Last TimeTodayIrradiance from Radiance (PBR Sect. 5.3)Integration MethodsProbability Theory OverviewDiscrete Random VariablesDiscrete Sampling (1)Discrete Sampling (2)Continuous Random VariablesExpected ValueVariance and Standard DeviationSamplingMonte Carlo IntegrationSimple ExampleOutputStandard Deviation of the EstimateRadiometric Integrals (PBR 5.3)Computing IrradianceIn Spherical CoordinatesSolid angle to SphericalIrradiance Integral in Spherical CoordsSolid Angle to AreaIrradiance Arriving From SurfaceNext Time01/24/05 © 2005 University of WisconsinLast Time•Raytracing and PBRT Structure•Radiometric quantities01/24/05 © 2005 University of WisconsinToday•Radiometric Integrals•Monte Carlo integration•Section 5.3 and Chapter 14 of PBR01/24/05 © 2005 University of WisconsinIrradiance from Radiance (PBR Sect. 5.3)•Integrate radiance over directions in the upper hemisphere:–cos term deals with projected solid angle.  is angle between  and n (the normal)•We are converting “per unit solid angle per unit projected area” into “per unit solid angle per unit area” and then integrating over solid angle to get “per unit area”•Today: solving integrals like this   )(i2cos,nppHdLE01/24/05 © 2005 University of WisconsinIntegration Methods•Analytic: not tractable for most functions you want to integrate•(Numerical) Quadrature:–Break the domain of integration into pieces, evaluate the function once in each piece, and sum up value for all pieces, weighted by the “size” of each little area–A very poor strategy for high-dimensional integrals – we will have lots of these, even infinite dimensional•Monte Carlo integration:–Evaluate the function at random points in the domain, and sum up the answers–Error independent of dimensionality of problem01/24/05 © 2005 University of WisconsinProbability Theory Overview•The aim is to give you enough to survive, for more see a probability (not statistics) textbook•A random variable X is a value chosen by some random process–Rolling dice, nuclear decay, pseudo random number generator, …•We are interested in the properties of random variables01/24/05 © 2005 University of WisconsinDiscrete Random Variables•Consider rolling a die•Possible values for random variable are Xi={1,2,3,4,5,6}•Probability of seeing some value is pi=1/6•Sampling x according to pi means choosing a value for x such that the probability that x=Xi is pi•In rendering, the most common discrete case is choosing a light, Li{L1,…,Ln}, according to the power output:jjiip01/24/05 © 2005 University of WisconsinDiscrete Sampling (1)•Always assume we can sample a canonical uniform random variable [0,1)–In PBRT, function: genrand_real1()–Always get same sequence, which can be annoying•We want to use this to choose a light according to pi•Choose light Li if ijjijjpp11101/24/05 © 2005 University of WisconsinDiscrete Sampling (2)•Define–The cumulative distribution function, the probability that a variable chosen according to the distribution pi will be less than Li•To sample according to pi, sample  then choose Li such that–Build an array of Pi values (sorted), and then search it to find the index such that above equation is true (binary search for large arrays)ijjipP1iiPP 101/24/05 © 2005 University of WisconsinContinuous Random Variables•A random variable, X–Takes values from some domain, •Has an associated probability density function (pdf), p(x)•Methods for sampling continuous random variables according to various distributions on various domains are discussed in PBR Sect 14.3-14.5–Again, useful to know what is available and how to use it, but not strictly necessary to understand how they workAdyypxP )()(01/24/05 © 2005 University of WisconsinExpected Value•The expected value of a random variable, x, is defined as:•The expected value of a function, f(x), is defined as:•The sample mean, for samples xi is defined as: dxxxpxE )(][ dxxpxfxfE )()()]([   niiniixgnxgxnx111101/24/05 © 2005 University of WisconsinVariance and Standard Deviation•The variance of a random variable is defined as:•The standard deviation of a random variable is defined as the square root of its variance:•The sample variance is:222][][][]])[[(][xExExVxExExV][][ xvx  niixxn12101/24/05 © 2005 University of WisconsinSampling•A process samples according to the distribution p(x) if it randomly chooses a value for x such that:•Weak Law of Large Numbers: If xi are independent samples from p(x), then in the limit of infinite samples, the sample mean is equal to the expected value:Adyypx )(y that probabilit the,A11lim)(Pr11lim][Pr11niinxniinxndxxxpxnxE01/24/05 © 2005 University of WisconsinMonte Carlo Integration•Say we wish to integrate•Choose some pdf, p(x)•If we sample xi, i{1,…,N}, according to p(x), then:ydyyf )(yNiiidyyfxpxfNE )()()(1101/24/05 © 2005 University of WisconsinSimple Example•Compute•Sample xi uniform on interval [1,5), so p(x)=1/4–Sample canonical i then xi=4i + 1•Monte Carlo Estimate is512dxxNiiNiixNxN1212441101/24/05 © 2005 University of WisconsinOutput01/24/05 © 2005 University of WisconsinStandard Deviation of the Estimate•Expected error in the estimate after N samples is measured by the standard deviation of the estimate:•Note that error goes down with •Often, p(x) is the uniform distribution over the domain•If p(x) is something else, the technique is called importance sampling and p(x) is the importance function•p must be >0 whenever f>0, and should be as close as possible to f•Same principle for high dimensional integralspfNxpxfNNiii1)()(11N101/24/05 © 2005 University of WisconsinRadiometric Integrals (PBR 5.3)•Physically-Based rendering is all about solving integral equations involving radiometric terms•The domains of integration are areas, or regions of solid angle, or even more abstract spaces–Choosing the right domain is one consideration•The challenge is finding a way to reduce variance, which manifests itself as