Computer Graphics III – Monte Carlo integration Direct illumination Jaroslav Křivánek, MFF UK Jaroslav.Krivanek@mff.cuni.cz
Entire the lecture in 5 slides
𝑀 in 𝑀 out Reflection equation ◼ Total reflected radiance: integrate contributions of incident radiance, weighted by the BRDF, over the hemisphere 𝑀 out (𝜕 out ) = න 𝑀 in (𝜕 in ) ⋅ 𝑔 𝑠 (𝜕 in → 𝜕 out ) ⋅ cos 𝜄 in d𝜕 in 𝐼(𝐲) upper hemisphere over x = න CG III (NPGR010) - J. Křivánek 2015
Rendering = Integration of functions 𝑀 out (𝜕 out ) = න 𝑀 in (𝜕 in ) ⋅ 𝑔 𝑠 (𝜕 in → 𝜕 out ) ⋅ cos 𝜄 in d𝜕 in 𝐼(𝐲) ◼ Problems ❑ Discontinuous integrand (visibility) ❑ Arbitrarily large integrand values (e.g. light distribution in caustics, BRDFs of glossy Incoming radiance surfaces) L in ( x , w in ) for a point ❑ Complex geometry on the ceiling. Images: Greg Ward CG III (NPGR010) - J. Křivánek 2015 4
Monte Carlo integration ◼ General tool for estimating definite integrals Integral: g ( x ) 𝐽 = න(𝐲)d𝐲 Monte Carlo estimate I : 𝑂 (𝜊 𝑙 ) 𝐽 = 1 𝑂 𝑞(𝜊 𝑙 ) ; 𝜊 𝑙 ∝ 𝑞(𝐲) p ( x ) 𝑙=1 Works “on average” : 5 3 1 4 2 6 0 1 𝐹[ 𝐽 ] = 𝐽 CG III (NPGR010) - J. Křivánek 2015 5
Application of MC to reflection eq: Estimator of reflected radiance ◼ Integral to be estimated: න 𝑀 in (𝜕 in ) 𝑔 𝑠 (𝜕 in → 𝜕 out ) cos 𝜄 in d𝜕 in 𝐼(𝐲) integrand( w in ) ◼ pdf for cosine-proportional sampling: 𝑞(𝜕 in ) = cos 𝜄 in 𝜌 ◼ MC estimator (formula to use in the renderer): 𝑂 integrand 𝜕 in,𝑙 1 𝑀 out = 𝑂 pdf 𝜕 in,𝑙 𝑙=1 𝑂 𝜌 = 𝑂 𝑀 in 𝜕 in,𝑙 𝑔 𝑠 (𝜕 in,𝑙 → 𝜕 out ) 𝑙=1 CG III (NPGR010) - J. Křivánek 2015 6
Estimator of reflected radiance: Implementation // input variables x...shaded point on a surface normal...surface normal at x omegaOut...viewing (camera) direction estimatedRadianceOut := Rgb(0,0,0); for k = 1... N [omegaInK, pdf] := generateRndDirection (); // evaluate integrand radianceInEst := getRadianceIn (x, omegaInK); brdf := evalBrdf (x, omegaInK, omegaOut); cosThetaIn := dot (normal, omegaInK); integrand := radianceInEst * brdf * cosThetaIn; // evaluate contribution to the outgoing radiance estimatedRadianceOut += integrand / pdf; end for estimatedRadianceOut /= N; CG III (NPGR010) - J. Křivánek 2015 7
Variance => image noise CG III (NPGR010) - J. Křivánek 2015 8
… and now the slow way
Digression: Numerical quadrature
Quadrature formulas for numerical integration ◼ General formula in 1D: 𝑂 መ 𝐽 = 𝑥 𝑙 (𝑦 𝑙 ) 𝑙=1 g integrand (i.e. the integrated function) N quadrature order (i.e. number of integrand samples) x k node points (i.e. positions of the samples) g (x k ) integrand values at node points w k quadrature weights CG III (NPGR010) - J. Křivánek 2015 11
Quadrature formulas for numerical integration ◼ Quadrature rules differ by the choice of node point positions x k and the weights w k E.g. rectangle rule, trapezoidal rule, Simpson’s method, ❑ Gauss quadrature , … ◼ The samples (i.e. the node points) are placed deterministically CG III (NPGR010) - J. Křivánek 2015 12
Quadrature formulas in multiple dimensions ◼ General formula for quadrature of a function of multiple variables: 𝑂 𝑂 𝑂 መ 𝐽 = . . . 𝑥 𝑙 1 𝑥 𝑙 2 . . . 𝑥 𝑙 𝑡 𝑔(𝑦 𝑙 1 , 𝑦 𝑙 2 , . . . , 𝑦 𝑙 𝑒 ) 𝑙 1 =1 𝑙 2 =1 𝑙 𝑒 =1 ◼ Convergence speed of approximation error E for a d - dimensional integral is E = O( N -1/ d ) ❑ E.g. in order to cut the error in half for a 3-dimensional integral, we need 2 3 = 8 times more samples ◼ Unusable in higher dimensions ❑ Dimensional explosion CG III (NPGR010) - J. Křivánek 2015 13
Quadrature formulas in multiple dimensions ◼ Deterministic quadrature vs. Monte Carlo ❑ In 1D deterministic better than Monte Carlo ❑ In 2D roughly equivalent ❑ From 3D, MC will always perform better ◼ Remember, quadrature rules are NOT the Monte Carlo method CG III (NPGR010) - J. Křivánek 2015 14
Monte Carlo
History of the Monte Carlo method ◼ Atomic bomb development, Los Alamos 1940 John von Neumann, Stanislav Ulam, Nicholas Metropolis ◼ Further development and practical applications from the early 50’s CG III (NPGR010) - J. Křivánek 2015 16
