Scene Graphs & Modeling Transformations COS 426

Transcription

1 Scene Graphs & Modeling Transformations COS 426

2 3D Object Representations Points Range image Point cloud Surfaces Polgonal mesh Subdivision Parametric Implicit Solids Voels BSP tree CSG Sweep High-level structures Scene graph Application specific

3 3D Object Representations What object representation is best for this?

4 3D Object Representations Desirable properties of an object representation Eas to acquire Accurate Concise Intuitive editing Efficient editing Efficient displa Efficient intersections Guaranteed validit Guaranteed smoothness etc. (CS Building, Princeton Universit)

5 Overview Scene graphs Geometr & attributes Transformations Bounding volumes Transformations Basic 2D transformations Matri representation Matri composition 3D transformations

6 Scene Graphs Building Floor Floor 2 Floor 3 Floor 4 Floor5 Floor Furniture Office Office N Bookshelf Office Furniture Desk Desk 2 Chair Instances Chair K Bookshelf Desk Chair Definitions

7 Scene Graphs Hierarch of nodes, where each node ma have: Geometr representation Modeling transformation Parents and/or children Bounding volume Upper Arm [M 2 ] Base [M ] Lower Arm [M 3 ] Robot Arm Angel Figures 8.8 & 8.9

8 Scene Graphs Advantages Allows definitions of objects in own coordinate sstems Allows use of object definition multiple times in a scene z

9 Scene Graphs Advantages Allows definitions of objects in own coordinate sstems Allows use of object definition multiple times in a scene Scene Instance Instance 4 Level A Instance Instance 4 Instance Level B Instance 4 Cube Tetrahedron Octahedron Dodecahedron Icosahedron H&B Figure 9 Instances Definitions

10 Scene Graphs Advantages Allows definitions of objects in own coordinate sstems Allows use of object definition multiple times in a scene Allows hierarchical processing (e.g., intersections) Haverkort

11 Scene Graphs Advantages Allows definitions of objects in own coordinate sstems Allows use of object definition multiple times in a scene Allows hierarchical processing (e.g., intersections) Allows articulated animation Base [M ] Upper Arm [M 2 ] Lower Arm [M 3 ] Robot Arm Angel Figures 8.8 & 8.9

12 Scene Graph Eample Piar

13 Overview Scene graphs Geometr & attributes Transformations Bounding volumes Transformations Basic 2D transformations Matri representation Matri composition 3D transformations

14 2D Modeling Transformations Modeling Coordinates Scale Translate Scale Rotate Translate World Coordinates

15 2D Modeling Transformations Modeling Coordinates Let s look at this in detail World Coordinates

16 2D Modeling Transformations Modeling Coordinates

17 2D Modeling Transformations Modeling Coordinates Scale.3,.3 Rotate -9 Translate 5, 3

18 2D Modeling Transformations Modeling Coordinates Scale.3,.3 Rotate -9 Translate 5, 3

19 2D Modeling Transformations Modeling Coordinates Scale.3,.3 Rotate -9 Translate 5, 3 World Coordinates

20 Basic 2D Transformations Translation: + t + t Scale: * s * s Shear: + h* + h* Rotation: *cosθ - *sinθ *sinθ + *cosθ Transformations can be combined (with simple algebra)

21 Basic 2D Transformations Translation: + t + t Scale: * s * s Shear: + h* + h* Rotation: *cosθ - *sinθ *sinθ + *cosθ

22 Basic 2D Transformations Translation: + t + t Scale: * s * s (, ) (,) Shear: + h* + h* Rotation: *cosθ - *sinθ *sinθ + *cosθ *s *s

23 Basic 2D Transformations Translation: + t + t Scale: * s * s Shear: + h* + h* Rotation: *cosθ - *sinθ *sinθ + *cosθ (, ) (*s)*cosθ (*s)*sinθ (*s)*sinθ + (*s)*cosθ

24 Basic 2D Transformations Translation: + t + t Scale: * s * s (, ) Shear: + h* + h* Rotation: *cosθ - *sinθ *sinθ + *cosθ ((*s)*cosθ (*s)*sinθ) + t ((*s)*sinθ + (*s)*cosθ) + t

25 Basic 2D Transformations Translation: + t + t Scale: * s * s Shear: + h* + h* Rotation: *cosθ - *sinθ *sinθ + *cosθ ((*s)*cosθ (*s)*sinθ) + t ((*s)*sinθ + (*s)*cosθ) + t

26 Overview Scene graphs Geometr & attributes Transformations Bounding volumes Transformations Basic 2D transformations Matri representation Matri composition 3D transformations

27 Matri Representation Represent 2D transformation b a matri a c b d Multipl matri b column vector appl transformation to point a c b d a c + + b d

28 Matri Representation Transformations combined b multiplication a c be d g f i h k j l Matrices are a convenient and efficient wa to represent a sequence of transformations!

