Advanced Computer Graphics. Rendering Equation. Matthias Teschner. Computer Science Department University of Freiburg

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1 Advanced Computer Graphics Rendering Equation Matthias Teschner Computer Science Department University of Freiburg

2 Outline rendering equation Monte Carlo integration sampling of random variables University of Freiburg Computer Science Department Computer Graphics - 2

3 Reflection and Rendering Equation reflection equation at point p for reflective surfaces incident radiance - weighted with the BRDF - is integrated over the hemisphere to compute the outgoing radiance expresses energy balance between surfaces outgoing radiance from a surface can be incident to another surface rendering equation at point p for reflective surfaces adds emissive surfaces to the reflection equation exitant radiance is the sum of emitted and reflected radiance expresses the steady state of radiance in a scene including light sources University of Freiburg Computer Science Department Computer Graphics - 3

4 Ray-Casting Operator in general, the incoming radiance is not only determined by light sources, but also by outgoing radiance of reflective surfaces incident radiance can be computed by tracing a ray from p into direction ray-casting operator nearest hit point from p into direction if no surface is hit, radiance from a background or light source can be returned University of Freiburg Computer Science Department Computer Graphics - 4 [Suffern]

5 Rendering Equation with Ray-Casting Operator using the ray-casting operator, can be rewritten as goal: computation of the outgoing radiance at all points p into all directions towards the camera to compute the image towards other surface points to account for indirect illumination University of Freiburg Computer Science Department Computer Graphics - 5

6 Forms of the Rendering Equation hemisphere form area form p is a sample point on a surface da visibility function solid angle vs. area [Suffern] University of Freiburg Computer Science Department Computer Graphics - 6

7 Forms of the Rendering Equation the area form works with a visibility term useful for direct illumination from area lights the hemisphere form works with the ray-casting operator useful for indirect illumination hemisphere form area form [Suffern] University of Freiburg Computer Science Department Computer Graphics - 7

8 Solving the Rendering Equation recursively cast rays into the scene maximum recursion depth due to absorption of light for point lights, directional lights, perfect reflection and transmission, the integrals reduce to simple sums radiance from only a few directions contributes to the outgoing radiance for area lights and indirect illumination, i. e. diffusediffuse light transport, Monte Carlo techniques are used to numerically evaluate the multi-dimensional integrals University of Freiburg Computer Science Department Computer Graphics - 8

9 Outline rendering equation Monte Carlo integration sampling of random variables University of Freiburg Computer Science Department Computer Graphics - 9

10 Introduction approximately evaluate the integral by randomly sampling the hemisphere tracing rays into the sample directions computing the incoming radiance from the sample directions challenge approximate the integral as exact as possible trace as few rays as possible trace relevant rays for diffuse surfaces, rays in normal direction are more relevant than rays perpendicular to the normal for specular surfaces, rays in reflection direction are relevant rays to light sources are relevant University of Freiburg Computer Science Department Computer Graphics - 10

11 Properties benefits processes only evaluations of the integrand at arbitrary points in the domain works for a large variety of integrands, e.g., it handles discontinuities appropriate for integrals of arbitrary dimensions drawbacks using n samples, the scheme converges to the correct result with O (n ½ ), i.e. to half the error, 4n samples are required errors are perceived as noise, i.e. pixels are arbitrarily too bright or dark evaluation of the integrand at a point is expensive University of Freiburg Computer Science Department Computer Graphics - 11

12 Continuous Random Variables continuous random variables canonical uniform random variable samples from arbitrary distributions can be computed from probability density function (PDF) the probability of a random variable taking certain value ranges cumulative distribution function (CDF) describes the probability of a random variable to be less or equal to x infinite number of possible values (in contrast to discrete random variable) The probability, that the random variable has a certain exact value, is 0. The probability, that the random variable is in the specified domain, is 1. University of Freiburg Computer Science Department Computer Graphics - 12

13 Expected Value motivation: expected value of an estimator function is equal to the integral in the rendering equation expected value of a function is defined as the weighted average value of the function over a domain with processes an infinite number of samples x according to a PDF p(x) properties example for uniform for independent random variables X i University of Freiburg Computer Science Department Computer Graphics - 13

14 Variance motivation: quantifies the error of a Monte Carlo algorithm variance of a function is the expected deviation of the function from its expected value properties for independent random variables X i University of Freiburg Computer Science Department Computer Graphics - 14

15 Monte Carlo Estimator Uniform Random Variables motivation: approximation of the integral in the rendering equation goal: computation of uniformly distributed random variables probability density function Monte Carlo estimator expected value of is equal to the integral variance convergence rate of independent from the dimensionality appropriate for high-dimensional integrals constant and integration to one University of Freiburg Computer Science Department Computer Graphics - 15

