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MCMCベースレンダリング入門

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レイトレ合宿2!!(https://sites.google.com/site/raytracingcamp2/)で発表したMCMCベースレンダリング入門のスライドです.

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MCMCベースレンダリング入門

  1. 1. Introduction to MCMC based rendering techniques perim (@hi2p_perim)
  2. 2. Scene with complex occlusion
  3. 3. Scene with specular / glossy materials
  4. 4. Why difficult? このようなシーンは効率的 なレンダリングが難しい
  5. 5. Why difficult? Path space Contribution
  6. 6. Solution? Markov chain Monte Carlo MCMC
  7. 7. MCMC Path space Contribution High Low probability of sampling a path MCMCを用いることによりエネルギーの分布に 従う光路をサンプリングできる
  8. 8. MCMC 目的 ある分布に従うようなマルコフ連鎖 푋1, 푋2, 푋3, … を生成する
  9. 9. MCMC BASED RENDERING TECHNIQUES
  10. 10. Metropolis light transport (MLT) [Veach & Guibas 1997] technique with trans-dimensional mutation Changing sample space Advanced MCMC techniques Primary sample space MLT (PSSMLT) [Kelemen et al. 2002] Multiplexed MLT [Hachisuka et al. 2014] Manifold exploration (ME) [Jacob & Marschner 2012] ME with natural constraints [Kaplanyan et al. 2014] Gradient-domain MLT [Lehtinen et al. 2013] Energy redistribution PT (ERPT) [Cline et al. 2005] Population Monte Carlo ERPT [Lai et al. 2006] Replica exchange light transport [Kitaoka et al. 2009] Original technique
  11. 11. Metropolis light transport (MLT) [Veach & Guibas 1997]
  12. 12. MLT 状態空間: Path space
  13. 13. MLT 光路を直接変更することで 変異を行う
  14. 14. MLT Selected stocastically with acceptance ratio (採択確率) Metropolis-Hastings法 による変異
  15. 15. MLT 様々な変異手法 Bidirectional mutation Lens perturbation Caustic perturbation etc.
  16. 16. Primary sample space MLT (PSSMLT) [Kelemen et al. 2002]
  17. 17. PSSMLT 0,1 ∞ 퐮 状態空間: 一様乱数列 (primary sample space) 写像により 光路に変換 푆(퐮)
  18. 18. PSSMLT 0,1 ∞ 퐮 푆(퐮) Primary sample space内の変異 →写像を通じて 光路が変異される
  19. 19. Multiplexed MLT [Hachisuka et al. 2014]
  20. 20. Metropolis light transport (MLT) [Veach & Guibas 1997] technique with trans-dimensional mutation Changing sample space Advanced MCMC techniques Primary sample space MLT (PSSMLT) [Kelemen et al. 2002] Multiplexed MLT [Hachisuka et al. 2014] Manifold exploration (ME) [Jacob & Marschner 2012] ME with natural constraints [Kaplanyan et al. 2014] Gradient-domain MLT [Lehtinen et al. 2013] Energy redistribution PT (ERPT) [Cline et al. 2005] Population Monte Carlo ERPT [Lai et al. 2006] Replica exchange light transport [Kitaoka et al. 2009] Original technique
  21. 21. Multiplexed MLT 状態空間: Multiplexed primary sample space
  22. 22. Multiplexed MLT 풘ퟎ 풖 푪ퟎ 풖 풘ퟏ 풖 푪ퟏ 풖 풘ퟐ 풖 푪ퟐ 풖 풘ퟑ 풖 푪ퟑ 풖 各空間は 異なるtarget distribution を持つ
  23. 23. Multiplexed MLT 풘ퟎ 풖 푪ퟎ 풖 풘ퟏ 풖 푪ퟏ 풖 풘ퟐ 풖 푪ퟐ 풖 풘ퟑ 풖 푪ퟑ 풖 双方向パストレーシングで 用いられるMISの重み関数
  24. 24. Multiplexed MLT 풘ퟎ 풖 푪ퟎ 풖 풘ퟏ 풖 푪ퟏ 풖 풘ퟐ 풖 푪ퟐ 풖 풘ퟑ 풖 푪ퟑ 풖 Primary sample space Contribution
  25. 25. Multiplexed MLT Contribution Mixture distribution = Target distribution Primary for PSSMLT sample space
  26. 26. Multiplexed MLT 状態空間: 空間のID, 乱数列
