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Precomputed Atmospheric Scattering
1

Eric Bruneton, Fabrice Neyret? 2008 ??. [Bruneton08]
??? ?? ???? ??? ???( ???, ?? ??, ?? )? ??? ??? ?? ??
( Precomputed ) ?? ?? ???? ????? ??? ??? ?? ????.
UE4? Atmospheric Fog? ? ??? ??? ?????. ( ???? Atmospheric~
?? ???? ??? )
??? OpenGL? ??? ?? ??? ????? ??? ???? ? ??? ??.
UE4 ??? ?? ??? ?? ????. ( Precomputed ??? ?? ? ?? ?? ?? )
??? ??? ?? ???? ?. ?? ??? ?? ?? ??? ????.
? ppt? ?? ??? ???? ?? ?? ??? ?? ??? ??? ??? ????
??? ???? ???.
Precomputed Atmospheric Scattering
2

??? ?? ??? ?????.
?? ??? ???? ?? ??? ?????? ???? ?? ????.
Q. ?? ???? ???? ? ? ????
A. ? ? ??.
Q. ?? ???? ???? ? ? ?? ????
?? ??( Scattering )
???? ?? ?????? ??
???? ? ?? ?? ??? ??? ? ??.
3

A. ???? ?? ???? ??? ??? ????.
https://youtu.be/qybisPmc2xw?t=67
??? ?? ?? ???? ???? ???? ???? ??? ??( Scattering ) ??
??.
?? ?? ??? ? ? ?? ?? ?? ????.
?? ???? ??? ?? ???? ????? ?? ??? ?? ???? ??.
?? ??( Scattering )
4

??? ??? ???? ?? ?????.
??? ??? ??? ?? ??? ???? ??.
??? ?? ??? ??? ?? ???? ????? ???? ??? ??? ? ??
??? ?? ??? ??? ????.
??( Atmosphere )
?? ??
?? (?2) 78.084%
?? (?2) 20.946%
??? (??) 0.9340%
????? (??2) 0.04%
?? (??) 0.001818%
?? (??) 0.000524%
?? (??4) 0.000179%
??? (?2 ?) 0.001% ~ 5%
??
5

??? ??? ??? ??? ??? ?? 2?? ???? ??? ? ??.
???? ??? ??? ????? ???? ?? ????. ( Molecules )
??? ??? ??(?? ??, ??)? ???? ?? ???? ?? ??? ?? ???
? ? ?? ??? ??? ??( Rayleigh Scattering )?? ??.
??? ??? ?? ??? ??? ?? ??? ?? ??? 4 ??? ????? ????
?? ?? ????.
?, ??? ? ?? ??? ? ???? ??? ???.
?? ??? ????? ??? ??? ??? ????? ????? ? ?? ??? ??
?? ??? ? ???? ??.
?? ??( Atmospheric Scattering )
6

??? ??? ??? ??? ??? ??? ?? ??? ?? ? ?? ???? ???
???? ????.
???? ??? ?? ??? ??? ?? ???? ??? ??? ???? ???? ??
???? ??? ???? ?? ??? ?? ???? ??? ???? ??? ?? ??
??? ???? ????.
7

???? ??? ??? ????? ??? ??? ????. ( Aerosol )
???? ??, ?? ??? ?? ??? ??? ?? ??? ???? ? ? ?? ???
? ??( Mie Scattering ) ?? ??.
? ??? ?? ?? ????? ??? ??, ??, ??? ??? ???.
??? ??? ? ??? ?? ??? ? ??? ??? ?? ??? ?? ??? ???
?? ???. ( ?? ? ??? ???? ?? ?? ??? ?? ??? ?? )
????? ?? ??? ??? ??? ???? ???? ??? ???? ?? ???
?? ???. ??? ??? ?????? ??? ?? ? ? ?? ?? ???.
8

??? ??? ?? ?? ???.
?? ??? ??? ?? ? ??.
?? ??? ??? ??? ?? ???.
Out-Scattering
In-Scattering
??? ?
Absorption
9

???? ?? ?? ? ? ?? ?? ???.
??? ??? ??? ???? ?? ??? ??. ( Out-Scattering, Absorption )
???? ?? ??? ???? ??? ???? ??? ?? ?? ???? ????.
Transmittance ( ??? )
A
B
?? : A ???? ?? ?
? ? : B ???? ?? ?
Transmittance �� ?(??) =
? ?
? ? 10

T x, x0 = ?? ????
?0 �� ?��{?,?} ??
?
? ??
??
?
= extinction coefficient (?? ??)
: ?? 1m? ???? ?? ??? ?? ????? ??? ???? ?? (= ?? )
?, ?? ??? ??? ????? ????1 ????2 ??? ???? ???? ??? ??
?, ?0 = ?? ?? ??
? = Rayleigh (???)
? = Mie (?)
y = ??
2. Absorption(??)
1. Scattering(??)
??
?
= ??
?
+ ??
?
11

