Path Tracing. Monte Carlo Path Tracing

Transcription

1 Page 1 Monte Calo Path Tacing Today Path tacing Random wals and Maov chains Adjoint equations Light ay tacing Bidiectional ay tacing Next Heni Wann Jensen Iadiance caching Photon mapping Path Tacing

2 Page 2 The Measuement Equation ( x, ω, t, λ) (, x ω,, t λ) R = Px (, λ )(, Sxω,) t LTx ((, ω, λ),, tλ) da dωdtdλ Pixel esponse Lens optics Shutte Scene adiance P(, x λ) ( x, ω ) = T( x, ωλ, ) Sx (, ω,) t Lx (, ω, t, λ) Light Tanspot Integate ove all paths of all lengths Lx (, x) 1 = L ( x,, x, x, x ) da( x ) da( x ) = 1 M M 2 2 S Questions How to sample space of paths Find good estimatos

3 Page 3 Path Tacing: Fom Camea Step 1. Choose a ay given (x,y,u,v,t) weight = 1; Step 2. Tace ay Step 3. Randomly choose to compute Le o L Step 3a. If Le, etun weight * Le; Step 3b. If L, weight = weight * eflectance; Choose new ay ~ BRDF pdf Go to Step 2. Penumba: Tees vs. Paths 4 eye ays pe pixel 16 shadow ays pe eye ay 64 eye ays pe pixel 1 shadow ay pe eye ay

4 Page 4 Conell Box: Path Tacing 10 ays pe pixel 100 ays pe pixel Fom Jensen, Realistic Image Synthesis Using Photon Maps Discete Random Wal

5 Page 5 Discete Random Pocess Ceation States Temination Tansition Assign pobabilities to each pocess p 0 i : pobability of ceation in state pij, : pobability of tansition fom state i j p * i : pobability of temination in state * i i, j j i i p = 1 p Maov Chain Pobability of being in state i afte n tansitions P = p 0 0 j j P = p + p P j j i, j i i P = p + p P n 0 n 1 j j i, j i i P = p P = p + MP n 0 n 1 P = p + MP M = p i, j i, j

6 Page 6 Equilibium Distibution of States Total pobability of being in state i i = 0 i ( ) P = P = I + M + M + p 2 0 But this is the solution of the matix equation ( I M) P = p 0 Monte Calo Algoithm Ceation States Temination Tansition 1. Geneate andom paticle paths fom souce. 2. Undetae a discete andom wal. 3. Count how many teminate in state i [von Neumann and Ulam; Fosythe and Leible; 1950s]

7 Page 7 Monte Calo Algoithm: Analysis Define a andom vaiable on the space of paths Path: α = ( i, i,, i ) 1 2 Pobability: P( α ) = p p p p 0 * i i, i i, i i Monte Calo Algoithm Define a andom vaiable on the space of paths Path: Pobability: α = ( i, i,, i ) 1 2 P( α ) = p p p p 0 * i i, i i, i i Expectation: EW [ ] P( α ) W( α ) = = 1 α

8 Page 8 Estimato Count the numbe of paticles teminating in state j Estimato: Unbiased: δ Wj( α ) = * p i, j i δ EW [ ] ( p p p p ) 0 * i, j j = i1 i1, i 2 i 1, i i * = 1 i i pj p Mp M p j j j = + + Adjoint Fomulation

9 Page 9 Light Path Sx (, x) x 0 Gx (, x) Gx ( 1, x2) x 1 x 2 2 L ( S x, 0 x,, ) 1 x2 x3 L ( x, x, x, x ) = S( x, x) G( x, x) f ( x, x, x ) G( x, x ) f ( x, x, x ) S 2 3 o Light Paths Sx (, x) x 0 Gx (, x) Gx ( 1, x2) x 1 x 2 2 L ( S x, ) 2 x3 L ( x, x ) L ( x, x, x, x ) da( x ) da( x ) S = 2 3 S 2 3 M 2 2 M

