Lecture 9 Body Representations


 Allan Stokes
 1 years ago
 Views:
Transcription
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]
CMSC 425: Lecture 13 Animation for Games: Basics Tuesday, Mar 26, 2013
CMSC 425: Lecture 13 Animation for Games: Basics Tuesday, Mar 26, 2013 Reading: Chapt 11 of Gregory, Game Engine Architecture. Game Animation: Most computer games revolve around characters that move around
More informationanimation animation shape specification as a function of time
animation animation shape specification as a function of time animation representation many ways to represent changes with time intent artistic motion physicallyplausible motion efficiency control typically
More informationComputer Animation. Lecture 2. Basics of Character Animation
Computer Animation Lecture 2. Basics of Character Animation Taku Komura Overview Character Animation Posture representation Hierarchical structure of the body Joint types Translational, hinge, universal,
More informationCharacter Animation from 2D Pictures and 3D Motion Data ALEXANDER HORNUNG, ELLEN DEKKERS, and LEIF KOBBELT RWTHAachen University
Character Animation from 2D Pictures and 3D Motion Data ALEXANDER HORNUNG, ELLEN DEKKERS, and LEIF KOBBELT RWTHAachen University Presented by: Harish CS525 First presentation Abstract This article presents
More information3D Morphing for Triangle Meshes using Deformation Matrix
3D Morphing for Triangle Meshes using Deformation Matrix Tianwei Xing, Yipin Yang, Yao Yu, Yu Zhou, Xianglei Xing, Sidan Du NanJing University, School of Electronic Science and Engineering JiangSu Province,
More informationInteractive Computer Graphics
Interactive Computer Graphics Lecture 18 Kinematics and Animation Interactive Graphics Lecture 18: Slide 1 Animation of 3D models In the early days physical models were altered frame by frame to create
More informationanimation shape specification as a function of time
animation 1 animation shape specification as a function of time 2 animation representation many ways to represent changes with time intent artistic motion physicallyplausible motion efficiency typically
More informationIntroduction to Computer Graphics MariePaule Cani & Estelle Duveau
Introduction to Computer Graphics MariePaule Cani & Estelle Duveau 04/02 Introduction & projective rendering 11/02 Prodedural modeling, Interactive modeling with parametric surfaces 25/02 Introduction
More informationTwo hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. M.Sc. in Advanced Computer Science. Friday 18 th January 2008.
COMP60321 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE M.Sc. in Advanced Computer Science Computer Animation Friday 18 th January 2008 Time: 09:45 11:45 Please answer any THREE Questions
More informationSemantic Parametric Reshaping of Human Body Models
2014 Second International Conference on 3D Vision Semantic Parametric Reshaping of Human Body Models Yipin Yang, Yao Yu, Yu Zhou, Sidan Du School of Electronic Science and Engineering Nanjing University
More informationMetrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
More informationINTRODUCTION TO RENDERING TECHNIQUES
INTRODUCTION TO RENDERING TECHNIQUES 22 Mar. 212 Yanir Kleiman What is 3D Graphics? Why 3D? Draw one frame at a time Model only once X 24 frames per second Color / texture only once 15, frames for a feature
More informationCIS 536/636 Introduction to Computer Graphics. Kansas State University. CIS 536/636 Introduction to Computer Graphics
2 Lecture Outline Animation 2 of 3: Rotations, Quaternions Dynamics & Kinematics William H. Hsu Department of Computing and Information Sciences, KSU KSOL course pages: http://bit.ly/hgvxlh / http://bit.ly/evizre
More informationarxiv:1312.4967v2 [cs.cv] 26 Jun 2014
Estimation of Human Body Shape and Posture Under Clothing Stefanie Wuhrer a Leonid Pishchulin b Alan Brunton c Chang Shu d Jochen Lang e arxiv:1312.4967v2 [cs.cv] 26 Jun 2014 Abstract Estimating the body
More informationMath for Game Programmers: Dual Numbers. Gino van den Bergen gino@dtecta.com
Math for Game Programmers: Dual Numbers Gino van den Bergen gino@dtecta.com Introduction Dual numbers extend real numbers, similar to complex numbers. Complex numbers adjoin an element i, for which i 2
More informationGeometric algebra rotors for skinned character animation blending
Geometric algebra rotors for skinned character animation blending briefs_0080* QLB: 320 FPS GA: 381 FPS QLB: 301 FPS GA: 341 FPS DQB: 290 FPS GA: 325 FPS Figure 1: Comparison between animation blending
