Radiative Transfer and Volume Path Tracing
R
a
d
i
a
t
i
v
e
T
r
a
n
s
f
e
r
&
V
o
l
u
m
e
P
a
t
h
T
r
a
c
i
n
g
C
S
2
9
5
,
S
p
r
i
n
g
2
0
1
7
S
h
u
a
n
g
Z
h
a
o
C
o
m
p
u
t
e
r
S
c
i
e
n
c
e
D
e
p
a
r
t
m
e
n
t
U
n
i
v
e
r
s
i
t
y
o
f
C
a
l
i
f
o
r
n
i
a
,
I
r
v
i
n
e
Modified from the original slides
CS295, Spring
2017
Shuang
Zhao
1
CS295, Spring
2017
Shuang
Zhao
2
T
o
d
a
y
’
s
L
e
c
t
u
r
e
•
Radiative
transfer
•
The
mathematical model
to
simulate light scattering
in participating media
(e.g., smoke)
and translucent
materials
(e.g.,
marble and
skin)
•
Volume
path tracing
(VPT)
•
A
Monte Carlo solution
to the
radiative
transfer
problem
•
Similar
to the
normal
PT from
previous
lectures
CS295, Spring
2017
Shuang
Zhao
3
R
a
d
i
a
t
i
v
e
T
r
a
n
s
f
e
r
CS295: Realistic Image
Synthesis
P
a
r
t
i
c
i
p
a
t
i
n
g
M
e
d
i
a
[Kutz
et
al.
2017]
CS295, Spring
2017
Shuang
Zhao
4
T
r
a
n
s
l
u
c
e
n
t
M
a
t
e
r
i
a
l
s
[Gkioulekas
et
al.
2013]
CS295, Spring
2017
Shuang
Zhao
5
S
u
b
s
u
r
f
a
c
e
S
c
a
t
t
e
r
i
n
g
•
Light enters a material and scatters
around
before eventually leaving or
absorbed
X
A
b
s
o
r
b
e
d
CS295, Spring
2017
Shuang
Zhao
6
Participating
medium
S
u
b
s
u
r
f
a
c
e
S
c
a
t
t
e
r
i
n
g
•
Light enters a material and scatters
around
before eventually leaving or
absorbed
Scattered
Participating
medium
CS295, Spring
2017
Shuang
Zhao
7
S
u
b
s
u
r
f
a
c
e
S
c
a
t
t
e
r
i
n
g
•
Light enters a material and scatters
around
before eventually leaving or
absorbed
Participating
medium
CS295, Spring
2017
Shuang
Zhao
8
C
a
n
w
e
u
s
e
t
h
e
r
e
n
d
e
r
i
n
g
e
q
u
a
t
i
o
n
?
•
Radiance is not constant for each ray segment!
•
We need to consider how the radiance change
during the ray segment
Participating
medium
CS295, Spring
2017
Shuang
Zhao
9
CS295, Spring
2017
Shuang
Zhao
10
R
a
d
i
a
t
i
v
e
T
r
a
n
s
f
e
r
•
A mathematical model describing how
light
interacts with participating
media
•
Originated in
physics
•
Now used in many areas
•
Astrophysics (light transport in
space)
•
Biomedicine (light transport in human
tissue)
•
Graphics
•
Nuclear science & engineering (neutron
transport)
•
Remote
sensing
•
…
R
a
d
i
a
t
i
v
e
T
r
a
n
s
f
e
r
E
q
u
a
t
i
o
n
(
R
T
E
)
•
Focus on how radiance changes at each
point and direction
•
Consider , not
R
a
d
i
a
t
i
v
e
T
r
a
n
s
f
e
r
E
q
u
a
t
i
o
n
(
R
T
E
)
In-scattering
Ou
t
-
scatt
ering
&
absorption
Emission
D
i
f
fer
enti
a
l
radiance
In-scattering
CS295, Spring
2017
Shuang
Zhao
12
Out-scattering
Emission
&
absorption
R
a
d
i
a
t
i
v
e
T
r
a
n
s
f
e
r
E
q
u
a
t
i
o
n
(
R
T
E
)
•
The
RTE
is a
first-order
integro-differential
equation
•
For
a participating medium in
a
volume
with
boundary
, the
RTE
governs
the
radiance values
inside
this
volume
(i.e.,
for
all
)
•
T
h
e
b
o
u
n
d
a
r
y
c
o
n
d
i
t
i
o
n
i
s
t
h
e
r
a
d
i
a
n
c
e
f
i
e
l
d
o
n
t
h
e
b
o
u
n
d
a
r
y
(
i
.
e
.
