Wil Yegelwel on The Rendering Equation

Illumination in computer graphics deals with calculating the color of each pixel on the screen when trying to render a photorealistic scene. The problem is that generating increasingly realistic renderings requires a lot of processing and so a tradeoff must be made between compute time and image quality. We will first look at two foundational papers from the 70s that built “good enough” models that were fast to render. Finally, we look at The Rendering Equation that provides a single model to encompass many effects of lighting assuming you are willing to pay the processing cost.

Transcript

1. Three Papers on Illumination 1
2. Who am I? • Engineer at Two Sigma working on distributed systems problems • Studied computer graphics in school • Just got a kitten, she is adorable 2
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4. What we are trying to do By Gilles Tran - http://www.oyonale.com/modeles.php?lang=en&page=40, Public Domain, https://commons.wikimedia.org/w/index.php?curid=745313 4
5. Lighting effects – diffuse 5
6. Lighting effects – specular 6
7. Lighting effects – (soft) shadows 7
8. Lighting effects – refraction and reflection 8
9. Lighting effects – caustics http://madebyevan.com/webgl-water 9
10. Lighting effects – color bleed http://madebyevan.com/webgl-path-tracing/ 10
11. What we are going to talk about • Background • Continuous shading of curved surfaces – Henri Gouraud • An Improved Illumination Model for Shaded Display – Turner Whitted • The Rendering equation – James T. Kajiya 11
12. The model 12
13. 3d scenes Mesh of triangles Functions https://www.shadertoy.com/view/4sS3zG 13
14. Rendering Items first for each item for each pixel if pixel between eye and item draw something here Pixels first for each pixel for each item if ray from eye through pixel hits item draw something here 14
15. Some math, Vectors and Rays Direction and magnitude Starting point and direction Vector Ray 15
16. Some math, Normals N N 16
17. Lighting models (, %) X’ x L 17
18. Lighting models – Lambert cosine law Incoming light Outgoing light ∗ = cos ( ℎ ) Constant (viewer independent) 18
20. Machbanding By The original uploader was Aliwiki at French Wikipedia - Transferred from fr.wikipedia to Commons by Korrigan using CommonsHelper., FAL, https://commons.wikimedia.org/w/index.php?curid=4770182 20
21. Mach bands 21
22. Its 1971… We are here 22
23. Put it in hardware! 23
24. for each triangle for each row in image for each col in row if pixel between eye and item draw something here Hidden surface removal problem 1-100 ~400 ~600 O(T*H*W) 24
25. Hidden surface removal (simple) *Project triangles into screen space* for each row in image for each triangle if row intersects triangle line segments for col between intersected line segments draw something here* * Consider depth 25
26. Gouraud Illumination model = < + ? @ A B B∈EFGHIJ N L 26
27. The one cool trick… A B C row 27
28. How well does it work? 28
30. Reflection 30
31. Refraction By JrPol - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/i ndex.php?curid=38535231 By Josell7 - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curi d=21670922 31
32. The model = < + ? @ A B B∈EFGHIJ + J + I 32
33. The model 33
34. Ray tracing By Henrik - Own work, GFDL, https://commons.wikimedia.org/w/index.php?curid=3869326 34
39. The rendering equation , % = , % , % + N , %, %% %, %% ′′ Q 39
40. The rendering equation , % = , % , % + N , %, %% %, %% ′′ Q 40
41. , % = , % , % + N , %, %% %, %% ′′ Q The rendering equation 41
42. , % = , % , % + N , %, %% %, %% ′′ Q The rendering equation 42
43. The rendering equation , % = , % , % + N , %, %% %, %% ′′ Q 43
44. The rendering equation , % = , % , % + N , %, %% %, %% ′′ Q 44
45. Approximation 45
46. Approximation 46
47. Previous methods as approximation 47
48. Previous methods as approximation 48
49. Integration 49
50. Expectation = N () = N 50
51. Expectation = N ≈ 1 @ F W FXY () = N ≈ 1 @ (F ) W FXY 51
52. Monte Carlo integration N 52
53. Monte Carlo integration N = N 53
54. Monte Carlo integration N = N = 54
55. Monte Carlo integration N = N = () = N Where = () 55
56. Monte Carlo integration N = N = () = N Where = () ≈ 1 @ F (F ) W FXY 56
57. Monte Carlo integration example 57
58. Monte Carlo Integration + Rendering Equation N , %, %% %, %% ′′ Q X’ 58
59. Path Tracing X’ 59
60. Variance reduction 60
61. Path tracing example 61
62. Summary 1. Continuous shading of curved surfaces – Henri Gouraud 2. An Improved Illumination Model for Shaded Display – Turner Whitted 3. The Rendering equation – James T. Kajiya = < + ? @ A B B∈EFGHIJ , % = , % , % + N , %, %% %, %% ′′ Q = < + ? @ A B + J + I B∈EFGHIJ 62
63. Thank you Andy van Dam John “Spike” Hughes 63