Spring Conference on Computer Graphics 2002
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A Survey of Modelling and Rendering
of the Earth’s Atmosphere
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Jaroslav Sloup
Department of Computer Science and Engineering
Czech Technical University in Prague
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Talk Outline
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Introduction.
Models of the atmosphere.
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Atmosphere composition.
Light transfer through the atmosphere.
Rendering of the atmosphere.
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Motivation
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Photo-realistic image synthesis of outdoor scenes
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• most of scene objects illumination comes directly from the sun
and sky
• sky forms the background of the scene
• atmosphere causes change of colors of distant objects
Instance of participating media simulation problem
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• long distances make the effects of atmosphere visible
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Models of the Atmosphere
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Plane-parallel model
Klassen [15]
• approximates the atmosphere as a set of plane-parallel layers
• neglects the sphericity of Earth
• introduces large errors near the horizon
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Spherical model
Nishita et al.[20,21], Irwin [8], Jackel et al. [10]
• for each atmospheric component one submodel
• submodel = a set of non-interleaving spherical layers
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H1
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H2
ground
submodel 1
+
ground
=
ground
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i−th
spherical layer
submodel 2
combination of
submodel 1 and 2
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Composition of the Atmosphere
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Major atmospheric constituents
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clean dry air
aerosols
ozone
water vapour, rain drops and ice crystals
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Constituents characteristics at point P
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• extinction coefficient σeλ(P ) (losses by absorption and scattering)
• scattering function pλ(P, θ) (angular distribution of scattered light)
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• single scattering albedo $λ(P )
(relation between absorption and scattering $λ(P ) =
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R
Ω pλ (P,θ) dθ
σe
)
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Composition of the Atmosphere – cont.
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Constituents properties depend on
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particle size
wavelength of incident light λ
refractive index of the particle
number of particles in unit volume
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Sources of constituents properties
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Mie scattering theory
Rayleigh scattering theory
geometric optics
measured data
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Composition of the Atmosphere – cont.
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Mie Scattering Theory
developed by G.Mie(1908)
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general theory - valid for all particle sizes
strong forward scattering
absorption of energy occurs
scattering function depends on particle size and wavelength
rays of scattered light
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incident
light
θ
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scattering function
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Composition of the Atmosphere – cont.
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Rayleigh Scattering Theory
established by Lord Rayleigh (1871)
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approximation for particles smaller than wavelength of light
no absorption of light
simple scattering function
radiance of scattered light is proportional to the ratio ∼ λ14
→ blue color of the sky
→ color of the sun (yellow → orange → red)
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6e-05
extinction coefficient
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5e-05
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extinction [m^-1]
4e-05
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3e-05
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2e-05
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1e-05
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wavelength [nm]
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Light Transfer Equation
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L λ (P0 , s)
P0
τ(s0,sE)
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scattering
volumes
beams of sunlight
penetrating the atmosphere
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P
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τ(s0,s)
skylight
s
L λ(P(s0 ), s)
s’
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incident skylight radiance
L λ(P, s’)
skylight scattered
along direction s
towards the observer
phase function
θ
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atmosphere
J λ (P, s)
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s
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Lλ(P (s0), ~
s)
=
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Lλ(P0, ~
s) τ (s0, sE )
+
sRE
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Jλ(P (s), ~
s) τ (s0, s) σeλ(P (s))ds
s0
initial ray radiance atte-
integral of skylight radiance scattered to-
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nuated along the viewing
wards the observer by volume elements
ray by transmittance τ
along the viewing ray attenuated by τ
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Light Transfer Equation – cont.
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Source function
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Jλ(P, ~
s)
=
1
4π
RR
Ω
n
P
i=1
$λi (P ) piλ(P, θ) Lλ(P, s~0) dω 0
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Light Transfer Equation – cont.
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Boundary condition at the top of the atmosphere
h
i
q
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fλ 1 − u(1 − 1 − dr2 )
ray hits the solar disc
s
Lλ(P0, ~
s) = {
0.0
outside the solar disc
irradiance decrease factor
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irradiance
decrease
factor
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limb darkening
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0.9
0.8
2.1
0.7
solar irradiance
black body at 5777K
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0.6
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1.9
0.5
solar irradiance
0.4
0
100000 200000 300000 400000 500000 600000
fλ
distance from the center of solar disc [km]
distance from the center
of solar disc
P0
atmosphere
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1.7
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1.4
s
L λ (P0 , s)
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1.3
1.2
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1.1
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Light Transfer Equation – cont.
