1. 11 Introduction to Global Illumination Overview Overview Radiometry Radiometry The rendering equation The rendering equation Monte Carlo Monte Carlo Overview Overview Radiometry Radiometry The rendering equation The rendering equation Monte Carlo Monte Carlo

2. 22 Image synthesis scene description scene description surface radiance surface radiance Deterministic and/or stochastic simulation. Deterministic and/or stochastic simulation.

3. 33 Stages of light transport luminaire blocker Direct illumination Indirect illumination

4. 44 An example of global illumination Lischinski, Tampieri, and Greenberg 1993

5. 55 Photo-realistic rendering

6. 66

7. 77

8. 88 Types of surface scattering diffuse directional diffuse specular

9. 99 Directional dependence highly directional

10. 1010 Defining radiance   dAdA dAdA dd dd r r u u f (r,u) cos  dA d  power crossing surface f (r,u) cos  dA d  power crossing surface Classical Definition Measure-Theoretic Radiant energy defines a measure on R 3 x S 2. Radiant energy defines a measure on R 3 x S 2. The associated density function is radiance. The associated density function is radiance.

11. 1111 Definition of radiance x x   (x,   ) f f is a scalar density function is a scalar density function radiance

12. 1212 dA Definition of radiance watts m 2 sr dd dd watts m2m2 m2m2 sr (x,   ) dA d  f f power =

13. 1313 Power from radiance dA Integrate over solid angle... Integrate over solid angle... and surface

14. 1414 IrradianceIrradiance dA power per unit area power per unit area

15. 1515 IrradianceIrradiance dA weighted integral over solid angle weighted integral over solid angle watts m2m2 m2m2

16. 1616 Simulating reflected light irradiance dd dd x x

17. 1717 Simulating reflected light irradiance radiance dfdf dfdf dd dd x x

18. 1818 dfdf dfdf dd dd df  d    x x

19. 1919 dfdf dfdf dd dd df  (  ’,  ) d  x x ’’ ’’  

20. 2020 dfdf dfdf dd dd x x watt m2m2 m2m2 m 2 sr

21. 2121 dfdf dfdf dd dd df  (  ’,  ) d  x x BRDF [sr -1 ]

22. 2222 dfdf dfdf f ’f ’ f ’f ’ df =  (  ’,  ) f ’ cos  ’ d  ’ d’d’ d’d’ ’’ ’’ x x

23. 2323 f f f =   (  ’,  ) f ’ cos  ’ d  ’   x x

24. 2424 f f x x

25. 2525 Formulating a balance equation light leaving a surface light leaving a surface reflected light reflected light emitted light emitted light = = + + EasyEasyHardHard

26. 2626 r’r’ r’r’ r ’’ u’u’ u’u’ u u  Classical formulation Balance equation in terms of radiance [Polyak, 1960] solid angle source term f f ( ( r r ' ',, u u ) )   f f 0 0 ( ( r r ' ',, u u ) )   k k ( ( r r ' ' ; ; u u ' '   u u ) ) f f ( ( r r ' ' ' ',, u u ' ' ) ) cos   d d   ( ( u u ' ' ) )     measure on sphere

27. 2727 Classical formulation Important features of the classical formulation: f f ( ( r r ' ',, u u ) )   f f 0 0 ( ( r r ' ',, u u ) )   k k ( ( r r ' ' ; ; u u ' '   u u ) ) f f ( ( r r ' ' ' ',, u u ' ' ) ) cos   d d   ( ( u u ' ' ) )     new measure implicit function r’r’ r’r’ r ’’ u’u’ u’u’ u u  The point r ’’ depends on the point r ’ and the direction u ’. The point r ’’ depends on the point r ’ and the direction u ’. only part of the domain

28. 2828 Two linear operators f (r,u’) d d   ( ( u u ’)’) ’)’)     k  (r;u’ u) ( K f ) (r,u)  ( G f ) (r,u)  f (r’,u)   “cosine weighted” measure measure implicit function r’r’ r’r’ r ’’ u’u’ u’u’ u u 

29. 2929 Linear operators for global illumination Field Radiance Operator Operator GG KK Local Reflection Operator Operator surface radiance field radiance surface radiance

30. 3030 Another way to write the rendering equation f = s + KG f f = s + KG f Local Reflection Operator Operator SourceSource RadianceRadiance “Global”Operator“Global”Operator

31. 3131 Operator norms || K || p < 1 || G || p = 1 1) First law of thermodynamics 2) Second law of thermodynamics 3) Constancy of radiance along rays

32. 3232 IrradianceIrradiance dA weighted integral over solid angle weighted integral over solid angle watts m2m2 m2m2

33. 3333 A vector form of irradiance Integrate vectors over solid angle Integrate vectors over solid angle vector irradiance or light vector vector irradiance or light vector   r r

34. 3434 Lambert’s formula for irradiance ii ii ii ii   Vector Irradiance  r    M M 2 2     i i     i i r r polygonalLambertianluminairepolygonalLambertianluminaire P P

35. 3535 Ideal diffuse reflection Compute using Lambert’s formula

36. 3636 Ideal diffuse reflection   Boundary integral Boundary integral Eye

37. 3737 Ideal specular reflection Compute using ray tracing

38. 3838 Ideal specular reflection   Eye

39. 3939 Glossy reflection Use extended Lambert’s formula

40. 4040 Glossy reflection   Numerical quadrature Numerical quadrature Eye

41. 4141 Glossy reflection Monte Carlo   Eye

42. 4242 Boundary integral for glossy reflection   boundary integral boundary integral Eye

43. 4343 Applications of directional scattering glossytransmissionglossytransmission glossyreflectionglossyreflection directionalemissiondirectionalemission luminaireluminaire

