Lecture 9 Body Representations


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1 Lecture 9 Body Representations Leonid Sigal Human Motion Modeling and Analysis Fall 2012
2 Course Updates Capture Project  Now due next Monday (one week from today) Final Project  3 minute pitches are due next class  You should try to post your idea to the blog and get some feedback before then Reading Signup  Everyone has signed up by this point?
3 Plan for Today s Class  Review representation of skeleton motion  Modeling shape and geometry of the body  Skinning (rigid, linear, dual quaternion)  Datadriven body models (SCAPE)  Applications and discussion  Shape estimation from images  Image reshaping
4 Skeleton Animations Skeleton (tree hierarchy)  Nodes represent joints Hierarchical Structure Common Data structure for Body Pose  Joints are local coordinate systems (frames)  Edges represent bones Right Hand Root Head Left Hand Skin Right Foot Left Foot  3D model driven by the skeleton Source: Meredith and Maddock, Motion Capture File Formats Explained
5 Skeleton Animations Skeleton (tree hierarchy)  Nodes represent joints  Joints are local coordinate systems (frames)  Edges represent bones Skin  3D model driven by the skeleton [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
6 Skeleton Animations Skeleton (tree hierarchy)  Nodes represent joints  Joints are local coordinate systems (frames)  Edges represent bones Skin  3D model driven by the skeleton [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
7 Skeleton Animations Skeleton (tree hierarchy)  Nodes represent joints  Joints are local coordinate systems (frames)  Edges represent bones Skin  3D model driven by the skeleton Both are typically designed in a reference pose [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
8 Skeleton in Reference Posture R j p(j) j Njoint skeleton in reference frame is given by  Root frame expressed with respect to the world   Relative joint coordinate frames  R 1,R 2,R 3,...,R N Mapping from world to a coordinate frame of joint j A j = R 0 R p(j) R j R 0 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
9 Skeleton in Reference Posture R j p(j) j Njoint skeleton in reference frame is given by  Root frame expressed with respect to the world   Relative joint coordinate frames  Mapping from world to a coordinate frame of joint j R j = 0 r 11 r 12 r 13 t 1 r 21 r 22 r 23 t 2 r 31 r 32 r 33 t A j = R 0 R p(j) R j R 0 R 1,R 2,R 3,...,R N 1 C A [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
10 Skeleton in Reference Posture R j p(j) j Njoint skeleton in reference frame is given by  Root frame expressed with respect to the world   Relative joint coordinate frames  R 1,R 2,R 3,...,R N Mapping from world to a coordinate frame of joint j A j = R 0 R p(j) R j R 0 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
11 Skeleton in Reference Posture R j p(j) j Njoint skeleton in reference frame is given by  Root frame expressed with respect to the world   Relative joint coordinate frames  Mapping from world to a coordinate frame of joint j A j = R 0 R p(j) R j R 0 R 1,R 2,R 3,...,R N parent of joint j [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
12 Animating Skeleton Achieved by rotating each joint from it s reference posture  Note: joint rotation effects the entire subtree (e.g., rotation at the shoulder will induce motion of the whole arm) Rotation at joint j is described by: 0 r 11 r 12 r T j = r 21 r 22 r 23 0 r 31 r 32 r 33 0 C A [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
13 Animating Skeleton Mapping from world to a coordinate frame of joint j in reference pose: A j = R 0 R p(j) R j Mapping from world to a coordinate frame of joint j in animated pose: F j = R 0 T 0...R p(j) T p(j) R j T j [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