16 Monte Carlo Estimator Uniform Random Variables University of Freiburg Computer Science Department Computer Graphics - 16

17 Examples - Uniform Random Variables integral estimator for an increasing number of uniformly distributed random variables X i, the estimator converges to one uniformly distributed random samples [Suffern] University of Freiburg Computer Science Department Computer Graphics - 17

18 Monte Carlo Estimator Non-uniform Random Variables Monte Carlo estimator p (X i ) 0 University of Freiburg Computer Science Department Computer Graphics - 18

19 Monte Carlo Estimator Multiple Dimensions samples X i are multidimensional e.g. uniformly distributed random samples probability density function Monte Carlo estimator N can be arbitrary, N is independent from the dimensionality University of Freiburg Computer Science Department Computer Graphics - 19

20 Monte Carlo Estimator Integration over a Hemisphere approximate computation of the irradiance at a point estimator probability distribution should be similar to the shape of the integrand as incident radiance is weighted with cos, it is appropriate to generate more samples close to the top of the hemisphere University of Freiburg Computer Science Department Computer Graphics - 20

21 Monte Carlo Estimator Integration over a Hemisphere probability distribution (cont.) estimator University of Freiburg Computer Science Department Computer Graphics - 21

22 Monte Carlo Integration Steps choose an appropriate probability density function generate random samples according to the PDF evaluate the function for all samples average the weighted function values University of Freiburg Computer Science Department Computer Graphics - 22

23 Monte Carlo Estimator Error variance estimator for increasing N the variance decreases with O(N) the standard deviation decreases with O(N ½ ) variance is perceived as noise University of Freiburg Computer Science Department Computer Graphics - 23

24 Monte Carlo Estimator Variance Reduction / Error Reduction importance sampling motivation: contributions of larger function values are more important PDF should be similar to the shape of the function optimal PDF e.g., if incident radiance is weighted with cos, the PDF should choose more samples close to the normal direction stratified sampling domain subdivision into strata does not increase the variance multi-jittered sampling alternative to random samples for, e.g., uniform sampling of area lights University of Freiburg Computer Science Department Computer Graphics - 24 [Suffern]

25 Outline rendering equation Monte Carlo integration sampling of random variables inversion method rejection method transforming between distributions 2D sampling examples University of Freiburg Computer Science Department Computer Graphics - 25

26 Inversion Method mapping of a uniform random variable to a goal distribution discrete example four outcomes with probabilities and computation of the cumulative distribution function University of Freiburg Computer Science Department Computer Graphics - 26 [Pharr, Humphreys]

27 Inversion Method discrete example cont. take a uniform random variable has the desired distribution continuous case and are continuous functions start with the desired PDF compute compute the inverse obtain a uniformly distributed variable compute which adheres to University of Freiburg Computer Science Department Computer Graphics - 27 [Pharr, Humphreys]

28 Inversion Method Example 1 power distribution e.g., for sampling the Blinn microfacet model computation of the PDF PDF CDF inverse of the CDF sample generation generate uniform random samples are samples from the power distribution University of Freiburg Computer Science Department Computer Graphics - 28

29 Inversion Method Example 2 exponential distribution e.g., for considering participating media computation of the PDF PDF CDF inverse of the CDF sample generation generate uniform random samples are samples from the power distribution University of Freiburg Computer Science Department Computer Graphics - 29

30 Inversion Method Example 3 piecewise-constant distribution e.g., for environment lighting PDF with University of Freiburg Computer Science Department Computer Graphics - 30 [Pharr, Humphreys]

31 Inversion Method Example 3 CDF CDF is linear between and with slope sample generation generate uniform random samples compute with and compute with are samples from University of Freiburg Computer Science Department Computer Graphics - 31 [Pharr, Humphreys]

32 Outline rendering equation Monte Carlo integration sampling of random variables inversion method rejection method transforming between distributions 2D sampling examples University of Freiburg Computer Science Department Computer Graphics - 32

33 Rejection Method draws samples according to a function dart-throwing approach works with a PDF and a scalar with properties is not necessarily a PDF PDF, CDF and inverse CDF do not have to be computed simple to implement useful for debugging purposes a sample generation generate a uniform random sample generate a sample according to accept if University of Freiburg Computer Science Department Computer Graphics - 33 b [Pharr, Humphreys]

34 Outline rendering equation Monte Carlo integration sampling of random variables inversion method rejection method transforming between distributions 2D sampling examples University of Freiburg Computer Science Department Computer Graphics - 34