Monte Carlo method ◼ We simulate many random occurrences of the same type of events, e.g.: ❑ Neutrons – emission, absorption, collisions with hydrogen nuclei ❑ Behavior of computer networks, traffic simulation. ❑ Sociological and economical models – demography, inflation, insurance, etc. CG III (NPGR010) - J. Křivánek 2015 17
Monte Carlo – applications ◼ Financial market simulations ◼ Traffic flow simulations ◼ Environmental sciences ◼ Particle physics ◼ Quantum field theory ◼ Astrophysics ◼ Molecular modeling ◼ Semiconductor devices ◼ Optimization problems ◼ Light transport calculations ◼ ... CG III (NPGR010) - J. Křivánek 2015 18
Slide credit: Iwan Kawrakov CG III (NPGR010) - J. Křivánek 2015 19
Slide credit: Iwan Kawrakov CG III (NPGR010) - J. Křivánek 2015 20
Variance => image noise CG III (NPGR010) - J. Křivánek 2015 21
Monte Carlo integration Samples are placed randomly (or pseudo-randomly) ◼ Convergence of standard error: std. dev. = O( N -1/2 ) ◼ Convergence speed independent of dimension ❑ Faster than classic quadrature rules for 3 and more ❑ dimensions Special methods for placing samples exist ◼ Quasi-Monte Carlo ❑ Faster asymptotic convergence than MC for “smooth” ❑ functions CG III (NPGR010) - J. Křivánek 2015 22
Monte Carlo integration ◼ Pros ❑ Simple implementation ❑ Robust solution for complex integrands and integration domains ❑ Effective for high-dimensional integrals ◼ Cons ❑ Relatively slow convergence – halving the standard error requires four times as many samples ❑ In rendering: images contain noise that disappears slowly CG III (NPGR010) - J. Křivánek 2015 23
Review – Random variables
Random variable ◼ X … random variable ◼ X assumes different values with different probability ❑ Given by the probability distribution D ❑ X D CG III (NPGR010) - J. Křivánek 2015 25
Discrete random variable Probability mass function ◼ Finite set of values of x i ◼ Each assumed with prob. p i p n i = = p 1 p Pr( X x ) 0 x i i i i = i 1 Cumulative distribution func. ◼ Cumulative distribution 1 function P i ) i ( = = P Pr X x p P 1 i i j n = j 1 x i CG III (NPGR010) - J. Křivánek 2015 26
Continuous random variable ◼ Probability density function, pdf, p ( x ) ( ) = Pr X D p ( x ) d x D ◼ In 1D: ) ( b = Pr a X b p ( t ) d t a CG III (NPGR010) - J. Křivánek 2015 27
Continuous random variable ◼ Cumulative distribution function, cdf, P ( x ) V 1D: ) ( x = P ( x ) Pr X x p ( t ) d t − ( ) a = = = Pr X a p ( t ) d t 0 ! a CG III (NPGR010) - J. Křivánek 2015 28
Continuous random variable Example: Uniform distribution Probability density function (pdf) Cumulative distribution function (cdf) CG III (NPGR010) - J. Křivánek 2015 29
Continuous random variable Gaussian (normal) distribution Probability density function (pdf) Image: wikipedia Cumulative distribution function (cdf) CG III (NPGR010) - J. Křivánek 2015 30
Expected value and variance ◼ Expected value = E X x p ( x ) d x D ◼ Variance ❑ Properties of variance (if X i are independent) CG III (NPGR010) - J. Křivánek 2015 31
Transformation of a random variable 𝑍 = (𝑌) ◼ Y is a random variable ◼ Expected value of Y 𝐹[𝑍] = න 𝐲 𝑞 𝐲 𝑒𝐲 𝐸 CG III (NPGR010) - J. Křivánek 2015 32
Monte Carlo integration
Monte Carlo integration ◼ General tool for estimating definite integrals Integral: g ( x ) 𝐽 = න(𝐲)d𝐲 Monte Carlo estimate I : 𝑂 (𝜊 𝑙 ) 𝐽 = 1 p ( x ) 𝑂 𝑞(𝜊 𝑙 ) ; 𝜊 𝑙 ∝ 𝑞(𝐲) 𝑙=1 Works “on average” : 5 3 1 4 2 6 0 1 𝐹[ 𝐽 ] = 𝐽 CG III (NPGR010) - J. Křivánek 2015 34
Primary estimator of an integral ( ) = Integral to be estimated: I f x d x Let X be a random variable from the distribution with the pdf p ( x ), then the random variable F prim given by the transformation f ( X )/ p ( X ) is called the primary estimator of the above integral. f ( X ) = F prim p ( X ) CG III (NPGR010) - J. Křivánek 2015 35
Primary estimator of an integral f ( X ) f ( x ) X 0 1 CG III (NPGR010) - J. Křivánek 2015 36
Estimator vs. estimate ◼ Estimator is a random variable ❑ It is defined though a transformation of another random variable ◼ Estimate is a concrete realization (outcome) of the estimator ◼ No need to worry: the above distinction is important for proving theorems but less important in practice CG III (NPGR010) - J. Křivánek 2015 37
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