More informationgeometric transforms
geometric transforms 1 linear algebra review 2 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3 matrix operations addition
More informationHuman Skeletal and Muscle Deformation Animation Using Motion Capture Data
Human Skeletal and Muscle Deformation Animation Using Motion Capture Data Ali Orkan Bayer Department of Computer Engineering, Middle East Technical University 06531 Ankara, Turkey orkan@ceng.metu.edu.tr
More informationPartBased Recognition
PartBased Recognition Benedict Brown CS597D, Fall 2003 Princeton University CS 597D, PartBased Recognition p. 1/32 Introduction Many objects are made up of parts It s presumably easier to identify simple
More informationComputer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D Welcome everybody. We continue the discussion on 2D
More informationA Comparison of Linear Skinning Techniques for Character Animation
A Comparison of Linear Skinning Techniques for Character Animation David Jacka ahoy.dave@gmail.com Bruce Merry bmerry@gmail.com ABSTRACT Character animation is the task of moving a complex, artificial
More informationRepresenting and Manipulating Meshbased Character Animations
Representing and Manipulating Meshbased Character Animations Edilson de Aguiar DECOM / CEUNES / UFES São Mateus, ES, Brazil Email: edilson.de.aguiar@gmail.com Norimichi Ukita Nara Institute of Science
More information3D POINT CLOUD CONSTRUCTION FROM STEREO IMAGES
3D POINT CLOUD CONSTRUCTION FROM STEREO IMAGES Brian Peasley * I propose an algorithm to construct a 3D point cloud from a sequence of stereo image pairs that show a full 360 degree view of an object.
More informationMultilinear Pose and Body Shape Estimation of Dressed Subjects from Image Sets
Multilinear Pose and Body Shape Estimation of Dressed Subjects from Image Sets Nils Hasler, Hanno Ackermann, Bodo Rosenhahn, Thorsten Thormählen, HansPeter Seidel MaxPlanckInstitut Informatik (MPII),
More informationAffine Transformations. University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell
Affine Transformations University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell Logistics Required reading: Watt, Section 1.1. Further reading: Foley, et al, Chapter 5.15.5. David
More informationClassifying Manipulation Primitives from Visual Data
Classifying Manipulation Primitives from Visual Data Sandy Huang and Dylan HadfieldMenell Abstract One approach to learning from demonstrations in robotics is to make use of a classifier to predict if
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationThis week. CENG 732 Computer Animation. Challenges in Human Modeling. Basic Arm Model
CENG 732 Computer Animation Spring 20062007 Week 8 Modeling and Animating Articulated Figures: Modeling the Arm, Walking, Facial Animation This week Modeling the arm Different joint structures Walking
More informationFACIAL RIGGING FOR 3D CHARACTER
FACIAL RIGGING FOR 3D CHARACTER Matahari Bhakti Nendya 1, Eko Mulyanto Yuniarno 2 and Samuel Gandang Gunanto 3 1,2 Department of Electrical Engineering, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia
More informationComputer Graphics: Visualisation Lecture 3. Taku Komura Institute for Perception, Action & Behaviour
Computer Graphics: Visualisation Lecture 3 Taku Komura tkomura@inf.ed.ac.uk Institute for Perception, Action & Behaviour Taku Komura Computer Graphics & VTK 1 Last lecture... Visualisation can be greatly
More informationGoldsmiths, University of London. Computer Animation. Goldsmiths, University of London
Computer Animation Goldsmiths, University of London Computer Animation Introduction Computer animation is about making things move, a key part of computer graphics. In film animations are created offline,
More informationMotion Capture Technologies. Jessica Hodgins
Motion Capture Technologies Jessica Hodgins Motion Capture Animation Video Games Robot Control What games use motion capture? NBA live PGA tour NHL hockey Legends of Wrestling 2 Lords of Everquest Lord
More informationThe Trimmed Iterative Closest Point Algorithm
Image and Pattern Analysis (IPAN) Group Computer and Automation Research Institute, HAS Budapest, Hungary The Trimmed Iterative Closest Point Algorithm Dmitry Chetverikov and Dmitry Stepanov http://visual.ipan.sztaki.hu
More informationCS 4204 Computer Graphics
CS 4204 Computer Graphics Computer Animation Adapted from notes by Yong Cao Virginia Tech 1 Outline Principles of Animation Keyframe Animation Additional challenges in animation 2 Classic animation Luxo
