CSE168 Computer Graphics II, Rendering. Spring 2006 Matthias Zwicker

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1 CSE168 Computer Graphics II, Rendering Spring 2006 Matthias Zwicker

2 Last time Global illumination Light transport notation Path tracing Sampling patterns

3 Reflection vs. rendering equation Reflection equation Given a point and incident radiance, integrate over the hemisphere

4 Direct illumination Incident radiance given by light source [Wojciech Jarosz]

5 Reflection vs. rendering equation Rendering equation Find radiance function such that reflection equation is satisfied simultaneously at each point

6 Global illumination Reflection equation satisfied at each point Incident radiance [Wojciech Jarosz] Reflected radiance

7 Path tracing Based on a recursive expansion of the rendering equation Compute incident radiance as integral of radiance transported along all paths connecting and a point on a light source

8 Path tracing Integration of radiance over all paths Length 3 Length 4 Length 7

9 Path tracing Monte Carlo integration where are radiance, PDF of random path through,

10 Construct random path

11 Construct random path

12 Construct random path

13 Construct random path

14 Construct random path

15 Construct random path

16 Russian roulette Introduce probaility to terminate path Return 0 with probability Sample as usual otherwise

17 Russian roulette Introduce probaility to terminate path Return 0 with probability Sample as usual otherwise Re-weight samples

18 Contribution of a path PDF for path of length is Russian roulette for path termination Terminate path of length with probability Usually

19 Contribution of a path PDF for sampling directions Uniform Cosine weighted Importance sampling the BRDF PDF for sampling light sources Uniform Multiple importance sampling

20 Contribution of a path Path of length given by points

21 Contribution of a path Path of length given by points

22 Contribution of a path Notation Radiance transported along path

23 Path tracing Monte Carlo integration Denote

24 Measurement equation Value of a pixel where pixel filter

25 Measurement equation Monte Carlo integration

26 Measurement equation Monte Carlo integration Last equality holds because

27 Measurement equation Bottom line: pixel value is weighted sum of random paths through pixel

28 Measurement equation Generalization If we have random variable then we can compute any measurement for any importance function

29 Path tracing algorithm Construct paths incrementally starting at the eye, dozens of rays for each pixel Incrementally update [Cutler, Durand]

30 Path tracing algorithm Construct paths incrementally starting at the eye, dozens of rays for each pixel Incrementally update Shoot shadow rays at each path vertex [Cutler, Durand]

31 Path tracing algorithm Construct paths incrementally starting at the eye, dozens of rays for each pixel Incrementally update Shoot shadow rays at each path vertex [Cutler, Durand]

32 Path tracing Unbiased Expected value for each pixel is the correct solution of the rendering equation Consistent If we shoot infinitely many rays, we will get the correct solution

33 Light transport notation Light L Diffuse D Specular S Eye E Example

34 Sampling patterns Stratified sampling avoid sample clumping Jittered grids N-rooks sampling Quasi Monte Carlo

35 Today Photon mapping

36 Problems with path tracing 10 paths/pixel [Wann Jensen]

37 Problems with path tracing 10 paths/pixel [Wann Jensen]

38 Problems with path tracing 100 paths/pixel [Wann Jensen]

39 Problems with path tracing 1000 paths/pixel [Wann Jensen]

40 Removing noise More samples/rays per pixel (but slow convergence, ) Better sampling patterns (stratified sampling, importance sampling) Adaptive sampling

41 Photon mapping Bi-directional paths Construct paths not only from the eye, but also from the light sources Caching Cache photons distributed along paths from the light sources Interpolation Interpolate radiance from cached photons

42 Photon mapping Photon emission and transport [Cutler, Durand]

43 Photon mapping Photon caching [Cutler, Durand]

44 Photon mapping Spatial data structure for fast access [Cutler, Durand]

45 Photon mapping Radiance estimation [Cutler, Durand]

46 What is a photon? A photon has a location and a direction A photon stores such that the expected value is class photon { vector3 x vector3 w vector3 a }

47 Photon emission and transport Very much like path tracing, only starting at light sources instead of at the eye! [Cutler, Durand]

48 Photon emission Rectangular diffuse light Power Area Radiance Choose uniform PDF Initialize photon

49 Photon transport Russian roulette with probability to abort after bounces At each surface, choose random and update photon

50 Choosing random directions Sampling BRDFs Uniform Cosine weighted Importance sampling as in path tracing

51 Photon mapping Photon caching [Cutler, Durand]

52 Taking measurements Use the average incident radiance to estimate reflected radiance

53 Taking measurements Measurement equation with averaging kernel W Because we made sure that introduces bias

54 Taking measurements To measure reflected radiance, we choose for The estimate of the measurement is where

55 Direct illumination

56 Indirect illumination

57 Caustics

58 Next time Photon data structure and efficient access Photon mapping rendering algorithms

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