transformation that maps the triangle onto its image in Mi: ∀p ∈ R3, φj := Tj p + tj

linear (3x3 matrix) translation vector

• only the gradient of φj is encoded: Dφj(p) = Tj
• to make Tj unique, a 4th vertex is added to the jth triangle, along

the normal direction Tj = [v1 − v4, v2 − v4, v3 − v4] · [¯ v1 − ¯ v4, ¯ v2 − ¯ v4, ¯ v3 − ¯ v4]−1 tj = v4 − Tj ¯ v4

Feature vectors

Given i ≥ 1, fi encodes the deformation gradients of the piecewise affine map from M0 to Mi

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

For the jth triangle of M0, let φj be the affine transformation that maps the triangle onto its image in Mi: ∀p ∈ R3, φj := Tj p + tj

linear (3x3 matrix) translation vector

• only the gradient of φj is encoded: Dφj(p) = Tj
• to make Tj unique, a 4th vertex is added to the jth triangle, along

the normal direction Tj = [v1 − v4, v2 − v4, v3 − v4] · [¯ v1 − ¯ v4, ¯ v2 − ¯ v4, ¯ v3 − ¯ v4]−1 tj = v4 − Tj ¯ v4

Feature vectors

Given i ≥ 1, fi encodes the deformation gradients of the piecewise affine map from M0 to Mi

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Tj = [v1 − v4, v2 − v4, v3 − v4] · [¯ v1 − ¯ v4, ¯ v2 − ¯ v4, ¯ v3 − ¯ v4]−1

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

linear in v1, v2, v3, v4 (3 × 3)

Feature vectors

Given i ≥ 1, fi encodes the deformation gradients of the piecewise affine map from M0 to Mi

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Tj = [v1 − v4, v2 − v4, v3 − v4] · [¯ v1 − ¯ v4, ¯ v2 − ¯ v4, ¯ v3 − ¯ v4]−1

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

linear in v1, v2, v3, v4 (3 × 3) fi

j :=

  Tjx

T

Tjy

T

Tjz

T

  = Gj                  v1x . . . v4x v1y . . . v4y v1z . . . v4z                  Tj :=   Tjx Tjy Tjz   Gj :=   Gjx Gjy Gjz   ∈ R9 ∈ R12 (9 × 12)

Feature vectors

Given i ≥ 1, fi encodes the deformation gradients of the piecewise affine map from M0 to Mi

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Tj = [v1 − v4, v2 − v4, v3 − v4] · [¯ v1 − ¯ v4, ¯ v2 − ¯ v4, ¯ v3 − ¯ v4]−1

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

linear in v1, v2, v3, v4 (3 × 3) fi

j :=

  Tjx

T

Tjy

T

Tjz

T

  = Gj                  v1x . . . v4x v1y . . . v4y v1z . . . v4z                  ∈ R9 ∈ R12 (9 × 12) fi :=    f i

1

. . . f i

m

   = G v G :=   Gx Gy Gz  

Feature vectors

Given i ≥ 1, fi encodes the deformation gradients of the piecewise affine map from M0 to Mi

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Tj = [v1 − v4, v2 − v4, v3 − v4] · [¯ v1 − ¯ v4, ¯ v2 − ¯ v4, ¯ v3 − ¯ v4]−1

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

linear in v1, v2, v3, v4 (3 × 3) fi

j :=

  Tjx

T

Tjy

T

Tjz

T

  = Gj                  v1x . . . v4x v1y . . . v4y v1z . . . v4z                  ∈ R9 ∈ R12 (9 × 12) fi :=    f i

1

. . . f i

m

   = G v ∈ R9m (9m × 3(n + m)) ∈ R3(n+m) G :=   Gx Gy Gz   (3m × (n + m))

Feature vectors

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f = G v

∈ R9m ∈ R3(n+m)

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Inverse transformation

(9m × 3(n + m))

”v = G−1 f ”

Feature vectors

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f = G v

∈ R9m ∈ R3(n+m)

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Inverse transformation

(9m × 3(n + m))

”v = G−1 f ”

