Computer Graphics 2 Lecture 13: Ray-Tracing Techniques

1. Computer Graphics 2Lecture 13:Ray-Tracing Techniques Dr. Benjamin Mora University of Wales Swansea 1 Benjamin Mora

2. Content • What is Ray-Tracing? • Mathematics for ray-tracing intersections. • Common Ray-Tracing effects. • Other Ray-Tracing Algorithms. • Current Ray-Tracing Technology. University of Wales Swansea 2 Benjamin Mora

3. What Is Ray-Tracing ? University of Wales Swansea 3 Benjamin Mora

4. What is Ray Tracing ? • The alternative to rasterization-based methods (like OpenGL or DirectX). • Image-Order algorithm. • Every pixel is processed separately. • Slower than rasterization methods in the case of few primitives to render. • Rarely used in games, widely used for realistic computer graphics. • www.povray.org. University of Wales Swansea 4 Benjamin Mora

5. What is Ray Tracing ? Jaime Vives Piqueres (2004), http://www.povray.org/i/hof/office-13.jpg University of Wales Swansea 5 Benjamin Mora

6. What is Ray Tracing ? Jaime Vives Piqueres (2001), http://www.povray.org/i/hof/office-13.jpg University of Wales Swansea 6 Benjamin Mora

7. image plane primary ray viewlocation secondary rays What is Ray Tracing ? University of Wales Swansea 7 Benjamin Mora

8. What is Ray Tracing ? • Basic (or Naïve )Algorithm (Whitted 1980): FOR every pixel DO FOR every primitive in the scene DO IF there is a ray/primitive intersection test if this is the closest intersection computed so far. If this is the case, keep this primitive as the closest primitive for the next tests. • Basic Complexity: • O(number of rays*number of primitives). • Optimized algorithms can do much better (See the next lectures)! University of Wales Swansea 8 Benjamin Mora

9. Mathematics for Ray-Tracing intersections University of Wales Swansea 9 Benjamin Mora

10. Vz Vy Vx P0 Parametric Ray/Plane Formulation • A ray can be described with a parametric formulation from a (starting) point and a direction • A parametric plane can be described from one point and two vectors: • A plane can also be defined from a point and a normal vector (here P0 and Vz). University of Wales Swansea 10 Benjamin Mora

11. p0 n vdir p(t) h o Ray/Plane Intersection • We are looking for the point p(t) such that p(t) belong to the plane h. • h is defined with a point o and a normal n. • If vdirperpendicular to n, then either 0 solution, or an infinite number of solutions if p0 belongs to h (the vector p0o perpendicular to n). University of Wales Swansea 11 Benjamin Mora

12. p0 n vdir p(t) h o Ray/Plane Intersection • Otherwise, there is only one point that is solution. We have to find out t. • The solution comes from the fact that the vector p(t)o is perpendicular to n: University of Wales Swansea 12 Benjamin Mora

13. txmin txmax tymax p0 tymin tymin tymax t txmin txmax Ray/Cube Intersection p(t)= p0+v•t v The ray intersects the cell only if max(txmin, tymin)<min(txmax, tymax). Finding out the exit face consists of finding out the minimum of (txmax, tymax). University of Wales Swansea 13 Benjamin Mora

14. Ray/Cube Intersection • A cube can be defined with 6 planes (faces). • 3 pairs of opposite faces (let’s reference them as x,y,z) . • A parametric ray p(t) intersects 2 opposite faces at 2 locations, with 2 respective parameters tmin and tmax. (let’s suppose tmin<=tmax). • The 6 faces are intersected with 6 different parameters (txmin,txmax, tymin,tymax, tzmin,tzmax). • The ray intersects the cube only if: Max (txmin,tymin,tzmin)< Min(txmax,tymax,tzmax). University of Wales Swansea 14 Benjamin Mora

15. p(t) vdir p0 r o Ray/Sphere Intersection • A sphere can be defined by a center o and a radius r. • Looking for the points p(t) such as: norm(p(t)o)=r. University of Wales Swansea 15 Benjamin Mora

16. p0 vdir p(t) Ray/Triangle Intersections • Several ways to achieve this goal. • Badouel’s Algorithm. • Moller’s Algorithm (not shown). • Other Algorithm (close to Segura et al. 98). University of Wales Swansea 16 Benjamin Mora