,
L
(
x
,
ω
)
f
o
r
a
l
l
)
In-scattering
Out-scattering
Emission
&
absorption
CS295, Spring
2017
Shuang
Zhao
13
R
a
d
i
a
t
i
v
e
T
r
a
n
s
f
e
r
E
q
u
a
t
i
o
n
(
R
T
E
)
•
Differential
radiance
,
,
a probability density
over
•
Scattering
coefficient:
Phase
function:
g
i
v
e
n
x
a
n
d
ω
i
•
Extinction
coefficient:
•
Source
term:
In-scattering
Out-scattering
Emission
&
absorption
CS295, Spring
2017
Shuang
Zhao
14
R
a
d
i
a
t
i
v
e
T
r
a
n
s
f
e
r
E
q
u
a
t
i
o
n
(
R
T
E
)
•
σ
t
controls how frequently light scatters and is
also
known
as the
optical
density
•
The
ratio between
σ
s
and
σ
t
controls
the fraction of
radiant energy
not
being absorbed
at
each
scattering
and is also known as the
single-scattering
albedo
In-scattering
CS295, Spring
2017
Shuang
Zhao
15
Out-scattering
Emission
&
absorption
R
a
d
i
a
t
i
v
e
T
r
a
n
s
f
e
r
E
q
u
a
t
i
o
n
(
R
T
E
)
•
The
phase function
f
p
is usually parameterized as
a
f
u
n
c
t
i
o
n
o
n
t
h
e
a
n
g
l
e
b
e
t
w
e
e
n
ω
i
a
n
d
ω
.
N
a
m
e
l
y
,
•
Example:
the
Henyey-Greenstein
(HG)
phase
function
with
parameter -1
<
g
<
1
(more forward scattering)
:
In-scattering
Out-scattering
Emission
&
absorption
CS295, Spring
2017
Shuang
Zhao
16
T
h
e
I
n
t
e
g
r
a
l
F
o
r
m
o
f
t
h
e
R
T
E
Integro-differential
equation
Integral
equation
•
It
is desirable
to
rewrite
the
RTE
as an integral
equation
•
which can then be solved numerically using Monte
Carlo
methods
CS295, Spring
2017
Shuang
Zhao
17
I
n
t
e
g
r
a
l
F
o
r
m
o
f
t
h
e
R
T
E
•
For
any
,
l
e
t
h
(
x
,
ω
)
d
e
n
o
t
e
s
t
h
e
m
i
n
i
m
a
l
d
i
s
t
a
n
c
e
f
o
r
t
h
e
r
a
y
(
x
,
-
ω
)
t
o
h
i
t
t
h
e
b
o
u
n
d
a
r
y
.
I
n
o
t
h
e
r
w
o
r
d
s
,
•
W
h
e
n
(
x
,
-
ω
)
n
e
v
e
r
h
i
t
s
t
h
e
b
o
u
n
d
a
r
y
,
•
This can happen when the volume is
infinite
•
For
any
with
,
let
CS295, Spring
2017
Shuang
Zhao
18
I
n
t
e
g
r
a
l
F
o
r
m
o
f
t
h
e
R
T
E
•
For
any
,
t
h
e
a
t
t
e
n
u
a
t
i
o
n
b
e
t
w
e
e
n
x
a
n
d
y
i
s
•
A
l
i
n
e
i
n
t
e
g
r
a
l
b
e
t
w
e
e
n
x
a
n
d
y
•
f
o
r
a
l
l
x
a
n
d
y
•
For homogeneous media
with
,
CS295, Spring
2017
Shuang
Zhao
19
I
n
t
e
g
r
a
l
F
o
r
m
o
f
t
h
e
R
T
E
In-scattering
Em
i
s
s
ion
A
t
tenuation
(The second term vanishes
when
)
w
h
e
re
Attenuation
Boundary
cond.
