Ray Tracing COMP575/COMP770

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1 Ray Tracing COMP575/COMP770 1

2 Ray tracing idea 2

3 Ray Tracing: Example (from [Whitted80])

4 Ray Tracing: Example

5 Ray Tracing for Highly Realistic Images Volkswagen Beetle with correct shadows and (multi-)reflections on curved surfaces

6 Reasons for Using Ray Tracing Flexible Primitive Types Volume visualization using multiple iso-surfaces

7 Ray tracing algorithm for each pixel { compute viewing ray intersect ray with scene compute illumination at visible point put result into image } 7

8 Generating eye rays Use window analogy directly 8

9 Generating eye rays ORTHOGRAPHIC PERSPECTIVE 9

10 Vector math review Vectors and points Vector operations addition scalar product More products dot product cross product Bases and orthogonality 10

11 Generating eye rays orthographic Just need to compute the view plane point s: but where exactly is the view rectangle? 11

12 Generating eye rays orthographic 12

13 Generating eye rays perspective View rectangle needs to be away from viewpoint Distance is important: focal length of camera still use camera frame but position view rect away from viewpoint ray origin always e ray direction now controlled by s 13

14 Generating eye rays perspective Compute s in the same way; just subtract dw coordinates of s are (u, v, d) 14

15 Pixel-to-image mapping One last detail: (u, v) coords of a pixel j j = 2.5 i j =.5 i =.5 i =

16 Ray intersection 16

17 Ray: a half line Standard representation: point p and direction d this is a parametric equation for the line lets us directly generate the points on the line if we restrict to t > 0 then we have a ray note replacing d with ad doesn t change ray (a > 0) 17

18 Ray-sphere intersection: algebraic Condition 1: point is on ray Condition 2: point is on sphere assume unit sphere; see Shirley or notes for general Substitute: this is a quadratic equation in t 18

19 Ray-sphere intersection: algebraic Solution for t by quadratic formula: simpler form holds when d is a unit vector but we won t assume this in practice (reason later) I ll use the unit-vector form to make the geometric interpretation 19

20 Ray-sphere intersection: geometric 20

21 Ray-box intersection Could intersect with 6 faces individually Better way: box is the intersection of 3 slabs 21

22 Ray-slab intersection 2D example 3D is the same! 22

23 Intersecting intersections Each intersection is an interval Want last entry point and first exit point Shirley fig

24 Ray-triangle intersection Condition 1: point is on ray Condition 2: point is on plane Condition 3: point is on the inside of all three edges First solve 1&2 (ray plane intersection) substitute and solve for t: 24

25 Ray-triangle intersection In plane, triangle is the intersection of 3 half spaces 25

26 Inside-edge test Need outside vs. inside Reduce to clockwise vs. counterclockwise vector of edge to vector to x Use cross product to decide 26

27 Ray-triangle intersection 27

28 Ray-triangle intersection See book for a more efficient method based on linear systems (don t need this for Ray 1 anyhow but stash away for Ray 2) 28

29 Image so far With eye ray generation and sphere intersection Surface s = new Sphere((0.0, 0.0, 0.0), 1.0); for 0 <= iy < ny for 0 <= ix < nx { ray = camera.getray(ix, iy); hitsurface, t = s.intersect(ray, 0, +inf) if hitsurface is not null image.set(ix, iy, white); } 29

30 Intersection against many shapes Group.intersect (ray, tmin, tmax) { tbest = +inf; firstsurface = null; for surface in surfacelist { hitsurface, t = surface.intersect(ray, tmin, tbest); if hitsurface is not null { tbest = t; firstsurface = hitsurface; } } return hitsurface, tbest; } 30

31 Image so far With eye ray generation and scene intersection for 0 <= iy < ny for 0 <= ix < nx { ray = camera.getray(ix, iy); c = scene.trace(ray, 0, +inf); image.set(ix, iy, c); } Scene.trace(ray, tmin, tmax) { surface, t = surfs.intersect(ray, tmin, tmax); if (surface!= null) return surface.color(); else return black; } 31

