Course Note Credit: Some of slides are extracted from the course notes of prof. Mathieu Desburn USC and prof. Han-Wei Sh

2. Rasterization Okay, you have three points (vertices) at screen space. How can you fill the triangle?

5. Interpolater

7. Better & still simple approach Find left edge & right edge from v0 You can not tell when y==v0.y But you can tell by comparing x values when y == v0.1+1 You need to treat horizontal edges carefully (divide by zero).

10. Back-face Culling If a surface�s normal is pointing to the same direction as our eye direction, then this is a back face The test is quite simple: if N * V > 0 then we reject the surface

21. Rasterizing Polygons

22. Scan Line Algorithms

23. Scan Line Algorithms

24. Scan Line Algorithms

25. Scanline Rasterization Special Handling Intersection is an edge end point, say: (p0, p1, p2) ?? (p0,p1,p1,p2), so we can still fill pairwise In fact, if we compute the intersection of the scanline with edge e1 and e2 separately, we will get the intersection point p1 twice. Keep both of the p1.

26. Scanline Rasterization Special Handling But what about this case: still (p0,p1,p1,p2)

27. Rule Rule: If the intersection is the ymin of the edge�s endpoint, count it. Otherwise, don�t. Don�t count p1 for e2

28. Data Structure Edge table: all edges sorted by their ymin coordinates. keep a separate bucket for each scanline within each bucket, edges are sorted by increasing x of the ymin endpoint

29. Edge Table

30. Active Edge Table (AET) A list of edges active for current scanline, sorted in increasing x

32. Polygon Scan-conversion Algorithm Construct the Edge Table (ET); Active Edge Table (AET) = null; for y = Ymin to Ymax Merge-sort ET[y] into AET by x value Fill between pairs of x in AET for each edge in AET if edge.ymax = y remove edge from AET else edge.x = edge.x + dx/dy sort AET by x value end scan_fill

33. Scan Line Algorithms

34. Convex Polygons

35. Crow�s Algorithm

36. Crow�s Algorithm

37. Crow�s Algorithm

38. Crow�s Algorithm

39. Crow�s Algorithm

40. Crow�s Algorithm

41. Crow�s Algorithm

42. Scan Line Algorithm Low memory cost Uses scan-line coherence but not vertical coherence Has several side advantages: filling the polygons reflections texture mapping Renderman (Toy Story) = scan line

43. Visibility Clipping Culling

44. Clipping Objects may lie partially inside and partially outside the view volume We want to �clip� away the parts outside

45. Cohen-Sutherland Clipping Clip line against convex region Clip against each edge Line crosses edge replace outside vertex with intersection Both endpoints outside trivial reject Both endpoints inside trivial accept

46. Cohen-Sutherland Clipping

47. Cohen-Sutherland Clipping Store inside/outside bitwise for each edge Trivial accept outside(v1) | outside(v2) == 0 Trivial reject outside(v1) & outside(v2) Compute intersection (eg. x = a) (a, y1 + (a-x1) * (y2-y1)/(x2-x1))

48. Classifies each vertex of a primitive, by generating an outcode. An outcode identifies the appropriate half space location of each vertex relative to all of the clipping planes. Outcodes are usually stored as bit vectors. Outcode Algorithm

51. The Maybe case

52. The Maybe Case

53. Culling Removing completely invisible objects/polygons

54. Culling Check whether all vertices lie outside the clip plane

55. Backface Culling For closed objects, back facing polygons are not visible

56. Flood Fill

57. Flood Fill

58. Area Subdivision (Warnock)(mixed object/image space)

59. Area Subdivision (Warnock) Initialize the area to be the image plane Four cases: No polygons in area: done One polygon in area: draw it Pixel sized area: draw closest polygon Front polygon covers area: draw it Otherwise, subdivide and recurse

60. BSP (Binary Space Partition) Trees

61. BSP Trees

62. BSP Trees

63. BSP Trees

64. Spatial Data Structures Data structures for efficiently storing geometric information. They are useful for Collision detection (will the spaceships collide?) Location queries (which is the nearest post office?) Chemical simulations (which protein will this drug molecule interact with?) Rendering (is this aircraft carrier on-screen?), and more Good data structures can give speed up rendering by 10x, 100x, or more

