K. H. Ko School of Mechatronics Gwangju Institute of Science and Technology

Photo-realistic Rendering and Global Illumination in Computer Graphics

Spring 2012

Ultimate Realism and Speed

K H Ko

School of MechatronicsGwangju Institute of Science and Technology

2

Introduction The quest for realism and speed has not yet

come to an end Derivation of the rendering equation was made

based on several restrictions imposed on light transport

Wave effects could be ignored Radiance is conserved along its path between

mutually visible surfaces Light scattering happens instantaneously Scattered light has the same wavelength as the

incident beam It scatters from the same location where it hits a

surface

3

Introduction However such assumptions are not always

true

Phenomena such as polarization diffraction interference fluorescence and phosphorescence which do not fall within the assumptions for the rendering equation should be considered in order to obtain high realism

4

Beyond the Rendering Equation Participating Media It was assumed that radiance is conserved along

its path between unoccluded surfaces The underlying idea was that all photons leaving the

first surface needed to land on the second one because nothing could happen to them along their path of flight

This is not always true Photons reflected or emitted by a car in front of us on the road

will often not reach us They will rather be absorbed or scattered by billions of tiny water

or fog droplets immersed in the air Light coming from the sky above will be scattered towards us The net effect is that distant objects fade away in gray

Our assumption of radiance conservation between surfaces is only true in a vacuum

In this case the relation between emitted radiance and incident radiance at mutually visible surface points x and y along direction θ is given by L(x-gtθ) = L(ylt- -θ)

5

Beyond the Rendering Equation Participating Media If a vacuum is not filling the space between

object surfaces this will cause photons to change direction and to transform into other forms of energy

Four processes are considered volume emission absorption out-scattering and in-scattering

6

Beyond the Rendering Equation Participating Media

Considering such phenomena allows us to generalize the rendering equation

7

Beyond the Rendering Equation Volume Emission The intensity by which a medium like fire glows

can be characterized by a volume emittance function ε(z) (units [Wm3])

It tells us basically how many photons per unit of volume and per unit of time are emitted at a point z in three-dimensional space

There is a close relationship between the number of photons and energy

Each photon of a fixed wavelength λ carries an energy quantum equal to 2πhcλ h is Planckrsquos constant and c is the speed of light

Many interesting graphics effects are possible by modeling volume light sources

Usually volume emission is isotropic meaning that the number of photons emitted in any direction around z is equal to ε(z)4π (units [Wm3sr])

8

Beyond the Rendering Equation Volume Emission Consider a point z = x + sθ along a thin pencil

connecting mutually visible surface points x and y

The radiance added along direction θ due to volume emission in a pencil slice of infinitesimal thickness ds at z is

4

)()( ds

zzdL

9

Beyond the Rendering Equation Absorption Photons traveling along our pencil form x to y

will collide with the medium causing them to be absorbed or change direction (scattering)

Absorption means that their energy is converted into a different kind of energy for instance kinetic energy of the particles in the medium

Transformation into kinetic energy is observed at a macroscopic level as the medium heating up by radiation

10

Beyond the Rendering Equation Absorption The probability that a photon gets absorbed in a

volume per unit of distance along its direction of propagation is called the absorption coefficient σa(z) (units [1m])

A photon traveling a distance Δs in a medium has a chance σa Δs of being absorbed

The absorption coefficient can vary from place to place

Absorption is usually isotropic a photon has the same chance of being absorbed regardless of its direction of flight

Absorption causes the radiance along the thin pencil from x to y to decrease exponentially with distance

Consider a pencil slice of thickness Δs at z = x + sθ The number of photons entering the slice at z is

proportional to the radiance L(z-gt θ) along the pencil

11

Beyond the Rendering Equation Absorption Assuming that the absorption coefficient is

equal everywhere in the slice a fraction σa(z)Δs of these photons will be absorbed

The radiance coming out on the other side of the slice at z + sθ will be

In a homogeneous nonscattering and nonemissive medium the reduced radiance at z along the pencil will be

sxzzzLs

zLszL

or

szzLzLszL

a

a

)()()()(

)()()()(

saexLzL )()(

12

Beyond the Rendering Equation Absorption This exponential decrease of radiance with

distance is sometimes called Beerrsquos Law It is a good model for colored glass for

instance and has been used for many years in classic ray tracing

If the absorption varies along the considered photon path Beerrsquos Law looks like this