CS295, Spring
2017
Shuang
Zhao
20
K
e
r
n
e
l
F
o
r
m
o
f
t
h
e
R
T
E
where
Kernel
function
CS295, Spring
2017
Shuang
Zhao
21
Source
function
(known term)
O
p
e
r
a
t
o
r
F
o
r
m
o
f
t
h
e
R
T
E
•
Phase
space:
•
For any real-valued function
g
on
Γ
,
define
o
p
e
r
a
t
o
r
K
a
s
where
•
Then, the
RTE
becomes
•
Similar
to the
RE!
•
Yield
Neumann
series
CS295, Spring
2017
Shuang
Zhao
22
CS295, Spring
2017
Shuang
Zhao
23
V
o
l
u
m
e
P
a
t
h
T
r
a
c
i
n
g
CS295: Realistic Image
Synthesis
Start from here…
CS295, Spring
2017
Shuang
Zhao
24
S
o
l
v
i
n
g
t
h
e
R
T
E
•
Given the similarity between the
RTE
and the
RE,
Monte Carlo solutions to the
RE
can be
adapted to solve the
RTE
•
Volume
path
tracing
•
Volume
adjoint particle
tracing
•
Volume
bidirectional path
tracing
•
…
V
o
l
u
m
e
P
a
t
h
T
r
a
c
i
n
g
w
h
e
re
Known
•
Basic
idea
•
Draw
from
•
D
r
a
w
ω
i
f
r
o
m
p
(
ω
i
)
•
E
v
a
l
u
a
t
e
L
(
r
,
ω
i
)
r
e
c
u
r
s
i
v
e
l
y
CS295, Spring
2017
Shuang
Zhao
25
F
r
e
e
D
i
s
t
a
n
c
e
S
a
m
p
l
i
n
g
•
is called
the
“free distance” and is sampled
from
w
h
ere
with
λ
0
being an arbitrary positive
number
•
p
gives an exponential distribution with varying
parameters
CS295, Spring
2017
Shuang
Zhao
26
F
r
e
e
D
i
s
t
a
n
c
e
S
a
m
p
l
i
n
g
•
For
all
,
it holds
that
CS295, Spring
2017
Shuang
Zhao
27
F
r
e
e
D
i
s
t
a
n
c
e
S
a
m
p
l
i
n
g
whe
r
e
CS295, Spring
2017
Shuang
Zhao
28
F
r
e
e
D
i
s
t
a
n
c
e
S
a
m
p
l
i
n
g
•
By
applying Monte Carlo integration, we
have
•
Pseudocode:
•
D
r
aw
•
If
from
p
,
return
•
Otherwise,
return
CS295, Spring
2017
Shuang
Zhao
29
D
i
r
e
c
t
i
o
n
S
a
m
p
l
i
n
g
•
One
extra
integral
remains:
is usually a
valid
•
ω
i
c
a
n
b
e
s
a
m
p
l
e
d
b
a
s
e
d
o
n
•
In
practice,
p
r
o
b
a
b
i
l
i
t
y
d
e
n
s
i
t
y
o
n
ω
i
,
y
i
e
l
d
i
n
g
CS295, Spring
2017
Shuang
Zhao
30
V
o
l
u
m
e
P
a
t
h
T
r
a
c
i
n
g
radiance(
x
,
ω
):
compute
h
=
h
(
x
,
ω
)
#
using ray
tracing
draw
τ
if
τ
<
h
:
r
=
x
–
τ*
ω
draw
ω
i
return
σ
s
(
r
)/
σ
t
(
r
)*radiance(
r
,
ω
i
)
+
Q
(
r
,
ω
)/
σ
t
(
r
)
else:
return
boundaryRadiance(
x
–
h*
ω
,
ω
)
How
to
implement
this?