65. Bounding Volume Simple notion: wrap things that are hard to check for ray intersection in things that are easy to check. Example: wrap a complicated polygonal mesh in a box. Ray can�t hit the real object unless it hits the box Adds some overhead, but generally pays for itself . Can build bounding volume hierarchies

66. Bounding Volumes Choose Bounding Volume(s) Spheres Boxes Parallelepipeds Oriented boxes Ellipsoids Convex hulls

67. Quad-trees Quad-tree is the 2-D generalization of binary tree node (cell) is a square recursively split into four equal sub-squares stop when leaves get �simple enough�

68. Octrees Octree is the 3-D generalization of quad-tree node (cell) is a cube, recursively split into eight equal sub- cubes stop splitting when the number of objects intersecting the cell gets �small enough� or the tree depth exceeds a limit internal nodes store pointers to children, leaves store list of surfaces more expensive to traverse than a grid adapts to non-homogeneous, clumpy scenes better

69. K-D tree The K-D approach is to make the problem space a rectangular parallelepiped whose sides are, in general, of unequal length. The length of the sides is the maximum spatial extent of the particles in each spatial dimension.

70. K-D tree

71. K-D Tree in 3-D Similarly, the problem space in three dimensions is a parallelepiped whose sides are the greatest particle separation in each of the three spatial dimensions.

72. Motivation for Scene Graph Three-fold Performance Generality Ease of use How to model a scene ? Java3D, Open Inventor, Open Performer, VRML, etc.

73. Scene Graph Example

74. Scene Graph Example

75. Scene Graph Example

76. Scene Graph Example

77. Scene Description Set of Primitives Specify for each primitive � Transformation � Lighting attributes � Surface attributes Material (BRDF) Texture Texture transformation

78. Scene Graphs Scene Elements Interior Nodes Have children that inherit state transform, lights, fog, color, � Leaf nodes Terminal geometry, text Attributes Additional sharable state (textures)

79. Scene Element Class Hierarchy

80. Scene Graph Graph Representation What do edges mean? Inherit state along edges group all red object instances together group logical entities together parts of a car Capture intent with the structure

81. Scene Graph

82. Scene Graph (VRML 2.0)

83. Example Scene Graph

84. Simulation Animation Intersection Collision detection Picking Image Generation Culling Detail elision Attributes Scene Graph Traversal

85. Scene Graph Considerations Functional Organization Semantics Bounding Volumes Culling Intersection Levels of Detail Detail elision Intersection Attribute Management Eliminate redundancies

86. Functional Organization Semantics: Logical parts Named parts

87. Functional Organization Articulated Transformations Animation Difficult to optimize animated objects

88. Bounding Volume Hierarchies

89. View Frustum Culling

90. Level Of Detail (LOD) Each LOD nodes have distance ranges

91. Attribute Management Minimize transformations Each transformation is expensive during rendering, intersection, etc. Need automatic algorithms to collapse/adjust transform hierarchy.

92. Attribute Management Minimize attribute changes Each state change is expensive during rendering

93. Question: How do you manage your light sources? OpenGL supports only 8 lights. What if there are 200 lights? The modeler must �scope� the lights in the scene graph?

94. Sample Scene Graph

95. Think! How to handle optimization of scene graphs with multiple competing goals Function Bounding volumes Levels of Detail Attributes

96. Scene Graphs Traversal Perform operations on graph with traversal Like STL iterator Visit all nodes Collect inherited state while traversing edges Also works on a sub-graph

97. Typical Traversal Operations Typical operations Render Search (pick, find by name) View-frustum cull Tessellate Preprocess (optimize)

98. Scene Graphs Organization Tree structure best No cycles for simple traversal Implied depth-first traversal (not essential) Includes lists, single node, etc as degenerate trees If allow multiple references (instancing) Directed acyclic graph (DAG) Difficult to represent cell/portal structures

99. Portals

100. Portals

101. Portals