Einführung in Visual Computing. Graphics Pipeline. Werner Purgathofer. Graphics Pipeline information is transformed to an image in successive steps

Transcription

1 Einführung in Visual Computing Graphics Pipeline Werner Purgathofer Graphics Pipeline information is transformed to an image in successive steps object and scene creation definition of view (camera) projection rasterization this is called the graphics pipeline (also viewing pipeline, transformation pipeline, rendering pipeline, ) Werner Purgathofer 1 1

2 Transformation Pipeline object coordinates world coordinates camera coordinates normalized device coordinates pixel coordinates model transformation view transformation projection + homogenization viewport transformation Werner Purgathofer 2 Rendering Pipeline (Techn. Implement.) scene objects in object space transformed vertices in clip space scene in normalized device coordinates raster image in pixel coordinates Werner Purgathofer 3 object capture/creation modeling viewing projection vertex stage ( vertex shader ) clipping + homogenization viewport transformation shading rasterization pixel stage ( fragment shader ) 2

3 Reminder: Graphs and Trees root arbitrary graph implemented with pointers Werner Purgathofer / Einf. in Visual Computing 4 Reminder: Graphs and Trees root^.left root root^.rightright binary tree implemented with pointers Werner Purgathofer / Einf. in Visual Computing 5 3

4 Reminder: Graphs and Trees root^.left root root^.rightright binary tree implemented with pointers Werner Purgathofer / Einf. in Visual Computing 6 Einführung in Visual Computing D Object Representations ti 4

5 Object Repres. in the Rendering Pipeline scene objects in object space transformed vertices in clip space scene in normalized device coordinates raster image in pixel coordinates Werner Purgathofer 8 object capture/creation modeling viewing projection vertex stage ( vertex shader ) clipping + homogenization viewport transformation shading rasterization pixel stage ( fragment shader ) 3D Object Representations graphics scenes contain solid geometric objects trees, flowers, clouds, rocks, water, creation of models surface interior models explicit procedural models heuristically physically based models representations geometrical data structures data structure organization Werner Purgathofer 9 5

6 Polygon Surfaces (1) set of surface polygons enclose object interior = Boundary Representation ( B (B-Rep Rep ) example: machine part surface represented by triangles Werner Purgathofer 10 Polygon Surfaces (2) polygon tables (B-Rep lists) geometric and attribute tables vertex, edge, polygon tables consistency, completeness checks Werner Purgathofer 11 6

7 Polygon Surfaces: Data Structure V 1 E 1 S 1 E 3 E 6 E 2 3 V 2 S2 V 3 V 5 E 4 E 5 V 4 Werner Purgathofer 12 Lists for B-Reps E 1 E 3 surface list S 1 S 2 S 1 S 2 V 1 E 6 E 2 E 4 V 2 V 3 V 4 V 5 E5 edge list E 1 E 2 E 3 E 4 E 5 E 6 vertex list xyz V 1 xyz V 2 Werner Purgathofer 13 xyz V 3 xyz V 4 xyz V 5 7

8 Reminder: Product of Vectors V 1 = scalar product: a 1 b 1 c 1 V 2 = a 2 b 2 c 2 V 1 V 2 =? cross product (vector product): V 1 V 2 =? Werner Purgathofer / Computergraphik 1 14 Reminder: Product of Vectors scalar product: a 1 V 1 V 2 = b 1 b 2 = a 1 a 2 +bb 1 b 2 +cc 1 c 2 a 2 c 1 c 2 cross product (vector product): a 1 b 1 c 1 a 2 b 2 c 2 V 1 V 2 = = b 1 c 2 c 1 b 2 c 1 a 2 a 1 c 2 a 1 b 2 b 1 a 2 V 1 V 2 = V 1 V 2 cosϕ ϕ Werner Purgathofer / Computergraphik

9 Polygon Surfaces: Plane Equation Ax + By + Cz + D = 0 plane parameters A,B,C,D normal (A,B,C) y N=(A,B,C) y 1 example: x 1= 0 N=(1,0,0) z x z 1 1 x Werner Purgathofer 16 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 to the polygon front face Werner Purgathofer 17 9