9m > 3(n + m) ⇒ G is not a square matrix

By fixing one vertex, one reduces the dim. to zero

To a feature vector f ∈ R9m corresponds ≤ 1 tranformed mesh (up to translation)

Feature vectors

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f = G v

∈ R9m ∈ R3(n+m)

Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Inverse transformation

(9m × 3(n + m))

”v = G−1 f ”

9m > 3(n + m) ⇒ G is not a square matrix

By fixing one vertex, one reduces the dim. to zero

least squares problem: v∗ = arg min

v

G v − f

most ”plausible” positions

To a feature vector f ∈ R9m corresponds ≤ 1 tranformed mesh (up to translation) If too many constraints (or imprecisions), the set of solutions can be empty

Linear feature space

Σ := Span(f 1, · · · , f k) To every vector w of coordinates in the basis

• f1, · · · , fk

corresponds a feature vector fw =

i wi fi ∈ R9m

Once one or more vertex positions are set, compute v∗, w∗ = arg min

v,w

• G v − (f1, · · · , fk) w
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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Linear feature space

Σ := Span(f 1, · · · , f k) To every vector w of coordinates in the basis

• f1, · · · , fk

corresponds a feature vector fw =

i wi fi ∈ R9m

Once one or more vertex positions are set, compute v∗, w∗ = arg min

v,w

• G v − (f1, · · · , fk) w
• To penalize solutions that are far from the fi, take the mean ¯

f of the fi as the origin: fw = ¯ f + k

i=1 wi ¯

f

i

and compute: v∗, w∗ = arg min

v,w

• G v −¯

f − (¯ f

1, · · · ,¯

f

k) w

• + α w

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Linear feature space

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Linear feature space

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

f1 f2 f( 1

3 , 2 3 )

Nonlinear feature space

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

For each triangle j of M0, separate the rotational component from the scale and shear components of Tj by polar decomposition: Tj := Rj Sj Interpolate the Si

j linearly, but the T i j by composition:

”T i

j(w) := k

• i=1

Ri

j wi · k

• i=1

wi Si

j ”

R2

θ = R2θ

Nonlinear feature space

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

For each triangle j of M0, separate the rotational component from the scale and shear components of Tj by polar decomposition: Tj := Rj Sj Interpolate the Si

j linearly, but the T i j by composition:

”T i

j(w) := k

• i=1

Ri

j wi · k

• i=1

wi Si

j ”

R2

θ = R2θ

For non-integer wi, use the exponential map: T i

j(w) := exp

k

• i=1

wi log Ri

j

• ·

k

• i=1

wi Si

j

Nonlinear feature space

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

For each triangle j of M0, separate the rotational component from the scale and shear components of Tj by polar decomposition: Tj := Rj Sj Interpolate the Si

j linearly, but the T i j by composition:

”T i

j(w) := k

• i=1

Ri

j wi · k

• i=1

wi Si

j ”

R2

θ = R2θ

(∗)

non-linear

To retrieve v∗, w∗, solve the following least squares problem: v∗, w∗ = arg min

v,w G v − fw

For non-integer wi, use the exponential map: T i

j(w) := exp

k

• i=1

wi log Ri

j

• ·

k

• i=1

wi Si

j

Nonlinear feature space

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

linear interpolation exponential map scale/shear rotational

Numerics

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

• To solve (∗), use Gauss-Newton algorithm
• To accelerate the resolution of the linear least squares problem, use

Cholesky decomposition see details in the paper reduces (∗) to solving a linear least squares problem at each iteration of the algorithm (∼ 6 iterations in practice)

Experimental results

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

Pentium IV @ 3.4 GHz 2GB RAM Cholesky

• ne iteration

Conclusion

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Mesh-Based Inverse Kinematics — R. W. Sumner, M. Zwicker, C. Gotsman, J. Popovi´ c

• MeshIK is an easy-to-use mesh manipulation tool, based on an in-

tuitive interface

• the algorithm adapts to each model by learning from example meshes
• the method is effective in practice and provides interactive timings
• What about meshes of genus > 0?
• Try other feature spaces, to enhance the feature interpolation
• Try other numerical techniques, to enhance speed