More information3D Face Modeling. Vuong Le. IFP group, Beckman Institute University of Illinois ECE417 Spring 2013
3D Face Modeling Vuong Le IFP group, Beckman Institute University of Illinois ECE417 Spring 2013 Contents Motivation 3D facial geometry modeling 3D facial geometry acquisition 3D facial deformation modeling
More informationBinary Image Analysis
Binary Image Analysis Segmentation produces homogenous regions each region has uniform graylevel each region is a binary image (0: background, 1: object or the reverse) more intensity values for overlapping
More informationModelling 3D Avatar for Virtual Try on
Modelling 3D Avatar for Virtual Try on NADIA MAGNENAT THALMANN DIRECTOR MIRALAB UNIVERSITY OF GENEVA DIRECTOR INSTITUTE FOR MEDIA INNOVATION, NTU, SINGAPORE WWW.MIRALAB.CH/ Creating Digital Humans Vertex
More informationTutorial 9: Skeletal Animation
Tutorial 9: Skeletal Animation Summary In the last couple of tutorials, you ve seen how to create a scene graph, and implemented a simple animating cube robot using them. You re probably wondering how
More informationComputer Animation and Visualisation. Lecture 1. Introduction
Computer Animation and Visualisation Lecture 1 Introduction 1 Today s topics Overview of the lecture Introduction to Computer Animation Introduction to Visualisation 2 Introduction (PhD in Tokyo, 2000,
More informationPOSESPACE IMAGEBASED RENDERING
POSESPACE IMAGEBASED RENDERING Anna Hilsmann 1,2 Philipp Fechteler 2 Peter Eisert 1,2 1 Humboldt University of Berlin 2 Fraunhofer HHI The work presented in this paper is funded by the German Science
More informationDetailed Human Shape and Pose from Images
Detailed Human Shape and Pose from Images 1 Alexandru O. Bălan 1 Leonid Sigal 1 Michael J. Black 2 James E. Davis 3 Horst W. Haussecker 1 Department of Computer Science, Brown University, Providence, RI
More informationProject 2: Character Animation Due Date: Friday, March 10th, 11:59 PM
1 Introduction Project 2: Character Animation Due Date: Friday, March 10th, 11:59 PM The technique of motion capture, or using the recorded movements of a live actor to drive a virtual character, has recently
More informationGiven a point cloud, polygon, or sampled parametric curve, we can use transformations for several purposes:
3 3.1 2D Given a point cloud, polygon, or sampled parametric curve, we can use transformations for several purposes: 1. Change coordinate frames (world, window, viewport, device, etc). 2. Compose objects
More informationGeometry and Topology from Point Cloud Data
Geometry and Topology from Point Cloud Data Tamal K. Dey Department of Computer Science and Engineering The Ohio State University Dey (2011) Geometry and Topology from Point Cloud Data WALCOM 11 1 / 51
More informationBlender 3D Animation
Bachelor Maths/Physics/Computer Science University ParisSud Digital Imaging Course Blender 3D Animation Christian Jacquemin Introduction to Computer Animation Animation Basics animation consists in changing
More information3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala)
3D Tranformations CS 4620 Lecture 6 1 Translation 2 Translation 2 Translation 2 Translation 2 Scaling 3 Scaling 3 Scaling 3 Scaling 3 Rotation about z axis 4 Rotation about z axis 4 Rotation about x axis
More informationAutodesk Inventor Tutorial 3
Autodesk Inventor Tutorial 3 Assembly Modeling Ron K C Cheng Assembly Modeling Concepts With the exception of very simple objects, such as a ruler, most objects have more than one part put together to
More informationConstrained Tetrahedral Mesh Generation of Human Organs on Segmented Volume *
Constrained Tetrahedral Mesh Generation of Human Organs on Segmented Volume * Xiaosong Yang 1, Pheng Ann Heng 2, Zesheng Tang 3 1 Department of Computer Science and Technology, Tsinghua University, Beijing
More informationImproved Billboard Clouds for Extreme Model Simplification
Improved Billboard Clouds for Extreme Model Simplification I.T. Huang, K. L. Novins and B. C. Wünsche Graphics Group, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland,
More informationPerformance Driven Facial Animation Course Notes Example: Motion Retargeting
Performance Driven Facial Animation Course Notes Example: Motion Retargeting J.P. Lewis Stanford University Frédéric Pighin Industrial Light + Magic Introduction When done correctly, a digitally recorded
More informationTutorial: Biped Character in 3D Studio Max 7, Easy Animation
