Advanced Computer Graphics Rendering Equation · · 2015-07-13University of Freiburg –Computer...

Matthias Teschner

Computer Science DepartmentUniversity of Freiburg

Advanced Computer Graphics

Rendering Equation

University of Freiburg – Computer Science Department – Computer Graphics - 2

Outline

rendering equation

Monte Carlo integration

sampling of random variables


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


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 bytracing a ray from p into direction

ray-casting operator nearest hit point from p into direction

if no surface is hit, radiance from a back-ground or light source can be returned

[Suffern]


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


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]


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

[Suffern]hemisphere form area form


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 andtransmission, 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. diffuse-diffuse light transport, Monte Carlo techniques are used to numerically evaluate the multi-dimensional integrals


Outline

rendering equation


sampling of random variables


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

Introduction


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 (n½), 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


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

Continuous Random Variables

The probability, that the random variable is in the specified domain, is 1.

The probability, that the random variable has a certain exact value, is 0.

infinite number of possible values(in contrast to discrete random variable)


motivation: expected value of an estimator functionis 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

properties

example for uniform

Expected Value

for independent random variables Xi

processes an infinite number of samples x according to a PDF p(x)


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

Variance

for independent random variables Xi


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

Monte Carlo EstimatorUniform Random Variables

constant and integration to one


Monte Carlo EstimatorUniform Random Variables


integral

estimator

for an increasing number of uniformly distributed random variables Xi , the estimator converges to one

Examples - Uniform Random Variables

[Suffern]

uniformly distributedrandom samples


Monte Carlo estimator

Monte Carlo EstimatorNon-uniform Random Variables

p (Xi) 0


samples Xi are multidimensional

e.g.

uniformly distributed random samples

probability density function

Monte Carlo estimator

N can be arbitrary, N is independent from the dimensionality

Monte Carlo EstimatorMultiple Dimensions


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

Monte Carlo EstimatorIntegration over a Hemisphere


probability distribution (cont.)

estimator

Monte Carlo EstimatorIntegration over a Hemisphere


Monte Carlo IntegrationSteps

choose an appropriate probability density function

generate random samples according to the PDF

evaluate the function for all samples

average the weighted function values


variance

estimator

for increasing N the variance decreases with O(N)

the standard deviation decreases with O(N½)

variance is perceived as noise

Monte Carlo EstimatorError


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

Monte Carlo EstimatorVariance Reduction / Error Reduction

[Suffern]


Outline

rendering equation


sampling of random variables inversion method

rejection method

transforming between distributions

2D sampling

examples


mapping of a uniform random variable to a goal distribution

discrete example four outcomes with probabilities

and

computation of the cumulative distribution function

Inversion Method

[Pharr, Humphreys]


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

Inversion Method

[Pharr, Humphreys]


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

Inversion MethodExample 1


exponential distribution e.g., for considering participating media

computation of the PDF

PDF

CDF

inverse of the CDF


are samples from the power distribution



piecewise-constant distribution e.g., for environment lighting

PDFwith


[Pharr, Humphreys]

University of Freiburg – Computer Science Department – Computer Graphics - 31[Pharr, Humphreys]

CDF

CDF is linear between and with slope


compute with and

compute with

are samples from



Outline

rendering equation



rejection method


2D sampling

examples


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

sample generation generate a uniform random sample generate a sample according to accept if

Rejection Method

[Pharr, Humphreys]

a b


Outline

rendering equation



rejection method


2D sampling

examples


computation of a resulting PDF, when a function isapplied to samples from an arbitrary distribution random variables are drawn from bijective transformation (one-to-one mapping) How does the distribution look like?

example

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



example (spherical coordinates)

example (solid angle)



Outline

rendering equation



rejection method


2D sampling

examples


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


Outline

rendering equation



rejection method


2D sampling

examples


Uniform Sampling of a Hemisphere

PDF is constant with respect to a solid angle

marginal density function

conditional density for

inversion method

[Suffern]



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



illustration for

generate less samples for smaller angles


Uniform Sampling of a Unit Disk

PDF is constant with respect to area


conditional density

inversion method


Uniform Sampling of a Unit Disk




generate less samples for smaller radii

1

r


Uniform Sampling of a Cosine-Weighted Hemisphere

PDF is proportional to


conditional density for

inversion method

[Suffern]






conversion to Cartesian space

is a normalized direction

x- y- values uniformly sample a unitdisk, i. e., cosine-weighted samplesof the hemisphere can also be obtainedby uniformly sampling a unit sphereand projecting the samples onto thehemisphere



illustration for

generate less samples for smaller and larger angles

cosine-weightedhemisphere

(top view, side view)

uniformhemisphere(top view)

[Suffern]


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


conditional density

inversion method

u

v(u, 1-u)

[Pharr, Humphreys]


Uniform Sampling of a Triangle




u is generated between 0 and 1

v is generated between 0 and 1-u=½


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




piecewise-constant 1D function

defined by values

conditional density

piecewise-constant 1D function

sample generation see example 3 of the inversion method



environment map

low-resolution of the marginal density function and the con-ditional distributions for rows

first, a row is selected accordingto the marginal density function

then, a column is selected fromthe row's 1D conditional distribution

[Pharr, Humphreys]

Paul Debevec, Grace Cathedral

Advanced Computer Graphics Rendering Equation · · 2015-07-13University of Freiburg –Computer...

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