Lecture 10: Ray tracing

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1 Interactive Computer Graphics Lecture 10: Ray tracing Graphics Lecture 10: Slide 1 Some slides adopted from H. Pfister, Harvard Graphics Lecture 10: Slide 2 Direct and Global Illumination Direct illumination: A surface point receives light directly from all light sources in the scene. Computed by the direct illumination model. Global illumination: A surface point receives light after the light rays interact with other objects in the scene. Points may be in shadow. Rays may refract through transparent material. Computed by reflection and transmission rays. Graphics Lecture 10: Slide 3 Graphics Lecture 10: Slide 4 1

2 Albrecht Dürer s Ray Casting Machine Albrecht Dürer, 16 th century Arthur Appel, 1968 On calculating the illusion of reality, 1968 Cast one ray per pixel (ray casting). For each intersection, trace one ray to the light to check for shadows Only a local illumination model Developed for pen-plotters Graphics Lecture 10: Slide 5 Graphics Lecture 10: Slide 6 Ray casting Ray Casting cast ray Intersect all objects color = ambient term For every light cast shadow ray col += local shading term Graphics Lecture 10: Slide 7 Graphics Lecture 10: Slide 8 2

3 Turner Whitted, 1980 Turner Whitted, 1980 An Improved Illumination Model for Shaded Display, 1980 First global illumination model: An object s color is influenced by lights and other objects in the scene Simulates specular reflection and refractive transmission Graphics Lecture 10: Slide 9 Graphics Lecture 10: Slide 10 Recursive ray casting trace ray Intersect all objects color = ambient term For every light cast shadow ray col += local shading term If mirror col += k_refl * trace reflected ray If transparent col += k_trans * trace transmitted ray Does it ever end? Stopping criteria: Recursion depth: Stop after a number of bounces Ray contribution: Stop if reflected / transmitted contribution becomes too small Graphics Lecture 10: Slide 11 Graphics Lecture 10: Slide 12 3

4 Ray tracing: Primary rays For each ray we need to test which objects are intersecting the ray: If the object has an intersection with the ray we calculate the distance between viewpoint and intersection If the ray has more than one intersection, the smallest distance identifies the visible surface. Primary rays are rays from the view point to the nearest intersection point Local illumination is computed as before: Ray tracing: Secondary rays Secondary rays are rays originating at the intersection points Secondary rays are caused by rays reflected off the intersection point in the direction of reflection rays transmitted through transparent materials in the direction of refraction shadow rays Graphics Lecture 10: Slide 13 Graphics Lecture 10: Slide 14 Recursive ray tracing: Putting it all together Illumination can be expressed as Recursive Ray Tracing: Ray Tree primary ray primary ray secondary rays Graphics Lecture 10: Slide 15 Graphics Lecture 10: Slide 16 4

5 Precision Problems Precision Problems In ray tracing, the origin of (secondary) rays is often on the surface of objects Theoretically, the intersection point should be on the surface Practically, calculation imprecision creeps in, and the origin of the new ray is slightly beneath the surface Result: the surface area is shadowing itself Graphics Lecture 10: Slide 17 Graphics Lecture 10: Slide 18 ε to the rescue... Check if t is within some epsilon tolerance: if abs(µ) < ε point is on the surface else point is inside/outside Choose the ε tolerance empirically Move the intersection point by epsilon along the surface normal so it is outside of the object Check if point is inside/outside surface by checking the sign of the implicit (sphere etc.) equation Mirror reflection Compute mirror contribution Cast ray in direction symmetric wrt. normal Multiply by reflection coefficient (color) Graphics Lecture 10: Slide 19 Graphics Lecture 10: Slide 20 5

6 Mirror reflection Mirror reflection Don t for get to add epsilon to the ray primary ray n secondary ray v φ in φ out v Graphics Lecture 10: Slide 21 Graphics Lecture 10: Slide 22 Mirror reflection To calculate illumination as a result of reflections calculate the direction of the secondary ray at the intersection of the primary ray with the object. given that n is the unit surface normal v is the direction of the primary ray v is the direction of the secondary ray as a result of reflections v ' = v " ( 2v! n) n Mirror reflection The v, v and n are unit vectors and coplanar so: v = α v + β n Taking the dot product with n yields the eq.: n v = α v n + β = v n Requiring v to be a unit vector yields the second eq.: 1 = v v = α α β v n + β 2 Solving both equations yields: v ' = v " ( 2v! n) n Graphics Lecture 10: Slide 23 Graphics Lecture 10: Slide 24 6