29 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Identit? a c b d 2D Scale around (,)? s* s * as c b ds

30 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Rotate around (,)? cosθ* sin Θ* sin Θ* + cosθ* cos a Θb sin Θ sin c Θd cos Θ 2D Shear? sh sh + * * + a sh c bsh d

31 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Mirror over Y ais? a c b d 2D Mirror over (,)? a c b d

32 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Translation? + t + t NO! a c b d Onl linear 2D transformations can be represented with a 22 matri

33 Linear Transformations Linear transformations are combinations of Scale, Rotation, Shear, and Mirror a c Properties of linear transformations: Satisfies: T s p + s p ) s T ( p ) + s T ( ( p2 Origin maps to origin Points at infinit sta at infinit Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition b d )

34 2D Translation 2D translation represented b a 33 matri Point represented with homogeneous coordinates t t + + t t

35 Homogeneous Coordinates Add a 3rd coordinate to ever 2D point (,, w) represents a point at location (/w, /w) (,, ) represents a point at infinit (,, ) is not allowed 2 (2,,) or (4,2,2) or (6,3,3) 2 Convenient coordinate sstem to represent man useful transformations

36 Basic 2D Transformations Basic 2D transformations as 33 matrices Θ Θ Θ Θ cos sin sin cos t t sh sh Translate Rotate Shear s s Scale

37 Affine Transformations Affine transformations are combinations of Linear transformations, and Translations a d w c f w Properties of affine transformations: Origin does not necessaril map to origin Points at infinit remain at infinit Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition b e

38 Projective Transformations Projective transformations (homographies): Affine transformations, and Projective warps a d w g c f i w Properties of projective transformations: Origin does not necessaril map to origin Point at infinit ma map to finite point Lines map to lines Parallel lines do not necessaril remain parallel Ratios are not preserved (but cross-ratios are) Closed under composition b e h

39 Projective Transformations Will be useful to model (pinhole) cameras: can represent camera projection in same framework as modeling transformations

40 Cross-Ratio Definition: for 4 collinear points A, B, C, D ( A, B; C, D) AC AD BD BC Projective Invariant: (A,B;C,D) (A,B ;C,D ) Wikipedia

41 Overview Scene graphs Geometr & attributes Transformations Bounding volumes Transformations Basic 2D transformations Matri representation Matri composition 3D transformations

42 Matri Composition Transformations can be combined b matri multiplication Θ Θ Θ Θ w s s t t w cos sin sin cos p T(t,t) R(Θ) S(s,s) p

43 Matri Composition Matrices are a convenient and efficient wa to represent a sequence of transformations General purpose representation Hardware matri multipl Efficienc with premultiplication» Matri multiplication is associative p (T * (R * (S*p) ) ) p (T*R*S) * p

44 Matri Composition Be aware: order of transformations matters» Matri multiplication is not commutative p T * R * S * p Global Local

45 Matri Composition Rotate b Θ around arbitrar point (a,b) MT(a,b) * R(Θ) * T(-a,-b) The trick: First, translate (a,b) to the origin. Net, do the rotation about origin. Finall, translate back. (a,b) Scale b s,s around arbitrar point (a,b) MT(a,b) * S(s,s) * T(-a,-b) (Use the same trick.) (a,b)

46 Overview Scene graphs Geometr & attributes Transformations Bounding volumes Transformations Basic 2D transformations Matri representation Matri composition 3D transformations

47 3D Transformations Same idea as 2D transformations Homogeneous coordinates: (,,z,w) 44 transformation matrices w z p o n m l k j i h g f e d c b a w z

48 Basic 3D Transformations w z w z w z tz t t w z w z sz s s w z w z w z Identit Scale Translation Mirror over X ais

49 Basic 3D Transformations Θ Θ Θ Θ w z w z cos sin sin cos Rotate around Z ais: Θ Θ Θ Θ w z w z cos sin sin cos Rotate around Y ais: Θ Θ Θ Θ w z w z cos sin sin cos Rotate around X ais:

50 Transformations in Scene Graphs Root Chest LHip RHip Neck LCollar LCollar LKnee RKnee Head LShld LShld LAnkle RAnkle LElbow LElbow LWrist LWrist Rose et al. `96

51 Transformations in Scene Graphs Mihai Parparita, COS 426

52 Summar Scene graphs Hierarchical Modeling transformations Bounding volumes Coordinate sstems World coordinates Modeling coordinates 3D modeling transformations Represent most transformations b 44 matrices Composite with matri multiplication (order matters)