1. 111/17/2015 17:21 Graphics II 91.547 Global Rendering and Radiosity Session 9

2. 211/17/2015 17:21 A More Sophisticated View of The Nature of Light Ray Oriented View Used in Phong shading and in ray tracing Single direction Zero width Notion of intensity Flux intensity view Vector field represents energy flow per unit time per unit area Finite beam widths

3. 311/17/2015 17:21 Energy Flow across a Surface

4. 411/17/2015 17:21 Flux and Energy Conservation + + + = Emission + in scattering = streaming + outscattering + absorption

5. 511/17/2015 17:21 Radiance:Energy emitted from a surface per unit projected area, per unit solid angle of direction Energy emitted per surface area, per steradian (solid angle) Therefore: p dd

6. 611/17/2015 17:21 Radiosity: Energy emitted from a surface per unit area Therefore: p

7. 711/17/2015 17:21 Irradiance: Energy arriving at a surface per unit area p

8. 811/17/2015 17:21 Reflectance: Bi-Directional Reflectance Function (BDRF) n

9. 911/17/2015 17:21 The Radiance Equation At any surface: radiance = emitted radiance + total reflected radiance For any incoming direction the reflected radiance in direction is the Irradiance multiplied by the BRDF: Integrating over the hemisphere of all incoming directions at p gives: The radiance equation for outgoing radiance is therefore:

10. 1011/17/2015 17:21 The Radiance Equation: All we really need to know for rendering? Material Surface Properties Light Sources Plenoptic Function

11. 1111/17/2015 17:21 Types of Solution to Radiance Equation LocalGlobal View Dependent OpenGL Phong Lighting Recursive Ray Tracing Monte Carlo Ray Tracing View Independent Flat or Smooth Defined Color (No Lighting) Radiosity Monte Carlo Photon Tracing

12. 1211/17/2015 17:21 The Radiance Equation: Defined Color: No Lighting Solved at Vertices All objects are “emitters” according to glColor*() No reflections considered

13. 1311/17/2015 17:21 The Radiance Equation: OpenGL Lighting Model Objects can be emitters Restricted to Point Light Sources Phong BDRF Solved at Vertices

14. 1411/17/2015 17:21 The Radiance Equation: Recursive Ray Tracing Objects can be emitters Restricted to Point Light Sources + Single Reflected & Refracted Ray Phong BDRF Solved for Rays Through Pixels

15. 1511/17/2015 17:21 The Radiance Equation: Monte Carlo Ray Tracing Objects can be emitters Rays cast recursively, chosen according to BDRF Actual BDRF Solved for Rays Through Pixels

16. 1611/17/2015 17:21 The Radiance Equation: Radiosity Objects can be emitters – emission assumed constant and independent of angle Constant reflectivity Assumed constant over surface “patches” independent of angle Assumed constant over surface “patches” independent of angle

17. 1711/17/2015 17:21 Perfectly Diffuse Reflectivity Energy is reflected uniformly in all directions

18. 1811/17/2015 17:21 Radiosity 0 Based on the theory of heat transfer (energy) between surfaces (Siegel 1984) 0 Adapted to computer graphics by Goral et al. (Goral 1984) 0 Based upon conservation of energy 0 Surfaces are assumed to be perfectly diffuse (lambertian) reflectors 0 Environment is divided into “patches” 0 Radiosity of a patch is the total rate of energy leaving the patch -Assumed constant over the patch -Equal to sum of emitted and reflected energy 0 Interaction among patches modeled by unitless form factors -F ij defined as the fraction of energy leaving dA i that arrives at dA j

19. 1911/17/2015 17:21 The Basic Radiosity Relationship Radiosity x area = emitted energy + reflected energy For an environment divided into n patches: (reciprocity)

20. 2011/17/2015 17:21 Resulting System of Equations

21. 2111/17/2015 17:21 Stages in Radiosity Solution Discretized environment Form factor calculations Full matrix solution Standard renderer Change scene geometry Change colors or lighting Change view Specific View

22. 2211/17/2015 17:21 Calculating the Form Factors: Energy reaching A j from A i Differential energy leaving Ai that reaches Aj is given by: Solid angle subtended by Aj at Ai can be expressed: Substituting gives:

23. 2311/17/2015 17:21 Calculating the Form Factors: Calculating the Energy Fraction

24. 2411/17/2015 17:21 The Nusselt Analogue (Siegel 1984) Patch Projection onto surface of hemisphere Projection onto base of circle

25. 2511/17/2015 17:21 Equivalent Projection Areas

26. 2611/17/2015 17:21 Hemicube Patch i Patch j Projection of patch j onto hemicube “pixels”

27. 2711/17/2015 17:21 Summing Delta Form Factors on Pixels onto which A j Projects Patch i Patch j

28. 2811/17/2015 17:21 Hemicube Algorithm Handling Occlusion Issue Patch i Patch j Patch k Hemicube

29. 2911/17/2015 17:21 Stages in Radiosity Solution Discretized environment Form factor calculations Full matrix solution Standard renderer Change scene geometry Change colors or lighting Change view Specific View N x N Computation Cost Storage N x N

30. 3011/17/2015 17:21 “Gathering”

31. 3111/17/2015 17:21 “Shooting”

32. 3211/17/2015 17:21 Progressive Refinement Radiosity Algorithm repeat for (each patch i) [Position a hemicube on patch I and calculate form factors Fij for the first iteration] for (each patch j ( j!=I )) do  rad =  j  Bi Fij Ai/Aj  Bj =  Bj +  rad Bj=Bj+  rad  Bi=0 until convergence

33. 3311/17/2015 17:21 Number of Patches = 124

34. 3411/17/2015 17:21 Number of Patches = 829

35. 3511/17/2015 17:21 Number of Patches = 124 Number of Elements = 829

36. 3611/17/2015 17:21 Number of Patches = 58 Number of Elements = 1135

37. 3711/17/2015 17:21 Comparison of Images 124 Patches 5.36 min. 829 Patches 96.46 min. 124 Patches, 829 Elements 32.6 min.58 Patches, 1135 elements 23.59 min

38. 3811/17/2015 17:21 Rendering Caustics Arvo, 1986

39. 3911/17/2015 17:21 Combining Radiosity and Ray Tracing

40. 4011/17/2015 17:21 Combined Radiosity and Ray Tracing