14 Animating Skeleton Mapping from world to a coordinate frame of joint j in reference pose: A j = R 0 R p(j) R j Mapping from world to a coordinate frame of joint j in animated pose: F j = R 0 T 0...R p(j) T p(j) R j T j Note: if T j = I 4 4 then F j = A j [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
15 Character Rigging 1. Embed the skeleton into a 3D mesh (skin) 2. Assign vertices of the mesh to one or more bones to allow skin to move with the skeleton [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
16 Rigid Skinning 1. Embed the skeleton into a 3D mesh (skin) 2. Assign vertices of the mesh to one or more bones to allow skin to move with the skeleton Assign each vertex to one bone/joint  j v i ˆv i = F j (A j ) 1 v i v i A j F j ˆv i  position of vertex in reference mesh  joint j in reference mesh  joint j in animated mesh  position of vertex in animated mesh
17 Rigid Skinning Requires assignment of vertices on 3D mesh to joints on skeleton  Often done manually  Assigning to joint influencing the closest bone is usually a good automatic guess Assign each vertex to one bone/joint  j v i ˆv i = F j (A j ) 1 v i v i A j F j ˆv i  position of vertex in reference mesh  joint j in reference mesh  joint j in animated mesh  position of vertex in animated mesh
18 Rigid Skinning Limitations In reference pose: In animated pose:  Works well away from the ends of the bone (joints)  Leads to unrealistic nonsmooth deformations near joints [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
19 Linear Blend Skinning Each vertex is assigned to multiple bone/joints v i ˆv i = NX j=1 w ji F j (A j ) 1 v i v i A j F j ˆv i w ji  position of vertex i in reference mesh  joint j in reference mesh  joint j in animated mesh  position of vertex i in animated mesh  influence of joint j on the vertex
20 Linear Blend Skinning Each vertex is assigned to multiple bone/joints v i ˆv i = NX j=1 w ji F j (A j ) 1 v i v i A j F j ˆv i w ji  position of vertex i in reference mesh  joint j in reference mesh  joint j in animated mesh  position of vertex i in animated mesh  influence of joint j on the vertex Weights need to be convex: NX j=1 w ji =1,w ji 0
21 Linear Blend Skinning Each vertex is assigned to multiple bone/joints v i ˆv i = NX j=1 w ji F j (A j ) 1 v i v i A j F j ˆv i w ji  position of vertex i in reference mesh  joint j in reference mesh  joint j in animated mesh  position of vertex i in animated mesh  influence of joint j on the vertex NX joints Weights need to be convex: w ji =1,w ji 0 j=1 painted on or based on distance to
22 Linear Blend Skinning Weights w larm + w uarm =1 w larm =1 w uarm =1 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
23 Linear Blend Skinning Weights w larm + w uarm =1 w larm =1 w uarm =1 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
24 Linear Blend Skinning Weights w larm + w uarm =1 w larm =1 w uarm =1 Why weights must convex? [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
25 Linear Blend Skinning Weights ˆv i = NX j=1 w ji F j (A j ) 1 v i w larm + w uarm =1 w larm =1 w uarm =1 Why weights must convex? [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
26 Linear Blend Skinning Example [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
27 Linear Blend Skinning Limitations Joint twisting 180 degrees produces a collapsing effect where skin collapses to a single point ( candywrapper artifact) [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
28 Linear Blend Skinning Limitations [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
29 Linear Blend Skinning Limitations [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
30 Why LBS Produce Artifacts? 0 1 NX NX ˆv i = w ji F j (A j ) 1 v i ˆv i w ji M A j v i j=1 j=1 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