Tutorial: Biped Character in 3D Studio Max 7, Easy Animation Written by: Ricardo Tangali 1. Introduction:... 3 2. Basic control in 3D Studio Max... 3 2.1. Navigating a scene:... 3 2.2. Hide and Unhide
More informationSampling via Moment Sharing: A New Framework for Distributed Bayesian Inference for Big Data
Sampling via Moment Sharing: A New Framework for Distributed Bayesian Inference for Big Data (Oxford) in collaboration with: Minjie Xu, Jun Zhu, Bo Zhang (Tsinghua) Balaji Lakshminarayanan (Gatsby) Bayesian
More informationComputer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science  Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: WireFrame Representation Object is represented as as a set of points
More informationSimFonIA Animation Tools V1.0. SCA Extension SimFonIA Character Animator
SimFonIA Animation Tools V1.0 SCA Extension SimFonIA Character Animator Bring life to your lectures Move forward with industrial design Combine illustrations with your presentations Convey your ideas to
More informationOffline Model Simplification for Interactive Rigid Body Dynamics Simulations Satyandra K. Gupta University of Maryland, College Park
NSF GRANT # 0727380 NSF PROGRAM NAME: Engineering Design Offline Model Simplification for Interactive Rigid Body Dynamics Simulations Satyandra K. Gupta University of Maryland, College Park Atul Thakur
More informationA. OPENING POINT CLOUDS. (Notepad++ Text editor) (Cloud Compare Point cloud and mesh editor) (MeshLab Point cloud and mesh editor)
MeshLAB tutorial 1 A. OPENING POINT CLOUDS (Notepad++ Text editor) (Cloud Compare Point cloud and mesh editor) (MeshLab Point cloud and mesh editor) 2 OPENING POINT CLOUDS IN NOTEPAD ++ Let us understand
More informationFinite Elements for 2 D Problems
Finite Elements for 2 D Problems General Formula for the Stiffness Matrix Displacements (u, v) in a plane element are interpolated from nodal displacements (ui, vi) using shape functions Ni as follows,
More informationComputer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)
Computer Graphics CS 54 Lecture 1 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include
More informationThis week. CENG 732 Computer Animation. The Display Pipeline. Ray Casting Display Pipeline. Animation. Applying Transformations to Points
This week CENG 732 Computer Animation Spring 20062007 Week 2 Technical Preliminaries and Introduction to Keframing Recap from CEng 477 The Displa Pipeline Basic Transformations / Composite Transformations
More informationThe Essentials of CAGD
The Essentials of CAGD Chapter 2: Lines and Planes Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/essentialscagd c 2000 Farin & Hansford
More informationDifferential Geometry
)NPUT $ATA 2ANGE 3CAN #!$ 4OMOGRAPHY 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS!NALYSIS OF SURFACE QUALITY 3URFACE SMOOTHING FOR NOISE REMOVAL Differential Geometry 0ARAMETERIZATION 3IMPLIFICATION FOR
More informationWe can display an object on a monitor screen in three different computermodel forms: Wireframe model Surface Model Solid model
CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant
More informationAnimation. Persistence of vision: Visual closure:
Animation Persistence of vision: The visual system smoothes in time. This means that images presented to the eye are perceived by the visual system for a short time after they are presented. In turn, this
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationEstimating Body Shape of Dressed Humans
Estimating Body Shape of Dressed Humans Nils Hasler a, Carsten Stoll a, Bodo Rosenhahn b, Thorsten Thormählen a, HansPeter Seidel a a MPI Informatik b Hannover University Abstract The paper presents a
More informationFeature Point Selection using Structural Graph Matching for MLS based Image Registration
Feature Point Selection using Structural Graph Matching for MLS based Image Registration Hema P Menon Department of CSE Amrita Vishwa Vidyapeetham Coimbatore Tamil Nadu  641 112, India K A Narayanankutty
More informationLecture 7. Matthew T. Mason. Mechanics of Manipulation. Lecture 7. Representing Rotation. Kinematic representation: goals, overview
Matthew T. Mason Mechanics of Manipulation Today s outline Readings, etc. We are starting chapter 3 of the text Lots of stuff online on representing rotations Murray, Li, and Sastry for matrix exponential
More informationWorking with Wireframe and Surface Design
Chapter 9 Working with Wireframe and Surface Design Learning Objectives After completing this chapter you will be able to: Create wireframe geometry. Create extruded surfaces. Create revolved surfaces.