44. 4444 A range of glossy reflections 10th-order moment 10th-order moment 45th-order moment 45th-order moment 400th-order moment 400th-order moment

45. 4545 Comparison with Monte Carlo Region used for comparison Region used for comparison order 65 order 300 order 1000

46. 4646 Comparison with Monte Carlo order 65 order 300 order 1000

47. 4747 Monte Carlo integration estimateirradianceestimateirradiance luminaireluminaire blockerblocker

48. 4848 Advantages of Monte Carlo Arbitrarily complex environments Arbitrarily complex environments Arbitrary reflectance functions Arbitrary reflectance functions Small memory requirements Small memory requirements Easily to distribute Easily to distribute Relatively easy to implement Relatively easy to implement Arbitrarily complex environments Arbitrarily complex environments Arbitrary reflectance functions Arbitrary reflectance functions Small memory requirements Small memory requirements Easily to distribute Easily to distribute Relatively easy to implement Relatively easy to implement

49. 4949 Monte Carlo sampling methods HemisphereHemispherePolygonPolygon PhongdistributionPhongdistribution

50. 5050 Light-ray tracing luminaire Rays represent photons that deposit energy on surfaces. No inverse-square law here! Rays represent photons that deposit energy on surfaces. No inverse-square law here!

51. 5151 Path tracing At each scattering event, estimate indirect irradiance with a single ray; continue recursively. At each scattering event, estimate indirect irradiance with a single ray; continue recursively. luminaire eye

52. 5252 Bidirectional path tracing Simultaneously follow paths from the light and the eye, looking for points that can “see” each other. Simultaneously follow paths from the light and the eye, looking for points that can “see” each other. luminaire eye

53. 5353 Metropolis path tracing Start with a path from eye to luminaire, then integrate over others by perturbing to nearby paths. Start with a path from eye to luminaire, then integrate over others by perturbing to nearby paths. luminaire eye

54. 5454 A Taxonomy of Errors Discrete Equation ApproximationApproximation Exact Equation Perturbed Equation PerturbationsPerturbations DiscretizationDiscretization ComputationComputation Radiance Function Space

55. 5555 Features of surface illumination luminaireluminaire blockerblocker gradientsgradients isolux contours extremaextrema

56. 5656 An example of meshing A simple environment A simple environment The underlying mesh The underlying mesh

57. 5757 Classical balance equation   (x,  ’  ) f(x’,  ’) cos  d  ’ ∫ ∫ f(x,  ) = s(x,  ) + A point on a distant visible surface A point on a distant visible surface radianceradiance

58. 5858 The change is a “pullback” The 2-form on the sphere is pulled back to the surface The 2-form on the sphere is pulled back to the surface x x

59. 5959 Change of variables d  = d A cos  ’ r 2r 2 r 2r 2 differential solid angle differential differentialareadifferentialarea

60. 6060 Kajiya’s rendering equation e(x,x’) +   (x,x’,x’’) I(x’,x’’) dx’’ ∫ ∫ I(x,x’) = g(x,x’) S S x, x’, x’’ are points on surfaces I = unknown intensity function

61. 6161 Kajiya’s rendering equation e(x,x’) +   (x,x’,x’’) I(x’,x’’) dx’’ ∫ ∫ I(x,x’) = g(x,x’) transport intensity geometry term transportemittancetransportemittance scattering function

62. 6262 Power from transport intensity dx Integrate over two surfaces Integrate over two surfaces dx’ source receiver

63. 6363 Radiance & transport intensity radiance transport intensity watts m 2 sr watts m4m4 m4m4

64. 6464 Radiance & transport intensity radiance transport intensity invariant along lines in free space invariant along lines in free space obeys inverse square law obeys inverse square law defined everywhere defined everywhere defined only at surfaces defined only at surfaces

65. 6565 Another way to write the rendering equation RadianceRadiance SourceSource TransportOperatorTransportOperator f = s + M f f = s + M f

66. 6666 The formal “solution” to the rendering equation f = ( I - M ) s f = ( I - M ) s IdentityoperatorIdentityoperator

67. 6767 The Neumann series f = s + M s + M 2 s + f = s + M s + M 2 s +...

68. 6868 L p -norms for radiance functions d d m m ( ( r r ) ) | f (r,u) | p s2s2 s2s2     M M d d   ( ( u u ) ) || f || p = [ [ ] ] “cosine weighted” measure measure 1 1 p p The collection of all functions with finite L p -norm is a Banach space The collection of all functions with finite L p -norm is a Banach space L L p p     m m    

69. 6969 Significance of the L p -norms total power watts symbol meaning units 1 1 f f rms radiance watts m sr 2 2 f f maximum radiance watts m 2 sr   f f

70. 7070 The L 1 -norm of K K K 1 1   max r r u u ' ' k k ( ( r r ; ; u u ' '     u u ) ) d d   ( ( u u ) ) maximal directional-hemispherical reflectance over all r and u ' maximal directional-hemispherical reflectance over all r and u ' dd dd u'u' u'u'   r r

71. 7171 The L -norm of K   max r r u u ' ' k k ( ( r r ; ; u u ' '     u u ) ) d d   ( ( u u ) ) maximal hemispherical-directional reflectance over all r and u ' maximal hemispherical-directional reflectance over all r and u ' dd dd   K K     r r u'u' u'u'

72. 7272 The L p -norms of K energy conservation K K 1 1  K K p p K K 1 1 K K     reciprocity K K p p K K   K K 1 1  ,, max { } } convexity

73. 7373 The G operator Surface radiance function Surface radiance function An enclosure. Equivalent flow through fictitious boundary Equivalent flow through fictitious boundary

74. 7474 Hilbert adjoint operators K* = K K* = K G* = G G* = G M = I - KG M = I - KG M* = I - GK M* = I - GK SinceSinceandand it follows that