CS295, Spring
2017
Shuang
Zhao
31
F
r
e
e
D
i
s
t
a
n
c
e
S
a
m
p
l
i
n
g
M
e
t
h
o
d
s
•
How to draw samples from this
distribution?
•
Homogeneous
media
•
•
Let
,
then
and
can be drawn using
the
inversion
•
In
this
case,
method:
CS295, Spring
2017
Shuang
Zhao
32
F
r
e
e
D
i
s
t
a
n
c
e
S
a
m
p
l
i
n
g
M
e
t
h
o
d
s
•
Heterogeneous
media
•
v
a
r
i
e
s
w
i
t
h
x
,
c
a
u
s
i
n
g
t
o
v
a
r
y
w
i
t
h
•
p
does
not
have a close-form expression in
general
•
Common sampling
methods
•
Ray
marching
•
Delta
tracking
CS295, Spring
2017
Shuang
Zhao
33
R
a
y
M
a
r
c
h
i
n
g
•
One can apply the inversion method
by
1.
Drawing
ξ
from
U(0,
1)
2.
Finding
satisfying
•
This is usually achieved numerically by
iteratively
increasing
with some
fixed
step
size
until
reaches
ξ
•
The
step
size
is
generally picked according to
the
u
n
d
e
r
l
y
i
n
g
r
e
p
r
e
s
e
n
t
a
t
i
o
n
o
f
σ
t
(
x
)
(
e
.
g
.
,
v
o
x
e
l
s
i
z
e
)
CS295, Spring
2017
Shuang
Zhao
34
R
a
y
M
a
r
c
h
i
n
g
•
Pros
•
For
each
sample
,
can be obtained easily
•
Cons
•
Biased
(for
any finite
step
size
)
•
Resolution
dependent
•
needs to be picked based on the resolution of
the
density (
σ
t
)
field
•
Slow
for
high-resolution density
fields
CS295, Spring
2017
Shuang
Zhao
35
D
e
l
t
a
T
r
a
c
k
i
n
g
•
Also known as
Woodcock
tracking
•
Basic
idea
•
Consider the medium to have homogeneous
density
, and use it to draw free
distances
•
To
compensate
the fact that
“phantom” densities have
been
introduced, the sampling process continues with
probability
a
t
e
a
c
h
r
i
CS295, Spring
2017
Shuang
Zhao
36
CS295, Spring
2017
Shuang
Zhao
37
D
e
l
t
a
T
r
a
c
k
i
n
g
•
Pseudocode:
deltaTracking(
x
,
ω
,
σ
t
)
max
compute
h
using ray
tracing
τ
=
0
while
τ
<
h
:
τ
+=
-log(rand())/
σ
t
max
r
=
x
-
τ
*
ω
if rand()
<
σ
t
(
r
)/
σ
t
:
max
break
return
τ
D
e
l
t
a
T
r
a
c
k
i
n
g
•
Pros
•
Unbiased
•
Resolution
independent
,
is
not
immediately
•
Cons
•
For
each
sample
available
•
Slow
for
density fields with widely varying
σ
t
values
(
i
.
e
.
,
σ
t
m
a
x
>
>
σ
t
(
x
)
f
o
r
m
a
n
y
x
)
CS295, Spring
2017
Shuang
Zhao
38
V
o
l
u
m
e
P
a
t
h
T
r
a
c
i
n
g
(
V
P
T
)
radiance(
x
,
ω
):
compute
h
=
h
(
x
,
ω
)
draw
τ
if
τ
<
h
:
r
=
x
–
τ*
ω
draw
ω
i
return
σ
s
(
r
)/
σ
t
(
r
)*radiance(
r
,
ω
i
)
+
Q
(
r
,
ω
)/
σ
t
(
r
)
else:
return
boundaryRadiance(
x
–
h*
ω
,
ω
)
This basic version can be improved
using techniques we have seen
earlier:
•
Russian
roulette
•
Next-event
estimation
•
Multiple
importance
sampling
CS295, Spring
2017
Shuang
Zhao
39
V
P
T
w
i
t
h
N
e
x
t
-
E
v
e
n
t
E
s
t
i
m
a
t
i
o
n
•
The
R
TE
impl
i
es
that
.