More informationExpression Invariant 3D Face Recognition with a Morphable Model
Expression Invariant 3D Face Recognition with a Morphable Model Brian Amberg brian.amberg@unibas.ch Reinhard Knothe reinhard.knothe@unibas.ch Thomas Vetter thomas.vetter@unibas.ch Abstract We present an
More informationSummary: Transformations. Lecture 14 Parameter Estimation Readings T&V Sec 5.15.3. Parameter Estimation: Fitting Geometric Models
Summary: Transformations Lecture 14 Parameter Estimation eadings T&V Sec 5.15.3 Euclidean similarity affine projective Parameter Estimation We will talk about estimating parameters of 1) Geometric models
More informationRealTime Dynamic Wrinkles of Face for Animated Skinned Mesh
RealTime Dynamic Wrinkles of Face for Animated Skinned Mesh L. Dutreve and A. Meyer and S. Bouakaz Université de Lyon, CNRS Université Lyon 1, LIRIS, UMR5205, F69622, France PlayAll Abstract. This paper
More informationLecture 3: Coordinate Systems and Transformations
Lecture 3: Coordinate Systems and Transformations Topics: 1. Coordinate systems and frames 2. Change of frames 3. Affine transformations 4. Rotation, translation, scaling, and shear 5. Rotation about an
More informationToday. Keyframing. Procedural Animation. PhysicallyBased Animation. Articulated Models. Computer Animation & Particle Systems
Today Computer Animation & Particle Systems Some slides courtesy of Jovan Popovic & Ronen Barzel How do we specify or generate motion? Keyframing Procedural Animation PhysicallyBased Animation Forward
More informationIntroduction to Computer Graphics
Introduction to Computer Graphics Torsten Möller TASC 8021 7787822215 torsten@sfu.ca www.cs.sfu.ca/~torsten Today What is computer graphics? Contents of this course Syllabus Overview of course topics
More informationIntroduction to the Finite Element Method (FEM)
Introduction to the Finite Element Method (FEM) ecture First and Second Order One Dimensional Shape Functions Dr. J. Dean Discretisation Consider the temperature distribution along the onedimensional
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationA Learning Based Method for SuperResolution of Low Resolution Images
A Learning Based Method for SuperResolution of Low Resolution Images Emre Ugur June 1, 2004 emre.ugur@ceng.metu.edu.tr Abstract The main objective of this project is the study of a learning based method
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationCHAPTER 4 4. METHODS FOR MEASURING DISTANCE IN IMAGES
50 CHAPTER 4 4. METHODS FOR MEASURING DISTANCE IN IMAGES 4.1. INTRODUCTION In image analysis, the distance transform measures the distance of each object point from the nearest boundary and is an important
More informationChapter 1. Introduction. 1.1 The Challenge of Computer Generated Postures
Chapter 1 Introduction 1.1 The Challenge of Computer Generated Postures With advances in hardware technology, more powerful computers become available for the majority of users. A few years ago, computer
More informationMocap class 8  Creating an interactive 3D garden JMG Spring 2011
Mocap class 8  Creating an interactive 3D garden JMG Spring 2011 INTRODUCTION: The interactive 3D garden project is a first step for creating a dynamic ecosystem, a natural environment where several animations
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationKIN Biomechanics
KIN 335  Biomechanics LAB: Center of Mass (Center of Gravity) of the Human Body Reading Assignment: Bishop, R.D. & Hay, J.G. (1979). Basketball: the mechanics of hanging in the air. Medicine and Science
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationVolume visualization I Elvins
Volume visualization I Elvins 1 surface fitting algorithms marching cubes dividing cubes direct volume rendering algorithms ray casting, integration methods voxel projection, projected tetrahedra, splatting
More informationAdvanced visualization with VisNow platform Case study #2 3D scalar data visualization
Advanced visualization with VisNow platform Case study #2 3D scalar data visualization This work is licensed under a Creative Commons Attribution NonCommercialNoDerivatives 4.0 International License.
