Complex Filtering and Integration via Sampling

Transcription

1 Overvew Complex Flterng and Integraton va Samplng Sgnal processng Sample then flter (remove alases) then resample onunform samplng: jtterng and Posson dsk Statstcs Monte Carlo ntegraton and probablty theory Samplng from dstrbutons Samplng from shapes Sequental, adaptve and stratfed samplng CS348B Lecture 8 Pat Hanrahan, Sprng Cameras R Lx (, ω, t) PxSt ( ) ( )cosθ dadωdt T Ω A Moton Blur Depth of FIeld Source: Cook, Porter, Carpenter, 984 Source: Mtchell, 99 CS348B Lecture 8 Pat Hanrahan, Sprng Page

2 Lghtng and Soft Shadows Ex ( ) Lx (, xvx ) (, x)cosθdω H Vsblty CS348B Lecture 8 Pat Hanrahan, Sprng Renderng Operators The measurement equaton R L( x, ω, t) P( x) S( t)cosθ dadωdt Lghtng equaton Lots of darn ntegrals: T Ω A E( x) L( x, x) V( x, x)cosθdω H Integrate over t: Moton blur Integrate over pxel x, y: Antalasng Integrate over lens u, v: Depth of feld Integrate over ω (Α): Penumbras Integrate over paths wth k bounces: Lght transport CS348B Lecture 8 Pat Hanrahan, Sprng Page

3 Random Varables Random varables X s chosen by some random process X ~ p( x) probablty dstrbuton Y f( X) s also a random varable Expected values E[ f] f( x) p( x) CS348B Lecture 8 Pat Hanrahan, Sprng Monte Carlo Integraton Defnte ntegral Expectaton of f I f( x) E[ f] f( x) p( x) Random varables X ~ p( x) Y f( X ) Estmator F Y Unbased E[ F ] I( f) CS348B Lecture 8 Pat Hanrahan, Sprng Page 3

4 Unbased Estmator E[ F ] I( f) Propertes E[ f] f( x) p( x) Y ] E [ E[ Y ] E [ ay ] ae[ Y ] EF [ ] E[ Y] E[ Y] E[ f( X)] f( x) p( x) f( x) CS348B Lecture 8 Pat Hanrahan, Sprng f( x) Assume unform probablty dstrbuton for now Monte Carlo Algorthms Advantages Easy to mplement Easy to thnk about (but be careful of basng!) Robust when used wth complex ntegrands and domans (shapes, lghts, ) Effcent for hgh dmensonal ntegrals Effcent soluton method for a few selected ponts Dsadvantages Slow (many samples needed for convergence) osy CS348B Lecture 8 Pat Hanrahan, Sprng Page 4

5 Dscrete Probablty Dstrbutons Random varables chosen by some random process Dscrete events X wth probablty p p p n p Constructon of samples To randomly select an event, Select X f p + p < U p + p + p Unform random varable CS348B Lecture 8 Pat Hanrahan, Sprng P j U j p Examples Samplng a set of pont lght sources Assocate a probablty wth each lght p n Φ j Φ j CS348B Lecture 8 Pat Hanrahan, Sprng Page 5

6 Contnuous Probablty Dstrbutons Cumulatve probablty dstrbuton functon P( x) Pr( X < x) Probablty densty functon Constructon of samples Solve for XP - (U) Must know:. The ntegral of p(x) Pr( α X β) p( x) P( β) P( α) dp( x) px ( ). The nverse functon P - (x) β α U X CS348B Lecture 8 Pat Hanrahan, Sprng Example: Power Functon Assume p( x) ( n+ ) x n n+ n x n x P( x) x + n+ n+ n X ~ p( x) + U Trck: Y max( U, U,, Un, U n + ) n+ Pr( Y < x) Pr( U < x) x n+ CS348B Lecture 8 Pat Hanrahan, Sprng Page 6

