Motivation. Motivation. Monte Carlo. Example: Soft Shadows. Outline. Monte Carlo Algorithms. Advanced Computer Graphics (Fall 2009)

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1 Advanced Comuter Grahcs Fall 29 CS 294, Renderng Lecture 4: Monte Carlo Integraton Rav Ramamoorth htt://nst.eecs.berkeley.edu/~cs294-3/a9 Motvaton Renderng = ntegraton Relectance equaton: Integrate over ncdent llumnaton Renderng equaton: Integral equaton Many sohstcated shadng eects nvolve ntegrals Antalasng Sot shadows Indrect llumnaton Caustcs Acknowledgements and many sldes courtesy: Thomas Funkhouser, Szymon Rusnkewcz and Pat Hanrahan Eamle: Sot Shadows Monte Carlo Algorthms based on statstcal samlng and random numbers Coned n the begnnng o 94s. Orgnally used or neutron transort, nuclear smulatons Von eumann, Ulam, Metrools, Canoncal eamle: D ntegral done numercally Choose a set o random onts to evaluate uncton, and then average eectaton or statstcal average Monte Carlo Algorthms Advantages Robust or comle ntegrals n comuter grahcs rregular domans, shadow dscontnutes and so on Ecent or hgh dmensonal ntegrals common n grahcs: tme, lght source drectons, and so on Qute smle to mlement Work or general scenes, suraces Easy to reason about but care taken re statstcal bas Dsadvantages osy Slow many samles needed or convergence ot used alternatve analytc aroaches est but those are rare Outlne Motvaton Overvew, D ntegraton Basc robablty and samlng Monte Carlo estmaton o ntegrals

2 Integraton n D d? We can aromate d g d g Standard ntegraton methods lke traezodal rule and Smsons rule Advantages: Converges ast or smooth ntegrands Determnstc = = Dsadvantages: Eonental comlety n many dmensons ot rad convergence or dscontnutes Slde courtesy o Peter Shrley Slde courtesy o Peter Shrley Or we can average Estmatng the average d E d E Monte Carlo methods random choose samles Advantages: E Robust or dscontnutes Converges reasonably or large dmensons Can handle comle geometry, ntegrals Relatvely smle to mlement, reason about = Slde courtesy o Peter Shrley Slde courtesy o Peter Shrley Other Domans b a b a d < > ab Multdmensonal Domans Same deas aly or ntegraton over Pel areas Suraces Projected areas Drectons Camera aertures Tme Paths Eye Pel UGL d =a =b Slde courtesy o Peter Shrley Surace 2

3 Outlne Motvaton Overvew, D ntegraton Basc robablty and samlng Monte Carlo estmaton o ntegrals Random ables Descrbes ossble outcomes o an eerment In dscrete case, e.g. value o a dce roll [ = -6] Probablty assocated wth each /6 or dce Contnuous case s obvous etenson Eected Value Eectaton Dscrete: E Contnuous: E d n For Dce eamle: n E Samlng Technques Problem: how do we generate random onts/drectons durng ath tracng? on-rectlnear domans Imortance BRDF Strated Eye Generatng Random Ponts Unorm dstrbuton: Use random number generator Probablty Surace 3

4 Generatng Random Ponts Secc robablty dstrbuton: Functon nverson Rejecton Metrools Probablty Common Oeratons Want to samle robablty dstrbutons Draw samles dstrbuted accordng to robablty Useul or ntegraton, ckng mortant regons, etc. Common dstrbutons Dsk or crcle Unorm Uer hemshere or vsblty Area lumnare Comle lghtng lke an envronment ma Comle relectance lke a BRDF Generatng Random Ponts Cumulatve Probablty 4

5 Rejecton Samlng Probablty Outlne Monte Carlo Path Tracng Motvaton Overvew, D ntegraton Basc robablty and samlng Monte Carlo estmaton o ntegrals Bg duse lght source, 2 mnutes Motvaton or renderng n grahcs: Covered n detal n net lecture Jensen 5

6 Monte Carlo Path Tracng Estmatng the average d Monte Carlo methods random choose samles Advantages: E Robust or dscontnutes Converges reasonably or large dmensons Can handle comle geometry, ntegrals Relatvely smle to mlement, reason about aths/el Jensen Slde courtesy o Peter Shrley Other Domans b a b a d More ormally < > ab =a =b Slde courtesy o Peter Shrley 6

7 ance 2 [ E ] E ance or Dce Eamle? Work out on board varance or sngle dce roll ance E ance Problem: varance decreases wth / Increasng # samles removes nose slowly E ance decreases as / Error decreases as /sqrt E 7

8 ance Reducton Technques Imortance samlng Strated samlng d Imortance Samlng Imortance Samlng Put more samles where s bgger E d Ths s stll unbased E E d d d or all Imortance Samlng Zero varance ~ Strated Samlng Estmate subdomans searately E c c E k Arvo Less varance wth better mortance samlng 8

9 Strated Samlng Strated Samlng Ths s stll unbased F M k F Less overall varance less varance n subdomans M F F 2 k E k E k More Inormaton Veach PhD thess chater lnked to rom webste Course otes lnks rom webste Mathematcal Models or Comuter Grahcs, Stanord, Fall 997 State o the Art n Monte Carlo Methods or Realstc Image Synthess, Course 29, SIGGRAPH 2 9