4. Interpolation-Based Animation

Transcription

1 4. Interpolation-Based Animation Methods for precisely specifying the motion of objects using basics introduced in previous chapter : Key-frame animation systems Animation languages Object deformation 1

2 Key-Frame Systems (good reference: Maya) 2 Relatively simple in computer animation if the shapes are defined by polygons and the correspondence between the vertices is known. Key frame 1 Key frame 2

3 Key-Frame Systems Order of correspondence is important 3

4 Key-Frame Systems Order of correspondence is important Starting frame ending frame 4

5 Key-Frame Systems (i) Interpolating control points of curves in the key frames 5

6 Key-Frame Systems Q(u) P(u) S(u, 1/3) S(u, 2/3) 6 e g., b( v) b( v). 2 3 v 3v (1 v) v

7 S(u, 0) S(u, v) S(u,1) 7

8 4.2 Animation Languages What are animation languages? - Structured commands used to encode the information necessary to produce animations Why do we need animation languages? - To avoid overhead in producing motion sequences 8

9 4.2 Animation Languages Script-based or graphical Script-based: composed of text instructions, earlier approaches such as ANIMA II, AL, 9

10 4.2 Animation Languages 10

11 4.2 Animation Languages graphical : encoding relationships between objects and procedures using acyclic graphs An animation is represented by a dataflow network 11 such as Houdini, Maya and most of recent animation languages

12 Houdini dataflow network The object it generates 12

13 Cartoon face prototype Operator node collection of nodes connected together that describe the steps needed to accomplish a task Container node Hierarchy of nodes is created using container nodes 13

14 Advantages of Animation Languages Bigger reusebility, mobility, and alterability: The script can be used at any time to regenerate it, can be copied and transmitted easily, allows the animation to be iteratively refined (because the script can be incremently changed and a new animation generated) 14 The availability of programming constructs allows an algorithmic approach to motion control

15 4.3 Deforming Objects You don t have much choice, but to follow the physics Physically based deformation simulating the flexing of objects undergoing forces Bending Dripping 15 Cloth modeling: mass-spring system

16 4.3 Deforming Objects Physically based deformation 1. Subdividing surface S(u,v) 2. Scaling S(u,v) with Ts 3. Relocating components that do not change shape S C i (t) 4. Set up feature preserving objective function 5. Set up constraints 6. Performe constrained shape deformation 7. Rendering 16

17 4.3 Deforming Objects Physically based deformation The deformation process requires the construction of a feature-preserving objective function. This function is used to determine the shape of the deformed object in an optimization process. Hence, the objective function should be defined as the difference of these two surfaces. 17 V( u, v) ( S T S)( u, v) s New surface External force

18 4.3 Deforming Objects Physically based deformation V( u, v) ( S T S)( u, v) s The goal is to minimize the energy of the displacement function E( V) E bending E gravity E spring E moment E edgeforce 18 Set to zero

19 4.3 Deforming Objects Physically based deformation V( u, v) ( S T S)( u, v) s E 2 bending ( Vuu Vvv ) (1 )( VuuVvv Vuv ) dudv 2 D E E 1 2 (2G )( Vu V 2 v 2 D stretching spring 1 2 D K( )[ V (, )] Substitute and minimize i i i i 2 ) dudv 2( V V u v ) dudv

20 Imaging your image is made of rubber and then warp your rubber 4.3 Deforming Objects 20 User defined distortion simulator deforms the object directly and defines key shapes usually needs to use non-affine transformations - warping - coordinate grid deformation an affine transformation is a linear mapping from an affine space to an affine space

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25 Or, Warping an Object: picking and pulling D: displacement vector for seed vertex n: range of propagation 25

26 Warping an Object: picking and pulling For a vertex i units away from the seed vertex, the displacement is 26 k = 1: linear attenuation k > 1: create a more elastic impression

27 Deforming an embedding space (free-form deformation (FFD)) include the object to be distorted in a local grid and then distort the local grid easier to manipulate the local grid than to manipulate the vertices of the object directly more powerful than affine transformations because the distortion of the local grid can be nonlinear f : X Y 27 x Mx b

28 2D grid deformation A B P C D A B P C D 28

29 2D grid deformation 29 A = (5.0, 12.0) B = (6.0, 12.0) C = (5.0, 11.0) D = (6.0, 11.0) P = (5.7, 11.6)

30 2D grid deformation E F E = 0.3 A B F = 0.3 C D 30 P = 0.4 F E

31 Polyline deformation (2D) draw a polyline thru the object to be distorted map object vertices to the polyline modify the polyline map object vertices to the same relative location of the polyline suitable for serpentine objects 31 of the shape of a snake

32 Polyline deformation (2D) draw a polyline thru the object to be distorted map object vertices to the polyline 32

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34 Free-Form Deformation (FFD): Sederberg 34

35 Free-Form Deformation (FFD): 35 3D extension of 2D grid deformation superimpose a localized coordinate grid over the object register vertices of the object to the grid (local coordinate system) manipulate the grid map object vertices back into the modified grid, then relocate them in global space

36 actually S, T and U do not have to be unit vectors either. If the local coordinate system is defined by ( S, T, U) Relationship between local and global coordinate systems: 36 s? t? u?

37 37 ) ) /(( (*) ) ) ) /(( ( ) ( ) ) /(( U T S P P T S u T S U P P S U t S U T P P U T s Why?

38 Here is why: Since T U T and T U U we have 38 P P ( T U ) ss tt uu ( T U ) ss ( T ) 0 U P P ( T U ) /( S ( T )) Hence s 0 U

39 Creation of the localized coordinate grid 39 P ijk P 0 i l S l : # of control pts in m : # of control pts in n : # of control pts in S T U j m T k n U direction direction direction

40 Animator adjusts the locations of the control points to deform the object P ijk Deformed position of a vertex of the object is determined through a trivariate Bezier interpolation process: 40

41 new location 41

42 FFDs can be composed - sequentially vs hierarchically Detail elements can be added in stages can be sub-dimensional or muli-dimensional 42 usually in the same direction Allows the user to work at various levels of detail

43 Free form deformation through control of parametric surfaces 43

44 Free form deformation through control of parametric surfaces 44

45 End of Interpolation IV 45