H331: Computer Graphics

1. H331: Computer Graphics http://www.cs.kuleuven.ac.be/~graphics/H331/Philip Dutré Department of Computer Science Wednesday, March 19

2. Today • Ray Tracing(Chapter 14) http://www.oyonale.com/ressources/english/mkofwetbird1.htm

3. How did we generate pictures so far? • Viewing transform • camera in origin • Perspective transform • Clipping • Scan-conversion • Z-buffer for visibility

4. Ray tracing • RT is a more powerful technique to render scenes • For each pixel: • What surface point is visible through this pixel? (“trace a ray …”) • What color does this point have? • No perspective transform, no clipping, …

5. Basic RT algorithm: set-up • Camera defined by eyepoint and view plane in world space screen environment eyepoint

6. Basic RT algorithm: viewing rays • Send a ray from the eye through each pixel • Intersect ray with all geometry in the scene screen environment eyepoint

7. Basic RT algorithm: viewing rays • If a ray hits an object, that object is shown • … otherwise, background color is shown screen environment eyepoint

8. Viewing rays screenv yres pixels (0..yres-1) ray origin = eye ray direction = d - screenh - screenv + (xres+.5)*screenh/||screenh|| + (yres+.5)*screenv/||screenv|| d screenh eye xres pixels (0..xres-1)

9. Basic RT algorithm • Evaluate shading model for each point, e.g.:

10. RT: skeleton • For each pixel (x,y): • Construct the ray for pixel (x,y) • Intersect the ray with all objects in the scene • Pick the closest intersection, in front of the eye • Compute the color of the hitpoint, w.r.t. lightsources • Color pixel

11. Ray-Sphere intersection • Sphere is centered at c, radius = R • Surface intersection point p defined twice: • (px-cx)2 + (py-cy)2 + (pz-cz)2 = R2 (equation for sphere) • e + t.dir = p (equation of ray) • Combining equations: • (ex-cx+t.dirx)2 + (ey-cy +t.diry)2 + (ez-cz +t.dirz)2 = R2 • Solving quadratic for t yields 0, 1, 2 intersections

12. Ray-triangle intersection

13. Ray-Transformed object intersection • Apply inverse transform to ray • Intersect transformed ray with original object • Apply transform to intersection point

14. Ray-Transformed object intersection • Transform ray only once  transform all objects • Exact location of hitpoint & normal need to be transformed back!

15. Ray – CSG Intersections: Easy!!! • Find intersections with operand objects • Apply operand to intersection spans Objects Union Intersection Difference

16. Ray / Object intersection questions • Closest hit: t>= 0, or t>= epsilon ? • Backface culling? • Normal vector at hit point: • Needed for shading • Transformed objects? • Local parametrization (u,v)for intersection point • Needed for texture mapping, bump-mapping etc.

17. Texture mapping

18. Texture mapping v y t u surface x s pixel on screen texture map

19. Bump mapping (u,v) used to perturbate normal  shading looks different

20. Bump mapping Color Plate from “Advanced Animation and Rendering Technique” by Alan Watt and Mark Watt

21. Bump mapping

22. RT and shadows • Trace shadow-ray to light source • If shadow-ray hits something  shadow screen environment eyepoint

23. RT and shadows • Trace shadow-ray to light source • If shadow-ray hits something  shadow screen environment eyepoint

24. RT and shadows No shadow test Shadow-ray intersection test

25. RT and shadows • Shadow Ray: • Origin = point to be shaded • Direction: towards the (point) light source • Actual intersection point not important • If we find ANY point between point to be shaded and light source  shadow • We do not have to find the closest one • Can be made slightly more effective

26. Reflections and transparancy • So far: • Direct illumination (shading model) • Shadows • One viewing ray for each pixel • Indirect light? • Reflections: light coming from perfect specular direction • Refractions: light coming from perfect refractive direction

27. Reflections and transparancy • Shoot perfect reflected ray, find closest hitpoint • Add color of this hitpoint to current hitpoint screen environment eyepoint

28. Reflections and transparancy

29. Reflections and transparancy • Reflected ray: • Origin: hitpoint • Direction: perfect reflected direction • Shading model: Recursive ray tracing

30. Reflections and transparancy • Shoot perfect refracted ray, find closest hitpoint • Add color of this hitpoint to current hitpoint screen environment eyepoint

31. Reflections and transparancy

32. Reflections and transparancy • Refracted ray: • Origin: hitpoint • Direction: perfect refracted direction (Snell’s law) • Shading model:

33. Very first raytraced picture (Whitted 1980)

34. Comparisons (Cornell PCG)

35. Examples

36. Examples

37. Recursive Ray Tracing T 2 N Surface normal R i N 2 R T 2 3 1 R Reflected ray i N 3 R L Shadow ray 1 i T Transmitted ray L 3 i L L 2 1 N 1 Point light source Viewpoint

38. Recursive Ray Tracing

39. Recursive Ray Tracing: how to stop? • Fixed depth • When contribution to original pixel falls below threshold • Successive reflections/refractions diminish contribution of ray • Adaptive tree pruning

40. Is ray tracing slow or fast? • Traditional ray tracing is O(N*P) • N objects • P pixels • One shadow ray for each light source • Recursive rays • … many, many rays!!!

41. Making RT Faster • Faster ray-object intersections • Bounding volumes • Fewer ray-object intersections • Bounding volumes hierarchies • Spatial subdivision techniques

42. Bounding Volumes • Put a simple enclosure (BV) around an object • Test ray against BV • If hit, then test ray against object • If miss, we know object is missed

43. Bounding Volumes • BV should be simple to test • BV should be tight

44. Bounding Volumes • Cost per ray = intersect_cost_BV + intersect_cost_object * prob_ray_hits_BV Keep this cheap Keep this small (very tight) (simple BV)

45. Bounding Volumes • Can be organised in hierarchy

46. Bounding Volumes • Most popular • Spheres • Axis-aligned cuboids • Projection extents • 2D rectangular bounding volume around projection of object on screen (p. 788)

47. Space Subdivision • Basic idea: • Divide space into cells • Store relevant objects in each cell • Trace ray from cell to cell, only intersect with objects in current cell

48. Space Subdivision

49. Space Subdivision

50. Space subdivision • Efficient for all rays • Uses a lot of memory • Bad for irregular spaced models in the scene (many empty cells) • Overhead: trace ray from cell-to-cell