Namely,
•
By
drawing
from
the aforementioned
exponential
distribution, we have
w
he
re
CS295, Spring
2017
Shuang
Zhao
40
V
P
T
w
i
t
h
N
e
x
t
-
E
v
e
n
t
E
s
t
i
m
a
t
i
o
n
•
The
remaining integral is then split into
two:
E
s
t
i
m
a
t
e
r
e
c
u
r
s
i
v
e
l
y
b
y
d
r
a
w
i
n
g
ω
i
b
a
s
e
d
o
n
f
p
“
i
n
d
i
r
e
c
t
i
l
l
u
m
i
n
a
t
i
o
n
”
Estimate directly by
area sampling or
MIS
“direct
illumination”
CS295, Spring
2017
Shuang
Zhao
41
V
P
T
w
i
t
h
N
e
x
t
-
E
v
e
n
t
E
s
t
i
m
a
t
i
o
n
•
Pseudocode:
scatteredRadiance(
x
,
ω
):
compute
h
=
h
(
x
,
ω
)
#
using ray
tracing
draw
τ
if
τ
<
h
:
r
=
x
–
τ*
ω
rad
=
directIllumination(
r
,
ω
)
draw
ω
i
rad
+=
scatteredRadiance(
r
,
ω
i
)
return
σ
s
(
r
)/
σ
t
(
r
)*rad
else:
return
0
CS295, Spring
2017
Shuang
Zhao
42
D
i
r
e
c
t
I
l
l
u
m
i
n
a
t
i
o
n
f
o
r
V
P
T
•
Recall
that
•
For
non-emissive materials,
Q
vanishes
and
•
In
this
case,
•
The area integral can be further restricted to the subset
of
where the boundary radiance is
non-zero
Change
of
measure
B
o
u
n
d
a
ry
radiance
CS295, Spring
2017
Shuang
Zhao
43
D
i
r
e
c
t
I
l
l
u
m
i
n
a
t
i
o
n
f
o
r
V
P
T
•
Phase function
sampling:
D
r
a
w
ω
i
b
a
s
e
d
o
n
f
p
•
Area
sampling:
D
r
a
w
y
f
r
o
m
•
The
two strategies can be combined using
MIS
CS295, Spring
2017
Shuang
Zhao
44
Explore the mathematical model of radiative transfer for simulating light scattering in participating media and translucent materials through volume path tracing. Learn about subsurface scattering and how radiance changes along ray segments in participating media. Delve into the origins and applications of radiative transfer in various fields, including astrophysics.