10 Page 10 Measuement S( x, x ) x 0 Gx (, x) Gx ( 1, x2) x 1 2 x 2 L ( S x, ) 2 x3 Gx (, x) 2 3 x 3 R( x, x ) 2 3 M L x x G x x R x x da x da x = M 2 2 M (, ) (, ) (, ) ( ) ( ) S Symmetic Light Path S( x, x ) x 0 Gx (, x) Gx ( 1, x2) x 1 x 2 Gx (, x) 2 3 x 3 2 R( x, x ) 2 3 M = S( x, x) G( x, x) f ( x, x, x ) G( x, x ) f ( x, x, x ) G( x, x ) R( x, x ) o

11 Page 11 Symmety L ( R x, 0 x,, ) 1 x2 x x 0 Gx ( 1, x2) x 2 Gx (, x) 2 3 x 1 x 3 2 R( x, x ) 2 3 L ( x, x, x, x ) = f ( x, x, x ) G( x, x ) f ( x, x, x ) G( x, x ) R( x, x ) R Symmety L ( R x, 0 x,, ) 1 x2 x3 = x 0 Gx ( 1, x2) x 2 Gx (, x) = Gx (, x) x 1 x 3 R( x2, x3) 2 L ( x, x, x, x ) = f ( x, x, x ) G( x, x ) f ( x, x, x ) G( x, x ) R( x, x ) R = Rx (, x) Gx (, x) f( x, x, x) Gx (, x) f( x, x, x)

12 Page 12 Impotance S( x, x ) x 0 Gx (, x) L ( R x,, ) 1 x2 x3 x 1 2 x 2 Gx ( 1, x2) L ( S x,, ) 0 x1 x2 Gx (, x) 2 3 x 3 R( x, x ) 2 3 M L ( x, x, x ) G( x, x ) L ( x, x, x ) da( x) da( x ) = 2 2 M M S R Adjoint Equations Oiginal equation K f = K( x, y) f( y) dy Fowad diection Out-Scatte Adjoint equation + K f = K( x, y) f( x) dx Bacwad diection In-Scatte

13 Page 13 Self-Adjoint Equations Self-adjoint K + = K Fowad = bacwad opeato Out-Scatte In-Scatte Fowad = Bacwad Estimate < f, g>= f( x) g( x) dx ( ) ( ( ) (, ) ) ( ) < f, K g>= f( x) K( x, y) g( y) dy dx = f xkxydx gydy + =< K f, g>

14 Page 14 Thee Consequences 1. Fowad estimate equal bacwad estimate - May use fowad o bacwad ay tacing 2. Adjoint solution - Impotance sampling paths 3. Solve fo small subset of the answe Example: Linea Equations Solve a linea system Mx = b Solve fo a single x i? Solve the adjoint equation Souce Estimato x i ( 2 i i i ), < x + Mx + M x + b> Moe efficient than solving fo all the unnowns [von Neumann and Ulam]

15 Page 15 Light Ray Tacing Path Tacing: Fom Lights Step 1. Choose a light ay Step 2. Tace ay Step 3. Randomly decide whethe to absob o eflect the ay ~ eflectance if eflected goto Step 2

16 Page 16 Path Tacing: Fom Lights Step 1. Choose a light ay Choose a light souce accoding to the light souce powe distibution function. Choose a ay fom the light souce adiance (aea) o intensity (point) distibution function weight = 1; Step 2. Tace ay Step 3. Randomly decide whethe to absob o eflect the ay if eflected goto Step 2 Path Tacing: Fom Lights Step 1. Choose a light ay Step 2. Tace ay Step 3. Randomly decide whethe to absob o eflect the ay ~ eflectance Step 3a. If eflected, Randomly scatte ~ BRDF pdf Go to Step 2. Step 3b. If absobed at the camea, Recod weight at x, y Go to Step 1;

17 Page 17 Bidiectional Ray Tacing Delta Functions Mios (Caustics) Eye ay tacing: ES*DL Light ay tacing: LS*DE ES*DS*L poblematic (vitual point light souce) Even moe poblematic is LS*DS*DS*E

18 Page 18 Bidiectional Ray Tacing = l+ e l = 0, e= 3 l = 1, e= 2 l = 2, e= 1 l = 3, e= 0 = 3 Path Pyamid =3 ( l = 2, e = 1) = 4 =5 =6 ( l = 5, e = 1) l Fom Veach and Guibas e

19 Page 19 Compaison Bidiectional ay tacing Path tacing Fom Veach and Guibas