TIEA311 Tietokonegrafiikan perusteet kevät 2018

Transcription

1 TIEA311 Tietokonegrafiikan perusteet kevät 2018 ( Principles of Computer Graphics Spring 2018) Copyright and Fair Use Notice: The lecture videos of this course are made available for registered students only. Please, do not redistribute them for other purposes. Use of auxiliary copyrighted material (academic papers, industrial standards, web pages, videos, and other materials) as a part of this lecture is intended to happen under academic fair use to illustrate key points of the subject matter. The lecturer may be contacted for take-down requests or other copyright concerns ( paavo.j.nieminen@jyu.fi).

2 TIEA311 Tietokonegrafiikan perusteet kevät 2018 ( Principles of Computer Graphics Spring 2018) Adapted from: Wojciech Matusik, and Frédo Durand: Computer Graphics. Fall Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA Original license terms apply. Re-arrangement and new content copyright by Paavo Nieminen and Jarno Kansanaho Frontpage of the local course version, held during Spring 2018 at the Faculty of Information technology, University of Jyväskylä: nieminen/tgp18/

3 Simple Example with s { numobjects 3 { numobjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } { numobjects 2 { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } } Text format is fictitious, better to use XML in real applications Durand 32

4 Simple Example with s { numobjects 3 { numobjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } { numobjects 2 { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } } Here we have only simple shapes, but easy to add a Mesh node whose parameters specify an.obj to load (say) Durand 34

5 Adding Attributes (Material, etc.) 35 { numobjects 3 Material { <BLUE> } { numobjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } { numobjects 2 Material { <BROWN> } { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } { Material { <GREEN> } Box { <BOX PARAMS> } Material { <RED> } Sphere { <SPHERE PARAMS> } Material { <ORANGE> } Sphere { <SPHERE PARAMS> } } } Material { <BLACK> } Plane { <PLANE PARAMS> } }

6 Adding Transformations 36

7 Questions? 37

8 Scene Graph Traversal 38 Depth first recursion Visit node, then visit subtrees (top to bottom, left to right) When visiting a geometry node: Draw it! How to handle transformations? Remember, transformations are always specified in coordinate system of the parent

9 Scene Graph Traversal 39 How to handle transformations? Traversal algorithm keeps a transformation state S (a 4x4 matrix) from world coordinates Initialized to identity in the beginning Geometry nodes always drawn using current S When visiting a transformation node T: multiply current state S with T, then visit child nodes Has the effect that nodes below will have new transformation When all children have been visited, undo the effect of T!

10 Traversal Example 43 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit)

11 Traversal Example 44 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = I

12 Traversal Example 45 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1

13 Traversal Example 46 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1

14 Traversal Example 47 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1 T2

15 Traversal Example 48 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1 T2

16 Traversal Example 49 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1 T2

17 Traversal Example 50 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1

18 Traversal Example 51 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1 R1

19 Traversal Example 52 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1 R1

20 Traversal Example 53 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1 R1

21 Traversal Example 54 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1

22 Traversal Example 55 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1

23 Traversal Example 56 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = I

24 Traversal Example 57 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = R2

25 Traversal Example 58 Root Translate T1 Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = R2

26 Traversal Example Root Translate T1 Rotate R2 (table, fruits) Translate T2 Rotate R1 (chair, legs)... (tabletop, legs) (basket, fruit) S = R2 59

27 Traversal Example Translate T1 Root At each node, the current object-to-world transformation is the matrix product of all transformations found on the way from the node to the root. Rotate R2 (table, fruits) (chair, legs) Translate T2 Rotate R1 (tabletop, legs) (basket, fruit) S = T1R1 60

28 Traversal State 61 The state is updated during traversal Transformations But also other properties (color, etc.) Apply when entering node, undo when leaving How to implement? Bad idea to undo transformation by inverse matrix (Why?)

29 Traversal State 62 The state is updated during traversal Transformations But also other properties (color, etc.) Apply when entering node, undo when leaving How to implement? Bad idea to undo transformation by inverse matrix Why I? T*T-1 = I does not necessarily hold in floating point even when T is an invertible matrix you accumulate error Why II? T might be singular, e.g., could flatten a 3D object onto a plane no way to undo, inverse doesn t exist!

30 Traversal State 63 The state is updated during traversal Transformations But also other properties (color, etc.) Apply when entering node, undo when leaving How to implement? Bad idea to undo transformation by inverse matrix Why I? T*T-1 = I does not necessarily hold in floating point even when T is an invertible matrix you accumulate error Why II? T might be singular, e.g., could flatten a 3D object onto a plane no way to undo, inverse doesn t exist! Can you think of a data structure suited for this?