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Presentation Transcript
Radiative Transfer & Volume Path Tracing CS295, Spring 2017 Shuang Zhao Computer Science Department University of California, Irvine Modified from the original slides CS295, Spring2017 ShuangZhao 1
Todays Lecture Radiative transfer The mathematical model to simulate light scattering in participating media (e.g., smoke) and translucent materials (e.g., marble and skin) Volume path tracing (VPT) A Monte Carlo solution to the radiative transfer problem Similar to the normal PT from previous lectures CS295, Spring2017 ShuangZhao 2
Radiative Transfer CS295: Realistic Image Synthesis CS295, Spring2017 ShuangZhao 3
Participating Media [Kutz et al. 2017] CS295, Spring2017 ShuangZhao 4
Translucent Materials [Gkioulekas et al. 2013] CS295, Spring2017 ShuangZhao 5
Subsurface Scattering Light enters a material and scatters around before eventually leaving or absorbed X Absorbed Participating medium CS295, Spring2017 ShuangZhao 6
Subsurface Scattering Light enters a material and scatters around before eventually leaving or absorbed Scattered Participating medium CS295, Spring2017 ShuangZhao 7
Subsurface Scattering Light enters a material and scatters around before eventually leaving or absorbed Participating medium CS295, Spring2017 ShuangZhao 8
Can we use the rendering equation? Radiance is not constant for each ray segment! We need to consider how the radiance change during the ray segment Participating medium CS295, Spring2017 ShuangZhao 9
Radiative Transfer A mathematical model describing how light interacts with participating media Originated in physics Now used in many areas Astrophysics (light transport in space) Biomedicine (light transport in human tissue) Graphics Nuclear science & engineering (neutrontransport) Remote sensing CS295, Spring2017 ShuangZhao 10
Radiative Transfer Equation (RTE) Focus on how radiance changes at each point and direction Consider , not
Radiative Transfer Equation (RTE) Differential radiance In-scattering Out-scattering & absorption Emission Out-scattering Emission & absorption In-scattering CS295, Spring2017 ShuangZhao 12
Radiative Transfer Equation (RTE) Out-scattering Emission & absorption In-scattering The RTE is a first-order integro-differential equation For a participating medium in a volume boundary , the RTE governs the radiance values inside this volume (i.e., for all with ) The boundary condition is the radiance field on the boundary (i.e., L(x, ) for all ) CS295, Spring2017 ShuangZhao 13
Radiative Transfer Equation (RTE) Out-scattering Emission & absorption In-scattering Differential radiance Scattering coefficient: Phase function: given x and i Extinction coefficient: Source term: , , a probability density over CS295, Spring2017 ShuangZhao 14
Radiative Transfer Equation (RTE) Out-scattering Emission & absorption In-scattering t controls how frequently light scatters and is also known as the optical density The ratio between s and t controls the fraction of radiant energy not being absorbed at each scattering and is also known as the single-scattering albedo CS295, Spring2017 ShuangZhao 15
Radiative Transfer Equation (RTE) Out-scattering Emission & absorption In-scattering The phase function fp is usually parameterized as a function on the angle between i and .Namely, Example: the Henyey-Greenstein (HG) phase function with parameter -1 < g < 1 (more forward scattering): CS295, Spring2017 ShuangZhao 16
The Integral Form of the RTE Integro-differential equation Integral equation It is desirable to rewrite the RTE as an integral equation which can then be solved numerically using Monte Carlo methods CS295, Spring2017 ShuangZhao 17
Integral Form of the RTE For any distance for the ray (x, - ) to hit the boundary other words, , let h(x, ) denotes the minimal . In When (x, - ) never hits the boundary, This can happen when the volume is infinite For any , let with CS295, Spring2017 ShuangZhao 18
Integral Form of the RTE For any , the attenuation between x and y is A line integral between x andy For homogeneous media with for all x andy , CS295, Spring2017 ShuangZhao 19
Integral Form of the RTE Attenuation In-scattering Emission where Attenuation Boundary cond. (The second term vanishes when ) CS295, Spring2017 ShuangZhao 20
Kernel Form of the RTE Kernel function Source function (known term) where CS295, Spring2017 ShuangZhao 21
Operator Form of the RTE Phase space: For any real-valued function g on , define operator K as where Then, the RTE becomes Similar to the RE! Yield Neumann series CS295, Spring2017 ShuangZhao 22