More informationGeometric Image Transformations
Geometric Image Transformations Part One 2D Transformations Spatial Coordinates (x,y) are mapped to new coords (u,v) pixels of source image > pixels of destination image Types of 2D Transformations Affine
More informationINTRODUCTION TO NEURAL NETWORKS
INTRODUCTION TO NEURAL NETWORKS Pictures are taken from http://www.cs.cmu.edu/~tom/mlbookchapterslides.html http://research.microsoft.com/~cmbishop/prml/index.htm By Nobel Khandaker Neural Networks An
More informationVisionbased Control of 3D Facial Animation
Eurographics/SIGGRAPH Symposium on Computer Animation (2003) D. Breen, M. Lin (Editors) Visionbased Control of 3D Facial Animation Jinxiang Chai, 1 Jing Xiao 1 and Jessica Hodgins 1 1 The Robotics Institute,
More informationART Extension for Description, Indexing and Retrieval of 3D Objects
ART Extension for Description, Indexing and Retrieval of 3D Objects Julien Ricard, David Coeurjolly, Atilla Baskurt LIRIS, FRE 2672 CNRS, Bat. Nautibus, 43 bd du novembre 98, 69622 Villeurbanne cedex,
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1  LOADING SYSTEMS
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1  LOADING SYSTEMS TUTORIAL 1 NONCONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects
More informationLimits and Possibilities of Markerless Human Motion Estimation
Limits and Possibilities of Markerless Human Motion Estimation Bodo Rosenhahn Universität Hannover Motion Capture Wikipedia (MoCap): Approaches for recording and analyzing Human motions Markerless Motion
More informationRecovering Primitives in 3D CAD meshes
Recovering Primitives in 3D CAD meshes Roseline Bénière a,c, Gérard Subsol a, Gilles Gesquière b, François Le Breton c and William Puech a a LIRMM, Univ. Montpellier 2, CNRS, 161 rue Ada, 34392, France;
More informationAnimation (4, 2, 0 ) + (( 2, 6, 4 )  (4, 2, 0 ))*.75 = (4, 2, 0 ) + ( 6, 8, 4)*.75 = (.5, 4, 3 ).
Animation A Series of Still Images We Call Animation Animation needs no explanation. We see it in movies and games. We grew up with it in cartoons. Some of the most popular, longestrunning television
More informationLevel Set Framework, Signed Distance Function, and Various Tools
Level Set Framework Geometry and Calculus Tools Level Set Framework,, and Various Tools Spencer Department of Mathematics Brigham Young University Image Processing Seminar (Week 3), 2010 Level Set Framework
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationCS445 Exam 2 Solutions
November 20, 2014 Name CS445 Exam 2 Solutions Fall 2014 1. (max = 15) 5. (max = 21) 2. (max = 8) 6. (max = 16) 3. (max = 10) 7. (max = 16) 4. (max = 14) Final Score: (max=100) Please try to write legibly.
More informationWednesday, March 30, 2011 GDC 2011. Jeremy Ernst. Fast and Efficient Facial Rigging(in Gears of War 3)
GDC 2011. Jeremy Ernst. Fast and Efficient Facial Rigging(in Gears of War 3) Fast and Efficient Facial Rigging in Gears of War 3 Character Rigger: creates animation control rigs creates animation pipeline
More informationRotations and Quaternions
Uniersity of British Columbia CPSC 414 Computer Graphics Rotations and Quaternions Week 9, Wed 29 Oct 2003 Tamara Munzner 1 Uniersity of British Columbia CPSC 414 Computer Graphics Clipping recap Tamara
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More information