)(exp)()(0

s

a dttxxLzL

13

Beyond the Rendering Equation Out-Scattering Extinction Coefficient and Albedo The radiance along the pencil will not only

reduce because of absorption but also because photons will be scattered into other directions by the particles along their path

The effect of out-scattering is almost identical to that of absorption

One just needs to replace the absorption coefficient by the scattering coefficient σs(z) (units[1m]) which indicates the probability of scattering per unit of distance along the photon path

Rather than using σa(z) and σs(z) it is sometimes more convenient to describe the processes in a participating medium by means of the total extinction coefficient σt(z) and the albedo α(z)

14

Beyond the Rendering Equation Out-Scattering Extinction Coefficient and Albedo The extinction coefficient σt(z) = σa(z)+σs(z)

(units[1m]) gives us the probability per unit distance along the path of flight that a photon collides with the medium

Then the reduced radiance at z is given by

In a homogeneous medium the average distance between two subsequent collisions can be shown to be 1σt (units[m])

The average distance between subsequent collisions is called the mean free path

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15

Beyond the Rendering Equation Out-Scattering Extinction Coefficient and Albedo The albedo α(z)= σs(z)σt(z) (dimensionless) describes

the relative importance of scattering versus absorption

It gives us the probability that a photon will be scattered rather than absorbed when colliding with the medium at z

The albedo is the volume equivalent of the reflectivity ρ at surfaces

Note that the extinction coefficient was not needed for describing surface scattering since all photons hitting a surface are supposed to scatter or to be absorbed

In the absence of participating media one could model the extinction coefficient by means of a Dirac delta function along the photon path

It is zero everywhere except at the first surface boundary met where scattering or absorption happens for sure

16

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function The out-scattered photons change direction

and enter different pencils between surface points In the same way photons out-scattered from other pencils will enter the pencil between the x and y we are considering This entry of photons due to scattering is called in-scattering

Similar to volume emission the intensity of in-scattering is described by a volume density Lvi(z-gtθ) (units [Wm3sr])

The amount of in-scattered radiance in a pencil slice of thickness ds will be

)()( dszLzdL vii

17

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function A first condition for in-scattering at a location z

is that there is scattering at z at all σs(z)=α(z)σt(z)ne0

The amount of in-scattered radiance further depends on the field radiance L(zΨ) along other direction Ψ at z and the phase function p(zΨlt-gtθ)

Field radiance is the usual concept of radiance It describes the amount of light energy flux in a given direction per unit of solid angle and per unit area perpendicular to that direction

The product of field radiance with the extinction coefficient Lv(zΨ)=L(zΨ)σt(z) describes the number of photons entering collisions with the medium at z per unit of time

18

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function Being a volume density it is sometimes called

the volume radiance (units [1m3sr]) Note that the volume radiance will be zero in

empty space The field radiance however does not need to

be zero and fulfills the law of radiance conservation in empty space

For surface scattering no distinction is needed between field radiance and surface radiance since all photons interact at a surface

Of these photons entering collision with the medium at z a fraction α(z) will be scattered

Unlike emission and absorption scattering is usually not isotropic

19

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function Photons may scatter with higher intensity in

certain directions than in others The phase function p(zΨlt-gtθ) plays the role

of the BSDF for volume scattering Just like BSDFs it is reciprocal and energy

conservation must be satisfied It is convenient to normalize the phase function so

that its integral over all possible directions is one

Energy conservation is then clearly satisfied since α(z) lt 1

1)(

dzp

20

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function Putting this together we arrive a the following

volume scattering equation

The volume scattering equation is the volume equivalent of the surface scattering equation

It describes how scattered volume radiance is the integral over all directions of the volume radiance Lv(z-gtΨ) weighted with α(z)p(zΨlt-gtθ)

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function Lv(z-gtΨ) is the volume equivalent of surface

radiance α(z)p(zΨlt-gtθ) is the equivalent of the BSDF

22

Beyond the Rendering Equation The Rendering Equation in the Presence of Participating Media