  27. 27. Multiplexed MLT 状態空間: 空間のID, 一様乱数列のペア
  28. 28. Multiplexed MLT 空間内の変異+ 空間を超える変異
  29. 29. Multiplexed MLT cf. Serial tempering 緩和された分布をいくつか導入し Mixingを向上させるMCMC法の一種
  30. 30. Energy redistribution PT (ERPT) [Cline et al. 2005]
  31. 31. Metropolis light transport (MLT) [Veach & Guibas 1997] technique with trans-dimensional mutation Changing sample space Advanced MCMC techniques Primary sample space MLT (PSSMLT) [Kelemen et al. 2002] Multiplexed MLT [Hachisuka et al. 2014] Manifold exploration (ME) [Jacob & Marschner 2012] ME with natural constraints [Kaplanyan et al. 2014] Gradient-domain MLT [Lehtinen et al. 2013] Energy redistribution PT (ERPT) [Cline et al. 2005] Population Monte Carlo ERPT [Lai et al. 2006] Replica exchange light transport [Kitaoka et al. 2009] Original technique
  32. 32. ERPT 푥 푦 すべての푥, 푦 ∈ Ωに対して, 퐸 퐾 푥 → 푦 = 퐸 퐾 푦 → 푥 Detailed balance (詳細つりあい条件)
  33. 33. ERPT Markov chain satisfying detailed balance (reversible Markov chain) Markov chain is stationary Detailed balanceは必要条件
  34. 34. ERPT 푥 すべての푥 ∈ Ωに対して, 퐸 퐾 푥 → 푦 푑휇 푦 = 퐸 퐾 푦 → 푥 푑휇 푥 General balance (一般つりあい条件)
  35. 35. ERPT 푥 光路のサンプリング
  36. 36. ERPT 光路の変異 & エネルギーの分配 푥
  37. 37. Population Monte Carlo ERPT [Lai et al. 2006]
  38. 38. Metropolis light transport (MLT) [Veach & Guibas 1997] technique with trans-dimensional mutation Changing sample space Advanced MCMC techniques Primary sample space MLT (PSSMLT) [Kelemen et al. 2002] Multiplexed MLT [Hachisuka et al. 2014] Manifold exploration (ME) [Jacob & Marschner 2012] ME with natural constraints [Kaplanyan et al. 2014] Gradient-domain MLT [Lehtinen et al. 2013] Energy redistribution PT (ERPT) [Cline et al. 2005] Population Monte Carlo ERPT [Lai et al. 2006] Replica exchange light transport [Kitaoka et al. 2009] Original technique
  39. 39. PMC-ERPT PMC Population Monte Carlo 通常のMCMC (e.g. Metropolis-Hastings)
  40. 40. PMC-ERPT Adapt kernels Mutate Resample D-Kernel PMC
  41. 41. Replica exchange light transport [Kitaoka et al. 2009]
  42. 42. Metropolis light transport (MLT) [Veach & Guibas 1997] technique with trans-dimensional mutation Changing sample space Advanced MCMC techniques Primary sample space MLT (PSSMLT) [Kelemen et al. 2002] Multiplexed MLT [Hachisuka et al. 2014] Manifold exploration (ME) [Jacob & Marschner 2012] ME with natural constraints [Kaplanyan et al. 2014] Gradient-domain MLT [Lehtinen et al. 2013] Energy redistribution PT (ERPT) [Cline et al. 2005] Population Monte Carlo ERPT [Lai et al. 2006] Replica exchange light transport [Kitaoka et al. 2009] Original technique
  43. 43. RELT Replica Exchange (a.k.a. parallel tempering) 通常のMCMC (e.g. Metropolis-Hastings)
  44. 44. RELT 状態空間: サンプルの積空間
  45. 45. BPTの重み関数 RELT 푓 푆 풖 푝푠,1 푆 풖 푓 푆 풖 푝푠,1 푆 풖 푤푠,푡 푆 풖 푓 푆 풖 푝푠,푡 푆 풖 空間のTargetの分布は 푠≥0 푤푠,0 푆 풖 푠≥0 푤푠,1 푆 풖 푠,푡≥2 BPTの重み付きcontributionから決める
  46. 46. Gradient domain MLT [Lehtinen et al. 2013]