31 Why LBS Produce Artifacts? 0 1 NX NX ˆv i = w ji F j (A j ) 1 v i ˆv i w ji M A j v i j=1 j=1 M 1 M 2 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
32 Why LBS Produce Artifacts? 0 1 NX NX ˆv i = w ji F j (A j ) 1 v i ˆv i w ji M A j v i j=1 j=1 M 1 M 2 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
33 Why LBS Produce Artifacts? 0 1 NX NX ˆv i = w ji F j (A j ) 1 v i ˆv i w ji M A j v i j=1 j=1 M 1 M blend M 2 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
34 Why LBS Produce Artifacts? M 1 M blend M 2 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
35 SE(3) Intrinsic Blending M 1 M blend M 2 [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
36 Dual Quaternion Skinning [Kavan et al., ACM TOG 2008] Closed form approximation of SE(3) blending [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
37 Regular Quaternions Remember: Euler angles Y Z X 2 Z Z X X Y Y Rotation in 3D X 0 Y 0 Z 0 X 0 Y 0 Z 0 X 0 Y 0 Z = = = 4 cos( ) sin( ) 0 sin( ) cos( ) R z cos( ) 0 sin( ) sin( ) 0 cos( ) R y cos( ) sin( ) 0 sin( ) cos( ) R x Note: I ve overloaded the use of \phi in these slides. Earlier I used \phi to denote the original orientation. Here I am using it to denote the rotation about the Zaxis. X Y Z X Y Z X Y Z Interpolating Euler angles has similar issues as LBS skinning
38 Regular Quaternions  Quaternions are alternative representation for orientations (defined using complex algebra)  Represents orientation using 4 tuples (roughly speaking one for amount of rotation and 3 for the axis) q = w + xi + yi + zi  However, there are only 3 degrees of freedom for a rotation  Hence, to be a valid rotation, quaternion must be unit norm q =1
39 Regular Quaternions  Quaternions are alternative representation for orientations (defined using complex algebra)  Represents orientation using 4 tuples (roughly speaking one for amount of rotation and 3 for the axis) q = w + xi + yi + zi  However, there are only 3 degrees of freedom for a rotation  Hence, to be a valid rotation, quaternion must be unit norm q =1 Interpretation: Quaternions live on a sphere in a 4D space
40 Dual Quaternions [Clifford 1873]  Dual quaternions are able to model rigid transformations (rotation + translation)  Map a 6 dimensional manifold in an 8 dimensional space  Need to be unit length to represent a valid rigid transform [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
41 Dual Quaternions [Clifford 1873]  Dual quaternions are able to model rigid transformations (rotation + translation)  Map a 6 dimensional manifold in an 8 dimensional space  Need to be unit length to represent a valid rigid transform [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
42 Dual Quaternion Skinning [Kavan et al., ACM TOG 2008] Closed form approximation of SE(3) blending [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
43 Comparison: Linear Blend Skinning [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
44 Comparison: Dual Quaternion [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
45 Dual Quaternion Skinning [Slide content and illustrations from Ladislav Kavan and Olga Sorkine]
46 Skinning Limitations  All skinning methods assume a fixed relationship between skeleton motion and the mesh  Humans are more complex (e.g., muscles lead to local deformations)  Skinning only allows animation of predefined body geometry (created by an animator), do not help us create this geometry
47 Datadriven Body Shape Models [ Cyberware ] Idea: Let s scan real people and figure out how their body deforms and what body types are possible
48 Datadriven Body Shape Models Pipeline  Register all the scans [ Cyberware ]  Create (typically parametric) model of shape  Use that model for an interesting application
