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Graphics & Visualization

Chapter 16

GLOBAL ILLUMINATION ALGORITHMS

Graphics & Visualization: Principles & Algorithms Chapter 16

Graphics & Visualization: Principles & Algorithms Chapter 16

• Global Illumination Algorithms:



Deal with the realistic computation of light transport in a scene



Estimate indirect illumination, i.e. light reaching a point through multiple surface

inter-reflections



Compute both direct illumination and indirect light

• Images are radiometrically

accurate and photorealistic

• All aspects of the image-generator pipeline are based on physics:



The reflection properties of all materials are described by BRDFs



The light sources are radiometrically modeled



The transport of light through the scene is computed accurately



The display of the image uses accurate tone-mapping operators

Introduction

2

Graphics & Visualization: Principles & Algorithms Chapter 16

• The rendering equation:



Is the most fundamental equation for photorealistic synthesis



Expresses the equilibrium of light distribution in a 3D scene



Takes into account:

• The radiometric specification of the light sources
• The BRDF specifications of all materials
• It is an energy balance that expresses how much excitant radiance is

present at a given surface point in a certain direction

The Physics of Light-Object Interaction II

3

Graphics & Visualization: Principles & Algorithms Chapter 16

• BRDF at a surface point x:



Expresses excitant radiance Lr in direction (φr , θr ) versus incident irradiance Ei from direction (φi , θi ): (16.1)

• Integrate Eq(16.1) over the hemisphere Ωi
• f all possible

differential solid angles dωi :

Rendering Equation: Hemispherical Integration

( , ) ( , ) ( , , , ) ( , ) ( , )cos( )

r r r r r r r r r i i i i i i i i i i

dL dL f dE L d                

( , ) ( , ) ( , , , )cos( ) ( , ) ( , ) ( , , , )cos( )

i

r r r i i i r r r i i i i r r r i i i r r r i i i i

dL L f d L L f d                    



  

4

Graphics & Visualization: Principles & Algorithms Chapter 16

• Add a constant term, which corresponds to the self-emitted radiance

Le (φr , θr ) of point x

• The complete rendering equation is:

Rendering Equation: Hemispherical Integration (2)

( , ) ( , ) ( , ) ( , , , )cos( )

i

r r r e r r i i i r r r i i i i

L L L f d            



 

5

(16.2)

Graphics & Visualization: Principles & Algorithms Chapter 16

• We can move from solid angle to surface integration domain
• The integral is taken over all visible surfaces
• Transform dωi to the corresponding differential surface dA
• Let



y: the first visible surface point seen from point x in direction (φi,θi)



(φi yθy ) : the direction pointing from y towards x



rxy: the distance between x and y



Svisible: the set of all visible surfaces as seen from x

• Then:
• Then Lr

becomes:

Rendering Equation: Surface-Area Integration

6

2

cos( )

y i

dA d r   

xy

2

cos( )cos( ) ( , ) ( , ) ( , ) ( , , , )

Visible

i y r r r e r r S i i i r r r i i

L L L f dA r              

xy

(16.3)

Graphics & Visualization: Principles & Algorithms Chapter 16

• Radiance remains constant along a straight line (no scattering),

so we may replace received radiance with emitted radiance at the source:

• In equation Eq(16.3):



the product of both cosine terms divided by rxy

2

• is a geometric coupling term
• is dependent on the geometrical relationship between x and y
• is independent of the actual radiance distribution or BRDF's

defined on the surfaces:

Rendering Equation: Surface-Area Integration (2)

7

( , ) ( , , ) ( , , )

i i i i i i r y y

L L L         x y

2

cos( )cos( ) ( , )

i y

G r   

xy

x y

Graphics & Visualization: Principles & Algorithms Chapter 16

• Substituting all of the above in Eq[16.3]:

Rendering Equation: Surface-Area Integration (3)

8

( , , ) ( , , ) ( , , ) ( , , , ) ( , )

Visible

r r r e r r S r y y r r r i i

L L L f G dA             x x y x y

Graphics & Visualization: Principles & Algorithms Chapter 16

• Integral over all surfaces is preferable to integral over visible

surfaces only:



Advantage: Single integration domain, identical for all points x

• We introduce a visibility term V(x, y):
• The rendering equation becomes:

S: the integration domain indicating all surface point y

9

1 , and are mutually invisible V( ) 0 , otherwise     x y x,y

( , , ) ( , , ) ( , , ) ( , , , ) ( , ) ( , )

r r r e r r r y y r r r i i S

L L L f G V dA             x x y x y x y

Rendering Equation: Surface-Area Integration (4)

Graphics & Visualization: Principles & Algorithms Chapter 16

• A special case:



Direct illumination from one light source S1 : the surface area domain of the light source



Direct illumination from more light sources:

• Split the integral in a sum of integrals for each light source

10

1

( , , ) ( , , ) ( , , , ) ( , ) ( , )

r r r S e y y r r r i i

L L f G V dA           x y x y x y

1

( , , ) ( , , ) ( , , , ) ( , ) ( , )

j

L r r r S e y y r r r i i j

L L f G V dA        





 

x y x y x y

Rendering Equation: Surface-Area Integration (5)

Graphics & Visualization: Principles & Algorithms Chapter 16 

Direct illumination from one light source

11

Rendering Equation: Surface-Area Integration (6)

Graphics & Visualization: Principles & Algorithms Chapter 16

• The light source is encoded as a (hemi)-spherical environment map
• An emitted radiance Le(φi,θi) is defined for each incoming direction
• Usually is given as a high dynamic range image
• Can contain more than a million pixels

Environment Map Illumination

12

( , ) ( , ) ( , , , )cos( )

i

r r r e i i r r r i i i i

L L f d          



 

Graphics & Visualization: Principles & Algorithms Chapter 16

• For some applications, it is useful to express:



Light energy per surface patch (usually individual polygons)



