Graphics Pipeline. Einführung in Visual Computing Werner Purgathofer. object coordinates. world coordinates. camera coordinates

Transcription

1 Einführung in Visual Computing object coordinates world coordinates Graphics Pipeline camera coordinates normalized device coordinates Werner Purgathofer pixel coordinates

2 Graphics Pipeline information is transformed toan image in successive steps object and scene creation definition of view (camera) projection rasterization this is called the graphics pipeline (l (also viewing i pipeline, transformation pipeline, rendering pipeline, ) Werner Purgathofer 2

3 Transformation Pipeline object coordinates world coordinates camera coordinates normalized device coordinates pixel coordinates model transformation view transformation projection + homogenization viewport transformation Werner Purgathofer 3

4 Rendering Pipeline (Technical Implementation) scene objects in object space transformed vertices in clip space scene in normalized device coordinates Werner raster Purgathofer image in pixel coordinates4 object capture/creation modeling viewing projection vertex stage ( vertex shader ) clipping + homogenization viewport transformation rasterization shading pixel stage ( fragment shader )

5 Rasterization in the Rendering Pipeline scene objects in object space transformed vertices in clip space scene in normalized device coordinates Werner raster Purgathofer image in pixel coordinates5 object capture/creation modeling viewing projection vertex stage ( vertex shader ) clipping + homogenization viewport transformation rasterization shading pixel stage ( fragment shader )

6 Reminder: Graphs and Trees 2 root arbitrary graph implemented with pointers Werner Purgathofer 6

7 Reminder: Graphs and Trees root^.left root root^.right binary tree implemented with pointers + recursive evaluation Werner Purgathofer 7

8 Einführung in Visual Computing D Object Representations

9 Object Representation in the Rendering Pipeline scene objects in object space transformed vertices in clip space scene in normalized device coordinates Werner raster Purgathofer image in pixel coordinates9 object capture/creation modeling viewing projection vertex stage ( vertex shader ) clipping + homogenization viewport transformation rasterization shading pixel stage ( fragment shader )

10 3D Object Representations graphics scenes contain solid geometric objects trees, flowers, clouds, rocks, water, creation of models surface interior models explicit procedural models heuristically i physically based models dl representations geometrical data structures data structure organization Werner Purgathofer 10

11 Polygon Surfaces (1) set of surface polygons enclose object interior i = Boundary Representation ( B Rep ) p) example: machine part surface represented by triangles Werner Purgathofer 11

12 Polygon Surfaces (2) polygon tables (B Rep lists) geometric and attribute t tables vertex, edge, polygon tables consistency, completenesschecks checks Werner Purgathofer 12

13 Polygon Surfaces: Data Structure V 1 E 1 E S 1 E E 6 3 S 2 E 2 V 3 V 5 V 2 2 E 4 E 5 Werner Purgathofer 13 V 4

14 Lists for B Reps V 1 E 1 E 3 E 6 surface list S S 1 1 S 2 S 2 V 2 E 2 V 3 E 4 E5 E V 5 V 4 V 5 edge list E 1 E 2 E 3 E 4 E 5 E 6 vertex list x y z x y z x y z x y z x y z V Werner Purgathofer 1 V 2 V14 3 V 4 V 5

15 Reminder: Product of Vectors V 1 = a 1 a 2 b 1 V 2 = b 2 scalar product: c 1 c 2 cross product (vector product): V 1 V 2 =? V 1 V 2 =? Werner Purgathofer 15

16 Reminder: Product of Vectors scalar product: a 1 a 2 V 1 V 2 = b 1 b 2 = a 1 a 2 + b 1 b 2 + c 1 c 2 ϕ c 1 c 2 V 1 V 2 = V 1 V 2 cosϕ cross product (vector product): a 1 a a 2 b 1 c 2 c 1 b 2 V 1 V 2 = b 1 b 2 = c 1 a 2 a 1c 2 c 1 c 2 a 1 b 2 b 1 a 2 V 1 V 2 = V 1 V 2 sinϕ Werner Purgathofer 16

17 Polygon Surfaces: Plane Equation plane parameters A,B,C,D normal (A,B,C) BC) y Ax + By + Cz +D = 0 N=(A,B,C) example: example: x 1= y x 1= N=(100) N=(1,0,0) x z z 1 1 x Werner Purgathofer 17

18 Front and Back Polygon Faces back face = polygon side that faces into the object interior front face = polygon side that faces outward behind a polygon plane = visible to the polygon back face in front of a polygon plane = visible iibl to thepolygon front tface [front = the side to which (A,B,C) points] Werner Purgathofer 18

19 Front and Back Polygon Faces Ax + By + Cz + D = 0 for points on the surface < 0 for points behind > 0 for points in front V if (1) right-handed coordinate system 4 (2) polygon points ordered V counterclockwise V 5 V 3 V 1 V 2 1 V 1, V 2, V 3 counterclockwise normal vector N=(V 2 V 1 ) x (V 3 V 1 ) N Werner Purgathofer 19

