Monte Calo Techniques fo Rendeing CS 517 Fall 2002 Compute Science Conell Univesity Announcements No ectue on Thusday Instead, attend Steven Gotle, Havad Upson Hall B17, 4:15-5:15 (efeshments ealie) Geomety Images, econstuction of 3D Geomety Inteesting talk by Jim Gay 2:30-3:30, Wednesday, 255 Olin Hall Can Databases Help Science? Theefoe, Office hous this week by appointment
MC Advantages Convegence ate of O( ) Simple Sampling Point evaluation Can use black boes 1 N Geneal Woks fo high dimensions Deals with discontinuities, cazy functions, Octee Popeties Font to back tavesal Poblem: Same object in multiple cells Could epeatedly intesect Use mailboes
Intesection Acceleation Intesect ay with oot: p oot.intesect(ay) If no intesection, done Find p in tee (node j oot.find(p)) Test ay against elements in node j If intesection found, done Else find eit point (q) fom node j, p q, goto 2 Rendeing Equation ( Θ) ( Θ) + e Ω f ( Ψ Θ) ( Ψ) cos( Ψ, n ) dω function to integate ove all incoming diections ove the hemisphee aound Ψ Value we want + e f Ω cos
How to compute? ( Θ)? Check fo e ( Θ) Now add ( Θ)? Ω f ( Ψ Θ) ( Ψ) cos( Ψ, n ) dω Ψ Monte Calo! How to compute? Geneate andom diections on hemisphee Ω, using pdf p(ψ) ( Θ) Ω f ( Ψ Θ) ( Ψ) cos( Ψ, n ) dω Ψ ( Θ) 1 N f ( Ψ Θ) ( Ψ ) cos( Ψ, n ) N i i 1 p( Ψi ) i i
How to compute? evaluate ( Ψ i )? Radiance is invaiant along staight paths vp(, Ψ i ) fist visible point ( Ψ i ) (vp(, Ψ i ) Ψ i ) Russian Roulette Integal I 1 1 f ( ) P f ( ) d Pd 0 0 P 0 f ( y / P) dy P f ( y / P) P f () Estimato I oulette f ( i ) P 0 if if i P, > P. i 0 P 1 Vaiance σ > σ oulette
Russian Roulette Pick some absoption pobability α pobability 1-α that ay will bounce estimated adiance becomes / (1-α) E.g. α 0.9 only 1 chance in 10 that ay is eflected estimated adiance of that ay is multiplied by 10 instead of shooting 10 ays, we shoot only 1, but count the contibution of this one 10 times Stochastic Ray Tacing Paametes? # stating ays pe piel # andom ays fo each suface point (banching facto) Banching facto 1: path tacing
Algoithm so fa... Shoot # viewing ays though each piel Shoot # indiect ays, sampled ove hemisphee Teminate ecusion using Russian Roulette Algoithm?
Algoithm e 0? Algoithm in?
Algoithm in? Algoithm?
Algoithm e 0? Algoithm in?
Algoithm? Algoithm e 1.234?
Algoithm Path Tacing
Pefomance/Eo Want bette quality with smalle numbe of samples Fewe samples/bette pefomance Well-distibuted samples Statified sampling Faste convegence Impotance sampling: net-event estimation Statified Sampling Samples could be abitaily close Split Integal in subpats I f ( ) d + K+ f ( ) d Estimato I X1 stat 1 N X N f ( ) N i i 1 p( i ) f () 0 1 Vaiance: σ stat σ sec
Numeical eample Statified Sampling 9 shadow ays not statified 9 shadow ays statified
Statified Sampling 36 shadow ays not statified 36 shadow ays statified Statified Sampling 100 shadow ays not statified 100 shadow ays statified
2 Dimensions N 2 samples Poblem fo highe dimensions Sample points can still be abitaily close to each othe Highe Dimensions Statified gid sampling: N d samples N-ooks sampling: N samples
N-Rooks Sampling - 9 ays not statified statified N-Rooks N-Rooks Sampling - 36 ays not statified statified N-Rooks
Othe types of Sampling How does it elate to Regula Sampling Random sampling Regula sampling Typically impotance sample then statify Quasi Monte Calo Use of low discepancy sequences (these ae not andom numbes)
Quasi Monte Calo Quasi Monte Calo Conveges as fast as statified sampling Does not equie knowledge about how many samples will be used Using QMC diections evenly spaced no matte how many samples ae used Samples popely statified-> bette than pue MC
Pefomance/Eo Want bette quality with smalle numbe of samples Fewe samples/bette pefomance Well-distibuted samples Statified sampling Faste convegence Impotance sampling: net-event estimation Sample hemisphee Path Tacing 1 sample/piel 16 samples/piel 256 samples/piel Impotance Sampling: compute diect illumination sepaately!
