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The Beam Radiance Estimate for Volumetric Photon Mapping Wojciech - PDF document

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The Beam Radiance Estimate for Volumetric Photon Mapping Wojciech Jarosz in collaboration with Matthias Zwicker and Henrik Wann Jensen University of California, San Diego April 17, 2008 Thursday, 6 September 12 Motivation

  1. Measurement Equation • Radiance, , is a 5D function over position, , and direction, . 27 Thursday, 6 September 12 * concisely written as an inner product between the radiance field and a weighting function * the weighting function is typically non-zero only within a small region of the whole domain

  2. Measurement Equation • Radiance, , is a 5D function over position, , and direction, . • A measurement is a weighted integral of radiance: 27 Thursday, 6 September 12 * concisely written as an inner product between the radiance field and a weighting function * the weighting function is typically non-zero only within a small region of the whole domain

  3. Measurement Equation • Radiance, , is a 5D function over position, , and direction, . • A measurement is a weighted integral of radiance: = � W e , L ⇥ 27 Thursday, 6 September 12 * concisely written as an inner product between the radiance field and a weighting function * the weighting function is typically non-zero only within a small region of the whole domain

  4. Measurement Equation = � W e , L ⇥ 27 Thursday, 6 September 12 * concisely written as an inner product between the radiance field and a weighting function * the weighting function is typically non-zero only within a small region of the whole domain

  5. Measurement Equation = � W e , L ⇥ • Many global illumination algorithms can be expressed this way 28 Thursday, 6 September 12 * concisely written as an inner product between the radiance field and a weighting function * the weighting function is typically non-zero only within a small region of the whole domain

  6. Measurement Equation = � W e , L ⇥ • Many global illumination algorithms can be expressed this way • path tracing 28 Thursday, 6 September 12 * concisely written as an inner product between the radiance field and a weighting function * the weighting function is typically non-zero only within a small region of the whole domain

  7. Measurement Equation = � W e , L ⇥ • Many global illumination algorithms can be expressed this way • path tracing • radiosity 28 Thursday, 6 September 12 * concisely written as an inner product between the radiance field and a weighting function * the weighting function is typically non-zero only within a small region of the whole domain

  8. Measurement Equation = � W e , L ⇥ • Many global illumination algorithms can be expressed this way • path tracing • radiosity • particle tracing [Veach98] 28 Thursday, 6 September 12 * concisely written as an inner product between the radiance field and a weighting function * the weighting function is typically non-zero only within a small region of the whole domain

  9. Photon Mapping as a Measurement • Photon tracing generates N weighted sample rays, or photons ( � i , x i , ⌅ ⇥ i ) • : ray ( x i , ⇤ � i ) • : corresponding weight α i 29 Thursday, 6 September 12 * Veach showed that given certain constraints on how the photons are distributed, unbiased measurements can be estimated as a weighted sum * Veach showed this for particle tracing on surfaces, and we extend his derivation to include participating media * Arbitrary measurements can be computed using the photon map

  10. Photon Mapping as a Measurement • Photon tracing generates N weighted sample rays, or photons ( � i , x i , ⌅ ⇥ i ) • : ray ( x i , ⇤ � i ) • : corresponding weight α i • Unbiased measurements can be estimated as a weighted sum of photons: � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 29 Thursday, 6 September 12 * Veach showed that given certain constraints on how the photons are distributed, unbiased measurements can be estimated as a weighted sum * Veach showed this for particle tracing on surfaces, and we extend his derivation to include participating media * Arbitrary measurements can be computed using the photon map

  11. Photon Mapping as a Measurement � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 29 Thursday, 6 September 12 * Veach showed that given certain constraints on how the photons are distributed, unbiased measurements can be estimated as a weighted sum * Veach showed this for particle tracing on surfaces, and we extend his derivation to include participating media * Arbitrary measurements can be computed using the photon map

  12. Conventional Radiance Estimate • Veach showed that the conventional radiance estimate (for surfaces) is a measurement, where blurs photon contributions across surfaces. • Also true for conventional volumetric radiance estimate, but blurs within volume. � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 30 Thursday, 6 September 12

  13. Arbitrary Measurements Using the Photon Map • However, any arbitrary weighting function can be used to compute a different measurement . � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 31 Thursday, 6 September 12 * if we can represent the quantity we want to compute as a measurement, then we can compute estimates of that quantity using the measurement equation

  14. Arbitrary Measurements Using the Photon Map • However, any arbitrary weighting function can be used to compute a different measurement . • If we can express our problem as a measurement, we can estimate it using the photon map. � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 31 Thursday, 6 September 12 * if we can represent the quantity we want to compute as a measurement, then we can compute estimates of that quantity using the measurement equation

  15. Volume Rendering Equation object x � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 32 Thursday, 6 September 12

  16. Volume Rendering Equation object � s 0                                                                                      x x s � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 32 Thursday, 6 September 12

  17. Volume Rendering Equation � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 32 Thursday, 6 September 12

  18. Volume Rendering Equation � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 33 Thursday, 6 September 12

  19. Volume Rendering Equation � where: L i ( x t , ⌃ � ) = p ( x t , ⌃ � t ) L ( x t , ⌃ � t ) d � t � , ⌃ Ω 4 π � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 33 Thursday, 6 September 12