Start from here Volume Path Tracing CS295: Realistic Image Synthesis CS295, Spring2017 ShuangZhao 23
Solving the RTE Given the similarity between the RTE and the RE, Monte Carlo solutions to the RE can be adapted to solve the RTE Volume path tracing Volume adjoint particle tracing Volume bidirectional path tracing CS295, Spring2017 ShuangZhao 24
Volume Path Tracing where Known Basic idea Draw Draw i fromp( i) Evaluate L(r, i) recursively from CS295, Spring2017 ShuangZhao 25
Free Distance Sampling is called the free distance and is sampled from where with 0 being an arbitrary positive number p gives an exponential distribution with varying parameters CS295, Spring2017 ShuangZhao 26
Free Distance Sampling For all , it holds that CS295, Spring2017 ShuangZhao 27
Free Distance Sampling where CS295, Spring2017 ShuangZhao 28
Free Distance Sampling By applying Monte Carlo integration, we have Pseudocode: Draw from p If , return Otherwise,return CS295, Spring2017 ShuangZhao 29
Direction Sampling One extra integral remains: i can be sampled basedon In practice, probability density on i, yielding is usually a valid CS295, Spring2017 ShuangZhao 30
Volume Path Tracing radiance(x, ): compute h = h(x, ) # using ray tracing How to implement this? draw if < h: r = x * draw i return s(r)/ t(r)*radiance(r, i) + Q(r, )/ t(r) else: return boundaryRadiance(x h* , ) CS295, Spring2017 ShuangZhao 31
Free Distance Sampling Methods How to draw samples from this distribution? Homogeneous media Let , then and can be drawn using the inversion In this case, method: CS295, Spring2017 ShuangZhao 32
Free Distance Sampling Methods Heterogeneous media varies with x, causing p does not have a close-form expression in general to vary with Common sampling methods Ray marching Delta tracking CS295, Spring2017 ShuangZhao 33
Ray Marching One can apply the inversion method by 1. Drawing from U(0, 1) 2. Finding satisfying This is usually achieved numerically by iteratively increasing with some fixed step size reaches The step size is generally picked according to the underlying representation of t(x) (e.g., voxelsize) until CS295, Spring2017 ShuangZhao 34
Ray Marching Pros For each sample , can be obtained easily Cons Biased (for any finite step size Resolution dependent needs to be picked based on the resolution ofthe density ( t)field Slow for high-resolution densityfields ) CS295, Spring2017 ShuangZhao 35
Delta Tracking Also known as Woodcock tracking Basic idea Consider the medium to have homogeneousdensity , and use it to draw freedistances To compensate the fact that phantom densities have been introduced, the sampling process continues with probability at each ri CS295, Spring2017 ShuangZhao 36
Delta Tracking Pseudocode: deltaTracking(x, , t ) max compute h using ray tracing = 0 while < h: += -log(rand())/ tmax r = x - * if rand() < t(r)/ t : max break return CS295, Spring2017 ShuangZhao 37
Delta Tracking Pros Unbiased Resolution independent Cons For each sample available Slow for density fields with widely varying tvalues (i.e., tmax >> t(x) for manyx) , is not immediately CS295, Spring2017 ShuangZhao 38
Volume Path Tracing (VPT) radiance(x, ): compute h = h(x, ) This basic version can be improved using techniques we have seen earlier: Russian roulette Next-event estimation Multiple importance sampling draw if < h: r = x * draw i return s(r)/ t(r)*radiance(r, i) + Q(r, )/ t(r) else: return boundaryRadiance(x h* , ) CS295, Spring2017 ShuangZhao 39
VPT with Next-Event Estimation The RTE Namely, implies that . where By drawing distribution, we have from the aforementioned exponential CS295, Spring2017 ShuangZhao 40
VPT with Next-Event Estimation The remaining integral is then split into two: Estimate directly by area sampling or MIS direct illumination Estimate recursively by drawing ibased on fp indirect illumination CS295, Spring2017 ShuangZhao 41
VPT with Next-Event Estimation Pseudocode: scatteredRadiance(x, ): compute h = h(x, ) # using ray tracing draw if < h: r = x * rad = directIllumination(r, ) draw i rad += scatteredRadiance(r, i) return s(r)/ t(r)*rad else: return 0 CS295, Spring2017 ShuangZhao 42
Direct Illumination for VPT Recall that For non-emissive materials, Q vanishes and In this case, Boundary radiance Change of measure The area integral can be further restricted to the subsetof where the boundary radiance is non-zero CS295, Spring2017 ShuangZhao 43
Direct Illumination for VPT Phase function sampling: Draw i based onfp Area sampling: Draw y from The two strategies can be combined using MIS CS295, Spring2017 ShuangZhao 44