Diect Illumination Paths of length 1 only, between eceive and light souce Diect ighting Global Illumination
Net Event Estimation ( Θ) ( Θ) + e Ω f ( Ψ Θ) ( Ψ) cos( Ψ, n ) dω Ψ Radiance fom light souces + adiance fom othe sufaces + e f Ω cos Net Event Estimation ( Θ) + + e diect indiect e + Ω f cos + Ω f cos So sample diect and indiect with sepaate MC integation
Algoithm? Algoithm e 0?
Algoithm?? 0 i d e?? 0 i d e Algoithm??? 2.345 0 i d e? 2.345 0 i d e
Algoithm Algoithm a vaiant of path tacing
Compaison Without N.E.E. With N.E.E. 16 samples/piel Rays pe piel 1 sample/ piel 4 samples/ piel 16 samples/ piel 256 samples/ piel
Diect Illumination ( Θ) f (, Ψ Θ) ( y Ψ) G(, y) A souce da y cos( n, Θ)cos( n G(, y) y 2 y, Ψ) Vis(, y) y n y Ψ Vis(,y)? Ψ n Θ Θ hemisphee integation aea integation y Ψ n diect Rendeing Equation ( Θ) Ω f ( Ψ Θ) ( Ψ) cosθ dω y vp(, Ψ) Ψ Ψ da y ( Ψ) ( vp(, Ψ) Ψ) dω Ψ da cosθ y dω Ψ y 2 y
Rendeing Equation: visible sufaces diect ( Θ) diect ( Θ) Ω Coodinate tansfom f ( Ψ Θ) ( Ψ) cosθ dω f ( Ψ Θ) ( y Ψ) all y lights on V (, y)cosθ y vp(, Ψ) Integation domain suface points y on lights Ψ cosθ y 2 y da y Geneating diect paths Pick suface points y i on light souce Evaluate diect illumination integal y θ y Vis(,y)? y ( Θ) 1 N f (...) (...) G(, y ) N i 1 p( yi ) i Θ θ
Applied to diect illumination p( y) 1 Aea souce cosθ cosθ y E( ) Aeasoucesouce Vis(, y) 2 y Moe points... 1 shadow ay 9 shadow ays E( ) Aea souce N souce N i 1 cosθ cosθ yi Vis(, y 2 i ) y i
Even moe points... 36 shadow ays 100 shadow ays E( ) Aea souce N souce N i 1 cosθ cosθ yi Vis(, y 2 i ) y i Paametes How many paths ( shadow-ays )? Total? Pe light souce? (~intensity, impotance, ) How to distibute paths within light souce? Distance fom point Unifom
Diect paths Diffeent path geneatos poduce diffeent estimatos and diffeent eo chaacteistics Diect illumination geneal algoithm: compute_adiance (point, diection) est_ad 0; fo (i0; i<n; i++) p geneate_path; est_ad + enegy_tansfe(p) / pobability(p); est_ad est_ad / n; etun(est_ad); Diect Paths: Using Aea Fom 1 path / souce 9 paths / souce 36 paths / souce
Altenative diect paths Shoot paths at andom ove hemisphee; check if they hit light souce paths not used efficiently noise in image might wok if light souce occupies lage potion on hemisphee Altenative diect paths 1 paths / point 16 paths / point 256 paths / point
Altenative diect paths Pick andom point on andom suface; check if on light souce and visible to taget point paths not used efficiently noise in image might wok fo lage suface light souces Diect path geneatos ight souce sampling - e non-zeo - 1 visibility tem in estimato Hemisphee sampling - e can be 0 - no visibility in estimato Suface sampling - e can be 0-1 visibility tem in estimato