?? ??( ??
?
)? ?? ???( scattering equation )?? ????.
??? ?? ??? ?(?, ?, ?)? ??? ??.
? : ??.
? : ?? ??.
? : ??.
? = 1.00029 : ??? ???.
? = 2.504 ? 1025 : ?? ??? ?? ?? ??. ( = ?3 ? ??? ? )
? ? : ?? ?.( density ratio ) ?????? 1?? ??? ??? ?? ?????
?? ????.
�� ??? ???? ??? ?? ???? Display of The Earth Taking into Account Atmospheric Scattering ???? ??? ?
??. ?????? ??? ? ? ?? ???.
? ?, ?, ? =
?2 ?2?1
2
2
? ?
?
1
?4 (1 + cos2 ?)
Scattering Equation ( ?? ??? )
12

??? ?? ??? ?(?, ?, ?)? ?? ??(?)?? ??? ?? ?? ????? ????.
??? ???? ?? ???? ??? ???? ?? ??? ??? ??? ??.
? ?, ? = ?
?
? ?, ?, ? ??
= ?
0
2?
?
0
?
? ?, ?, ? ????????
= ?
0
2?
?
0
?
?2 ?2 ? 1 2
2
? ?
?
1
?4 (1 + cos2 ?)????????
=
?2 ?2 ? 1 2
2
? ?
?
1
?4 ?
0
2?
?
0
?
(1 + cos2 ?)????????
=
?2 ?2 ? 1 2
2
? ?
?
1
?4 ?
0
2?
8
3
??
=
?2 ?2 ? 1 2
2
? ?
?
1
?4
16?
3
=
8?3 ?2 ? 1 2
3
? ?
?
1
?4 =
8?3 ?2 ? 1 2
3??4 ? ?
Scattering Equation ( ?? ??? )
13

??? ??? ? ?, ? ? ??? ?? ??( Rayleigh Scattering Coefficient )?
?? ? ?
?
? ????.
??? ??? ??? ?? ????? ? ?? ???? ??? ? ?
?
= ? ?
?
? ??.
? ??? ?? ?? ?? ? ?
?
? ? ?
?
?, ? = ? ?
?
(0, ?)? ? ? ??? ??? ???? ??
?? ?? ? ?
?
? ?? ??(? ?
?
)? ?? ??(? ?
?
)? ??? ?? ? ??.
???? ? ?? ? ?? ?? ?? ??? ??? ?? ?(? ? )? ????? ???
??? ???.
? ?? ?? ?? ?????? ??? ? ??.
? ?
?
= ? ?
?
+ ? ?
?
? ?
?
(?) =
8?3 ?2 ? 1 2
3??4 ? ?
? ?
?
(?) = ? ?
?
(0, ?)? ?
Scattering Coefficient ( ?? ?? )
14

??? ??? ?? ??? ????. ?? ?? ? ??? ??? ???? ??? ???
?? ??? ? ??.
?? ? ? ? ? ?? ???? ??? ?? 0??? ??? ??? ??? ?? ????.
??? ??? ?? ???? ??, ??, ??? ?? ?? ??? ???? ???
1976??? ???? ? ?? ?? ??? ??? ?? ??? ???? ??? ??.
Density ratio ( ?? ? )
? ? =
??????? ?
???????(0)
??(?) ??(??/?3)
0 1.2250
500 1.1673
1000 1.1117
1500 1.0581
2000 1.0066
2500 0.95695
3000 0.90912
3500 0.86323
4000 0.81913
4500 0.77677
?? : https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770009539.pdf
15

? ?? ?? ?? ?? ???? ??? 0 ? ?? 1, ??? 4000? ?? 0.66868? ?
?.
?? ??? ??? ?? ?? ?? ?? ???? ?? ?? ???? ?? ??? ???
? ???? ??.
??? ?? ??? ???? ???? ??? ?? ????? ???? ?? ?? ? ?
??.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5000 10000 15000 20000 25000 30000 35000
16

??? ???? ?? ??? ??? ??. ?? ?? ???? ??? ??.
? : ?? ??( scale height )
?? ??? ?? ?? ????( e = 2.71828 ) ? ??? ????? ???? ???
???, ? ??? ?? ?? ?? ????.
? ? = e?
?
?
?? ?? ?? ??(??)
Rayleigh 8.5
Mie 1.2
0
0.2
0.4
0.6
0.8
1
1.2
0 5000 10000 15000 20000 25000 30000 35000
??? ????? ?? ?
?? ?? ?
?? ?? ?
17