  20. Volume Rendering Equation � where: L i ( x t , ⌃ � ) = p ( x t , ⌃ � t ) L ( x t , ⌃ � t ) d � t � , ⌃ Ω 4 π � s � T r ( x ↔ x t ) � s ( x t ) p ( x t , ⇥ t ) L ( x t , ⇥ t ) d ⇥ t dt ⇥ , Ω 4 π 0 � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 33 Thursday, 6 September 12

  21. Beam Radiance � s � T r ( x ↔ x t ) � s ( x t ) p ( x t , ⇥ t ) L ( x t , ⇥ t ) d ⇥ t dt ⇥ , Ω 4 π 0 � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 34 Thursday, 6 September 12

  22. Beam Radiance � s � T r ( x ↔ x t ) � s ( x t ) p ( x t , ⇥ t ) L ( x t , ⇥ t ) d ⇥ t dt ⇥ , Ω 4 π 0 � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 34 Thursday, 6 September 12

  23. Beam Radiance � s � T r ( x ↔ x t ) � s ( x t ) p ( x t , ⇥ t ) L ( x t , ⇥ t ) d ⇥ t dt ⇥ , Ω 4 π 0 � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 34 Thursday, 6 September 12

  24. Beam Radiance � s � T r ( x ↔ x t ) � s ( x t ) p ( x t , ⇥ t ) L ( x t , ⇥ t ) d ⇥ t dt ⇥ , Ω 4 π 0 � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 34 Thursday, 6 September 12

  25. Beam Radiance � s � T r ( x ↔ x t ) � s ( x t ) p ( x t , ⇥ t ) L ( x t , ⇥ t ) d ⇥ t dt ⇥ , Ω 4 π 0 Change integration over t into integration over V . � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 34 Thursday, 6 September 12

  26. Beam Radiance is a Measurement Change integration over t into integration over V . Use delta function, , to limit integration to line. � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 35 Thursday, 6 September 12 * delta function means we only get a useable estimate if a photon falls directly on the line

  27. Beam Radiance (Bias) In practice, use cylindrical blurring kernel, K . � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 36 Thursday, 6 September 12 * so in practice we replace the delta function with a blurring kernel which blurs radiance from the line into a cylinder. * the kernel allows photons that are not directly on the line to be used in the estimate * we have the freedom to choose the exact form of this blurring kernel

  28. Beam Radiance (Bias) � N ⇥ 1 ⇤ W e ( x i , � ⇥ i ) � i = � W e , L ⇥ E N i =1 36 Thursday, 6 September 12 * so in practice we replace the delta function with a blurring kernel which blurs radiance from the line into a cylinder. * the kernel allows photons that are not directly on the line to be used in the estimate * we have the freedom to choose the exact form of this blurring kernel

  29. Volumetric Photon Mapping Conventional Radiance Estimate 37 Thursday, 6 September 12

  30. Volumetric Photon Mapping Conventional Radiance Estimate 37 Thursday, 6 September 12

  31. Volumetric Photon Mapping Conventional Radiance Estimate 38 Thursday, 6 September 12

  32. Volumetric Photon Mapping Beam Radiance Estimate 39 Thursday, 6 September 12

  33. Volumetric Photon Mapping Beam Radiance Estimate 39 Thursday, 6 September 12

  34. What is ? • A fixed-size kernel results in a uniform blur of the photon map. • In this case, we need to find photons in fixed-radius cylinder about ray. 40 Thursday, 6 September 12

  35. Fixed Radius Comparison Beam Estimate 41 Thursday, 6 September 12 * When using a constant blurring radius, in the limit the conventional and beam radiance estimates are equivalent. * uses exactly the same photon map

  36. Fixed Radius Comparison Conv. Estimate Beam Estimate 41 Thursday, 6 September 12 * When using a constant blurring radius, in the limit the conventional and beam radiance estimates are equivalent. * uses exactly the same photon map

  37. Fixed Radius Comparison Conv. Estimate Beam Estimate (4:21) (4:15) 41 Thursday, 6 September 12 * When using a constant blurring radius, in the limit the conventional and beam radiance estimates are equivalent. * uses exactly the same photon map

  38. Fixed Radius Comparison Conv. Estimate Conv. Estimate Beam Estimate ( ∞ ) (4:21) (4:15) 41 Thursday, 6 September 12 * When using a constant blurring radius, in the limit the conventional and beam radiance estimates are equivalent. * uses exactly the same photon map

  39. Conventional Radiance Estimate Fixed Radius Nearest Neighbor 42 Thursday, 6 September 12 * however, in practice a fixed radius is rarely used, and the nearest neighbors method is used to adapt the radius to the local density of photons

  40. Adaptive ? Conventional Beam 43 Thursday, 6 September 12 * The conventional radiance estimate uses the k-nearest neighbor method at a point. * How can we generalize this along a line?

