Introduction to Computer Graphics Animation (1) May 23, 2019 - - PowerPoint PPT Presentation

Introduction to Computer Graphics – Animation (1) –

May 23, 2019 Kenshi Takayama

Skeleton-based animation

• Simple
• Intuitive
• Low comp. cost

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https://www.youtube.com/watch?v=DsoNab58QVA

Representing a pose using skeleton

• Tree structure consisting of bones & joints
• Each bone holds relative rotation angle w.r.t. parent joint
• Whole body pose determined by the set of joint angles

(Forward Kinematics)

• Deeply related to robotics

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Inverse Kinematics

• Find joint angles s.t. an

end effector comes at a given goal position

• Typical workflow:
• Quickly create pose

using IK, fine adjustment using FK

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https://www.youtube.com/watch?v=e1qnZ9rV_kw

Simple method to solve IK: Cyclic Coordinate Descent

• Change joint angles one by one
• S.t. the end effector comes as close

as possible to the goal position

• Ordering is important! Leaf  root
• Easy to implement  Basic assignment
• More advanced
• Jacobi method (directional constraint)
• Minimizing elastic energy [Jacobson 12]

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https://mukai-lab.org/content/CcdParticleInverseKinematics.pdf

IK minimizing elastic energy

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Fast Automatic Skinning Transformations [Jacobson SIGGRAPH12] https://www.youtube.com/watch?v=PRcXy2LjI9I

Ways to obtain/measure motion data

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Optical motion capture

• Put markers on the actor, record video from many viewpoints (~48)

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https://www.youtube.com/watch?v=c6X64LhcUyQ from Wikipedia

Mocap using inexpensive depth camera

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https://www.youtube.com/watch?v=qC-fdgPJhQ8

Mocap designed for outdoor scene

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Motion Capture from Body-Mounted Cameras [Shiratori SIGGRAPH11] https://www.youtube.com/watch?v=xbI-NWMfGPs

Motion database

• http://mocap.cs.cmu.edu/
• 6 categories, 2605 in total
• Free for research purposes
• Interpolation, recombination, analysis, search, etc.

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Recombining motions

• Allow transition from one motion to

another if poses are similar in certain frame

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Motion Graphs [Kovar SIGGRAPH02] Motion Patches: Building Blocks for Virtual Environments Annotated with Motion Data [Lee SIGGRAPH06] http://www.tcs.tifr.res.in/~workshop/thapar_igga/motiongraphs.pdf

Pose similarity matrix

frame frame

Generating motion through simulation

• For creatures

unsuitable for mocap

• Too dangerous,

nonexistent, ...

• Natural motion

respecting body shape

• Can interact with

dynamic environment

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Generalizing Locomotion Style to New Animals With Inverse Optimal Regression [Wampler SIGGRAPH14] https://www.youtube.com/watch?v=KF_a1c7zytw

Creating poses using special devices

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Tangible and Modular Input Device for Character Articulation [Jacobson SIGGRAPH14] Rig Animation with a Tangible and Modular Input Device [Glauser SIGGRAPH16]

https://www.youtube.com/watch?v=vBX47JamMN0

Many topics about character motion

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Character motion synthesis by topology coordinates [Ho EG09] Aggregate Dynamics for Dense Crowd Simulation [Narain SIGGRAPHAsia09] Synthesis of Detailed Hand Manipulations Using Contact Sampling [Ye SIGGRAPH12] Space-Time Planning with Parameterized Locomotion Controllers.[Levine TOG11]

Interaction between multiple persons Crowd simulation Grasping motion Path planning

https://www.youtube.com/ watch?v=1S_6wSKI_nU https://www.youtube.com/ watch?v=pqBSNAOsMDc https://www.youtube.com/ watch?v=x8c27XYTLTo https://vimeo.com/33409868

Skinning

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𝐰𝑗

′ = blend

𝑥𝑗,1, 𝐔

1 , 𝑥𝑗,2, 𝐔2 , …

𝐰𝑗

• Input
• Vertex positions 𝐰𝑗

𝑗 = 1, … , 𝑜

• Transformation per bone 𝐔

𝑘

𝑘 = 1, … , 𝑛

• Weight from each bone to each vertex 𝑥𝑗,𝑘

𝑗 = 1, … , 𝑜 𝑘 = 1, … , 𝑛

• Output
• Vertex positions after deformation 𝐰𝑗

′

𝑗 = 1, … , 𝑜

• Main focus
• How to define weights 𝑥𝑗,𝑘
• How to blend transformations

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Simple way to define weights: painting

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https://www.youtube.com/watch?v=TACB6bX8SN0

Automatic weight computation

• Define weight 𝑥

𝑘 as a smooth scalar field that takes 1 on the j-th bone

and 0 on the other bones

• Minimize 1st-order derivative ׬

Ω

𝛼𝑥

𝑘 2𝑒𝐵

[Baran 07]

• Approximate solution only on surface  easy & fast
• Minimize 2nd-order derivative ׬

Ω Δ𝑥 𝑘 2𝑒𝐵 [Jacobson 11]

