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MCMCベースレンダリング入門

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Hisanari Otsu
Hisanari Otsu

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.

Technology
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Introduction to MCMC based rendering techniques perim (@hi2p_perim)
Scene with complex occlusion
Scene with specular / glossy materials
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
Why difficult? Path space Contribution
Solution? Markov chain Monte Carlo MCMC
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
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
MCMC BASED RENDERING TECHNIQUES
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
Metropolis light transport (MLT) [Veach & Guibas 1997]
MLT 状態空間: Path space
MLT 光路を直接変更することで 変異を行う
MLT Selected stocastically with acceptance ratio (採択確率) Metropolis-Hastings法 による変異
MLT 様々な変異手法 Bidirectional mutation Lens perturbation Caustic perturbation etc.
Primary sample space MLT (PSSMLT) [Kelemen et al. 2002]
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)
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
Multiplexed MLT [Hachisuka et al. 2014]
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
Multiplexed MLT 状態空間: Multiplexed primary sample space
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
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
Multiplexed MLT 풘ퟎ 풖 푪ퟎ 풖 풘ퟏ 풖 푪ퟏ 풖 풘ퟐ 풖 푪ퟐ 풖 풘ퟑ 풖 푪ퟑ 풖 Primary sample space Contribution
Multiplexed MLT Contribution Mixture distribution = Target distribution Primary for PSSMLT sample space
Multiplexed MLT 状態空間: 空間のID, 乱数列
Multiplexed MLT 状態空間: 空間のID, 一様乱数列のペア
Multiplexed MLT 空間内の変異+ 空間を超える変異
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
Energy redistribution PT (ERPT) [Cline et al. 2005]
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
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)
ERPT Markov chain satisfying detailed balance (reversible Markov chain) Markov chain is stationary Detailed balanceは必要条件
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)
ERPT 푥 光路のサンプリング
ERPT 光路の変異 & エネルギーの分配 푥
Population Monte Carlo ERPT [Lai et al. 2006]
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
PMC-ERPT PMC Population Monte Carlo 通常のMCMC (e.g. Metropolis-Hastings)
PMC-ERPT Adapt kernels Mutate Resample D-Kernel PMC
Replica exchange light transport [Kitaoka et al. 2009]
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
RELT Replica Exchange (a.k.a. parallel tempering) 通常のMCMC (e.g. Metropolis-Hastings)
RELT 状態空間: サンプルの積空間
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
Gradient domain MLT [Lehtinen et al. 2013]
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
Manifold exploration [Jacob & Marschner 2012]
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
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
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
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
Manifold exploration 푬 푳 푫 푺 푺 푫 푺 퐱0 퐱1 퐱2 퐱3 퐱4 퐱5 퐱6 푥 : 現在の光路
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
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′
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′
Manifold exploration 푆頂点の探索: WalkManifold
Manifold exploration 状態空間: Specular manifold 퐸 퐷 퐿/퐷 푆 푆
Manifold exploration Constraint : 入射角= 反射角 퐸 퐷 퐿/퐷 푆 푆
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
Manifold exploration
Manifold exploration with half vector space [Kaplanyan et al. 2014]
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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MCMCベースレンダリング入門

  • 1. Introduction to MCMC based rendering techniques perim (@hi2p_perim)
  • 2. Scene with complex occlusion
  • 3. Scene with specular / glossy materials
  • 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. Why difficult? Path space Contribution
  • 6. Solution? Markov chain Monte Carlo MCMC
  • 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. 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. MCMC BASED RENDERING TECHNIQUES
  • 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. Metropolis light transport (MLT) [Veach & Guibas 1997]
  • 14. MLT Selected stocastically with acceptance ratio (採択確率) Metropolis-Hastings法 による変異
  • 15. MLT 様々な変異手法 Bidirectional mutation Lens perturbation Caustic perturbation etc.
  • 16. Primary sample space MLT (PSSMLT) [Kelemen et al. 2002]
  • 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. 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
  • 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. Multiplexed MLT 状態空間: Multiplexed primary sample space
  • 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. 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. Multiplexed MLT 풘ퟎ 풖 푪ퟎ 풖 풘ퟏ 풖 푪ퟏ 풖 풘ퟐ 풖 푪ퟐ 풖 풘ퟑ 풖 푪ퟑ 풖 Primary sample space Contribution
  • 25. Multiplexed MLT Contribution Mixture distribution = Target distribution Primary for PSSMLT sample space
  • 26. Multiplexed MLT 状態空間: 空間のID, 乱数列
  • 27. Multiplexed MLT 状態空間: 空間のID, 一様乱数列のペア
  • 28. Multiplexed MLT 空間内の変異+ 空間を超える変異
  • 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. Energy redistribution PT (ERPT) [Cline et al. 2005]
  • 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. 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. ERPT Markov chain satisfying detailed balance (reversible Markov chain) Markov chain is stationary Detailed balanceは必要条件
  • 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)
  • 36. ERPT 光路の変異 & エネルギーの分配 푥
  • 37. Population Monte Carlo ERPT [Lai et al. 2006]
  • 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. PMC-ERPT PMC Population Monte Carlo 通常のMCMC (e.g. Metropolis-Hastings)
  • 40. PMC-ERPT Adapt kernels Mutate Resample D-Kernel PMC
  • 41. Replica exchange light transport [Kitaoka et al. 2009]
  • 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. RELT Replica Exchange (a.k.a. parallel tempering) 通常のMCMC (e.g. Metropolis-Hastings)
  • 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. Gradient domain MLT [Lehtinen et al. 2013]
  • 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. Manifold exploration [Jacob & Marschner 2012]
  • 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. 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. 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. 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. Manifold exploration 푬 푳 푫 푺 푺 푫 푺 퐱0 퐱1 퐱2 퐱3 퐱4 퐱5 퐱6 푥 : 現在の光路
  • 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. 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. 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′
  • 58. Manifold exploration 状態空間: Specular manifold 퐸 퐷 퐿/퐷 푆 푆
  • 59. Manifold exploration Constraint : 入射角= 反射角 퐸 퐷 퐿/퐷 푆 푆
  • 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
  • 62. Manifold exploration with half vector space [Kaplanyan et al. 2014]
  • 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

Editor's Notes

  1. In order to get a stationary distribution
  2. 反射とする