?? ???? ???? ?? ?? ????
???? ???? ?? ?? ?? ??
?
? ??? ?? ??? ? ? ??.
???? ?? ??? ?? ????
???? ?? ?? ?? ( x ~ x0 ) ? ????? ??? ?? ???? ???? ?? ?
? ?? ????. ??? ??? ?? ??? ?0, ? ?? ?? ? ?? ??? ?1?? ??
??.
? ? ???? ?? ??? ? ?? ??? ?????
T x, x0 = ?? ????
?0 �� ?��{?,?} ??
?
? ??
?1 = ?0(1 ? ? ?)
?1 = ?0(1 ?
? ?
2
)
?2 = ?1 1 ?
? ?
2
= ?0 1 ?
? ?
2
2
18

??? ???? ??? ?????
? ?? ?? ??? ?? 1m ? ????? ??? ????? ??? ??( ? )? ??
???? ??? ?? ????.
??? ?? ??? ?? ??? ???? ??? ? ? ???
? ? ?? ??? ??( Optical depth ) ? ??.
? = ?0 lim
?��
1 ?
? ?
?
?
?
?0
= ? = ??? ?
? = ?0 ??? ?
�� lim
?��
1 +
1
?
?
= e
? = ??? ? ?
? = ??? ????????
? ? ? ?
19

? = ???��
(�� : ??? ?????? ???( zenith angle ))
? �� ??
r
?
Transmittance Table
20

Compute Shader? ???? Transmittance Table? ?????. cs_5_0? ???
??( Maximum Threads = 1024 ) ??? ???? ??? Dispatch ???.
??? ?? ?? T x, x0 = ?? ????
?0 �� ?��{?,?} ??
?
? ??
? ??? ???? ?? ? ? ??.
GetTransmittanceRMu ??? ??? UV? ?? ??(R)? ??? ???(Mu)? ??
?.
Transmittance Table
[numthreads(TRANSMITTANCE_W / TRANSMITTANCE_GROUP_X, TRANSMITTANCE_H / TRANSMITTANCE_GROUP_Y, 1)]
void main( uint3 DTid : SV_DispatchThreadID )
{
float r, muS;
GetTransmittanceRMu( DTid, r, muS );
float3 depth = betaR * OpticalDepth( HR, r, muS ) + betaMEx * OpticalDepth( HM, r, muS );
transmittanceBuffer[DTid.xy] = float4( exp( -depth ), 0.f );
}
void GetTransmittanceRMu( uint3 DTid, out float r, out float muS )
{
r = DTid.y / float( TRANSMITTANCE_H );
muS = DTid.x / float( TRANSMITTANCE_W );
#ifdef TRANSMITTANCE_NON_LINEAR
r = Rg + ( r * r ) * ( Rt - Rg );
muS = -0.15f + tan( 1.5f * muS ) / tan( 1.5f ) * ( 1.f + 0.15f );
#else
r = Rg + r * ( Rt - Rg );
muS = -0.15f + muS * ( 1.f + 0.15f ); // 0 ~ ( pi/2 + epsilon )
#endif
}
21

OpticalDepth? ??? ??? ????.
?? limit ??? ??? ??? ???? ??? ??? ??? ???? ???
????? ???.
Transmittance Table
float OpticalDepth( float H, float r, float mu )
{
float result = 0.f;
float dx = limit( r, mu ) / float( TRANSMITTANCE_INTEGRAL_SAMPLES );
float xi = 0.f;
float yi = exp( -( r - Rg ) / H );
for ( int i = 1; i <= TRANSMITTANCE_INTEGRAL_SAMPLES; ++i )
{
float xj = float( i ) * dx;
float yj = exp( -( sqrt( r * r + xj * xj + 2.f * xj * r * mu ) - Rg ) / H ); // law of cosines
result += ( yi + yj ) * 0.5f * dx;
xi = xj;
yi = yj;
}
return mu < -sqrt( 1.f - ( Rg / r ) * ( Rg / r ) ) ? 1e9f : result;
}
t
??? ???? ?? ?? ???? ???? ?? 22

?? ??? ???? ??? ???? ??? ? ??.
Transmittance Table
float limit( float r, float mu )
{
float dout = -r * mu + sqrt( r * r * ( mu * mu - 1.f ) + RL * RL );
float delta2 = r * r * ( mu * mu - 1.f ) + Rg * Rg;
if ( delta2 >= 0.f )
{
float din = -r * mu - sqrt( delta2 );
if ( din > 0.f )
{
dout = min( dout, din );
}
}
return dout;
}
float dout = -r * mu + sqrt( r * r * ( mu * mu - 1.f ) + RL * RL );
23

????? ?? ???? ????? ? ? ? ? ? ??? ?? ???? ?? ???? ??
?? ??? ?? ??? ??? ????.
??? ??? ??? ?? ??? ??? ? 2 ??? ??(?2 = ?2 + ?2 ? 2??????)?
?? ??? ??.
Transmittance Table
a
r * r + xj * xj + 2.f * xj * r * mu
24