• Introduce inequality constraints 0 ≤ 𝑥

𝑘 ≤ 1

• Quadratic Programming over the volume  high-quality

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Automatic rigging and animation of 3d characters [Baran SIGGRAPH07] Bounded Biharmonic Weights for Real-Time Deformation [Jacobson SIGGRAPH11]

Pinocchio demo

Simple way to blend transformations: Linear Blend Skinning

• Represent rigid transformation 𝐔

𝑘 as a 3×4 matrix consisting of

rotation matrix 𝐒𝑘 ∈ ℝ3×3 and translation vector 𝐮𝑘 ∈ ℝ3 𝐰𝑗

′ =

෍

𝑘

𝑥𝑗,𝑘 𝐒𝑘 𝐮𝑘 𝐰𝑗 1

• Simple and fast
• Implemented using vertex shader: send 𝐰𝑗 & 𝑥𝑗,𝑘 to GPU at initialization,

send 𝐔

𝑘 to GPU at each frame

• Standard method

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Artifact of LBS: ”candy wrapper” effect

• Linear combination of rigid transformation is

not a rigid transformation!

• Points around joint concentrate when twisted

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Initial shape & two bones Twist one bone Deformation using LBS

Alternative to LBS: Dual Quaternion Skinning

• Idea
• Quaternion (four numbers)  3D rotation
• Dual quaternion (two quaternions)  3D rigid motion (rotation + translation)

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Initial shape & two bones Deformation using LBS Deformation using DQS

Dual number & dual quaternion

• Dual number
• Introduce dual unit 𝜁 & its arithmetic rule 𝜁2 = 0 (cf. imaginary unit 𝑗)
• Dual number is sum of primal & dual components:
• Dual conjugate:

ത ො 𝑏 = 𝑏0 + 𝜁𝑏𝜁 = 𝑏0 − 𝜁𝑏𝜁

• Dual quaternion
• Quaternion whose elements are dual numbers
• Can be written using two quaternions
• Dual conjugate:

ഥ ෝ 𝐫 = 𝐫0 + 𝜁𝐫𝜁 = 𝐫0 − 𝜁𝐫𝜁

• Quaternion conjugate:

ෝ 𝐫∗ = 𝐫0 + 𝜁𝐫𝜁 ∗ = 𝐫0

∗ + 𝜁𝐫𝜁 ∗

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Geometric Skinning with Approximate Dual Quaternion Blending [Kavan TOG08]

ෝ 𝐫 ≔ 𝐫0 + 𝜁𝐫𝜁 ො 𝑏 ≔ 𝑏0 + 𝜁𝑏𝜁

𝑏0, 𝑏𝜁 ∈ ℝ

Arithmetic rules for dual number/quaternion

• For dual number ො

𝑏 = 𝑏0 + 𝜁𝑏𝜁 :

• Reciprocal

1 ො 𝑏 = 1 𝑏0 − 𝜁 𝑏𝜁 𝑏0

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• Square root

ො 𝑏 = 𝑏0 + 𝜁

𝑏𝜁 2 𝑏0

• Trigonometric

sin ො 𝑏 = sin 𝑏0 + 𝜁𝑏𝜁 cos 𝑏0 cos ො 𝑏 = cos 𝑏0 − 𝜁𝑏𝜁 sin 𝑏0

• For dual quaternion ෝ

𝐫 = 𝐫0 + 𝜁𝐫𝜁 :

• Norm

ෝ 𝐫 = ෝ 𝐫∗ෝ 𝐫 = 𝐫0 + 𝜁

𝐫0,𝐫𝜁 𝐫0

• Inverse

ෝ 𝐫−1 =

ෝ 𝐫∗ ෝ 𝐫 2

• Unit dual quaternion satisfies ෝ

𝐫 = 1

• ⟺ 𝐫0

= 1 & 𝐫0, 𝐫𝜁 = 0

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Dot product as 4D vectors

Easily derived by combining usual arithmetic rules with new rule 𝜁2 = 0 From Taylor expansion

Geometric Skinning with Approximate Dual Quaternion Blending [Kavan TOG08]

Rigid transformation using dual quaternion

• Unit dual quaternion representing rigid motion of

translation Ԧ 𝐮 = 𝑢𝑦, 𝑢𝑧, 𝑢𝑨 and rotation 𝐫0 (unit quaternion) : ෝ 𝐫 = 𝐫0 + 𝜁 2 Ԧ 𝐮𝐫0

• Rigid transformation of 3D position 𝐰 = 𝑤𝑦, 𝑤𝑧, 𝑤𝑨 using

unit dual quaternion ෝ 𝐫 : ෝ 𝐫 1 + 𝜁𝐰 ෝ 𝐫∗ = 1 + 𝜁𝐰′

• 𝐰′ : 3D position after transformation

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Note: 3D vector is considered as quaternion with zero real part

Geometric Skinning with Approximate Dual Quaternion Blending [Kavan TOG08]