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Boundary condition for ground reflection
ZZ
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Lλ(P0, ~
s) =
ρλ(~
s, s~0) Lλ(P0, s~0) cos ϑ dω,
π
Ω
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dω
s
ϑ
L λ (P0, s)
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s’
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L λ(P0, s’)
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P0
brfd
ρλ
Earth’s surface
x
y
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Rendering of the Atmosphere
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For rendering we need
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~ date, atm cond) → spectral radiance
• function F (P, dir,
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to simulate the color of the sky and sun
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~ date, atm cond, Lr (Pr )) → spectral radiance
• function G(P, dir,
to simulate the change of color of distant objects observed through the
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atmosphere
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atm_cond
dir
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F(...)
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Lr (Pr )
P
G(...)
Pr
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F and G are based on the light transfer equation
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Rendering of the Atmosphere – cont.
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Rendering of the sky color
For every pixel covered by the sky do
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evaluate LTE to determine amount of spectral radiance
convert spectral radiance → CIE XYZ color space
apply tone mapping techniques
convert CIE XYZ color → RGB color to be displayed
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Simulation of the aerial perspective
For all pixels not covered by the sky
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find intersection with the scene object
calculate amount of light reflected towards the observer
evaluate LTE to calculate gains and losses of reflected light
tone mapping and conversion to the RGB color
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Rendering of the Atmosphere – cont.
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Methods to solve the light transfer equation
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• analytic methods
+ fast and easy to use
– hardly extendable
• numerical methods
Nishita et al.[13,20,21], Jackel et al.[9,10]
– still very difficult to solve
– require a lot of computational time
+ work with arbitrarily complex atmospheric conditions
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Simplifying assumptions
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• uniform medium, simple geometry or very small albedo
• restriction of order of scattering → single scattering methods
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Rendering of the Atmosphere – cont.
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Hybrid methods
Dobashi et al.[3,4], Preetham et al.[23]
• numerical solution ⇒ analytical approximation
• sky approximated by hemisphere with a large radius
• numerical solution
⇒ sky spectral radiance distribution for several sun altitudes
⇒ approximation by a series of basis functions / simple
parametrically fitted formulas
⇒ weights / parameters values are stored in tables
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Rendered Images
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Preetham et al.[23]
Walter and Jackel [9,10]
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Summary
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We discussed
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• two models of the atmosphere
• atmosphere composition and properties of the major
atmospheric constituents
• light transfer equation
• overview of methods used for rendering of the atmosphere
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Spring Conference on Computer Graphics 2002
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QUESTIONS ?
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H1
H2
ground
submodel 1
+
ground
=
ground
i−th
spherical layer
submodel 2
combination of
submodel 1 and 2
rays of scattered light
incident
light
θ
scattering function
6e-05
extinction coefficient
extinction [m^-1]
5e-05
4e-05
3e-05
2e-05
1e-05
0
350
400
450
500 550 600 650
wavelength [nm]
700
750
L λ (P0 , s)
P0
τ(s0,sE)
scattering
volumes
beams of sunlight
penetrating the atmosphere
P
τ(s0,s)
skylight
s
L λ(P(s0 ), s)
s’
incident skylight radiance
L λ(P, s’)
skylight scattered
along direction s
towards the observer
phase function
θ
J λ (P, s)
s
atmosphere
irradiance decrease factor
1
irradiance
decrease
factor
limb darkening
0.9
0.8
0.7
0.6
0.5
solar irradiance
0.4
0
100000 200000 300000 400000 500000 600000
fλ
distance from the center of solar disc [km]
distance from the center
of solar disc
P0
atmosphere
s
L λ (P0 , s)
2.1
solar irradiance
black body at 5777K
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
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N
dω
s
ϑ
L λ (P0, s)
P0
brfd
ρλ
Earth’s surface
x
s’
L λ(P0, s’)
y
atm_cond
dir
F(...)
Lr (Pr )
P
G(...)
Pr
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