# Computer Graphics. Bing-Yu Chen National Taiwan University The University of Tokyo

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1 Computer Graphics Bing-Yu Chen National Taiwan University The University of Tokyo

2 Hidden-Surface Removal Back-Face Culling The Depth-Sort Algorithm Binary Space-Partitioning Trees The z-buffer Algorithm Visible-Surface Ray Tracing (Ray Casting) Space Subdivision Approaches

3 Hidden-Surface Removal =Visible-Surface Determination Determining what to render at each pixel. A point is visible if there exists a direct line-of-sight to it, unobstructed by another other objects (visible surface determination). Moreover, some objects may be invisible because there are behind the camera, outside of the field-ofview, too far away (clipping) or back faced (backface culling).

4 Hidden Surfaces: why care? Occlusion: Closer (opaque) objects along same viewing ray obscure more distant ones. Reasons for removal Efficiency: As with clipping, avoid wasting work on invisible objects. Correctness: The image will look wrong if we don t model occlusion properly.

5 Back-Face Culling = Front Facing B x A D C H E F z G

6 Back-Face Culling = Front Facing use cross-product to get the normal of the face (not the actual normal) use inner-product to check the facing N v 1 N = ( v v ) ( v v ) v 2 V v 3

7 Clipping (View Frustum Culling) view frustum eye back face occlusion view frustum

8 List-Priority Algorithms The Painter s Algorithm The Depth-Sort Algorithm Binary Space-Partitioning Trees

9 The Painter s Algorithm Draw primitives from back to front need for depth comparisons.

10 The Painter s Algorithm for the planes with constant z not for real 3D, just for 2½D sort all polygons according to the smallest (farthest) z coordinate of each scan convert each polygon in ascending order of smallest z coordinate (i.e., back to front)

11 The Depth-Sort Algorithm sort all polygons according to the smallest (farthest) z coordinate of each resolve any ambiguities that sorting may cause when the polygons z extents overlap, splitting polygons if necessary scan convert each polygon in ascending order of smallest z coordinate (i.e., back to front)

12 Overlap Cases x y y R P Q Q P P Q z x x

13 Binary Space-Partitioning Trees An improved painter s algorithm Key observation: f( p ) < 0 f( p ) > 0 T 3 T 4 T 2 T 1 T 5 Ax + By + Cz + D = 0 r f( p): n( p a) = 0

14 Binary Space-Partitioning Trees - T 1 + T 1 T 2 T 3 T 2 T 3

15 Binary Space-Partitioning Trees - T 1 + T 1 - T 2 + T 3 T 2 T 3

16 Splitting triangles a a A c A c B B b b

17 BSP Tree Construction BSPtree makebsp(l: list of polygons) { if (L is empty) { return the empty tree; } Choose a polygon P from L to serve as root; Split all polygons in L according to P return new TreeNode ( P, makebsp(polygons on negative side of P), makebsp(polygons on positive side of P)) } Splitting polygons is expensive! It helps to choose P wisely at each step. Example: choose five candidates, keep the one that splits the fewest polygons.

18 BSP Tree Display void showbsp(v: Viewer, T: BSPtree) { if (T is empty) return; P = root of T; if (viewer is in front of P) { showbsp(back subtree of T); draw P; showbsp(front subtree of T); } else { showbsp(front subtree of T); draw P; showbsp(back subtree of T); } } 2D BSP demo

19 Binary Space-Partitioning Trees Same BSP tree can be used for any eye position, constructed only once if the scene if static. It does not matter whether the tree is balanced. However, splitting triangles is expensive and try to avoid it by picking up different partition planes.

20 BSP Tree

21 BSP Tree inside ones outside ones 2 4 3

22 BSP Tree

23 BSP Tree a 11a 9 9b 11b a 10 11a 5 8 9b 11b 4 3

24 BSP Tree b a 11a 11 11b b 8 4 9a 11b a

25 BSP Tree b a 11a 11 11b b point 10 9a 11b 11a

26 BSP Tree Traversal b a 11a 11 11b b point 10 9a 11b 11a

27 BSP Tree Traversal b a 11a 11 11b b point 10 9a 11b 11a

28 The z-buffer Algorithm Resolve depths at the pixel level Idea: add Z to frame buffer, when a pixel is drawn, check whether it is closer than what s already in the frame buffer

29 The z-buffer Algorithm + = + =

30 The z-buffer Algorithm void zbuffer() { int pz; for (each polygon) { for (each pixel in polygon s projection) { pz=polygon s z-value at (x,y); if (pz>=readz(x,y)) { WriteZ(x,y,pz); WritePixel(x,y,color); } } } }

31 The z-buffer Algorithm y y 1 y 2 y 3 y s z 1 z 2 z 3 z a z p z b Scan line a b p b a b b p s b s a x x x x z z z z y y y y z z z z y y y y z z z z = = = ) ( ) ( ) (

32 z-buffer: Example color buffer depth buffer

33 The z-buffer Algorithm Benefits Easy to implement Works for any geometric primitive Parallel operation in hardware independent of order of polygon drawn Limitations Memory required for depth buffer Quantization and aliasing artifacts Overfill Transparency does not work well

34 Ray Tracing = Ray Casting select center of projection and window on viewplane; for (each scan line in image) { for (each pixel in scan line) { determine ray from center of projection through pixel; for (each object in scene) { if (object is intersected and is closest considered thus far) record intersection and object name; } set pixel s color to that at closest object intersection; } }

35 Ray Casting Center of projection Window

36 Ray Casting (Appel, 1968)

37 Ray Casting (Appel, 1968)

38 Ray Casting (Appel, 1968)

39 Ray Casting (Appel, 1968) k a I a + nls i= 1 I i ( ( ) + ( ) ) n k L N k R V d i s i

40 Ray Casting (Appel, 1968) direct illumination

41 Spatial Partitioning

42 Spatial Partitioning R C A B

43 Spatial Partitioning 1 A B 3 2

44 Space Subdivision Approaches Uniform grid K-d tree

45 Space Subdivision Approaches Quadtree (2D) Octree (3D) BSP tree

46 Uniform Grid

47 Uniform Grid Preprocess scene 1. Find bounding box

48 Uniform Grid Preprocess scene 1. Find bounding box 2. Determine grid resolution

49 Uniform Grid Preprocess scene 1. Find bounding box 2. Determine grid resolution 3. Place object in cell if its bounding box overlaps the cell

50 Uniform Grid Preprocess scene 1. Find bounding box 2. Determine grid resolution 3. Place object in cell if its bounding box overlaps the cell 4. Check that object overlaps cell (expensive!)

51 Uniform Grid Traversal Preprocess scene Traverse grid 3D line = 3D-DDA

52 From Uniform Grid to Quadtree

58 From Quadtree to Octree y x z

59 Leaf nodes correspond to unique regions in space K-d Tree A A

60 K-d Tree A B A

61 K-d Tree A B B A

62 K-d Tree C A B B A

63 K-d Tree C A B B C A

64 K-d Tree C A D B B C A

65 K-d Tree C A D B B A D C

66 K-d Tree A D B B C C D A Leaf nodes correspond to unique regions in space

67 K-d Tree Traversal A D B B C C D A Leaf nodes correspond to unique regions in space

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