The hemisphere of all outgoing directions

• Achieved by discretizing

the rendering equation

• We solve a linear system: each equation describes the energy

balance of a single patch Radiosity algorithms

• Note that not all of these algorithms use the radiosity

B radiometric quantity

Discretized Form of the Rendering Equation

13

Graphics & Visualization: Principles & Algorithms Chapter 16

Assumptions for the formulation of radiosity equations:

• All surfaces in the scene are subdivided in surface patches
• The outgoing radiance is similar for all surface points on the patch
• The algorithm will compute only the average radiance of all surface

points

• All surface patches have diffuse reflectance characteristics
• Each patch has only 1 radiance value as a final solution
• The light sources are considered to be diffuse as well (although

this is not strictly necessary)

Discretized Form of the Rendering Equation (2)

14

Graphics & Visualization: Principles & Algorithms Chapter 16

• Radiosity

B for a single point x



Is the flux per surface area, or



Radiance integrated over the hemisphere of outgoing directions at x

• Average radiosity

Bi emitted by a surface patch i with area Ai:

• On purely diffuse surfaces:



self-emitted radiance Le and the BRDF fr do not depend on incoming or

• utgoing directions
• Rendering equation for a surface point x:

Discretized Form of the Rendering Equation (3)

15

1 ( , , )cos( )

i

i S r r r r i i

B L d dA A    





 

x

x ( ) ( ) ( , , ) ( )cos( )

x

r e i i i r i i

L L L f d    



  x x x x

Graphics & Visualization: Principles & Algorithms Chapter 16

• The incident radiance Li(x,φi,θi):



depends on incident direction



corresponds to the exitant radiance Lr(y) emitted towards x by the point y visible from x along the direction (φi,θi)

• The integral equation without any directions present:
• In a diffuse environment:



Radiosity and radiance are related



B(x) = π Lr(x) and Be(x) = π Le(x)

Discretized Form of the Rendering Equation (4)

16

( ) ( ) ( ) ( , ) ( , ) ( )

r e r r S

L L f G V L dA  



y

x x x x y x y y

Graphics & Visualization: Principles & Algorithms Chapter 16

• Radiosity

Integral equation:



Multiply by π

• f the light-

and left- hand side of the above equation where

ρ(x) = π fr(x) :the diffuse hemispherical reflectance

and

K(x,y) = G(x, y)V(x, y)

• The rendering equation becomes:

Discretized Form of the Rendering Equation (5)

17

( ) ( ) ( ) ( , ) ( )

e S

B B K B dA    



y

x x x x y y

1 1 1 ( ) cos( ) ( ) ( )

i i i

i S r r i S r S i i i

B L d dA L dA B dA A A A   



  

   

x

x x x

16.4

Graphics & Visualization: Principles & Algorithms Chapter 16

• Assume the radiosity

B(x) is constant over each surface element i:



B(x) = Bi for all x Si

• Eq[16.4] can be converted into a linear system as follows:

Discretized Form of the Rendering Equation (6)

18



( ) ( ) ( ) ( , ) ( ) 1 1 1 ( ) ( ) ( ) ( , ) ( ) 1 1 1 ( ) ( ) ( ) ( , ) ( ) 1 ( ) ( , )

i i i i i i j i j

e y S S x S e x S y x S i i i S x S e x S S y x j i i i i ei j S S y x j i

B x B x K x y B y dA B x dA B x dA K x y B y dA dA A A A B x dA B x dA K x y B y dA dA A A A B B B K x y dA dA A                     

            

x x x x

Graphics & Visualization: Principles & Algorithms Chapter 16

• Assume the hemispherical diffuse reflectivity is constant over the

surface patch:



i.e. ρ(x) = ρi for all x Si



the following classical radiosity system of equations results:

• Patch-to-patch form factors Fij

:

• The form factors:



represent the amount of energy transfer between 2 surface patches i and j



are nontrivial 4D integrals



depend on the geometry of the scene



do not depend on any specific configuration of light sources

Discretized Form of the Rendering Equation (7)

19



i ei i ij j j

B B F B      

1 ( , )

i j

ij S S y x i

K x y F dA dA A  

 

Graphics & Visualization: Principles & Algorithms Chapter 16

• Is an important tool for evaluating the global illumination equation
• Evaluates integrals based on a selection of random samples in the

integration domain

• The Monte Carlo method:



Generates random points in the integration domain



Evaluates the integrand in each random sample



Averages these evaluations

• Works no matter how complex the function to be integrated is
• Drawback:



Slow converge rate: when drawing N samples, we expect a converge rate of

Monte Carlo Integration

1/ N

20

Graphics & Visualization: Principles & Algorithms Chapter 16

Verification of the method:

• Evaluate the following 1D integral:
• Draw uniformly N samples x1

, x2 ,…, xN from the domain [0,1]

• Average the function evaluations f(xi

) to obtain an estimation for I:

• The expected value of equals the value of I:
• Every different computation of will yield a different result
• But on average, we will get the same answer.

