Computer Graphics (fall 2009)

1. Computer Graphics(fall 2009) School of Computer Science University of Seoul

2. Topics • Graphics pipeline • Algorithms for the tasks in the pipeline

3. Chap 7: From Vertices to Fragments • Basic Implementation Strategies • Four Major Tasks • Clipping • Line-Segment Clipping • Polygon Clipping • Clipping of Other Primitives • Clipping in Three Dimensions • Rasterization • Bresenham’s Algorithm • Polygon Rasterization • Hidden-Surface Removal • Antialiasing • Display Considerations

4. 7.1 Basic Implementation Strategies

5. Big Picture • Input • Geometric objects • Attributes: color, material, normal, etc. • Lights • Camera specifications • Etc. • Output • Arrays of colored pixels in the framebuffer

6. Big Picture (cont’d) • Tasks by graphics system • Transformations • Clipping • Shading • Hidden-surface removal • Rasterization

7. Object-Oriented Approach • for(each_object) render(object); • Pipeline renderer • Same operation on every primitive (independently, in arbitrary order) SIMD (Single Instruction, Multiple Data) • Cannot handle global calculations (exception: hidden-surface removal)

8. Image-Oriented Approach • for(each_pixel) assign_a_color(pixel); • To determine which geometric primitives can contribute to its color • Coherency  incremental implementation • Complex data structure • Can handle global effects • Example: raytracing

9. 7.2 Four Major Tasks

10. Four Major Tasks • Modeling • Geometry processing • Rasterization • Fragment processing

11. 1. Modeling • Output: set of vertices

12. 2. Geometry Processing • Model-view transformation • To camera (eye) coordinate • Projection transformation • To a normalized view volume • Vertices represented in clip coordinate • Primitive assembly • Clipping • Shading • Modified Phong model • Perspective division

13. 3. Rasterization • A.k.a. scan conversion • “How to approximate a line segment with pixels?” • “Which pixels lie inside a 2D polygon?” • Viewport transformation • Fragments in window coordinates • Vs. screen coordinates?

14. 4. Rasterization • Color assigned by linear interpolation • Hidden-surface removal on a fragment-by-fragment basis • Blending • Antialiasing

15. 7.3 Clipping

16. Clipping • Before perspective division • Normalized device coordinates

17. 7.4 Line-Segment Clipping

18. Clipper • Clipper accepts, rejects (or culls), or clips primitives against the view volume • Before rasterization • Four cases in 2D • Two algorithms • Cohen-Sutherland clipping • Liang-Barsky clipping

19. Cohen-Sutherland Clipping • Intersection calculation replaced by membership test • Intersection calculation: FP mul & div • Membership test: FP sub & bit operations • Intersection calculation only when needed • The whole 2D space decomposed into 9 regions • Membership test against each plane • outcodes computed • “0000” for inside of the volume

20. Cohen-Sutherland Clipping (cont’d) • Four cases associated with the outcodes of endpoints • o1==o2==0: both inside (AB) • o1<>0, o2==0 (or vide versa): one inside and the other outsideintersection calculation required (CD) • o1&o2<>0: outside of the common plane(edge)can be discarded (EF) • o1&o2==0: outside of the different planemore computation required (GH & IJ)

21. Cohen-Sutherland Clipping (cont’d) • Works best when many line segments are discarded • Can be extended to three dimension • Must be recursive

22. Liang-Barsky Clipping • Parametric form of line segment • Four parameter values computed associated with the intersections with four planes • Example • 1:bottom, 2: left, 3: top, 4: right • (a) 0<1< 2< 3< 4<1 • (b) 0<1< 3< 2< 4<1

23. Liang-BarskyClipping (cont’d) • Intersection calculation (against top plane) • Simpler form used for clipping decision • FP div only when required • Multiple shortening not required • Not extend to three dimension

24. More on Line Clipping • Line clipping by Wikipedia

25. 7.5 Polygon Clipping

26. Applications • Non-rectangular window • Shadow generation • Hidden-surface removal • Antialiasing

27. Concave & Convex Polygons • Clipping concave polygon is complex • Clipping convex polygon is easysingle clipped polygon • Concave polygon is tessellated into convex polygons

28. Algorithm • Sutherland-Hodgeman • Any line segment clipper can be applied blackboxed • Convex polygon (including rectangle) as the intersection of half-spaces • Intersection test against each plane

29. Algorithm (cont’d) • Pipelined

30. 7.6 Clipping of Other Primitives

31. Bounding Volume • Early clipping can improve performancebounding boxes & volumes • AABB (Axis-Aligned Bounding Box) • Bounding sphere • OBB(Oriented Bounding Box) • DOP (Discrete Oriented Polytop) • Convex hull • …and many more

32. Bounding Volume (cont’d) (image courtesy of http://www.ray-tracing.ru)

33. Curves, Surfaces and Texts • Approximated with line segments (or triangles/quads) • “Convex hull property” for parametric curves & surfaces • Texts • Texts as bit patterns  clipping in framebuffer • Texts as geometric objects  polygon clipping • OpenGL allows both • Scissoring: clipping in the framebuffer

34. 7.7 Clipping in Three Dimensions

35. 3D Clipping • Clipping against 3D view volume • Extension of Cohen-Sutherland

36. 3D Clipping (cont’d) • Extension of Liang-Barsky • Intersection calculation is simple due to normalization • Additional clipping planes with arbitrary orientations supported

37. 7.8 Rasterization

38. Pixels • Square-shaped • Integer coordinates • In OpenGL center is located at the halfway between integers

39. DDA Algorithm • Rasterization of line segment • Only for small slopes • FP addition for each pixel

40. 7.9 Bresenham’s Algorithm

41. Bresenham’s Algorithm • No FP calculation! • Standard algorithm • For integer endpoints (x1,y1)-(x2,y2)

42. Bresenham’sAlgorithm (cont’d) • How it works: • With slope 0<=m<=1 • Assume we just colored the pixel (i+1/2,j+1/2) • We need to color either (i+3/2,j+1/2) or (i+3/2,j+3/2)depending on d=a-b • (x2-x1)(a-b) is integer  simpler calculation • d can be computed incrementally(next page)

43. Bresenham’s Algorithm (cont’d) • d can be computed incrementally • If a_k > b_k (left) • a_{k+1}+m=a_k a_{k+1}=a_k-m • b_k= b_{k+1}-m  b_{k+1}=b_k+m • If a_k<b_k (right) • 1+a_k=a_{k+1}+m  a_{k+1}=a_k-(m-1) • 1-b_k=m-b_{k+1}  b_{k+1}=b_k+(m-1)

44. 7.10 Polygon Rasterization

45. Polygon Rasterization • Inside-outside testing • Crossing (or odd-even test) • Winding test – how to compute?

46. Tessellation • Supported by GLU functions • Triangles generated based on given contour • Different tessellation depending on the winding number (gluTessProperty)

47. Polygon Fill • “How to fill the interior of a polygon?” • Three algorithms • Flood fill – starts with “seed point” • Scanline fill • Odd-Even fill • Singularity • Handle separately • Perturb the position • Different values for pixels and vertices

48. 7.11 Hidden-Surface Removal

49. Hidden-Surface Removal • Object-space approach • For each object, determine & render the visible parts • Pairwise comparison  O(k^2) • Image-space approach • For each pixel, determine the closest polygon • O(k)

50. Scanline Algorithm • “Spans” processed independently for lighting and depth calculations • Overhead to generate spans y-x algorithm