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# CS 354 Global Illumination

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April 19, 2012; CS 354 Computer Graphics; University of Texas at Austin

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### CS 354 Global Illumination

1. 1. CS 354Global IlluminationMark KilgardUniversity of TexasApril 19, 2012
2. 2. CS 354 2 Today’s material  In-class quiz  On ray casting & tracing lecture  Lecture topic  Project 4  Global illumination
3. 3. CS 354 3 My Office Hours  Tuesday, before class  Painter (PAI) 5.35  8:45 a.m. to 9:15  Thursday, after class  ACE 6.302  11:00 a.m. to 12  Randy’s office hours  Monday & Wednesday  11 a.m. to 12:00  Painter (PAI) 5.33
4. 4. CS 354 4 Last time, this time  Last lecture, we discussed  Ray casting and tracing  This lecture  Global illumination  Projects  Project 3 due yesterday  Project 4 on ray tracing on Piazza  Due May 2, 2012
5. 5. CS 354 5 On a sheet of paper Daily Quiz • Write your EID, name, and date • Write #1, #2, #3 followed by its answer  Multiple choice: determining if a ray intersects a sphere  Multiple choice: With involves solving a distribution ray tracing, one can accomplish a) linear equation a) depth-of-field b) quadratic equation b) soft shadows c) system of linear equations c) motion blur d) cubic equation d) a. and b. but not c.  True or False: With ray tracing, the problem of anti- e) a., b., and c. aliasing doesn’t apply.
6. 6. CS 354 6 Solving the Rendering Equation with Ray Tracing  Ray “tracing” or “shooting”  Given a point and direction, sample radiance  Building block for global illumination algorithms  Use ray traces to approximate integrals  Conceptually, solving the rendering equation  Intractable to shoot every ray  Instead use Monte Carlo techniques
7. 7. CS 354 7 Aspects of Modeling Light  Geometric optics  Our concern  Assume light travels instantaneously  Light travels in straight lines  Light not influenced by gravity or magnetic fields  Wave optics  Diffraction, etc.  Quantum optics  Sub-microscopic
8. 8. CS 354 8 Rendering Equation  Theory for light-surface interactions Lo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′ Ω
9. 9. CS 354 9 Rendering Equation PartsLo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′ Ω  Lo = outgoing light from x in direction ω  x = point on a surface  ω = normalized outgoing light vector  λ = wavelength of light  t = time  Le = emitted light at x going towards from ω  n = surface normal at x
10. 10. CS 354 10 Rendering Equation Integral  ∫ = integrate over a region  Ω = region of a hemisphere  Together: “integrate overall the incoming directions for a hemisphere at a point x”  ωˊ = normalized incoming light vector  Li = incoming light at x coming from ωˊ  n = surface normal at x  (-ωˊ • n) = cosine of angle between incoming light and surface normal  dωˊ = differential of incoming angle
11. 11. CS 354 11 Bidirectional Reflectance Distribution Function  Ratio of differential outgoing (reflected) radiance to differential incoming irradiance dLr (x, ω , λ , t ) dLr (x, ω , λ , t ) f r (x, ω ′, ω , λ , t ) = = dEi (x, ω ′, λ , t ) dLi (x, ω ′, λ , t ) (−ω ′ • n) dω ′  Physically based BRDF properties L = radiance  Must be non-negative E = irradiance  Must be reciprocal  Swap incoming & relected directions generates same ratio  Conserves energy  Integrating over entire hemisphere must be ≤ 1
12. 12. CS 354 12 Cosine Weighting for Irradiance  At shallow angles, incoming light spreads over a wider area of the surface  Thus spreading the energy  Thus dimming the received light
13. 13. CS 354 13 Measuring BRDFs Empirically  Gonioreflectometer is device to measure BRDFs
14. 14. CS 354 14 BRDF Decomposition  Broad characterization of BRDF appearance Pure diffuse Pure specular Glossy
15. 15. CS 354 15 Common BRDFs  Diffuse model, a.k.a. Lambertian f diffuse (x, Ψ ↔ Θ, λ , t ) = k d ( R • Θ) n  Phongphong (x, Ψ ↔a.k.a.=Reflection dSpecular f model, Θ, λ , t ) k s +k N •Ψ ( N • H )n f blinn (x, Ψ ↔ Θ, λ , t ) = k s + kd  Blinn model, a.k.a. Half-angle Specular N •Ψ
16. 16. CS 354 16 Hemisphere over a Surface  Incident radiance is integrated over the hemisphere
17. 17. CS 354 17 Rendering Equation Interpretation  Recursive equation Lo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′ Ω outgoing light “sums up” all incoming light for all directions on hemisphere  Integral over hemisphere (Ω)  Limitations  Treats wavelengths all independent  Treats time in Newtonian way  Ignores volumetric and subsurface scattering  More complex models can incorporate these aspects of light  Impractical for actual computer graphics rendering