Rigid transformation using dual quaternion

• ෝ

𝐫 = 𝐫0 +

𝜁 2 Ԧ

𝐮𝐫0

• ෝ

𝐫 1 + 𝜁𝐰 ෝ 𝐫∗ = 𝐫0 +

𝜁 2 Ԧ

𝐮𝐫0 1 + 𝜁𝐰 𝐫0

∗ + 𝜁 2 𝐫0 ∗ Ԧ

𝐮 = 𝐫0 +

𝜁 2 Ԧ

𝐮𝐫0 𝐫0

∗ + 𝜁𝐰𝐫0 ∗ + 𝜁 2 𝐫0 ∗ Ԧ

𝐮 = 𝐫0𝐫0

∗ + 𝜁 2 Ԧ

𝐮𝐫0𝐫0

∗ + 𝜁𝐫0𝐰𝐫0 ∗ + 𝜁 2 𝐫0𝐫0 ∗ Ԧ

𝐮 = 1 + 𝜁 Ԧ 𝐮 + 𝐫0𝐰𝐫0

∗

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0 + Ԧ 𝐮 𝐫0

∗

= 𝐫0

∗ 0 + Ԧ

𝐮

∗

= −𝐫0

∗ Ԧ

𝐮 𝐫0 2 = 1 3D position 𝐰 rotated by quaternion 𝐫0

Geometric Skinning with Approximate Dual Quaternion Blending [Kavan TOG08]

Rigid transformation as “screw motion”

• Any rigid motion is uniquely described as screw motion
• (Up to antipodality)

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Conventional notion: rotation + translation Screw motion Axis direction

Geometric Skinning with Approximate Dual Quaternion Blending [Kavan TOG08]

Screw motion & dual quaternion

• Unit dual quaternion ෝ

𝐫 can be written as: ෝ 𝐫 = cos መ 𝜄 2 + ො 𝐭 sin መ 𝜄 2

• ෠

𝜄 = 𝜄0 + 𝜁𝜄𝜁

𝜄0, 𝜄𝜁 : real number

• ො

𝐭 = 𝐭0 + 𝜁𝐭𝜁

𝐭0, 𝐭𝜁 : unit 3D vector

• Geometric meaning
• 𝐭0 : direction of rotation axis
• 𝜄0 : amount of rotation
• 𝜄𝜁 : amount of translation parallel to 𝐭0
• 𝐭𝜁 : when rotation axis passes through Ԧ

𝐬, it satisfies 𝐭𝜁 = Ԧ 𝐬 × 𝐭0

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Geometric Skinning with Approximate Dual Quaternion Blending [Kavan TOG08]

Interpolating two rigid transformations

• Linear interpolation + normalization (nlerp)

nlerp ෝ 𝐫1, ෝ 𝐫2, 𝑢 ≔

1−𝑢 ෝ 𝐫1+𝑢ෝ 𝐫2 1−𝑢 ෝ 𝐫1+𝑢ෝ 𝐫2

• Note: ෝ

𝐫 & −ෝ 𝐫 represent same transformation with opposite path

• If 4D dot product of non-dual

components of ෝ 𝐫1 & ෝ 𝐫2 is negative, use −ෝ 𝐫2 in the interpolation

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Blending rigid motions using dual quaternion

blend 𝑥1, ෝ 𝐫1 , 𝑥2, ෝ 𝐫2 , … ≔ 𝑥1ෝ 𝐫1 + 𝑥2ෝ 𝐫2 + ⋯ 𝑥1ෝ 𝐫1 + 𝑥2ෝ 𝐫2 + ⋯

• Akin to blending rotations using quaternion
• Same input format as LBS & low computational cost
• Standard feature in many

commercial CG packages

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Geometric Skinning with Approximate Dual Quaternion Blending [Kavan TOG08]

Artifact of DQS: ”bulging” effect

• Produces ball-like shape around the joint when bended

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LBS DQS

Elasticity-Inspired Deformers for Character Articulation [Kavan SIGGRAPHAsia12] Bulging-free dual quaternion skinning [Kim CASA14]

Overcoming DQS’s drawback

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Decompose transformation into bend & twist, interpolate them separately [Kavan12]

Elasticity-Inspired Deformers for Character Articulation [Kavan SIGGRAPHAsia12] Bulging-free dual quaternion skinning [Kim CASA14]

LBS DQS [Kavan12]

After deforming using DQS,

• ffset vertices along normals [Kim14]

LBS DQS [Kavan12]

Limitation of DQS: Cannot represent rotation by more than 360°

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Differential Blending for Sketch-based Expressive Posing [Oztireli SCA13]

Skinning for avoiding self-intersections

• Make use of implicit functions

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Implicit Skinning; Real-Time Skin Deformation with Contact Modeling [Vaillant SIGGRAPH13]

https://www.youtube.com/watch?v=RHySGIqEgyk

Other deformation mechanisms than skinning

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Bounded Biharmonic Weights for Real-Time Deformation [Jacobson SIGGRAPH11]

https://www.youtube.com/watch?v=P9fqm8vgdB8

Unified point/cage/skeleton handles [Jacobson 11]

https://www.youtube.com/watch?v=BFPAIU8hwQ4

BlendShape

References

• http://en.wikipedia.org/wiki/Motion_capture
• http://skinning.org/
• http://mukai-lab.org/category/library/legacy

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