Stochastic Ray Tacing Sample aea of light souce fo diect tem Sample hemisphee with andom ays fo indiect tem Optimizations: Statified sampling Impotance sampling Combine multiple pobability density functions into a single PDF Hemisphee integation Ψ n Θ ( Θ) 1 N f (...) (...) G(, y ) N i 1 p( yi ) i
Sampling stategies Unifom sampling ove the hemisphee ( Θ) Ω ( Ψ) f ( Ψ Θ) cos( Ψ, n ) dω Ψ p( Θ) 1/(2π ) Sampling stategies Sampling accoding to the cosine facto ( Θ) Ω ( Ψ) f ( Ψ Θ) cos( Ψ, n ) dω Ψ p( Θ) cosθ /π
Sampling stategies Sampling accoding to the BRDF ( Θ) Ω ( Ψ) f ( Ψ Θ) cos( Ψ, n ) dω Ψ p( Θ) ~ f ( Θ Ψ) Sampling stategies Sampling accoding to the BRDF times the cosine ( Θ) Ω ( Ψ) f ( Ψ Θ) cos( Ψ, n ) dω Ψ p( Θ) ~ f ( Θ Ψ) cosθ
Multi-Impotance-Sampling ( Θ) Radiance Ω f ( Ψ Θ) ( Ψ) cos( Ψ, n BRDF Iadiance ) dω Ψ Compaison With impotance sampling (bdf on sphee) Without impotance sampling (bdf on sphee)
Stochastic Ray Tacing Sample aea of light souce fo diect tem Sample hemisphee with andom ays fo indiect tem Optimizations: Statified sampling Impotance sampling Combine multiple pobability density functions into a single PDF Indiect Illumination Paths of length > 1 Many diffeent path geneatos possible Efficiency dependent on: BRDFs along the path Visibility function...
Indiect paths - suface sampling Simple geneato (path length 2): select point on light souce select andom point on sufaces pe path: 2 visibility checks Indiect paths - suface sampling Indiect illumination (path length 2): ( Θ) ( y Ψ1 ) f ( z, Ψ1 Ψ2 ) G( z, y) f ( z, Ψ2 Θ Asouce A ) G( y, ) da da z y ( Θ) 1 N N i 1 ( yi Ψ1 i ) f ( zi, Ψ1 i Ψ2i ) G( zi, yi ) f ( zi, Ψ2i Θ) G( yi, ) p ( y ) p ( z ) y i z i 2 visibility values (which might be 0) cause noise
Indiect paths - souce shooting Shoot ay fom light souce, find hit location Connect hit point to eceive pe path: 1 ay intesection 1 visibility check Indiect paths - eceive gatheing Shoot ay fom eceive point, find hit location Connect hit point to andom point on light souce pe path: 1 ay intesection 1 visibility check
Indiect paths Suface sampling - 2 visibility tems; can be 0 Souce shooting - 1 visibility tem - 1 ay intesection Receive gatheing - 1 visibility tem - 1 ay intesection Moe vaiants... Shoot ay fom eceive point, find hit location Shoot ay fom hit point, check if on light souce pe path: 2 ay intesections e might be zeo
Indiect paths Same pinciples apply to paths of length > 2 geneate multiple suface points geneate multiple bounces fom light souces and connect to eceive geneate multiple bounces fom eceive and connect to light souces Estimato and noise chaacteistics change with path geneato Geneal algoithm: Indiect paths compute_adiance (point, diection) est_ad 0; fo (i0; i<n; i++) p geneate_indiect_path; est_ad + enegy_tansfe(p) / pobability(p); est_ad est_ad / n; etun(est_ad);