10 Front and Back Polygon Faces Ax + By + Cz + D = 0 for points on the surface < 0 for points behind > 0 for points in front if (1) right-handed coordinate system (2) polygon points ordered V 4 counterclockwise V 3 V 5 V 1,V 2,V 3 counterclockwise normal vector N = (V 2 -V 1 ) x (V 3 -V 1 ) V V 2 1 N Werner Purgathofer 18 Triangle Meshes most polygons are triangles triangle mesh = connected triangles triangle-strip = successive triangles (1 additional point per triangle) C.Schlick Werner Purgathofer 19 10

11 Constructive Solid Geometry Constructive Solid 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 20 CSG: Different Set Operations Werner Purgathofer 21 11

12 CSG Data Structure Every object is assembled from simple solids with set operations data structure: binary tree recursive evaluation Werner Purgathofer 22 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 op A op B: A B Werner Purgathofer 23 12

13 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 24 Ray-Casting Method (1) line-of-sight of each pixel is intersected with all surfaces take closest intersected surface viewing direction Werner Purgathofer / Computergraphik 1 25 closest intersection point 13

14 Ray-Casting Method (2) based on geometric optics, tracing paths of light rays backward tracing of light rays suitable for complex, curved surfaces special case of ray-tracing algorithms efficient ray-surface intersection techniques necessary intersection point normal vector Werner Purgathofer / Computergraphik 1 26 Ray-Casting Methods for CSG (1) visibility processing pixel plane y ray x -z Werner Purgathofer 27 14

15 Ray-Casting Methods for CSG (2) determining surface limits obj 1 obj 2 Operation pixel A C z B D obj 1 obj 2 plane {A, B} {C, D} Werner Purgathofer 28 Operation Result Union {A, D} Intersection {C, B} Difference {A, C} Ray-Casting Methods for CSG (3) volume determination V ij A ij z ij pixel plane y V ΣV ij ray A ij z ij x -z Werner Purgathofer 29 15

16 Properties of CSG advantages exact representation low memory cost combinations and transformations trivial disadvantages rendering effort is high Werner Purgathofer 30 Quadtrees hierarchical enumeration of objects in 2D: quadtree hierarchical subdivision i i until a region is homogeneous Quadrant 0 region of a 2-dim. space Quadrant Quadrant 3 Quadrant 2 data elements in the representative quadtree node Werner Purgathofer 31 16

17 Quadtrees area with 2 n by 2 n pixels quadtree with n levels storage efficiency Werner Purgathofer 32 Quadtrees area with 2 n by 2 n pixels quadtree with n levels storage efficiency Werner Purgathofer 33 17

18 Quadtrees quadtree representation for a region containing one foreground-color pixel on a solid background Werner Purgathofer 34 Quadtree Example suitable for representing (2D) images Werner Purgathofer 35 18

19 Octree = extension to 3D regular space subdivision: simple (empty or uniform) leaf node complex (other cases) divide further Werner Purgathofer 36 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 representative octree node Werner Purgathofer 37 19

20 Octree Simple Example G (W W W W S G (W W W W W W W S) S S) Werner Purgathofer 38 Operations with Octrees transformations very complicated except for a few special 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 39 20

21 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 40 Octree Examples Yoshifumi Kitamura (c) Yoshifumi Kitamura Michael Goodfellow GRAVIR/IMAG INRIA Werner Purgathofer 41 E.Strasser 21

22 Other 3D Object Representations BSP trees fractal geometry methods shape grammars, procedural models particle systems physically based modeling visualization of data sets... Werner Purgathofer 42 Scene Graphs object-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 din most graphics systems, e.g. OpenSceneGraph VRML X3D Werner Purgathofer 43 22

23 Scene Graph Example sun world door walls house car tree roof windows body engine wheels transf. transf. transf. transf. wheel Werner Purgathofer 123RF 44 Scene Graph Example hypercosm Werner Purgathofer 45 23

24 Scene Graph Example Werner Purgathofer 46 jmonkeyengine.org Scene Graph Example Werner Purgathofer 47 Garret Foster 24

25 Scene Graph Example Werner Purgathofer 48 B.B.Christensen Scene Graph Example Werner Purgathofer 49 Francois Faure 25

26 Scene Graph Example Werner Purgathofer 50 S.Teichmann End of 3D Object Representations 26