18. 18. CS 354 18 Alternate Rendering Equation  Integrate over all points in the scene  Instead of all directions for a hemisphere Lo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f (x, ω y∈Γ r yx , ω , λ , t ) L(y, ωyx , λ , t ) G (x, y ) dy outgoing light “sums up” all incoming light for all surfaces, modulated by visibility  Notes  G(x,y) returns the visibility (occlusion) between points x & y  Integral over all surface points (Γ) in the scene
19. 19. CS 354 19 Two Versions of Rendering Equation Occlusion (G) is zero o Le (x, ω , λ , t ) + Le (x, ω , λ , t ) +Lo (x, ω , λ , t ) = ∫ Ω f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′ Lo (x, ω , λ , t ) = ∫ f r (x, ωyx , ω , λ , t ) L(y, ωyx , λ , t ) G (x, y ) dy y∈Γ Integrate over hemisphere Integrate over all surface points
20. 20. CS 354 20 Discrete Integral Approximation  Uniform sampling  Shown in 1D, but we need to integrate over hemisphere (solid angles)
21. 21. CS 354 21 Hemisphere Sampling  Vary Euler angles
22. 22. CS 354 22 Monte Carlo Integrations  Randomly sample the rendering equation integrals  Sample space based on probability distribution function  Compute estimated variance to know when to stop  Importance sampling  Try to weight probability function to match variance
23. 23. CS 354 23 Ray Casting for Initial Samples
24. 24. CS 354 24 Path Tracing via Simple Stochastic Ray Tracing
25. 25. CS 354 25 Two Versions of Rendering Equation (again) Occlusion (G) is zero o Le (x, ω , λ , t ) + Le (x, ω , λ , t ) +Lo (x, ω , λ , t ) = ∫ Ω f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′ Lo (x, ω , λ , t ) = ∫ f r (x, ωyx , ω , λ , t ) L(y, ωyx , λ , t ) G (x, y ) dy y∈Γ Integrate over hemisphere Integrate over all surface points
26. 26. CS 354 26 Sufficient Shadow Ray Sampling
27. 27. CS 354 27 Light Tracing  Trace rays from the light  Contribution rays accumulate image samples
28. 28. CS 354 28 Light Tracing Needs Lots of Samples
29. 29. CS 354 29 How many rays to shoot?  Two approaches  Fixed number of ray evaluations  Limits the depth of recursion  Means some paths can never be explored  Implies bias in result, since favors short paths  Russian Roulette approach  Roll dice to see if ray tracing should continue  Reduces bias in rendering result  Allows “obscure” paths to be explored
30. 30. CS 354 30 Radiosity History  Used for thermal engineering since 1960s  Introduced to computer graphics in 1984  SIGGRAPH paper: “Modeling the Interaction of Light Between Diffuse Surfaces”  Goral, Torrance, Greenberg, Battaile (Cornell)  Reduces to solving a linear system of equations  In simple formulation, just works for diffuse surfaces  OK for building interiors
31. 31. CS 354 31 Radiosity Example
32. 32. CS 354 32 Scene Diced into Uniform Patches
33. 33. CS 354 33 Radiosity Method Self-emitted reflectivity total radiosity radiosity Form factor Approximation to general Rendering Equation
34. 34. CS 354 34 Form Factors  Form factor between i,j is Fij  Tells how much of patch A is visible from patch B  Indicates how much radiance can be transferred 1 Fij = ∫ ∫ K ( x, y ) dAy dAx Ai Si Si K ( x , y ) = G ( x, y ) V ( x, y ) cos(Θ xy , N x ) cos(−Θ xy , N y ) G ( x, y ) = π rxy 2 V ( x, y ) =< visibility between x & y >
35. 35. CS 354 35 Form Factor Visualized  Mutual visibility of two patches  Don’t have to be same size
36. 36. CS 354 36 Differential Patches and Foreshortening  Cosine fall-off applies  And assuming Lambertian surfaces foreshortening
37. 37. CS 354 37 Hemi-cube Approach to Sampling Form Factors  Think about drawing hemi-cube at every point  What you see is an input for radiosity  Known as “instant radiosity”
38. 38. CS 354 38 Interpolation of Radiosity Solution Constant Approximation “true” solution Quadratic Approximation flat smooth
39. 39. CS 354 39 Iterative Radiosity Solver 1 iteration 4 iterations 16 iterations 64 iterations 252 iterations
40. 40. CS 354 40 Participating Media  Light doesn’t just interact with surfaces  Gaseous phenomena affects lighting  Usually handled statistically emission in-scattering [Jarosz et.al. 2008] absorption out-scattering [FogShop 2007]
41. 41. CS 354 41 Participating Media Assumptions  Simpler  Single scattering  Uniform  Ignore shadowing  More complex  Multiple scatterings  Non-uniform  Account for shadowing
42. 42. CS 354 42 Photon Mapping  Two-pass global illumination algorithm  Developed by Henrick Jensen (1996)  Two passes  Randomly distribute photons around the scene  Called “photon map construction”  Render treating photons as mini-light sources  Capable of efficiently generating otherwise very expensive effects  Caustics  Diffuse inter-reflections, such as color bleed  Sub-surface scattering
43. 43. CS 354 43 Photon Mapping Examples without diffuse photo map caustics interreflection visualization sub-surface scattering diffuse interreflection
44. 44. CS 354 44 Next Class  Next lecture  Acceleration structures  How do we make the ray trace operation faster?  Reading  Chapter 5, 297-298  Chapter 11, 569-577  Project 4  Project 4 is a simple ray tracer  Due Wednesday, May 2, 2012