7 Transparency Compute transmitted contribution Cast ray in refracted direction Multiply by transparency coefficient Refraction The angle of the refracted ray can be determined by Snell s law: k 1 sin(! 1) = k2 sin(! 2) η 1 is a constant for medium 1 η 2 is a constant for medium 2 φ 1 is the angle between the incident ray and the surface normal φ 2 is the angle between the refracted ray and the surface normal v Medium 2 (eg glass) n φ 1 Medium 1 (eg air) φ 2 v Graphics Lecture 10: Slide 25 Graphics Lecture 10: Slide 26 Refraction Refraction In vector notation Snell s law can be written: k1( v! n) = k2( v'! n) The direction of the refracted ray is v " = # %, 1 (n\$ v) 2 + # 2 % ( / ( '. 2 ' * +1 + n\$ v1 # 2. & # - 1 ) 1 \$ n + v * ' * & 0 ) This equation only has a solution if % (n" v) 2 >1# \$ ( 2 ' * & ) This illustrates the physical phenomenon of the limiting angle: if light passes from one medium to another medium whose index of refraction is low, the angle of the refracted ray is greater than the angle of the incident ray if the angle of the incident ray is large, the angle of the refracted ray is larger than 90 o the ray is reflected rather than refracted \$ 1 2 Graphics Lecture 10: Slide 27 Graphics Lecture 10: Slide 28 7

8 Refraction Make sure you know whether you are entering or leaving the transmissive material Amount of reflection and refraction Traditional (hacky) ray tracing Constant coefficient reflectioncolor Component per component multiplication Better: Mix reflected and refracted light according to the Fresnel factor. L = k fresnel L reflected + (1" k fresnel )L refracted Graphics Lecture 10: Slide 29 Graphics Lecture 10: Slide 30 Fresnel factor Fresnel factor More reflection at grazing angle Graphics Lecture 10: Slide 31 Graphics Lecture 10: Slide 32 8

9 Schlick s Approximation Example Schlick s approximation k fresnel (") = k fresnel (0) + (1# k fresnel (0))(1# (n\$ l)) 5 k fresnel (0) = Fresnel factor at zero degrees Choose k fresnel (0) = 0.8, this will look like stainless steel Graphics Lecture 10: Slide 33 Graphics Lecture 10: Slide 34 How do we add shadows? How do we add shadows? Graphics Lecture 10: Slide 35 Graphics Lecture 10: Slide 36 " s = 0 if light source is obscured # \$ 1 if light source is not obscured 9

10 Shadows: Problems? Example Make sure to avoid self-shadowing Graphics Lecture 10: Slide 37 Graphics Lecture 10: Slide 38 Shadows One shadow ray per intersection per point light source Soft shadows Multiple shadow rays to sample area light source Graphics Lecture 10: Slide 39 Graphics Lecture 10: Slide 40 10

11 Reflection: Conventional ray tracing One reflection per intersection Reflection: Conventional ray tracing How can we create effects like this? Graphics Lecture 10: Slide 41 Graphics Lecture 10: Slide 42 Reflection: Monte Carlo ray tracing Glossy surfaces Random reflection rays around mirror direction Graphics Lecture 10: Slide 43 Graphics Lecture 10: Slide 44 11

12 Ray tracing Cast a ray from the eye through each pixel Trace secondary rays (light, reflection, refraction) Monte-Carlo Ray Tracing Cast a ray from the eye through each pixel Cast random rays from the visible point Accumulate radiance contribution Graphics Lecture 10: Slide 45 Graphics Lecture 10: Slide 46 Monte-Carlo Ray Tracing Cast a ray from the eye through each pixel Cast random rays from the visible point Recurse Monte-Carlo Ray Tracing Cast a ray from the eye through each pixel Cast random rays from the visible point Recurse Graphics Lecture 10: Slide 47 Graphics Lecture 10: Slide 48 12

13 Monte Carlo Path Tracing Trace only one secondary ray per recursion But send many primary rays per pixel Monte Carlo Ray Tracing Send rays to light Graphics Lecture 10: Slide 49 Graphics Lecture 10: Slide 50 Example Example 1 sample per pixel 256 samples per pixel Graphics Lecture 10: Slide 51 Graphics Lecture 10: Slide 52 13

14 Some cool pictures Graphics Lecture 10: Slide 53 Some cool pictures Graphics Lecture 10: Slide 55 Some cool pictures Graphics Lecture 10: Slide 54 Some cool pictures Graphics Lecture 10: Slide 56 14

15 Some cool pictures Graphics Lecture 10: Slide 57 took 4.5 days to render! 15

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