32 Shading Compute light reflected toward camera Inputs: eye direction light direction (for each of many lights) surface normal surface parameters (color, shininess, ) 32

33 Diffuse reflection Top face of cube receives a certain amount of light Top face of 60º rotated cube intercepts half the light In general, light per unit area is proportional to cos θ = l n 33

34 Lambertian shading illumination from source diffusely reflected light diffuse coefficient 34

35 Lambertian shading Produces matte appearance [Foley et al.] 35

37 Image so far Scene.trace(Ray ray, tmin, tmax) { surface, t = hit(ray, tmin, tmax); if surface is not null { point = ray.evaluate(t); normal = surface.getnormal(point); return surface.shade(ray, point," normal, light); } else return backgroundcolor; } Surface.shade(ray, point, normal, light) { v = normalize(ray.direction); l = normalize(light.pos point); // compute shading } 37

38 Shadows Surface is only illuminated if nothing blocks its view of the light. With ray tracing it s easy to check just intersect a ray with the scene! 38

39 Image so far Surface.shade(ray, point, normal, light) { shadray = (point, light.pos point); if (shadray not blocked) { v = normalize(ray.direction); l = normalize(light.pos point); // compute shading } return black; } 39

40 Shadow rounding errors Don t fall victim to one of the classic blunders: What s going on? hint: at what t does the shadow ray intersect the surface you re shading? 40

41 Shadow rounding errors Solution: shadow rays start a tiny distance from the surface Do this by moving the start point, or by limiting the t range 41

42 Multiple lights Important to fill in black shadows Just loop over lights, add contributions Ambient shading black shadows are not really right one solution: dim light at camera alternative: add a constant ambient color to the shading 42

43 Image so far shade(ray, point, normal, lights) { result = ambient; for light in lights { if (shadow ray not blocked) { result += shading contribution; } } return result; } 43

44 Specular shading (Blinn-Phong) Intensity depends on view direction bright near mirror configuration 44

45 Specular shading (Blinn-Phong) Close to mirror half vector near normal Measure near by dot product of unit vectors 3 specularly reflected light specular coefficient 45

46 Phong model plots Increasing n narrows the lobe [Foley et al.] 46

47 Specular shading [Foley et al.] 47

48 Diffuse + Phong shading 48

49 Ambient shading Shading that does not depend on anything add constant color to account for disregarded illumination and fill in black shadows ambient coefficient reflected ambient light 49

50 Putting it together Usually include ambient, diffuse, Phong in one model The final result is the sum over many lights 50

51 Mirror reflection Consider perfectly shiny surface there isn t a highlight instead there s a reflection of other objects Can render this using recursive ray tracing to find out mirror reflection color, ask what color is seen from surface point in reflection direction already computing reflection direction for Phong Glazed material has mirror reflection and diffuse where L m is evaluated by tracing a new ray 51

52 Mirror reflection Intensity depends on view direction reflects incident light from mirror direction 3 52

53 Diffuse + mirror reflection (glazed) (glazed material on floor) 53

54 Ray tracer architecture 101 You want a class called Ray point and direction; evaluate(t) possible: tmin, tmax Some things can be intersected with rays individual surfaces groups of surfaces (acceleration goes here) the whole scene make these all subclasses of Surface limit the range of valid t values (e.g. shadow rays) Once you have the visible intersection, compute the color may want to separate shading code from geometry separate class: Material (each Surface holds a reference to one) its job is to compute the color 54

55 Architectural practicalities Return values surface intersection tends to want to return multiple values t, surface or shader, normal vector, maybe surface point in many programming languages (e.g. Java) this is a pain typical solution: an intersection record a class with fields for all these things keep track of the intersection record for the closest intersection be careful of accidental aliasing (which is very easy if you re new to Java) Efficiency what objects are created for every ray? try to find a place for them where you can reuse them. Shadow rays can be cheaper (any intersection will do, don t need closest) but: First Get it Right, Then Make it Fast 55

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