31 Traversal State Stack The state is updated during traversal Transformations But also other properties (color, etc.) Apply when entering node, undo when leaving How to implement? Bad idea to undo transformation by inverse matrix Why I? T*T-1 = I does not necessarily hold in floating point even when T is an invertible matrix you accumulate error Why II? T might be singular, e.g., could flatten a 3D object onto a plane no way to undo, inverse doesn t exist! Solution: Keep state variables in a stack Push current state when entering node, update current state Pop stack when leaving state-changing node See what the stack looks like in the previous example! 64

32 Questions? 65

33 Plan 66 Hierarchical Modeling, Scene Graph OpenGL matrix stack Hierarchical modeling and animation of characters Forward and inverse kinematics

34 Hierarchical Modeling in OpenGL 67 The OpenGL Matrix Stack implements what we just did! Commands to change current transformation gltranslate, glscale, etc. Current transformation is part of the OpenGL state, i.e., all following draw calls will undergo the new transformation Remember, a transform affects the whole subtree Functions to maintain a matrix stack glpushmatrix, glpopmatrix Separate stacks for modelview (object-to-view) and projection matrices

35 When You Encounter a Transform Node 68 Push the current transform using glpushmatrix() Multiply current transform by node s transformation Use glmultmatrix(), gltranslate(), glrotate(), glscale(), etc. Traverse the subtree Issue draw calls for geometry nodes Use glpopmatrix() when done. Simple as that!

36 More Specifically An OpenGL transformation call corresponds to a matrix T The call multiplies current modelview matrix C by T from the right, i.e. C = C * T. This also works for projection, but you often set it up only once. This means that the transformation for the subsequent vertices will be p = C * T * p Vertices are column vectors on the right in OpenGL This implements hierarchical transformation directly!

37 More Specifically An OpenGL transformation call corresponds to a matrix T The call multiplies current modelview matrix C by T from the right, i.e. C = C * T. This also works for projection, but you often set it up only once. This means that the transformation for the subsequent vertices will be p = C * T * p Vertices are column vectors on the right in OpenGL This implements hierarchical transformation directly! At the beginning of the frame, initialize the current matrix by the viewing transform that maps from world space to view space. For instance, glloadidentity() followed by glulookat()

38 TIEA311 - Code! Let us revisit the Assignment 0 example that draws a teapot. Can we use the theory just presented to make a controlled scene of a couple of more teapots in their own object coordinates wrt. the world and the camera? Does real OpenGL code look the way it was just promised? [live, on-screen, if Visual Studio is working also in today s lecture room]

39 Questions? 71 Further reading on OpenGL Matrix Stack and hierarchical model/view transforms It can be a little confusing if you don t think the previous through, but it s really quite simple in the end. I know very capable people who after 15 years of experience still resort to brute force (trying all the combinations) for getting their transformations right, but it s such a waste :)

40 Plan 72 Hierarchical Modeling, Scene Graph OpenGL matrix stack Hierarchical modeling and animation of characters Forward and inverse kinematics

41 Animation 73 Hierarchical structure is essential for animation Eyes move with head Hands move with arms Feet move with legs Without such structure the model falls apart.

42 Articulated Models 74 Articulated models are rigid parts connected by joints each joint has some angular degrees of freedom Articulated models can be animated by specifying the joint angles as functions of time.

43 Joints and bones 75 Describes the positions of the body parts as a function of joint angles. Body parts are usually called bones Each joint is characterized by its degrees of freedom (dof) Usually rotation for articulated bodies 1 DOF: knee 2 DOF: wrist 3 DOF: arm

44 Skeleton Hierarchy 76 Each bone position/orientation described relative to the parent in the hierarchy: hips... left-leg r-thigh For the root, the parameters include a position as well Joints are specified by angles. z y vs x r-calf r-foot

45 Draw by Traversing a Tree 77 hips... left-leg r-thigh r-calf r-foot Assumes drawing procedures for thigh, calf, and foot use joint positions as the origin for a drawing coordinate frame glloadidentity(); glpushmatrix(); gltranslatef( ); glrotate( ); drawhips(); glpushmatrix(); gltranslate( ); glrotate( ); drawthigh(); gltranslate( ); glrotate( ); drawcalf(); gltranslate( ); glrotate( ); drawfoot(); glpopmatrix(); left-leg