49 Registration / Correspondence [Allen et al., 2003] Markerbased Nonrigid Iterative Closest Point Registration Goal: Fit a template mesh to triangulated 3D point cloud E = ÆE d + ØE s + E m Amounts to estimating a 4x4 transform for every vertex through optimization of above
50 Registration / Correspondence [Allen et al., 2003] Markerbased Nonrigid Iterative Closest Point Registration Goal: Fit a template mesh to triangulated 3D point cloud E = ÆE d + ØE s + E m data term VX E d = w i gap 2 (T i ~v i, D) i=1 tices for the template mesh T Distance from the reference mesh point to closest compatible vertex on observed surface  Angle between surface normals  Robust measure for dealing with holes
51 Registration / Correspondence [Allen et al., 2003] Markerbased Nonrigid Iterative Closest Point Registration Goal: Fit a template mesh to triangulated 3D point cloud E = ÆE d + ØE s + E m E s = data term X smoothness term T i T j 2 F {i,j (~v i,~v j )2edges(T )} Ensures that transforms on near by vertices are similar (induces local smoothness)
52 Registration / Correspondence [Allen et al., 2003] Markerbased Nonrigid Iterative Closest Point Registration Goal: Fit a template mesh to triangulated 3D point cloud E = ÆE d + ØE s + E m data term smoothness term marker anchor term MX E m = T i ~v i ~m i 2 i=1 Ensures that transforms on near by vertices s from the template mesh th are similar (induces local smoothness)
53 Registration / Correspondence [Allen et al., 2003] Markerbased Nonrigid Iterative Closest Point Registration Goal: Fit a template mesh to triangulated 3D point cloud E = ÆE d + ØE s + E m data term smoothness term marker anchor term Solved using gradient descent Initialized by aligning centers of mass
54 Registration [Image from Alexandru Balan]
55 Mesh Deformation Gradients [Sumner and Popovic, 2004] 3x3 transform for every triangle A t [ ~x t,2, ~x t,3 ]=[ ~y t,2, ~y t,3 ] ~x t,2 ~y t,2 ~x t,3 ~y t,3 Source Mesh Target Mesh [Image from Alexandru Balan]
56 Mesh Deformation Gradients [Sumner and Popovic, 2004] 3x3 transform for every triangle A t [ ~x t,2, ~x t,3 ]=[ ~y t,2, ~y t,3 ] ~x t,2 ~y t,2 ~x t,3 ~y t,3 Source Mesh Target Mesh Estimation: arg min {A 1,,A T } TX t=1 X k=2,3 A t ~x t,k ~y t,k 2 + w s [Image from Alexandru Balan] X A t1 A t2 2 F t 1,t 2 adj
57 Mesh Deformation Gradients [Sumner and Popovic, 2004] 3x3 transform for every triangle A t [ ~x t,2, ~x t,3 ]=[ ~y t,2, ~y t,3 ] ~x t,2 ~y t,2 ~x t,3 ~y t,3 Source Mesh Target Mesh Estimation: arg min {A 1,,A T } TX t=1 X k=2,3 A t ~x t,k ~y t,k 2 + w s X underconstrained A t1 A t2 2 F t 1,t 2 adj [Image from Alexandru Balan]
58 Mesh Deformation Gradients [Sumner and Popovic, 2004] 3x3 transform for every triangle A t [ ~x t,2, ~x t,3 ]=[ ~y t,2, ~y t,3 ] ~x t,2 ~y t,2 ~x t,3 ~y t,3 Source Mesh Target Mesh Estimation: arg min {A 1,,A T } TX t=1 X k=2,3 A t ~x t,k ~y t,k 2 + w s [Image from Alexandru Balan] X A t1 A t2 2 F t 1,t 2 adj
59 Mesh Deformation Gradients [Sumner and Popovic, 2004] Applying deformation gradients typically leads to inconsistent edges and structures ~x t,2 ~y t,2 ~x t,3 ~y t,3 Source Mesh Target Mesh [Image from Alexandru Balan]
60 Mesh Deformation Gradients [Sumner and Popovic, 2004] Applying deformation gradients typically leads to inconsistent edges and structures ~x t,2 ~y t,2 ~x t,3 ~y t,3 Source Mesh Target Mesh [Image from Alexandru Balan] Reconstruction: arg min {~y 1,,~y V } TX t=1 X k=2,3 A t ~x t,k ~y t,k 2
61 SCAPE: Shape Completion and Animation of PEople [Angelov et al., ACM TOG, 2005] Key Idea: factor mesh deformation for a person into: (1) Articulated rigid deformations (2) Nonrigid deformations (3) Body shape deformations
62 SCAPE: Shape Completion and Animation of PEople [Angelov et al., ACM TOG, 2005] Key Idea: factor mesh deformation for a person into: (1) Articulated rigid deformations (2) Nonrigid deformations (3) Body shape deformations Gives a parametric model of any person in any pose!