The above two equations model how volume emission and in-scattering add radiance along a ray from x to y

This equation describes how radiance is reduced due to absorption and out-scattering

4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function Not only the surface radiance L(x-gtθ) inserted

into the pencil at x is reduced in this way but also all radiance inserted along the pencil due to in-scattering and volume emission is reduced

The combined effect is

For compactness let z = x+rθ and

)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function The transmittance τ(zy) indicates how

radiance is attenuated between z and y

The equation below replaces the law of radiance conservation in the presence of participating media

)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function The rendering equation was obtained by using

the law of radiance conservation in order to replace the incoming radiance L(xlt-Ψ) in

by the outgoing radiance L(y-gt-Ψ) at the first surface point y seen from x in the direction Ψ

Then the following rendering equation is obtained in the presence of participating media

26




27

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function Then the following rendering equation is

obtained in the presence of participating media

28

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function The in-scattered radiance Lvi in L+(z-gt-

Ψ)=ε(z)4π+Lvi(z-gt-Ψ) is expressed in terms of field radiance

Field radiance is expressed in terms of surface radiance and volume emitted and in-scattered radiance elsewhere in the volume

Participating media can be handled by extending the rendering equation in two ways

Attenuation of radiance received from other surfaces the factor τ(xy) in the former integral

A volume contribution the latter integral

29

Beyond the Rendering Equation In-Scattering Field-and Volume-Radiance and the Phase Function Spatial Formulation

In order to better understand how to trace photon trajectories in the presence of participating media it is instructive to transform the integrals to a surface and volume integral

The relation between differential solid angle and surface is available rxy

2dωΨ=V(xy)cos(-ΨNy)dAy A similar relationship exists between drdωΨ and

differential volume rxz2drdωΨ=V(xz)dVz

30


31

Beyond the Rendering Equation Global Illumination Algorithms for Participating Media There exist two types of methods

deterministic and stochastic Deterministic Approaches

Classic and hierarchical radiosity methods have been extended to handle participating media by discretizing the volume integral above into volume elements and assuming that the radiance in each volume element is isotropic

Many other deterministic approaches have been proposed based on spherical harmonics and discrete ordinates methods

Deterministic approaches are valuable in relatively easy settings for instance homogeneous media with isotropic scattering or simple geometries

32

Beyond the Rendering Equation Global Illumination Algorithms for Participating Media Stochastic Approaches

Various authors have proposed extensions to path tracing to handle participating media

The extension of bidirectional path tracing to handle participating media has been proposed

These path-tracing approaches are flexible and accurate

But they become enormously costly in optically thick media where photons suffer many collisions and trajectories are long

A good compromise between accuracy and speed is offered by volume photon density estimation methods

In particular the extension of photon mapping to participating media is a reliable and affordable method capable of rendering highly advanced effects such as volume caustics

33

Beyond the Rendering Equation Global Illumination Algorithms for Participating Media Monte Carlo and volume photon density

estimation are methods of choice for optically thin media in which photons undergo only relatively few collisions

For optically thick media they become intractable

Highly scattering optically thick media can be handled with the diffusion approximation

34

Beyond the Rendering Equation Tracing Photon Trajectories in Participating Media Photon trajectory tracing can be extended to deal

with participating media

Sampling volume emission Light particles may not only be emitted at surfaces but

also in midspace Frist a decision needs to be made whether to sample

surface emission or volume emission This decision can be a random decision based on the

relative amount of self-emitted power by surfaces and volumes

If volume emission is to be sampled a location somewhere in midspace needs to be selected based on the volume emission density ε(z)

Bright spots in the participating media shall give birth to more photons than dim regions

Finally a direction needs to be sampled at the chosen location

Since volume emission is usually isotropic a direction can be sampled with uniform probability across a sphere

35

Beyond the Rendering Equation Tracing Photon Trajectories in Participating Media Sampling a next collision location

In the absence of participating media a photon emitted from point x into direction θ always collides on the first surface seen from x along the direction θ

With participating media however scattering and absorption may also happen at every location in the volume along a line to the first surface hit

A good way to handle this problem is to sample a distance along the ray from x into θ based on the transmittance factor