7 Samplng a Crcle π π r π A rdr dθ r dr dθ θ π r p(, r θ) dr dθ r dr dθ p(, r θ) A π pr (, θ) pr () p( θ) pr () r P() r r p( θ ) P( θ) θ π r U θ πu rdθ dr CS348B Lecture 8 Pat Hanrahan, Sprng Samplng a Crcle WROG! <> Equ-Areal RIGHT Equ-Areal r U θ πu r U θ πu CS348B Lecture 8 Pat Hanrahan, Sprng Page 7

8 Rejecton Methods I Algorthm f( x) y< f ( x) dy Pck U and U Accept U f U < f(u ) Wasteful? Effcency Area / Area of rectangle CS348B Lecture 8 Pat Hanrahan, Sprng Samplng a Crcle: Rejecton do { X-U Y-U whle( X + Y > ) May be used to pck random D drectons Crcle technques may also be appled to the sphere CS348B Lecture 8 Pat Hanrahan, Sprng Page 8

9 Samplng a Trangle u v u+ v v ( u, u) u u+ v u ( u) A dvdu ( u) du puv (, ) CS348B Lecture 8 Pat Hanrahan, Sprng Samplng a Trangle Here u and v are not ndependent! Condtonal probablty p( uv, ) p( u) p( u, v) dv pu ( v) p( u) Trangle puv (, ) u pu ( ) dv ( u) pv ( u) ( u) u vo vo v ( u) ( u) P( u ) ( u) du ( u ) P( v u ) p( v u ) dv dv CS348B Lecture 8 Pat Hanrahan, Sprng u v U U U Page 9

10 Drect Lghtng θ x E( x) Ls ( x, x) V( x, x)cosθ dω H da θ x x cosθ dω cos θ da' x x cosθ cosθ E( x) Ls ( x, x) V( x, x) da x x A cosθ cosθ Ls ( x, x) V( x, x) da x x Convert from an sold angle ntegral to an area ntegral CS348B Lecture 8 Pat Hanrahan, Sprng Varance Defnton Propertes [ ] [( [ ]) ] VY E Y EY E Y YEY + EY [ [ ] [ ] ] EY [ ] EY [ ] V[ Y ] V[ Y] VaY [ ] avy [ ] CS348B Lecture 8 Pat Hanrahan, Sprng Page

11 Stratfed Samplng Stratfed samplng lke jttered samplng Allocate samples per regon m m F F ew varance V[ F ] V[ F] Thus, f the varance n regons s less than the overall varance, there wll be a reducton n resultng varance For example: An edge through a pxel V[ F ] R V[ F] V[ Fj].5 CS348B Lecture 8 Pat Hanrahan, Sprng Hgh-dmensonal Samplng Stratfed samplng (also numercal quadrature) For a gven error E ~ d Random samplng For a gven varance E / ~ V ~ / Monte Carlo much better for ntegraton n hgh dmensonal spaces CS348B Lecture 8 Pat Hanrahan, Sprng Page

12 Shrley s Mappng r U π U θ 4 U CS348B Lecture 8 Pat Hanrahan, Sprng Convergence Mean and standard devaton µ Y E[ µ ] E[ Y] EY [ ] EY [ ] V[ µ ] V[ Y] V[ Y ] V[ Y] σ µ ( Y ) CS348B Lecture 8 Pat Hanrahan, Sprng Page

13 Sequental Samplng Central Lmt Theorem t tσ µ / π x / lm Pr E[ Y] e Sample rays untl confdence n the estmate s hgh Student t-dstrbuton Purgathofer Ch-squared dstrbuton Lee, Redner, Uselton (985) Essentally the procedures used by pollsters CS348B Lecture 8 Pat Hanrahan, Sprng Adaptve Samplng Whtted Recurse 4 9 Kajya Splt node Mtchell Shotgun blast Contrast I I Max Max I + I Mn Mn Other crtera CS348B Lecture 8 Pat Hanrahan, Sprng Page 3