63 Articulated Rigid Deformation This is basically rigid skinning [Angelov et al., ACM TOG, 2005] R p (θ) Articulated Rigid Deformation θ joint angles [Image from Alexandru Balan]
64 Nonrigid Deformations [Angelov et al., ACM TOG, 2005] From a dataset of one person in different poses, learn residual arg min {Q i 1,,Qi T } Y TX t=1 X k=2,3 deformations Ø ØØR i p[t] Qi t ~x t,k ~y i t,k ØØ 2 + w s X t 1,t 2 adj p[t 1 ]=p[t 2 ] Q i t 1 Q i t 2 2 F [Image from Alexandru Balan]
65 Nonrigid Deformations [Angelov et al., ACM TOG, 2005] Let nonrigid deformations be linear functions of pose Models bulging of muscles, etc. [Image from Alexandru Balan]
66 Body Shape [Angelov et al., ACM TOG, 2005] From a dataset of many people in one pose learn variations  Extract mesh deformation gradients  Do PCA on them (vectorizing them first)  6 PCA dimensions can capture > 80% of variation [Image from Alexandru Balan]
67 Body Shape [Image from Peng Guan]
68 Body Shape It is also possible to create semantic parameters and regress from them to PCA coefficients [Image from Peng Guan]
69 Body Shape It is also possible to create semantic parameters and regress from them to PCA coefficients Demo [Image from Peng Guan]
70 SCAPE: Putting it all together [Angelov et al., ACM TOG, 2005] R p (θ) Q t (R p (θ)) S t (v) θ joint angles v shape parameters We can simply concatenate all the deformations by multiplying them together
71 SCAPE Model [Video curtesy of Peng Guan]
72 Applications: Shape Estimation Goal: learn functional mapping from image features to pose and shape parameters of the SCAPE model (from synthesized inputoutput pairs) [Sigal et al., NIPS 2007]
73 Applications: Shape Estimation Goal: learn functional mapping from image features to pose and shape parameters of the SCAPE model (from synthesized inputoutput pairs) [Sigal et al., NIPS 2007] Remember how Kinect works? Let s try that for shape
74 Applications: Shape Estimation Goal: learn functional mapping from image features to pose and shape parameters of the SCAPE model (from synthesized inputoutput pairs) [Sigal et al., NIPS 2007]
75 Applications: Shape Estimation Goal: learn functional mapping from image features to pose and shape parameters of the SCAPE model (from synthesized inputoutput pairs) [Sigal et al., NIPS 2007] feature space SCAPE space
76 Applications: Shape Estimation Goal: learn functional mapping from image features to pose and shape parameters of the SCAPE model (from synthesized inputoutput pairs) [Sigal et al., NIPS 2007] feature space SCAPE parameters = g ( features ) SCAPE space
77 Proof of concept results [Sigal et al., NIPS 2007] Can we estimate weight loss discriminatively from monocular images? After Before
78 Proof of concept results [Sigal et al., NIPS 2007] Can we estimate weight loss discriminatively from monocular images? After Before
79 Proof of concept results [Sigal et al., NIPS 2007] Can we estimate weight loss discriminatively from monocular images? After Before
80 Proof of concept results [Sigal et al., NIPS 2007] Can we estimate weight loss discriminatively from monocular images? Before Assume density of water After
81 Proof of concept results [Sigal et al., NIPS 2007] Can we estimate weight loss discriminatively from monocular images? Before Assume density of water After
82 Proof of concept results [Sigal et al., NIPS 2007] Can we estimate weight loss discriminatively from monocular images? Before Assume density of water After Estimated Weight Loss: 22lb Reported Weight Loss: 24lb
83 Proof of concept results [Sigal et al., NIPS 2007] Can we estimate weight loss discriminatively from monocular images? Before Assume density of water After Estimated Weight Loss: 32lb Reported Weight Loss: 64lb
84 Silhouettes is not enough [Guan et al., ICCV, 2009] Silhouettes are ambiguous
85 Silhouettes is not enough [Guan et al., ICCV, 2009] Silhouettes are ambiguous
86 Silhouettes is not enough [Guan et al., ICCV, 2009] Silhouettes are ambiguous Adding edges
87 Silhouettes is not enough [Guan et al., ICCV, 2009] Silhouettes are ambiguous Adding edges They also add shading cues
88 Estimating Pose and Shape from Image [Guan et al., ICCV, 2009]
89 Fun Results [Guan et al., ICCV, 2009] Internet Images: Paintings:
90 Parametric Body Reshaping [Zhou et al., ACM TOG, 2010]
91 Parametric Body Reshaping [Zhou et al., ACM TOG, 2010]
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