One draws a uniformly distributed random number ζisin[01) and finds the distance τ corresponding to

36


In a homogeneous medium r = -log(1-ζ)σt In a heterogeneous medium sampling such a

distance is less trivial It can be done exactly if the extinction coefficient is

given as a voxel grid by extending ray-grid traversal algorithms

For procedurally generated media one can step along the ray in small possibly adaptively chosen intervals

If the selected distance becomes greater than or equal to the distance to the first surface hit point of the ray surface scattering shall be selected as the next event

If the sampled distance is nearer volume scattering is chosen

37


38

Beyond the Rendering Equation Tracing Photon Trajectories in Participating Media Sampling scattering or absorption

Sampling scattering or absorption in a volume is pretty much the same as for surfaces

The decision whether to scatter or absorb will be based on the albedo α(z) for volumes just like the reflectivity ρ(z) is used for surfaces

Sampling a scattered direction is done by sampling the phase function p(zθlt-gtΨ) for volumes

39

Beyond the Rendering Equation Tracing Photon Trajectories in Participating Media Connecting path vertices

Algorithms such as path tracing and bidirectional path tracing require us to connect path vertices for instance a surface or volume hit with a light source position for a shadow ray

Without participating media the contribution associated with such a connection between points x and y is

In the presence of participating media the contribution shall be

With Cx(θ)=cos(Nxθ) if x is a surface point or 1 if it is a volume point

40

Beyond the Rendering Equation Volume Photon Density Estimation In order to render participating media it is necessary

to estimate the volume density of photons colliding with the medium in midspace

A histogram method would discretize the space into volume bins and count photon collisions in each bin

The ratio of the number of photon hits over the volume of the bin is an estimate for the volume radiance in the bin

Photon mapping has been extended with a volume photon map a third kd-tree for storing photon hit points in the same spirit as the caustic and global photon map

The volume photon map contains the photons that collide with the medium in midspace

Rather than finding the smallest disc containing a given number N of photon hit points as was done on surfaces one will search for the smallest sphere containing volume photons around a query location

Again the ratio of the number of photons and the volume of the sphere yields an estimate for the volume radiance

41

Beyond the Rendering Equation Volume Photon Density Estimation Viewing Precomputed Illumination in

Participating Media

42

Beyond the Rendering Equation Light Transport as a Diffusion Process The volume rendering equation is not the only

way light transport can be described mathematically

An interesting alternative is to consider the flow of light energy as a diffusion process

This point of view has been shown to result in efficient algorithms for dealing with highly scattering optically thick participating media such as clouds

43

Beyond the Rendering Equation Light Transport as a Diffusion Process Diffuse Approximation Method

The idea is to split field radiance in a participating media into two contributions which are computed separately

The first part Lr called reduced radiance is the radiance that reaches point x directly from a light source or from the boundary of the participating medium It needs to be computed first

The second part Ld called diffuse radiance is radiance scattered one or more times in the medium

In a highly scattering optically thick medium the computation of diffuse radiance is hard to do according to the rendering equation

Multiple scattering however tends to smear out the angular dependence of diffuse radiance

Indeed each time a photon scatters in the medium its direction is randomly changed as dictated by the phase function

After many scattering events the probability of finding the photon traveling in any direction will be nearly uniform

)x(L)x(L)x(L dr

44

Beyond the Rendering Equation Light Transport as a Diffusion Process One approximates diffuse radiance by the

following function which varies only a little with direction

Ud(x) represents the average diffuse radiance at x

))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45

Beyond the Rendering Equation Light Transport as a Diffusion Process The diffuse flux vector models the direction

and magnitude of the multiple scattered light energy flow through x

The average diffuse radiance Ud(x) fulfills a steady-state diffusion equation

d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46

Beyond the Rendering Equation Light Transport as a Diffusion Process Once the diffusion equation has been solved

the reduced radiance and the gradient of Ud(x) allows us to compute the flux vector wherever it is needed

The flux vector in turn yields the radiosity flowing through any given real or imaginary surface boundary

