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1. Last Time • Raytracing and PBRT Structure • Radiometric quantities © 2005 University of Wisconsin

2. Today • Radiometric Integrals • Monte Carlo integration • Section 5.3 and Chapter 14 of PBR © 2005 University of Wisconsin

3. Irradiance 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 © 2005 University of Wisconsin

4. Integration 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 problem © 2005 University of Wisconsin

5. Probability 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 variables © 2005 University of Wisconsin

6. Discrete 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 topimeans 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: © 2005 University of Wisconsin

7. Discrete 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 © 2005 University of Wisconsin

8. Discrete 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) © 2005 University of Wisconsin

9. Continuous 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 © 2005 University of Wisconsin

10. Expected 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: © 2005 University of Wisconsin

11. Variance 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: © 2005 University of Wisconsin

12. Sampling • 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: © 2005 University of Wisconsin

13. Monte Carlo Integration • Say we wish to integrate • Choose some pdf, p(x) • If we sample xi, i{1,…,N}, according to p(x), then: © 2005 University of Wisconsin

14. Simple 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 © 2005 University of Wisconsin

15. Output © 2005 University of Wisconsin

16. Standard 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 © 2005 University of Wisconsin

17. Radiometric 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 noise in images • More on this later, after we have some more background © 2005 University of Wisconsin

18. Computing Irradiance • This integral is expressed in terms of solid angle within the upper hemisphere • To solve it, we need to sample directions • We can’t represent  with just one number • It’s a multi-dimensional integral • How do we parameterize directions? © 2005 University of Wisconsin

19. In Spherical Coordinates • Note that =0 is normal vector direction • We need a basis to define  • The tangent vectors • How do we convert an angle expressed in terms of solid angle, to one in terms of spherical coordinates? • Convert domain to range of , • Convert d to dd © 2005 University of Wisconsin

20. Solid angle to Spherical • d is projected area • Recall the definition of solid angle • What area goes with dd? © 2005 University of Wisconsin

21. Irradiance Integral in Spherical Coords • In general, the incoming radiance varies over the scene • It depends on what is “seen” in each direction • If incoming radiance is constant, then • Conversion functions for unit vectors =(x,y,z) to spherical coordinates are available in PBRT © 2005 University of Wisconsin

22. Solid Angle to Area • Solid angle is defined in terms of area projected onto the unit sphere © 2005 University of Wisconsin

23. Irradiance Arriving From Surface • We want to integrate the irradiance due to an area light source • Note there are two  now • Convert integral over area into integral over (s,t) (parameters for surface) © 2005 University of Wisconsin

24. Next Time • Cameras and Film Plane Sampling © 2005 University of Wisconsin