35 Transforming Between Distributions computation of a resulting PDF, when a function is applied to samples from an arbitrary distribution random variables are drawn from bijective transformation (one-to-one mapping) How does the distribution look like? example University of Freiburg Computer Science Department Computer Graphics - 35

36 Transforming Between Distributions multiple dimensions is an n-dimensional random variable is a bijective transformation transformation of the PDF example (polar coordinates) samples with density corresponding density with and University of Freiburg Computer Science Department Computer Graphics - 36

37 Transforming Between Distributions example (spherical coordinates) example (solid angle) University of Freiburg Computer Science Department Computer Graphics - 37

38 Outline rendering equation Monte Carlo integration sampling of random variables inversion method rejection method transforming between distributions 2D sampling examples University of Freiburg Computer Science Department Computer Graphics - 38

39 Concept generation of samples from a 2D joint density function general case compute the marginal density function compute the conditional density function generate a sample according to generate a sample according to marginal density function integral of for a particular over all -values conditional density function density function for given a particular University of Freiburg Computer Science Department Computer Graphics - 39

40 Outline rendering equation Monte Carlo integration sampling of random variables inversion method rejection method transforming between distributions 2D sampling examples University of Freiburg Computer Science Department Computer Graphics - 40

41 Uniform Sampling of a Hemisphere PDF is constant with respect to a solid angle marginal density function conditional density for inversion method University of Freiburg Computer Science Department Computer Graphics - 41 [Suffern]

42 Uniform Sampling of a Hemisphere inversion method cont. inverse functions of the cumulative distribution functions generating uniformly sampled random values and applying the inverse CDFs to obtain and conversion to Cartesian space is a normalized direction University of Freiburg Computer Science Department Computer Graphics - 42

43 Uniform Sampling of a Hemisphere illustration for generate less samples for smaller angles University of Freiburg Computer Science Department Computer Graphics - 43

44 Uniform Sampling of a Unit Disk PDF is constant with respect to area marginal density function conditional density inversion method University of Freiburg Computer Science Department Computer Graphics - 44

45 Uniform Sampling of a Unit Disk inversion method cont. inverse functions of the cumulative distribution functions generating uniformly sampled random values and applying the inverse CDFs to obtain and r 1 generate less samples for smaller radii University of Freiburg Computer Science Department Computer Graphics - 45

46 Uniform Sampling of a Cosine-Weighted Hemisphere PDF is proportional to marginal density function conditional density for inversion method University of Freiburg Computer Science Department Computer Graphics - 46 [Suffern]

47 Uniform Sampling of a Cosine-Weighted Hemisphere inversion method cont. inverse functions of the cumulative distribution functions generating uniformly sampled random values and applying the inverse CDFs to obtain and conversion to Cartesian space is a normalized direction x- y- values uniformly sample a unit disk, i. e., cosine-weighted samples of the hemisphere can also be obtained by uniformly sampling a unit sphere and projecting the samples onto the hemisphere University of Freiburg Computer Science Department Computer Graphics - 47

48 Uniform Sampling of a Cosine-Weighted Hemisphere illustration for generate less samples for smaller and larger angles cosine-weighted hemisphere (top view, side view) uniform hemisphere (top view) University of Freiburg Computer Science Department Computer Graphics - 48 [Suffern]

49 Uniform Sampling of a Triangle sampling an isosceles right triangle of area 0.5 u, v can be interpreted as Barycentric coordinates can be used to generate samples for arbitrary triangles marginal density function conditional density inversion method v u (u, 1-u) University of Freiburg Computer Science Department Computer Graphics - 49 [Pharr, Humphreys]

50 Uniform Sampling of a Triangle inversion method cont. inverse functions of the cumulative distribution functions u is generated between 0 and 1 v is generated between 0 and 1-u= ½ generating uniformly sampled random values and applying the inverse CDFs to obtain and University of Freiburg Computer Science Department Computer Graphics - 50

51 Piecewise-Constant 2D Distribution samples defined over e.g., an environment map is defined by a set of values is the value of in the range with and integral over the domain PDF University of Freiburg Computer Science Department Computer Graphics - 51

52 Piecewise-Constant 2D Distribution marginal density function piecewise-constant 1D function defined by values conditional density piecewise-constant 1D function sample generation see example 3 of the inversion method University of Freiburg Computer Science Department Computer Graphics - 52

53 Piecewise-Constant 2D Distribution environment map low-resolution of the marginal density function and the conditional distributions for rows first, a row is selected according to the marginal density function then, a column is selected from the row's 1D conditional distribution Paul Debevec, Grace Cathedral University of Freiburg Computer Science Department Computer Graphics - 53 [Pharr, Humphreys]

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