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) - Data-driven 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 N-joint 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 N-joint 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 N-joint 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 N-joint 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 sub-tree (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 non-smooth 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 ( candy-wrapper 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 Z-axis. 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 Data-driven Body Shape Models [ Cyberware ] Idea: Let s scan real people and figure out how their body deforms and what body types are possible

48 Data-driven 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] Marker-based Non-rigid 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] Marker-based Non-rigid 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] Marker-based Non-rigid 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] Marker-based Non-rigid 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] Marker-based Non-rigid 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 under-constrained 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) Non-rigid 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) Non-rigid 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 Non-rigid 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 Non-rigid Deformations [Angelov et al., ACM TOG, 2005] Let non-rigid 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 input-output 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 input-output 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 input-output 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 input-output 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 input-output 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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