Graphics Reference
In-Depth Information
coecient over the path, which is given by the following equation:
B
)=
B
A
(
β
R
e
−h
(
t
)
/H
R
+
β
e
M
e
−h
(
t
)
/H
M
)
dt.
T
(
A
→
(2.1)
Scattering by molecules and aerosols happens independently and is propor-
tional to the corresponding scattering coecient (
β
R/M
)atsealevelandtothe
particle density scale factor (
e
−h/H
R/M
)at
P
. The fraction of scattered light
going in the view direction is given by the phase function
p
R/M
(
θ
)foreachtype.
Light in-scattered at
P
is attenuated on the way to the camera by a factor of
e
−T
(
P→C
)
. Finally, to account for shadowing, we also need to introduce a visi-
bility term
V
(
P
), which equals 1 if
P
is visible from light and 0 otherwise. Total
in-scattering along the view ray is thus given by the following integral:
L
In
=
O
C
e
−T
(
A
(
s
)
→P
(
s
))
e
−T
(
P
(
s
)
→C
)
L
Sun
·
·
·
V
(
P
(
s
))
(2.2)
β
R
e
−h
(
s
)
/H
R
p
R
(
θ
)+
β
s
M
e
−h
(
s
)
/H
M
p
M
(
θ
)
ds.
·
Initial object radiance
L
O
is attenuated in the atmosphere before it reaches
the camera by a factor of
e
−T
(
O→C
)
. The final radiance measured at the camera
is a sum of the attenuated object radiance and in-scattered light, which stands
for a phenomenon called
aerial perspective
:
e
−T
(
O→C
)
+
L
In
.
L
=
L
O
·
(2.3)
2.5 Computing Scattering Integral
Integral (2.2) cannot be solved analytically. Nor it is feasible to compute it
directly with numerical integration, because at each step we will have to solve two
optical depth integrals (2.1). Numerical integration can be significantly optimized
using a number of tricks. To begin, we can eliminate computation of the optical
depth integrals
T
(
A
C
). O'Neil noticed that equation (2.1)
depends on two parameters: the altitude
h
and the angle
ϕ
between the vertical
direction and the light direction [O'Neil 04]. Optical depth
T
(
A
→
P
)and
T
(
P
→
P
)canthus
be pre-computed and stored in a lookup table. We follow the same idea, but
use slightly different implementation. First we rewrite the optical depth integral
(2.1) as follows:
→
B
)=
β
R
B
A
e
−h
(
t
)
/H
R
dt
+
β
e
M
B
A
e
−h
(
t
)
/H
M
dt.
T
(
A
→
(2.4)
Now one can see that while
β
R
and
β
e
M
are 3-component vectors, both integrals
in equation (2.4) are scalar. We can use a two-channel lookup table storing total