20 Triangle Meshes most polygons are triangles triangle mesh = connected triangles triangle strip = successive triangles (1 additional point per triangle) C.Schlick Werner Purgathofer 20

21 Constructive Solid Geometry ConstructiveSolid Geometry (CSG) boolean set operations on 3D objects union, intersection, difference operation combining 2 objects with a union operation, producing a single composite object Werner Purgathofer 21

22 CSG: Different Set Operations Werner Purgathofer 22

23 CSG Data Structure Every object is assembled from simple solids with set operations data structure: binary tree recursive evaluation Werner Purgathofer 23

24 Operations with CSG Trees transformations multiplication of all transformation matrices with the matrix of this transformation combinations generate a new node with the desired operator and link the operands as subtrees to it A op B: op A B Werner Purgathofer 24

25 Rendering of CSG Trees transform into B Rep and use normal hidden surface algorithm or render directly with ray casting (or with ray tracing) Werner Purgathofer 25

26 Ray Casting Method (1) line of sight of each pixel is intersected with all surfaces take closest intersected surface viewing direction closest intersection point Werner Purgathofer 26

27 Ray Casting Method (2) based on geometric optics, tracing paths of light rays backward tracing of light rays suitable forcomplex, curved surfaces special case of ray tracing algorithms efficient ray surface intersection techniques necessary intersection point normal vector Werner Purgathofer 27

28 Ray Casting Methods for CSG (1) visibility processing pixel plane y ray x -z Werner Purgathofer 28

29 Ray Casting Methods for CSG (2) determining surface limits Operation pixel plane obj 1 obj 2 A C B D z obj 1 obj 2 {A, B} {C, D} Operation Result Union {A, D} Intersection {C, B} Difference {A, C} Werner Purgathofer 29

30 Ray Casting Methods for CSG (3) volume determination pixel plane V ij A ij z ij y V ΣV ij ray A ij z ij x -z Werner Purgathofer 30

31 Properties of CSG advantages exact representation low memory cost combinations and transformations trivial disadvantages rendering effort is high Werner Purgathofer 31

32 Quadtrees hierarchical enumeration of objects in 2D: quadtree hierarchical subdivision until a region is homogeneous Quadrant Quadrant 0 1 Quadrant Quadrant 3 2 region of a 2 dim. space data elements in the representative quadtree node Werner Purgathofer 32

33 Quadtrees area with 2 n by 2 n pixels quadtree with n levels storage efficiency Werner Purgathofer 33

34 Quadtrees area with 2 n by 2 n pixels quadtree with n levels storage efficiency Werner Purgathofer 34

35 Quadtrees quadtree representation for a region containing one foregroundcolor pixel on a solid background Werner Purgathofer 35

36 Quadtree Example suitable for representing (2D) images Werner Purgathofer 36

37 Octree = extension to 3D regular space subdivision: simple (empty or uniform) leaf node complex (other cases) divide further Werner Purgathofer 37

38 Octrees octree divides 3D cube into octants volume elements (voxels) set operations easy on octrees geometric transformations difficult on octrees 7 5 region of a 3 dim. space data elements in the 4 representative octree node Werner Purgathofer 38

39 Octree Simple Example G (W W W W S G (W W W W W W W S) S S) Werner Purgathofer 39

40 Operations with Octrees transformations very complicated exceptfor a few specialcases cases, e.g. rotation by 90, mirroring at a subdivision plane, scalation by 2 n combinations very simple: if A or B homogeneous simple rules else combine recursively all 8 octants of A and B Werner Purgathofer 40

41 Properties of Octrees advantages combinations very simple fast rendering spatial search possible disadvantages inexact representation low image quality restricted transformations high memory cost Werner Purgathofer 41

42 Octree Examples Yoshifumi Kitamura E.Strasser GRAVIR/IMAG INRIA (c) Yoshifumi Kitamura Michael Goodfellow

43 Other 3D Object Representations BSP trees fractal geometrymethodsmethods shape grammars, procedural models particle systems physically based modeling visualization at of data sets... Werner Purgathofer 43

44 Scene Graphs object oriented oriented data structure directed acyclic graph describes logical and/or spatial relationship of scene objects describes groups of (groups of ) objects no exact definition used in most graphics systems, e.g. OpenSceneGraph VRML X3D Werner Purgathofer 44

45 Scene Graph Example sun world house car tree walls body engine wheels roof door windows transf. transf. transf. transf. wheel Werner Purgathofer RF

46 Scene Graph Example hypercosm Werner Purgathofer 46

47 Scene Graph Example Werner Purgathofer 47 jmonkeyengine.org

48 Scene Graph Example Garret Foster Werner Purgathofer 48

49 Scene Graph Example Werner Purgathofer 49 B.B.Christensen

50 Scene Graph Example Werner Purgathofer 50 Francois Faure

51 Scene Graph Example Werner Purgathofer 51 S.Teichmann