2

Introduction The quest for realism and speed has not yet

come to an end Derivation of the rendering equation was made

based on several restrictions imposed on light transport

Wave effects could be ignored Radiance is conserved along its path between

mutually visible surfaces Light scattering happens instantaneously Scattered light has the same wavelength as the

incident beam It scatters from the same location where it hits a

surface

3


true


4









5




6



7








8




4

)()( ds

zzdL

9





10









11





sxzzzLs

zLszL

or

szzLzLszL

a

a

)()()()(

)()()()(

saexLzL )()(

12





)(exp)()(0

s

a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







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30


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32








33





34











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37


38





39






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41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




3


true


4









5




6



7








8




4

)()( ds

zzdL

9





10









11





sxzzzLs

zLszL

or

szzLzLszL

a

a

)()()()(

)()()()(

saexLzL )()(

12





)(exp)()(0

s

a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







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30


31






32








33





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37


38





39






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41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

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d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




4









5




6



7








8




4

)()( ds

zzdL

9





10









11





sxzzzLs

zLszL

or

szzLzLszL

a

a

)()()()(

)()()()(

saexLzL )()(

12





)(exp)()(0

s

a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

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22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











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37


38





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Participating Media

42





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)x(L)x(L)x(L dr

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))x(F()x(U)x(Ldd

d

4

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d)cos()x(L))x(F(d

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d 22

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4

)()( ds

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9





10









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sxzzzLs

zLszL

or

szzLzLszL

a

a

)()()()(

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saexLzL )()(

12





)(exp)()(0

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a dttxxLzL

13






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)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

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)()( dszLzdL vii

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dzp

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dzLzpz

dzLzpzzL

s

vvi

21



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)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

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vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



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Participating Media

42





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)x(L)x(L)x(L dr

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))x(F()x(U)x(Ldd

d

4

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4

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d 22

46




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8




4

)()( ds

zzdL

9





10









11





sxzzzLs

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or

szzLzLszL

a

a

)()()()(

)()()()(

saexLzL )()(

12





)(exp)()(0

s

a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






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19







1)(

dzp

20





)()()(

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dzLzpz

dzLzpzzL

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vvi

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)()( ds

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)()()(

)()()()(

dzLzpz

dzLzpzzL

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vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







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30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









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d

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d)cos()x(L))x(F(d

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d 22

46




7








8




4

)()( ds

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9





10









11





sxzzzLs

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or

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a

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12





)(exp)()(0

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a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

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19







1)(

dzp

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)()()(

)()()()(

dzLzpz

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)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




8




4

)()( ds

zzdL

9





10









11





sxzzzLs

zLszL

or

szzLzLszL

a

a

)()()()(

)()()()(

saexLzL )()(

12





)(exp)()(0

s

a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




9





10









11





sxzzzLs

zLszL

or

szzLzLszL

a

a

)()()()(

)()()()(

saexLzL )()(

12





)(exp)()(0

s

a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




10









11





sxzzzLs

zLszL

or

szzLzLszL

a

a

)()()()(

)()()()(

saexLzL )()(

12





)(exp)()(0

s

a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




11





sxzzzLs

zLszL

or

szzLzLszL

a

a

)()()()(

)()()()(

saexLzL )()(

12





)(exp)()(0

s

a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




12





)(exp)()(0

s

a dttxxLzL

13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




13






14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




14






)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




15








16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




16





)()( dszLzdL vii

17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




17






18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




18








19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




19







1)(

dzp

20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




20





)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




21



22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




22




4

)()( ds

zzdL

)()()(

)()()()(

dzLzpz

dzLzpzzL

s

vvi

)(exp)()()()(0

xzr

tr dttxzxwithzxxLzL

23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




23





)()()()()(0 xyr

dryzzLzxxLyL

)(4)()( zLzzL vi

24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




24




)(exp)(0

zyr

t dsszyz

)()()()()(0 xyr

dryzzLzxxLyL

25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




25





26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




26




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




27



28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




28







29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




29






30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

)x(Q)x(U)x(U dtr

d 22

46




30


31






32








33





34











35






36








37


38





39






40









41


Participating Media

42





43









)x(L)x(L)x(L dr

44




))x(F()x(U)x(Ldd

d

4

3

d)x(L)x(U dd

4

1

45




d)cos()x(L))x(F(d

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Documents

K. H. Ko School of Mechatronics Gwangju Institute of Science and Technology