17. p2 p(t) p1 p0 Badouel’s Algorithm • First, find the intersection p(t) between the ray and the plane containing the triangle. • Then use the barycentric coordinates of the point to sort the intersection out (2D algorithm). University of Wales Swansea 17 Benjamin Mora

18. Other Ray/Triangle Algorithms • The edges of the triangles can be considered as oriented. • The Ray should always be “on the same side” of the edges • => Three triple products with the same sign. p vdir p0 p1 p(t) p2 Segura, R.J., Feito, F.R. Segura, R.J., Feito, F.R.: An algorithm for determining intersection segment-polygon in 3D. Computer & Graphics, Vol.22, No.5, pp.587-592, 1998. University of Wales Swansea 18 Benjamin Mora

19. p vdir p0 p1 p(t) p2 Other Ray/Triangle Algorithms • There is an intersection Ray/Triangle only if: • Fast algorithm. • Can take advantage of SIMD instructions on processor. • But: • Only in order to detect an intersection. • Extra work must be done in order to get the tri-linear interpolating weights (including a div) and the real intersection location if needed (but not always required). University of Wales Swansea 19 Benjamin Mora

20. Triangle/Box Intersection • Why? • Bounding boxes are used much to speed-up intersections. • Standalone boxes. • Spatial subdivision techniques. • Therefore needed to construct such acceleration structures. University of Wales Swansea 20 Benjamin Mora

21. Triangle/Box Intersection • Solution 1: Working in the triangle plane. • For every face of the cube: • Find the face/triangle plane intersection. • A line! • Clip the triangle/polygon with this line. (Sutherland ). • Intersection occurs if result not NULL! • Complex and Costly. University of Wales Swansea 21 Benjamin Mora

22. Triangle/Box Intersection • Solution 2: Use the Separating Axis Theorem. (Akenine-Moller) • No intersection occurs between 2 convex objects if there exists a separating plane. University of Wales Swansea 22 Benjamin Mora

23. Triangle/Box Intersection (2D) University of Wales Swansea 23 Benjamin Mora

24. Triangle/Box Intersection (2D) University of Wales Swansea 24 Benjamin Mora

25. Triangle/Box Intersection (3D) • 13 planes to be tested in 3D. • For each plane: • Compute distances to the triangle vertices • Using the plane equation. • Compute distances to the box vertices. • No intersection if MIN of one distance set greater than MAX of the other distance set. • If condition is false for all 13 cases, an intersection exists. University of Wales Swansea 25 Benjamin Mora

26. Ei X Tj Triangle/Box Intersection (3D) • Plane directions are determined from: Where Ei are the directions of the edges of the box, and Tj the directions of the edges of the triangle. (9 cases) • The faces of the box. (3 cases) • The triangle plane normal. (1 case) University of Wales Swansea 26 Benjamin Mora

27. Simple Ray-Tracing Effects University of Wales Swansea 27 Benjamin Mora

28. SimpleRay-Tracing Effects • The main operation is to find an intersection from a given ray. • This operation can be used (by casting secondary rays) to compute simple effects that are more difficult to implement with OpenGL like (but fog): • Reflection. • Refraction. • Shadows. • Fog. Source: Min Chen University of Wales Swansea 28 Benjamin Mora

29. object colour result of tracing reflective ray result of tracing refractive ray + + SimpleRay-Tracing Effects • Basic Lighting model: Rin kt Rfr km Rfl University of Wales Swansea 29 Benjamin Mora

30. SimpleRay-Tracing Effects • Shading can be done in the same way as to OpenGL. • But better methods exist. N V (-Rin) R L Source: Min Chen University of Wales Swansea 30 Benjamin Mora

31. SimpleRay-Tracing Effects • Reflection: University of Wales Swansea 31 Benjamin Mora

32. SimpleRay-Tracing Effects • Reflection: N Vincident Vreflected i i L University of Wales Swansea 32 Benjamin Mora

33. N V (-Rin) Rfl i i R L Snell’s Law r Rfr SimpleRay-Tracing Effects • Refraction: University of Wales Swansea 33 Benjamin Mora

34. N critical angle critical V (-Rin) Rfr SimpleRay-Tracing Effects • Total Internal Reflection: University of Wales Swansea 34 Benjamin Mora