Monte Carlo Integration (2)

1

( ) I f x dx  

1

1 ( )

N i i

I f x N



  



[ ] E I  

I  

1 1 1 1 1 1

1 1 1 [ ] [ ( )] [ ( )] ( ) 1 · · ( ) ( )

N N N i i i i i

E I E f x E f x f x dx N N N N f x dx f x dx I N

  

       

    

I  

21

Graphics & Visualization: Principles & Algorithms Chapter 16

• The variance σ2

indicates the spread of possible values of around the expected outcome I:

• As the number of N increases:



σ2 decreases linearly with N  σ decreases with

• To decrease the error by a factor of 2 need 4 times as many samples
• The converge speed:



Is lower than that of many other integration techniques, but



Is independent of the number of dimensions in the integral

Monte Carlo Integration (3)

I  

1/ N

2 2 2 2 1 1 1 1 2 2 2

1 1 [ ] [ ( )] [ ( )] 1 1 · · ( ( ) ) ( ( ) )

N N i i i i

I f x f x N N N f x I dx f x I dx N N   

 

       

   

22

Graphics & Visualization: Principles & Algorithms Chapter 16

Generalization:

• Generalize to the domain [a,b] and use a non-uniform probability

density p(x) to draw the samples:

• Again
• The pdf

p(x) has to:



Be strictly larger than 0 over the entire integration domain



Integrate to 1:

Monte Carlo Integration (4)

2 2 1

( ) 1 1 ( ) , [ ] ( ) ( ) ( )

N b i a i i

f x f x I I I dx N p x N p x 



      

 

( ) E I I    ( ) 1

b a p x dx 



23

Graphics & Visualization: Principles & Algorithms Chapter 16

• To draw samples distributed according to p(x):



Compute the cumulative distribution function P(x): P(x): is a monotonic increasing function over [a,b] P(a) = 0 and P(b) = 1



If a random number t is generated uniformly over [0,1], then x = P-1(t) is distributed according to p(x)

• In practice, to compute the inverse value quickly:



Compute the inverse cumulative function



Store it as a table



Use binary search to compute the inverse value

Monte Carlo Integration (5)

( ) ( )

x a

P x p y dy  

24

Graphics & Visualization: Principles & Algorithms Chapter 16

• Multidimensional Monte Carlo integration:



Works in the same way as 1D integration



If the integral is defined over a domain [a, b]×[c, d]:

• Advantages:



simple to implement



Very robust



Provides an answer independent of:

• The complexity of the function
• The dimensions of the domain
• Drawbacks:



The error is hard to control and cannot be expressed by explicit lower and upper bounds

Monte Carlo Integration (6)

25 1

( , ) 1 ( , ) , ( , )

N b d i i a c i i i

f x y I f x y dxdy I N p x y



   

  

Graphics & Visualization: Principles & Algorithms Chapter 16

• Usually involves applying Monte Carlo integration
• Assumptions:



Compute the reflected radiance value

• In a surface point x
• In a specific direction (usually the direction pointing towards the

camera)

Computing Direct Illumination

26

Graphics & Visualization: Principles & Algorithms Chapter 16

• Integration domain: the surface Sl
• f the light source
• Pdf

p(y): generates surface points yi

• ver the total light source area
• Use N

sample points {y1, y2,…, yN}

• The estimator for the radiance value Lr(x,φr, θr):
• Pdf

p(y):



Is a 2D pdf



Generates 2 coordinates u and v



Transformed to a 3D point on the surface with a proper mapping



The mapping is identical to the texture-mapping procedure

Single Light Source

27 1

( , , ) ( , , , ) ( , ) ( , ) 1 ( , , ) ( )

j j

N e j y y r r r i i j r r r j j

L f G V L N p        



  



j

y x y x y x y

Graphics & Visualization: Principles & Algorithms Chapter 16

• Each point on the light source has pdf

p(yi) ≠0, otherwise:



Some parts of the light source will not be sampled



The estimated radiance will be biased

Single Light Source (2)

28

Graphics & Visualization: Principles & Algorithms Chapter 16

Algorithm:

// direct illumination from a single light source // for a surface point x, direction phi, theta directIllumination (x, phi, theta) estimatedRadiance = 0; for all shadow rays generate point y on light source; estimatedRadiance += Le(y,phi_y,theta_y)*BRDF*radianceTransfer(x,y)/pdf(y); estimatedRadiance = estimatedRadiance / #shadowRays; return(estimatedRadiance);

Single Light Source (3)

29

Graphics & Visualization: Principles & Algorithms Chapter 16

Algorithm (cont.):

// transfer between x and y // 2 cosines, distance and visibility taken into account radianceTransfer(x,y) transfer = G(x,y)*V(x,y); return(transfer);

• Evaluating the visibility term V(x, yi):



Shoot a shadow ray from x towards y



Check whether any objects are blocking the visibility

Single Light Source (4)

30

Graphics & Visualization: Principles & Algorithms Chapter 16

Single Light Source (5)

31

Graphics & Visualization: Principles & Algorithms Chapter 16

Noise factors: 1. The visibility function V(x, yi)



V(x, yi) = 1 yi  light source fully visible to x



V(x, yi) = 0 yi  light source fully occluded



Otherwise, artifacts will occur



Increase the number of samples to smooth the soft shadow



The number of samples depends on the size of penumbra region

2. The geometric coupling term G(x, yi):



Not a problem when the light source is small



When it is large, for points x located close to it:

• 1/ rxyj

takes on arbitrary large values (instability)

• Results to very bright pixels

Single Light Source (6)

32

 

Graphics & Visualization: Principles & Algorithms Chapter 16

Noise factors (cont.): 3. BRDFs:



If the BRDF values vary largely, the picture has additional noise

4. Pdf p(y):



Can choose any pdf



Usually is uniform over the source light

Single Light Source (7)

33

Graphics & Visualization: Principles & Algorithms Chapter 16

• Two approaches:



Individual light sources



Combined light sources

• Individual light sources



All light sources are individual contributors to the illumination of a single point:

• Light is linear additive 

the separate contributions of each light source can be added

• A number of shadow rays is generated for each light source:

they can be chosen independently

Multiple Light Sources

34

Graphics & Visualization: Principles & Algorithms Chapter 16

• Group all light sources in a single integration domain:



Apply Monte Carlo integration to the combined integral



Shadow rays can be directed to any of the light sources



May even compute the direct illumination with a single shadow ray for each point (although this can be very noisy)



Still need access to any of the light sources

Multiple Light Sources (2)

35

Graphics & Visualization: Principles & Algorithms Chapter 16

• A two-step sampling process for each shadow ray:



Assign each light a probability value for it being chosen



A discrete pdf pL(k) generates a randomly selected light source ki among NL lights

• A surface point yi

,on the selected light source k is selected using a conditional pdf p(y|ki)

• pL

(k) p(y|k): the combined pdf

for the sampled point yi

• n the combined

area of all light sources



The total estimator, using N shadow-rays, is:

Multiple Light Sources - Combined

36 1

( , , ) ( , , , ) ( , ) ( , ) 1 ( , , ) ( ) ( | )

j j

N e i y y r r r i i i i r r r i L i i i

L f G V L N p k p k        



  



y x y x y x y

Graphics & Visualization: Principles & Algorithms Chapter 16

Algorithm:

// direct illumination from multiple light sources // for surface point x, direction phi, theta directIllumination (x, phi, theta) estimatedRadiance = 0; for all shadow rays select light source k; generate point y on light source k; estimatedRadiance += Le(y, phi_y, theta_y) * BRDF * radianceTransfer(x,y) / (pdf(k) * pdf(y|k)); estimatedRadiance = estimatedRadiance / #shadowRays; return(estimatedRadiance);

Multiple Light Sources - Combined (2)

37

Algorithm (cont.):

// transfer between x and y // 2 cosines, distance and // visibility taken into account radianceTransfer(x,y) transfer = G(x,y)*V(x,y); return(transfer);

Graphics & Visualization: Principles & Algorithms Chapter 16 38

Multiple Light Sources - Combined (3)

Graphics & Visualization: Principles & Algorithms Chapter 16

• Any valid pdfs

for ( pL(k) and p(y|k) ) will produce unbiased results

• The choice of pdfs

affects the variance of the estimators

• Two of the more common PDFs

( pL(k) and p(y|k) ):



Uniform source selection / uniform area sampling



Power-proportional source selection / uniform sampling

Choosing pdfs for MLS

39

Graphics & Visualization: Principles & Algorithms Chapter 16

Uniform source selection / uniform light source area sampling



Both PDFs are uniform: pL (k) = 1/NL and p(y|k) = 1/SLk



Every light source will receive an equal number of shadow rays



The shadow rays are distributed uniformly



Disadvantage Equal number of shadow rays for both:

• Bright and weak lights
• Near and far or invisible lights
• Estimator for the direct illumination:

Choosing pdfs for MLS – Uniform/uniform

40 1

( , , ) ( , , ) ( , , , ) ( , ) ( , )

k j j

N L r r r L e i y y r r r i i i i i

N L S L f G V N        



  



x y x y x y

Graphics & Visualization: Principles & Algorithms Chapter 16

Power-proportional source selection / uniform light area sampling



pL (k) = Pk /Ptotal with Pk: the radiant power of light source k Ptotal: the total power emitted by all light sources



Bright sources receive more shadow rays than dim sources



Estimator:



If all lights are diffuse:



Superior approach



Results in slower converge at pixels where:

• Bright lights are invisible
• Illumination comes from less bright lights

41

1

( , , ) ( , , , ) ( , ) ( , ) ( , , )

k j j

N L e i y y r r r i i i i total r r r i k

S L f G V P L N P        



  



y x y x y x

, 1

( , , ) so ( , , , ) ( , ) ( , )

N total k k e k r r r r r r i i i i i

P P S L L f G V N        



   



x x y x y

Choosing pdfs for MLS – power-driven/uniform

Graphics & Visualization: Principles & Algorithms Chapter 16

• We must be careful not to exclude any lights contribute to Lr

(x, φr, θr)

• 3 random numbers are needed to generate a shadow ray



One to select the light source k



2 to select a specific surface point

Drawbacks of the unified MLS

42

Graphics & Visualization: Principles & Algorithms Chapter 16

• Environment maps:



Encode the total illumination on the hemisphere of directions around a point



Are usually captured in natural environments with digital cameras



Mathematically: is a stepwise continuous function



Each pixel corresponds to a small solid angle ΔΩ around point x



Intensity of each pixel corresponds to incident radiance value

• Capturing environment maps:



Most common techniques: I. A light probe in conjunction with a digital camera II. A digital camera equipped with a fisheye lens

Environment Map Illumination

43

( , , ) , ( , )

i i i i

L with      x

Graphics & Visualization: Principles & Algorithms Chapter 16

I. A light probe in conjunction with a digital camera



Light probe: a specularly reflective ball



Probe is positioned at the point where the incident illumination needs to be captured

Capturing Environment Maps - Probe

44

Graphics & Visualization: Principles & Algorithms Chapter 16

Issues to be considered:

• The camera will be present in the photo

blocks light coming from behind the camera

• Not uniform sampling of directions over the hemisphere
• All directions at the edge of the image represent illumination

from the same direction

• the light probe has small radius
• The camera cannot capture all illumination levels
• Solution for some of the above problems:



Capture 2 photos 90 degrees apart

45

Capturing Environment Maps – Probe (2)

Graphics & Visualization: Principles & Algorithms Chapter 16 46

Capturing Environment Maps – Probe (3)

Graphics & Visualization: Principles & Algorithms Chapter 16

• 2 photos taken from opposite view directions
• Fisheye lenses are expensive and hard to calibrate
• Both images need to be taken in perfect opposite view directions

47

Capturing Environment Maps – Fisheye lens

Graphics & Visualization: Principles & Algorithms Chapter 16

• Environment maps need to be expressed in some parametric space

Environment Map Parameterization

48

• a. latitude-longitude
• b. Projected-disk
• c. Paraboloid
• d. Concentric-map

Graphics & Visualization: Principles & Algorithms Chapter 16

• The direct illumination of a surface point due to an environment

map:

• Lmap

(φi,θi) : the incident illumination on x, coming from (φi,θi)

• Add a visibility term V(x ,φi,θi) for cases where:



Other surfaces prevent the light from reaching x



The object to which x belongs can cast a self-shadow onto x

• Estimator from Monte Carlo integration:

Environment Map Sampling

49

( , , ) ( , ) ( , , , )cos( )

r r r map i i r r r i i i i

L L f d          



 

x

x

( , , ) ( , ) ( , , , ) ( , , )cos( )

r r r map i i r r r i i i i i i

L L f V d            



 

x

x x

, , , , , , , 1 , ,

( , , ) ( , , , ) ( , , )cos( ) 1 ( , , ) ( , )

N map i j i j r r r i j i j i j i j i j r r r j i j i j

L f V L N p             



  



x x x

Graphics & Visualization: Principles & Algorithms Chapter 16

• Various problems with Monte Carlo integration 

increased variance and noise:



Integration domain has a large extent



Environment map ≈ textured light source  high frequencies and discontinuities



Product of environment map and BRDF. Specular BRDF peaks and environment map peaks do not coincide in general, resulting in sharp variations of their dot product



Visibility discontinuities

50

Environment Map Sampling (2)

Graphics & Visualization: Principles & Algorithms Chapter 16

• Practical approaches try to address these problems:



Direct illumination map sampling

• Treat environment map as pdf

when sampling it

• Transform the environment map into a number of selected

point light sources



BRDF sampling

• BRDF-guided sampling requires the analytical sampling
• f the BRDF, or when this is not possible, the inverse

cumulative pdf technique



Sampling the product

• Rejection sampling is commonly used

51

Environment Map Sampling (3)

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• Stochastic Ray Tracing algorithm:



Extends the basic ray tracing algorithm



Does not limit itself to direct illumination only



Includes all possible indirect illumination effects



Applies Monte Carlo integration directly to the hemispherical rendering equation

• Steps:



Ray-Tracing Set-up



Generation of truly random paths



Recursive path tracing



Termination

Indirect Illumination: Stochastic Ray Tracing

52

Graphics & Visualization: Principles & Algorithms Chapter 16

• To compute a global illumination picture:



Attribute a radiance value Lpixel to each pixel



This value is a weighted measure of radiance values incidents on the image plane where p : a point on the imageplane h(p): a weighting or filtering function x : the visible point seen from the eye through p

• Evaluation of Lr(x,φr, θr):



Cast a ray from the eye through p, in order to find x



Evaluate the rendering equation.

Stochastic Ray Tracing: Set-up

53

( ) ( ) ( , , ) ( )

pixel r r r imageplane imageplane

L L h d L h d    

 

p p p x p p

Graphics & Visualization: Principles & Algorithms Chapter 16

Pixel-driven rendering algorithm // pixel-driven rendering algorithm computeImage(eye) for each pixel radiance = 0; H = integral(h(p)); for each viewing ray pick uniform sample point p such that h(p) <> 0; construct ray at origin eye, direction from eye to p; radiance = radiance + rad(ray)*h(p); radiance = radiance / (#viewingRays*H); rad(ray) find closest intersection point x of ray with scene; computeRadiance(x, direction eye to p);

54

Stochastic Ray Tracing: Set-up (2)

Graphics & Visualization: Principles & Algorithms Chapter 16

Pixel-driven rendering algorithm

• For each pixel computes the integral in the image plane
• Can use Monte Carlo sampling over the plane where h(p) ≠
• For each sample point p, construct a primary ray
• Function rad(ray):



Computes the radiance along the primary ray:



finds the intersection point x



computes the radiance leaving surface point x in the direction of the eye

• The final radiance for each pixel :



average over the total number of viewing rays



take into account the normalizing factor of the uniform PDF over the domain

55

Stochastic Ray Tracing: Set-up (3)

Graphics & Visualization: Principles & Algorithms Chapter 16

• The function computeRadiance(x, direction eye to p)



uses the rendering equation to evaluate the appropriate radiance value



applies a basic and straightforward Monte Carlo integration scheme to Eq(16.2)



evaluates the integral by using Monte Carlo integration:

• generates N random directions (φi, θi) over the hemisphere Ωx
• samples distributed according to some probability density p(φi

, θi )



The estimator for Lr(x, φr, θ

r) is then given by :

Stochastic Ray Tracing: Truly Random Paths

56

, , , , , 1 , ,

( , , ) ( , , , )cos( ) 1 ( , , ) ( , )

N i j i j r r r i j i j i j r r r j i j i j

L f L N p           



  



x x

Graphics & Visualization: Principles & Algorithms Chapter 16

• The cosine and BRDF terms in the integrand



evaluated by accessing the scene description

• L(x,φi,j,θi,j) is unknown
• Radiance remains invariant along straight lines
• Need to trace the ray leaving x in direction (φi,j

,θi,j ) through the environment to find the closest intersection point y

• At this point another radiance evaluation is needed



Have a recursive procedure to evaluate L(x,φi,j,θi,j)



A path, or a tree of paths, is traced through the scene

Truly Random Paths (2)

57

Graphics & Visualization: Principles & Algorithms Chapter 16

• The recursive path needs to hit one of the light sources in the scene:



does not occur very often because



the light sources usually are small compared to the other surfaces



very few of the paths will yield a contribution to the radiance value



the resulting image will mostly be black



The pixel will be attributed a color only when a path hits a light source

Truly Random Paths (3)

58

Graphics & Visualization: Principles & Algorithms Chapter 16

• Ray tracing algorithm needs a stopping condition
• Otherwise:



the generated paths would be of infinite length



the algorithm would not come to a halt

• Be careful not to introduce any bias to the final image
• Have to find a way:



to limit the length of the paths



but still be able to obtain a correct solution

Stochastic Ray Tracing: Terminating the Recursion

59

Graphics & Visualization: Principles & Algorithms Chapter 16

• Classic ray-tracing implementations use 2 techniques:
• First technique:



Is cutting off the recursive evaluations after a fixed number of evaluations



Puts an upper bound on the amount of rays that need to be traced



Important light transport might have been ignored



The image will be biased



A typical fixed path length is set at 4 or 5



The path length should be dependent on the scene to be rendered



Scenes with many specular surfaces will require a larger path length



Scenes with mostly diffuse surfaces can usually use a shorter path length

Terminating the Recursion (2)

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Graphics & Visualization: Principles & Algorithms Chapter 16

• Second technique:



Use an adaptive cut-off length



When a path hits a light source the radiance found at the light source is multiplied by all cosine factors and BRDFs at all previous intersection



Store the accumulating multiplication factor with the lengthening path



More efficient technique



Still the final image will still be biased

• Russian Roulette:



Addresses the problem of keeping the lengths of the paths manageable



Leaves room for exploring all possible paths of any length



Produce an unbiased image

Terminating the Recursion (3)

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Example

• Suppose we want to compute the one-dimensional integral:
• The standard Monte Carlo integration procedure:



generates random points xi in the domain [0,1]



computes the weighted average of all function values f(xi)

• Assume that:



f(x) is difficult or complex to evaluate



We want to limit the number of evaluations of f(x) necessary to estimate I

• Express the quantity I

as:



Scale f(x) by a factor P horizontally, and a factor 1/P vertically

Russian Roulette (1)

62 1

( ) I f x dx  

1 ( ) , 1

P RR

x I f dx P P P  



Graphics & Visualization: Principles & Algorithms Chapter 16

Example (cont.)

• Apply Monte Carlo integration to compute the new integral
• Use a uniform pdf

p(x)=1 to generate the samples over [0,1]

• So we get the following estimator for IRR:

Russian Roulette (2)

63

1 ( ) if x P if x P

RR

x f I P P          

Graphics & Visualization: Principles & Algorithms Chapter 16

• The expected value of
• The result of applying Russian roulette:



If f(x) be another recursive integral



Recursion stops with a probability equal to a = 1-P for each evaluation point



a: the absorption probability

• The overall estimator still remains unbiased:



in [P,1] will generate a function value equal to 0



but this is compensated by weighting the samples in [0,P] with a factor 1/P

• If a is small:



the recursion will continue many times



the final estimator will be more accurate.

• If a is large:



the recursion will stop sooner



the estimator will have a higher variance

Russian Roulette (3)

64

RR

I equals I  

Graphics & Visualization: Principles & Algorithms Chapter 16

computeRadiance(x, dir) find closest intersection point x of ray with scene; estimatedRadiance = simpleStochasticRT(x, phi, theta); return(estimatedRadiance);

Simple Stochastic Ray Tracing

65

Graphics & Visualization: Principles & Algorithms Chapter 16

simpleStochasticRT (x, phi, theta) estimatedRadiance = 0; if (no absorption) // Russian roulette for all paths // N rays sample direction phi_i, theta_i on hemisphere; y = trace(x, phi_i, theta_i); estimatedRadiance += simpleStochasticRT(y, phi_i, theta_i)*BRDF *cos(theta_i)/pdf(phi_i, theta_i); estimatedRadiance /= #paths; estimatedRadiance /= (1-absorption) estimatedRadiance += Le(x, phi_i, theta_i) return(estimatedRadiance);

Simple Stochastic Ray Tracing (2)

66

Graphics & Visualization: Principles & Algorithms Chapter 16

• Build a full global illumination renderer using stochastic path tracing

The efficiency& accuracy of the complete algorithm is determined by:

• Number of viewing rays per pixel

A higher number of viewing rays Np eliminates aliasing and decreases noise

• Direct illumination



The total number of shadow rays Nd cast from each point x



Selection of single source among all sources for each shadow ray



The distribution of the shadow rays over the area of a single light source

• Indirect illumination



Number of indirect illumination rays Ni distributed over Ωx



Exact distribution of these rays over the hemisphere



Absorption probabilities for Russian Roulette in order to stop the recursion

Putting All Together

67

Graphics & Visualization: Principles & Algorithms Chapter 16

The complete algorithm:

//stochastic ray tracing computeImage(eye) for each pixel radiance = 0; H = integral(h(p)); for each sample // Np viewing rays pick sample point p within support of h; construct ray at eye, direction eye to p; radiance = radiance + rad(ray)*h(p); radiance = radiance/(#samples*H);

Putting All Together (2)

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Graphics & Visualization: Principles & Algorithms Chapter 16

The complete algorithm (cont):

rad(ray) find closest intersection point x of ray with scene; return Le(x,dir) + computeRadiance(x, dir); computeRadiance(x, dir) estimatedRadiance += directIllumination(x, dir); estimatedRadiance += indirectIllumination(x, dir); return(estimatedRadiance);

Putting All Together (3)

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Graphics & Visualization: Principles & Algorithms Chapter 16

The complete algorithm (cont):

directIllumination (x, dir) estimatedRadiance = 0; for all shadow rays // Nd shadow rays select light source k; sample point y on light source k; estimated radiance += Le * BRDF * G(x,y) * V(x,y) /(pdf(k)pdf(y|k)); estimatedRadiance = estimatedRadiance / #paths; return(estimatedRadiance);

Putting All Together (4)

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Graphics & Visualization: Principles & Algorithms Chapter 16

The complete algorithm (cont):

indirectIllumination (x, dir) estimatedRadiance = 0; if (no absorption) // Russian roulette for all indirect paths // Ni indirect rays sample random direction on hemisphere; y = trace(x, random direction); estimatedRadiance += compute_radiance(y, random direction) * BRDF * cos / pdf(random direction); estimatedRadiance = estimatedRadiance / #paths; return(estimatedRadiance/(1-absorption));

Putting All Together (5)

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• Stochastic ray tracing:



Paths start at the surface point



End at the light sources

• Light tracing:



Paths start from the light sources



Record contributions to relevant pixels



Is the dual algorithm of stochastic ray-tracing

• Bidirectional ray tracing:



Combines both approaches



A two-pass algorithm



Generates paths starting at the light and at the surface point simultaneously



Connects both paths in the middle



Finds a contribution to the light transport between:

• The light source
• The point for which we need the radiance

Bidirectional Ray Tracing

72



Graphics & Visualization: Principles & Algorithms Chapter 16

• 2 different path generators:



An eye path

• Starts at a sampled surface point y0

visible through the pixel

• Has length k
• Consists of a series of surface points y0, y1,…, yk
• The length is controlled by Russian Roulette
• The probability of generating this path is composed of the individual pdf



A light path

• Starts at the light source
• Has length l
• Consists of points x0

, x1 ,…, xk

• Has its own probability density distribution

Bidirectional Ray Tracing (2)

73



Graphics & Visualization: Principles & Algorithms Chapter 16

• Connect



The endpoint yk

• f the eye with



The xl

• f the light path
• The probability density function is the product of the individuals pdfs
• An estimator for the flux Φ

through the pixel:

Bidirectional Ray Tracing (3)

74



1 1 1 1 1 1 1 1

, ( , , , , , , , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( )

k l e r l k l k r k r

K pdf with K L G V f G V f f G V h            y y y x x x x x x x x x x y x y y y y y y y p

Graphics & Visualization: Principles & Algorithms Chapter 16

• Paths of a certain length can be generated by different combinations
• E.g.: different combinations for a path of length 3:

Bidirectional Ray Tracing (4)

75



Graphics & Visualization: Principles & Algorithms Chapter 16

• An eye-
• r light-path of length k-1

can be extended to a path of k

• We use the same subpath

more than once



not only connect the endpoints



also connect all possible subpaths to each other

Bidirectional Ray Tracing (5)

76



Graphics & Visualization: Principles & Algorithms Chapter 16

Bidirectional Ray Tracing (6)

77



Stochastic ray tracing bidirectional tracing full bidirectional using only light paths ray tracing

Graphics & Visualization: Principles & Algorithms Chapter 16

Bidirectional Ray Tracing (7)

78



Graphics & Visualization: Principles & Algorithms Chapter 16

• Photon mapping:



A practical and robust two-pass algorithm



Traces illumination paths both from the lights and from the viewpoint



Caches and reuses illumination values in a scene for efficiency



In the first pass: Photons

• are traced from the light sources into the scene
• carry flux information
• are cached in a data structure, called the photon map



In the second pass:

• an image is rendered using the information stored in the photon

map

Photon Mapping

79



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• Decouples photon storage from surface parameterization
• Handles arbitrary geometry , including procedural geometry
• Increases the practical utility of the algorithm
• Not prone to meshing artifacts
• Making specialized photon maps:



By tracing or storing only particular types of photons



Example : the caustic map: captures photons that interact with one or more specular surfaces

• Photon mapping is biased

Photon Mapping (2)

80



Graphics & Visualization: Principles & Algorithms Chapter 16

Pass 1: Tracing photons

• Use of compact, point-based photons to propagate flux through the scene
• Photons :



are traced from the light sources and



are propagated through the scene just as rays are in ray tracing



i.e. they are reflected, transmitted, or absorbed

• To propagate photons use:



Russian roulette



Standard Monte Carlo sampling techniques

• When the photons hit non-specular

surfaces, store them in the photon map

• Use a balanced kd-tree to implement this data structure

Photon Mapping (3)

81



Graphics & Visualization: Principles & Algorithms Chapter 16

Pass 1: Tracing photons

• Photon mapping can be efficient for computing caustics:
• A caustic is formed when light is reflected or transmitted through
• ne or more specular

surface before reaching a diffuse surface

• For improvement:



separate out the computation of caustics from global illumination



For each scene compute 2 photon maps:

• a caustic photon map
• a global photon map

Photon Mapping (4)

82



Graphics & Visualization: Principles & Algorithms Chapter 16

Pass 1: Tracing photons

• Caustic photon maps:



Computed efficiently



Caustics occur when light is focused



Not too many photons are needed to get a good estimate of caustics

• Efficiency significantly improved when shooting photons only

towards the set of specular surfaces

• Compute the reflected radiance at each point in the scene:



Photon map represents incoming flux at each point in the scene



Photon density at a point estimates the irradiance at that point



Multiply the irradiance by the surface BRDF

Photon Mapping (5)

83



Graphics & Visualization: Principles & Algorithms Chapter 16

Pass 2: Computing Images

• Display the reflected radiance values computed above for each

visible point in an image.

• Direct illumination via photon mapping causes significant blurring
• f radiance
• Instead, integrate with a ray tracer that



computes direct illumination and



queries the photon map only after one diffuse or glossy bounce from the viewpoint is traced through the scene

Photon Mapping (6)

84



Graphics & Visualization: Principles & Algorithms Chapter 16

Pass 2: Computing Images

• The final rendering of images:



Rays are traced through each pixel to find the closest visible surface



The radiance for a visible point is split into

• direct illumination
• specular
• r glossy illumination,
• illumination due to caustics
• the remaining indirect illumination

Photon Mapping (7)

85



Graphics & Visualization: Principles & Algorithms Chapter 16

Pass 2: Computing Images

• Each of the above components is computed as follows:



Direct illumination for visible surfaces

• computed using regular Monte Carlo sampling



Specular reflections and transmissions are ray traced



Caustics are computed using the caustic photon map



The remaining indirect illumination

• computed by sampling the hemisphere
• the global photon map is used to compute radiance at the next recursion

step

Photon Mapping (8)

86



Graphics & Visualization: Principles & Algorithms Chapter 16

Photon Mapping (9)

87



Graphics & Visualization: Principles & Algorithms Chapter 16

• Steps:

1. Discretization

• f the input geometry in different patches i

2. For each resulting patch i a radiosity value Bi is computed 2. Computation of form factors Fij for every pair of patches i and j 3. Numerical solution of the radiosity system of linear equations 4. Display of the solution: use any rendering algorithm that displays patches with a given color value Bi

• In practical implementations, these steps are highly connected, e.g.:



Form factors are only computed when they are needed



Intermediate results can already be displayed during system solution



In adaptive or hierarchical radiosity, perform discretization during solution

• Each step is non trivial

Radiosity: Classic Radiosity

88

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• Main problems are related to:



Discretization

• f the scene into patches
• The patches should be small enough to capture illumination variations
• Radiosity

B(x) across each patch need to be constant

• A higher number of patches usually solves for artifacts caused by

discretization

• The number of patches shouldn’t be too large
• Compute a form factor between each pair of patches
• The number of form factors can be huge storage problem



Form factor computation

• Each form factor requires the solution of a non-trivial, 4D integral
• The integral will be singular for adjacent patches, where

rxy =0

• The integral can have discontinuities due to changing visibility

Radiosity: Classic Radiosity (2)

89

Graphics & Visualization: Principles & Algorithms Chapter 16

• Form factor difficulties:

Radiosity: Classic Radiosity (3)

90

Graphics & Visualization: Principles & Algorithms Chapter 16

• Proposed solutions:
• Specialized algorithms for form-factor integration:



The hemicube algorithm



Shaft culling ray- tracing acceleration



Discontinuity meshing



Adaptive and hierarchical subdivision



Clustering



Form-factor caching strategies



The use of view importance



Higher-order radiosity approximations

Radiosity: Classic Radiosity (4)

91

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• The radiosity
• f Bi
• f a single patch i:
• Bi

is the sum of 2 contributions:



The self-emitted radiosity Bei



The fraction of the incident irradiance at i that is reflected

• Fij

: the fraction of the irradiance on patch i that originates at patch i

• Rewrite the above equation by:



Transforming radiosity values to flux values



Pi = Ai Bi and Pei = Ai Bei

Radiosity: Form Factors

92

i ei i ij j j

B B F B    

ij j j

F B



Graphics & Visualization: Principles & Algorithms Chapter 16

• Obtain the following system of linear equations by:



Multiplying both sides of the equation by Ai



Using symmetry between Fij and Fji

• The total power Pi

emitted by patch i consists of 2 parts:



The self-emitted power Pei



The power received and reflected from all other patches j

• Fij

: the fraction of power emitted by j that arrives at i

• Important property of form factors:

Radiosity: Form Factors (2)

93

i ei i ij j i i i ei i i ij j j j i i i ei i j ji j j i ei j ji i j

B B F B A B A B A F B A B A B A F B P P P F               

   

1

ij j

F 



Graphics & Visualization: Principles & Algorithms Chapter 16

Radiosity: Form Factors (3)

94

Graphics & Visualization: Principles & Algorithms Chapter 16

• Estimation of form factors:
• Let i be the source of a number Ni
• f virtual particles originating on

a diffuse surface



Nij : the number of these particles that land on the 2nd patch

• Consider:



A particle originating at a uniformly chosen location x on Si



It is distributed over the hemisphere using a cosine-distributed direction



The pdf p(x, φ, θ) is:

Radiosity: Form Factors (4)

95

/

ij ij ij i

F F N N 

cos( ) ( , , )

i

p A      x

Graphics & Visualization: Principles & Algorithms Chapter 16

• Let
• The probability Pij

that the particle lands on patch j is:

• When generating Ni

particles from i  Ni Fij hits on patch j

• The more particles used the better tha

ratio Nij /Ni approximates Fij

• The variance of this estimator = Fij

(1-Fij ) / Ni

Radiosity: Form Factors (5)

96

j

1, the particle hits the patch χ ( , , ) 0, the particle doesn't hit the patch j j       x

2

cos( )cos( ) 1 ( , , ) ( ) ( ) , , ,

i x i j

i j ij S j x S S y x ij i xy

P p dA d V dA dA F A r         



  

   

x x x y

Graphics & Visualization: Principles & Algorithms Chapter 16

• A method to solve systems of linear equations x = e

+Ax

• Uses a simple iteration scheme:



A system with n equations and n unknowns



e, x, any approximation of x : n-dimensional vectors or points



Start with an arbitrary point x(0)



Transform x(k) into the next point x(k+1) by x(k+1) = e + Ax(k)

• The method converges if the matrix norm of A is >1
• In radiosity:



Vectors x and e correspond to a distribution of light power over the surfaces



Each Jacobi iteration:

• Computes an additional single bounce of light interreflection
• Re-adds self-emitted power



The solution is the equilibrium illumination distribution

Radiosity: The Jacobi Iterative Method

97

Graphics & Visualization: Principles & Algorithms Chapter 16

• 3 slightly different ways:

1. Regular gathering of radiosity 2. Regular shooting of power



j shoots its power towards all other patches i

3. Incremental shooting of power



Unshot power is propagated rather than total power

Radiosity: The Jacobi Iterative Method (2)

98

(0) ( 1) ( )

,

k k i ei i ei i ij j j

B B B B F B 



   

(0) ( 1) ( )

,

k k i ei i ei j ji i j

P P P P P F 



  

(0) ( 1) ( ) ( ) ( ) i ei k k i j ji i j k k l i i l

P P P P F P P 

 

      

 

Graphics & Visualization: Principles & Algorithms Chapter 16

• Global illumination algorithms are developing since 1979
• Its possible to generate an image like a photograph of a real scene
• We can implement and investigate the physical processes that form

a realistic rendering:



Light-material interaction



Light transport

• Global illumination:



Not yet in mainstream applications



Some use in

• Feature animation films
• Computer games



High-quality rendering of architectural designs becomes more common



Rendering of industrial products in realistic virtual environments

Conclusion

99