  41. Adaptive ? Conventional Beam Adaptive radius ? 43 Thursday, 6 September 12 * The conventional radiance estimate uses the k-nearest neighbor method at a point. * How can we generalize this along a line?

  42. Primal vs. Dual Primal 44 Thursday, 6 September 12 * in order to address this we turn to the primal vs. dual interpretation of density estimation * two di fg erent interpretations of density estimation * exactly equivalent for fixed-radius searches

  43. Primal vs. Dual Primal Dual 44 Thursday, 6 September 12 * in order to address this we turn to the primal vs. dual interpretation of density estimation * two di fg erent interpretations of density estimation * exactly equivalent for fixed-radius searches

  44. Primal vs. Dual Primal Dual allow radius to vary: adaptive kernel method 44 Thursday, 6 September 12 * in order to address this we turn to the primal vs. dual interpretation of density estimation * two di fg erent interpretations of density estimation * exactly equivalent for fixed-radius searches

  45. Volumetric Photon Mapping Beam Radiance Estimate 45 Thursday, 6 September 12

  46. Volumetric Photon Mapping Beam Radiance Estimate 46 Thursday, 6 September 12

  47. Volumetric Photon Mapping Beam Radiance Estimate 47 Thursday, 6 September 12

  48. Volumetric Photon Mapping Beam Radiance Estimate 48 Thursday, 6 September 12

  49. Adaptive Radius Comparison Conv. Estimate Conv. Estimate Beam Estimate 49 Thursday, 6 September 12

  50. Adaptive Radius Comparison Conv. Estimate Conv. Estimate Beam Estimate ( ∞ ) (6:38) (6:22) 49 Thursday, 6 September 12

  51. Algorithm 1) Shoot photons from light sources. 2) Construct a balanced kD-tree for the photons. 3) Assign a radius for each photon ( photon-discs ). 4) Create acceleration structure over photon-discs. 5) Render: • For each ray through the medium, accumulate all photon-discs that intersect ray. 50 Thursday, 6 September 12 * first two steps identical to regular photon mapping

  52. Algorithm Same as Regular Photon Mapping 1) Shoot photons from light sources. 2) Construct a balanced kD-tree for the photons. 3) Assign a radius for each photon ( photon-discs ). 4) Create acceleration structure over photon-discs. 5) Render: • For each ray through the medium, accumulate all photon-discs that intersect ray. 50 Thursday, 6 September 12 * first two steps identical to regular photon mapping

  53. Algorithm Same as Regular Photon Mapping 1) Shoot photons from light sources. 2) Construct a balanced kD-tree for the photons. 3) Assign a radius for each photon ( photon-discs ). 4) Create acceleration structure over photon-discs. 5) Render: • For each ray through the medium, accumulate all photon-discs that intersect ray. Our Method 50 Thursday, 6 September 12 * first two steps identical to regular photon mapping

  54. Algorithm object 1) Shoot photons from light sources. 51 Thursday, 6 September 12

  55. Algorithm 2) Construct a balanced kD-tree for the photons. 52 Thursday, 6 September 12

  56. Algorithm 2) Construct a balanced kD-tree for the photons. 53 Thursday, 6 September 12

  57. Algorithm 3) Assign a radius for each photon ( photon-discs ). Adaptive: perform k-NN search at each photon 54 Thursday, 6 September 12 * if we use a fixed kernel, then each radius is the same, otherwise the radius is computed from the local density of each photon

  58. Algorithm 4) Create a bounding-box hierarchy over photon-discs 55 Thursday, 6 September 12

  59. Algorithm kD-tree a g a b c f c f b d e g e d 4) Create a bounding-box hierarchy over photon-discs 56 Thursday, 6 September 12

  60. Algorithm kD-tree BBH a g a a b c f c f c f b d e g b d e g e d 4) Create a bounding-box hierarchy over photon-discs reuse hierarchical structure of kD-tree 56 Thursday, 6 September 12

  61. Algorithm 5) Render: For each ray through the medium, accumulate all photon-discs that intersect ray. 57 Thursday, 6 September 12

  62. Results • 1K horizontal resolution • 2.4 GHz Core 2 Duo (using one Core) • Comparing identical photon maps 58 Thursday, 6 September 12

  63. Smoky Cornell Box Conv. Estimate Beam Estimate 59 Thursday, 6 September 12

  64. Smoky Cornell Box Conv. Estimate Beam Estimate (4:03) (3:35) 59 Thursday, 6 September 12

  65. Lighthouse Beam Estimate Conventional Estimate 60 Thursday, 6 September 12

  66. Lighthouse Beam Estimate (1:05) Conventional Estimate (1:12) 60 Thursday, 6 September 12

  67. Cars on Foggy Street Beam Estimate Conventional Estimate 61 Thursday, 6 September 12

  68. Cars on Foggy Street Beam Estimate (1:53) Conventional Estimate (2:02) 61 Thursday, 6 September 12

  69. Summary • Theoretical reformulation of PM ( measurement equation ) • Beam radiance estimate • Eliminates ray-marching (and all high-frequency noise) in PM • Same photon map as conv. PM • Can handle adaptive search radius 62 Thursday, 6 September 12

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