35. SimpleRay-Tracing Effects • Taking into account reflection and refraction several times (recursively) requires an exponential number of rays! • The algorithm should stop after reaching a maximum number of consecutive reflections/refractions. • Only reduced to some specific lighting effects. University of Wales Swansea 35 Benjamin Mora

36. Simple Ray-Tracing Effects University of Wales Swansea 36 Benjamin Mora

37. Simple Ray-Tracing Effects • Fog effect (also possible with OpenGL). • Depth Cue: • Fog: or University of Wales Swansea 37 Benjamin Mora

38. non-adaptive supersampling stochastic sampling adaptive supersampling Simple Ray-Tracing Effects • Super Sampling can be used in order to improve Anti-Aliasing. University of Wales Swansea 38 Benjamin Mora

39. Simple Ray-Tracing Effects University of Wales Swansea 39 Benjamin Mora

40. Simple Ray-Tracing Effects • The final color is the average of all the sub-sampled rays. • Uniform Super-Sampling: • Nice but not always needed. • Adaptive Super-Sampling: • Useful for the regions with large color transitions (high frequencies), like object boundaries. • Trade-off between quality and speed. • Stochastic Super-Sampling: • Take advantage of the visual system limitations. University of Wales Swansea 40 Benjamin Mora

41. Simple Ray-Tracing Effects • Stochastic Super-Sampling: • Proposed by Robert L. Cook. • Stochastic Sampling in Computer Graphics, ACM transaction on Graphics No. 5(1), January 1986, pp. 51-72. • The ray location(s) are randomly chosen inside a pixel. • Our visual system is less sensitive to noise than to contours. University of Wales Swansea 41 Benjamin Mora

42. Other Ray-Tracing Algorithms. University of Wales Swansea 42 Benjamin Mora

43. Other Ray-Tracing Algorithms. • Pyramidal Ray-Tracing. • Cone Tracing. • Beam Tracing. University of Wales Swansea 43 Benjamin Mora

44. image plane primary ray viewlocation Pyramidal Ray-Tracing. • Proposed by: • And improved by: • Main Idea: Replacing Rays by Pyramids to improve aliasing. Whitted, Turner. An Improved Illumination Model for Shaded Display. C.A.C.M. vol. 23, no. 6, June 1980, pp. 343-3~19. J. Genetti, D. Gordon and G. Williams. Adaptive supersampling in object space using pyramidal rays. Computer Graphics Forum 17 (1998) 29-54. University of Wales Swansea 44 Benjamin Mora

45. The triangle will contribute here to 7/64 of the final pixel colour. Pyramidal Ray-Tracing. J. Genetti, D. Gordon and G. Williams. Adaptive supersampling in object space using pyramidal rays. Computer Graphics Forum 17 (1998) 29-54. University of Wales Swansea 45 Benjamin Mora

46. Pyramidal Ray-Tracing. • If a pyramid only partially intersects a graphics primitive (e.g., triangle), then the pyramid is subdivided until a given depth is reached. • The final pixel value is the (weighted) average of all the samples inside the triangle. • Non-intersecting pyramids keep on computing intersections with the rest of the scene. University of Wales Swansea 46 Benjamin Mora

47. Cone Tracing. • Proposed by: John Amanatides. Ray Tracing with Cones. Siggraph 1984, pp. 129-135. image plane primary ray viewlocation Partial intersection University of Wales Swansea 47 Benjamin Mora

48. Cone Tracing. • Similar to pyramidal ray-tracing, but the pyramid is now replaced by a cone. • Allows easier intersection computations. • 1 cone per pixel, no subdivision. • For a ray-triangle intersection: • The cone is projected onto the triangle plane. • Circular projection. • If there is a partial intersection detected, the colour of the triangle is weighted by the corresponding covering percentage, and computing intersections will carry on. University of Wales Swansea 48 Benjamin Mora

49. Cone Tracing: Result University of Wales Swansea 49 Benjamin Mora

50. Beam Tracing. • Proposed by: • Takes advantages of spatial coherency between rays to group them and save computations. • Rays are distributed after Reflections and Refractions according to the reflection/refraction coefficients. • Constructs a beam tree. • Outdated. Paul S. Heckbert and Pat Hanrahan. BEAM TRACING POLYGONAL OBJECTS. Siggraph 1984, pp. 119-127. University of Wales Swansea 50 Benjamin Mora