46 Forward Kinematics 78 vs How to determine the world-space position for point vs?

47 Forward Kinematics 79 Transformation matrix S for a point vs is a matrix composition of all joint transformations between the point and the root of the hierarchy. S is a function of all the joint angles between here and root. vs

48 Forward Kinematics 80 Transformation matrix S for a point vs is a matrix composition of all joint transformations between the point and the root of the hierarchy. S is a function of all the joint angles between here and root. Note that the angles have a non-linear effect. vs This product is S

49 Forward Kinematics Transformation matrix S for a point vs is a matrix composition of all joint transformations between the point and the root of the hierarchy. S is a function of all the joint angles between here and root. Note that the angles have a non-linear effect. vs This product is S parameter vector p Durand 81

50 Questions? 82

51 TIEA311 - Fast-forward just a bit Worry not, if you get an yli hilseen feeling about the following few things. On this course, we leave them on a nice to know level. That is, while learning basics, it is nice to know where the rabbit hole could lead to! Note to self: apply great speed with the next slide button on lecture.

52 Inverse Kinematics 83 Context: an animator wants to pose a character Specifying every single angle is tedious and not intuitive Simpler interface: directly manipulate position of e.g. hands and feet That is, specify vw, infer joint transformations vs

53 Inverse Kinematics 84 Forward Kinematics Given the skeleton parameters p (position of the root and the joint angles) and the position of the point in local coordinates vs, what is the position of the point in the world coordinates vw? Not too hard, just apply transform accumulated from the root. vs

54 Inverse Kinematics Forward Kinematics Given the skeleton parameters p (position of the root and the joint angles) and the position of the point in local coordinates vs, what is the position of the point in the world coordinates vw? Not too hard, just apply transform accumulated from the root. Inverse Kinematics Given the current position of the point and the desired new position in world coordinates, what are the skeleton parameters p that take the point to the desired position? vs ṽw 85

55 Inverse Kinematics Given the position of the point in local coordinates vs and the desired position in world coordinates, what are the skeleton parameters p? ṽw skeleton parameter vector p Requires solving for p, given vs and Non-linear and 86

56 It s Underconstrained 87 Count degrees of freedom: We specify one 3D point (3 equations) We usually need more than 3 angles p usually has tens of dimensions vs Simple geometric example (in 3D): specify hand position, need elbow & shoulder The set of possible elbow location is a circle in 3D

57 How to tackle these problems? 88 Deal with non-linearity: Iterative solution (steepest descent) Compute Jacobian matrix of world position w.r.t. angles Jacobian: If the parameters p change by tiny amounts, what is the resulting change in the world position vws? Then invert Jacobian. This says if vws changes by a tiny amount, what is the change in the parameters p? But wait! The Jacobian is non-invertible (3xN) Deal with ill-posedness: Pseudo-inverse Solution that displaces things the least See Deal with ill-posedness: Prior on good pose (more advanced) Additional potential issues: bounds on joint angles, etc. Do not want elbows to bend past 90 degrees, etc.

58 Example: Style-Based IK 89 Video Prior on good pose Link to paper: Grochow, Martin, Hertzmann, Popovic: Style-Based Inverse Kinematics, ACM SIGGRAPH 2004

59 Mesh-Based Inverse Kinematics 90 Video Doesn t even need a hierarchy or skeleton: Figure proper transformations out based on a few example deformations! Link to paper: Sumner, Zwicker, Gotsman, Popovic: Mesh-Based Inverse Kinematics, ACM SIGGRAPH 2005

60 TIEA311 - Was that cool or was that cool? Hyperlinks were broken in PDF translation but these were easily found in YouTube based on the titles: (original style IK) (later research from the same group, project page: ) Important note: You must also learn to find (and, the difficult part: understand) full text research articles, available from within the university IP address space, because the institute is paying a lot of tax payers money to let YOU do that. Would be a shame not to start using this privilege today!

61 TIEA311 - Was that cool or was that cool? Interested in this stuff? Words of advice: Learn about mathematics, statistics, optimization, and numerics. (programming comes for free at IT Faculty) Just a few keywords from the video narrative: local optimum, global optimum, probability density function, [machine] learning Cool stuff happens at the bleeding edge of science... Your first step can not be a sudden appearance at the top of this mountain (or any mountain)! At the Faculty, we support every step you choose to make towards being a graphics algorithm pro (e.g., BSc, MSc theses). Today, I offer the first piece of support by repeating the main words of advice: Get serious about your math, and keep applying it to graphics, if that makes you tick!