7M836 Animation & Rendering

1. 7M836 Animation & Rendering Global illumination, ray tracing Arjan Kok a.j.f.kok@tue.nl

2. Local illumination models • What is missing in local (Phong) illumination model • (Shadows) • Real mirrors • Transparency • Area light sources • Indirect diffuse reflection • Atmosphere

3. Light paths • Light path notation • L: light source • D: diffuse reflection • S: specular reflection • E: eye point • Local reflection models: L(D|S)E • Complete solution: L(D|S)*E

4. Global illumination models • Illumination for complete scene • All illumination, also indirect illumination • Several approaches: • Ray tracing • Radiosity • …

5. Brute force solutions • Trace photons from light source into scene • Follow paths of photons through reflections/transmissions • At each reflection/transmission “energy” of photon is modified (part of energy absorbed by surface) • Photons that go though image plane and reach eye contribute to image.

6. Trace photons

7. Trace photons • Light paths complete solution: L(D|S)*E • Forward ray tracing • Problem • Most photons will not contribute to image

8. Backward ray tracing • Only trace light that arrives at viewpoint through viewing plane • Trace light backwards from viewpoint until light source reached • (Backward) Ray tracing

9. eye pixel Ray tracing • Basic algorithm • for all pixels in image plane • create ray from eye point through pixel • trace this ray on his path(s) through scene until it reaches light source(s) and collect illumination encountered during travel • amount of collected lightdetermines color of pixel

10. S3 S4 V1 R1 S1 S2 Ray tracing – example 1 eye image plane 3 LA LB 2 1 4 Object 1 diffuse & specular

11. V1 S1 S2 S5 S6 T1 Ray tracing – example 2 eye image plane 3 LA LB 2 1 4 Object 1 diffuse & transparent

12. Ray tracing • Create viewing ray • Trace ray • At intersection point • Compute (local) illumination. Trace shadow rays to light sources to account for shadows. • If surface at intersection specularly reflecting, trace shadow ray and compute its contribution:continue at step 2 • If surface at intersection transparent, trace transparency ray and compute its contribution: continue at step 2 • Sum contribution of steps (a), (b) and (c)

13. Ray-tracing picture (Whitted 80)

14. Example using ray tracing

15. Computations in ray-tracing • Creation of viewing ray • Intersection computation • Find first intersected object (ray-scene intersection) • Point of intersection • Normal (and local coordinates) of intersection point • Creation shadow rays • Creation reflection rays • Creation transparency rays • Illumination

16. E P Creation viewing ray • Origin is eye point (E) • Through pixel (P) • R(t) = E + V * t t > 0

17. Ray-scene intersection • Compute intersection of ray R(t) with all objects in scene • Intersection with smallest positive intersection distance t gives intersected object t = 3 t = 2 t = 5

18. Ray-object intersection • Ray equation R(t) = Ro + Rd * t t > 0 • Ro = ray origin • Rd = ray direction • ||Rd|| = 1 • Object • Implicit surface: f(x) = 0 • Polygons • …

19. Ray-sphere intersection • Sphere ||P – C||2 – r2 = 0 • Substitution of R(t) for P gives t2 + 2bt + c = 0 • b = Rd• (Ro – C) • c = ||Ro – C||2 – r2 • Solution for t gives intersection points P’ N P r C Rd R0

20. t2 t1 = t2 t2 t1 t2 t1 t1 P’ N P r C Rd Ro Ray-sphere intersection • Solutions for t1 and t2: • Normal at intersection point N = (P – C) / r

21. Ray-polygon intersection • Two steps • Intersect ray with plane of polygon.Compute intersection point • Determine if intersection point within polygon • Normal of intersection point is plane normal R0

22. Ray-polygon intersection • Intersection ray-plane • Equation of plane: Ax + By + CZ + D = 0 • Substitution of R(t) for x, y, and z gives: where N is normal of plane: {A, B, C}

23. Ray-polygon intersection • Determine if intersection point within polygon (2D) • Draw line from intersection point in a direction • Count number of intersection with edges of polygon • If count even, than point outside polygon, otherwise inside

24. Shadow ray • Determine if point is illuminated by light source • Shadow ray: • R(t) = P + S * t t > 0 • Point P is illuminated by light source L if there is no intersection of shadow ray with scene for 0 < t < ||L – P|| L S P

25. N θr θi I R Reflection ray • Physical laws: • R, N, and I are in same plane: R = αI + βN • Angle of incidence = angle of reflection: θr = θi • Reflection ray direction

26. Reflection

27. N θi I medium i medium t T θt Transparency ray • Snell’s law: • ηi = index of refraction medium iwith respect to vacuum • ηit = index of refraction medium I with respect to medium t • Transparency ray direction

28. Transparency

29. Transparency

30. N θi Ri Li αi V Illumination • Phong illumination model for local illumination

31. Illumination for ray tracing • Extension of local model with shadow information, mirroring and transparency • Si = shadow factor (0, 1) • IR = intensity of reflection ray • IT = intensity of transmission ray

32. Ray tracing

33. Ray tracing • Light paths ray tracing: LS*E and LDS*E • Allows for transparency with refraction • Easy shadow computation • Easy to program • Inefficient

34. modelleer transformatie display ray tracing Ray tracing

35. Ray tracing extensions • More effects • Caustics • Area light sources, soft shadows • Depth of field • Motion blur

36. Two-pass ray tracing • Two-pass method • First pass: forward tracing (from lights into scene). • Limited to rays from light to reflective and transparent objects • When transparency ray hits surface, energy is stored at surface • Second pass: backward ray tracing • When local illumination applied, also check for stored intensity. Add this intensity to illumination • Light paths ray tracing with caustics: LS*E and LS*DS*E

37. entirely illuminated penumbra umbra penumbra entirely illuminated Area light sources • Area light sources generate soft shadows (penumbrae)

38. Area light source

39. Area light source • Subdivide area light source in number of point light sources • Use regular grid of points

40. Area light source • How many point light sources must be used to simulate area source? • Number depends on distance from point to light source • Regular pattern of point sources generates shadow bands

41. Area source: adaptive subdivision

42. Shadow bands

43. Area source: irregular pattern • Trace shadow ray to random point on sub light source • Degree of randomness often indicated with “jitter” • Regular shadow patterns replaced by noise

44. Area source: jittered subdivision

45. Ray tracing - efficiency • Ray tracing inefficient • Need for methods to improve performance • Efficiency ray tracing determined by • Number of rays • Number of light sources • Number of specular and transparent objects • Recursion depth • Efficiency ray-object intersections • Number of intersections to be computed

46. Ray tracing – bounding volume • Reduce number of complex ray-object intersection computations by providing complex objects with a simple geometry around complex geometry • Sphere • Cube • Do intersection with simple geometry • Only if intersection found, do intersection with complex geometry • Hierarchy of bounding volumes

47. Ray tracing – space subdivision • Partition scene into small cells • Store in each cell pointers to objects it contains • Trace ray through cells and only compute intersections with objects in visited cell • When intersection found, stop tracing rays through cells

48. Ray tracing: conclusion • Spectacular effects: • Shadows • Mirrors • Transparency, refraction • Simple implementation • Limitations • Not all light paths possible, missing diffuse interreflection • Area light sources possible, but at high price • => Radiosity method solves (parts of) limitations