  47. 47. Metropolis light transport (MLT) [Veach & Guibas 1997] technique with trans-dimensional mutation Changing sample space Advanced MCMC techniques Primary sample space MLT (PSSMLT) [Kelemen et al. 2002] Multiplexed MLT [Hachisuka et al. 2014] Manifold exploration (ME) [Jacob & Marschner 2012] ME with natural constraints [Kaplanyan et al. 2014] Gradient-domain MLT [Lehtinen et al. 2013] Energy redistribution PT (ERPT) [Cline et al. 2005] Population Monte Carlo ERPT [Lai et al. 2006] Replica exchange light transport [Kitaoka et al. 2009] Original technique
  48. 48. Manifold exploration [Jacob & Marschner 2012]
  49. 49. Metropolis light transport (MLT) [Veach & Guibas 1997] technique with trans-dimensional mutation Changing sample space Advanced MCMC techniques Primary sample space MLT (PSSMLT) [Kelemen et al. 2002] Multiplexed MLT [Hachisuka et al. 2014] Manifold exploration (ME) [Jacob & Marschner 2012] ME with natural constraints [Kaplanyan et al. 2014] Gradient-domain MLT [Lehtinen et al. 2013] Energy redistribution PT (ERPT) [Cline et al. 2005] Population Monte Carlo ERPT [Lai et al. 2006] Replica exchange light transport [Kitaoka et al. 2009] Original technique
  50. 50. Manifold exploration MLT [Veach 1997] 本手法(ME) 鏡面, 光沢の含まれるシーンの 効率的なレンダリング
  51. 51. Manifold exploration 푬 푳 푫 푺 푺 푫 푺 Mutation technique Supported paths Lens perturbation 퐸푆∗퐷(퐷|퐿) Caustic perturbation 퐸퐷푆∗(퐷|퐿) Multi-chain perturbation 퐸퐷푆∗퐷푆∗퐷(퐷|퐿) 既存手法では効率的に扱えない
  52. 52. Manifold exploration Mutation technique Supported paths Lens perturbation 퐸푆∗퐷(퐷|퐿) Caustic perturbation 퐸퐷푆∗(퐷|퐿) Multi-chain perturbation 퐸퐷푆∗퐷푆∗퐷(퐷|퐿) Manifold perturbation 퐄퐃푺∗퐃푺∗(퐃|퐋) 扱える光路のクラスが増える
  53. 53. Manifold exploration 푬 푳 푫 푺 푺 푫 푺 퐱0 퐱1 퐱2 퐱3 퐱4 퐱5 퐱6 푥 : 現在の光路
  54. 54. Manifold exploration 퐱2 퐱4 푬 푳 푫 푺 푺 푫 푺 퐱0 퐱푎 = 퐱1 퐱푏 = 퐱3 퐱5 퐱푐 = 퐱6 Step 1. 푥 から(퐷|퐿)の3頂点퐱푎, 퐱푏, 퐱푐 を選択する
  55. 55. Manifold exploration 푬 푳 Step 2. 퐱푎 → 퐱푎+1 の角度を変異させ푏 − 푎個の 푆の頂点を更新し, 新たに到達した퐷の頂点を퐱푏′ とする(赤いパス) 푫 푺 푺 푫 푺 퐱0 퐱푎 = 퐱1 퐱2 퐱푏 = 퐱3 퐱4 퐱5 퐱푐 = 퐱6 퐱푏′
  56. 56. Manifold exploration 푬 푳 Step 3. 퐱푏′ 푫 푺 푺 푫 푺 と퐱푐間の푏 − 푎 − 1個の푆頂点を探索し 接続する(青いパス) 퐱0 퐱푎 = 퐱1 퐱2 퐱푏 = 퐱3 퐱4 퐱5 퐱푐 = 퐱6 퐱푏′
  57. 57. Manifold exploration 푆頂点の探索: WalkManifold
  58. 58. Manifold exploration 状態空間: Specular manifold 퐸 퐷 퐿/퐷 푆 푆
  59. 59. Manifold exploration Constraint : 入射角= 反射角 퐸 퐷 퐿/퐷 푆 푆
  60. 60. Manifold exploration 푺 푫 푺 푫 푳 Specular manifoldに陰関数定理を適用 Dの微小変化から残りの頂点の微小変化がわかる
  61. 61. Manifold exploration
  62. 62. Manifold exploration with half vector space [Kaplanyan et al. 2014]
  63. 63. Metropolis light transport (MLT) [Veach & Guibas 1997] technique with trans-dimensional mutation Changing sample space Advanced MCMC techniques Primary sample space MLT (PSSMLT) [Kelemen et al. 2002] Multiplexed MLT [Hachisuka et al. 2014] Manifold exploration (ME) [Jacob & Marschner 2012] ME with natural constraints [Kaplanyan et al. 2014] Gradient-domain MLT [Lehtinen et al. 2013] Energy redistribution PT (ERPT) [Cline et al. 2005] Population Monte Carlo ERPT [Lai et al. 2006] Replica exchange light transport [Kitaoka et al. 2009] Original technique

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