?? ?? ?? ?? ??( Irradiance )? ???? ??? ????.
?? ??? ?? ??( ?? )? ??? ?? ?? ?( ? �� ??????? ???? ) ? ??? ???
??? ? =
?
??
? ??? ? ??.
? ???? ??? ?? ???? ???? ?? ??( ? : radiance )?? ??? ???
??? ?? ??? ?? ???? ?? ???? ??? ?, ? ???? ??? ?? ??
????, ?? ?? ? ? ???? ?? ?? ???.
?? ??? L =
��
????????
? ??? ? ?? ?? ??? ?? ??? ??? ? ??.
Irradiance
E = ?
��
???????
??
??
25

?? ?? ?? ?? ??? ??? ? ???? ???? ??? ?? ???? ????
??? ?? ???? ??? ? ??.
????? ? ???? ??? E = ?(?, ?0) ???? ? ???? ????.
Single Irradiance Table
E = ?(?, ?0)? ??? ????
[numthreads( IRRADIANCE_W / IRRADIANCE_GROUP_X, IRRADIANCE_H / IRRADIANCE_GROUP_Y, 1)]
{
float r, muS;
GetIrradianceRMuS( DTid, r, muS );
irradiance1Buffer[DTid.xy] = float4( Transmittance( r, muS ) * max( muS, 0.f ), 0.f );
}
? = ???��
(�� : ??? ????? ???( zenith angle ))
? �� ??
26

?? ??? ?? ??? ????? ??. ???? ???, ?? ?? ???? ?? ??
?? 3D ???? ????.
??? 3D ???? 4?? ??? LUT( Look Up Table )? ???? ?? ???
??? ??.
Single Inscatter Table
? �� ??
? = ???��
(�� : ??? ??????
???( zenith angle ))
? ? = ???�� ??? (�� ??? : ??? ???( zenith angle ))
? = ???? (? : ??? ??? ?? ?)
Rayleigh Inscatter Table? ?? ??? ??
27

?? ???? ?? ?? ? ?, ? ? ??? ??? ?? ??? ? ??.
? ? ? ?? ??? ?(?, ?, ?)? ?? ?? ? ?, ? ? ??? ??? ?? ? ??.
??? ?? ???? ?? ??? ?? ? ? ? ????
??? ???? ??? ?? ???? ?? ? ? ??.
? ? ? ?? ?? ?? ?? ??? ???? ?? ??( Phase Function )? ???.
Phase Function
?(?, ?, ?) = ? ?, ? ?(?)
? ? =
? ?, ?, ?
? ?, ?
? ? =
?2 ?2 ? 1 2
2
? ?
?
1
?4 1 + cos2 ?
3??4
8?3 ?2 ? 1 2 ? ?
=
3
16?
1 + cos2 ?
? ?, ?, ?
1
? ?,?
28

???, ? ?? ???? ?? ???? ??? ? ? ??.
?? ???? Inscatter ???? ????.
[numthreads( RES_MU_S, RES_MU / INSCATTER1_GROUP_Y, 1)]
{
float r;
float4 dhdH;
GetRdhdH( DTid.z, r, dhdH );
float mu, muS, nu;
GetMuMuSNu( DTid, r, dhdH, mu, muS, nu );
float3 ray, mie;
Inscatter( r, mu, muS, nu, ray, mie );
deltaSRBuffer[DTid] = float4( ray, 0.f );
deltaSMBuffer[DTid] = float4( mie, 0.f );
}
void Inscatter( float r, float mu, float muS, float nu, out float3 ray, out float3 mie )
{
ray = 0.f;
mie = 0.f;
float dx = limit( r, mu ) / float( INSCATTER_INTEGRAL_SAMPLES );
float3 rayi;
float3 miei;
Integrand( r, mu, muS, nu, 0.f, rayi, miei );
for ( int i = 1; i <= INSCATTER_INTEGRAL_SAMPLES; ++i )
{
float3 rayj;
float3 miej;
Integrand( r, mu, muS, nu, xj, rayj, miej );
ray += ( rayi + rayj ) * 0.5f * dx;
mie += ( miei + miej ) * 0.5f * dx;
rayi = rayj;
miei = miej;
}
ray *= betaR;
mie *= betaMSca;
}
?? ??? ????? ???? ????.
�� ?? ??? ???? ???
????? ???? ????. 29

??? ?? ?? ??? ??? ??? ?? ??? ?? ???? ??? ???? ??.
void Integrand( float r, float mu, float muS, float nu, float t, out float3 ray, out float3 mie )
{
ray = 0.f;
mie = 0.f;
float ri = sqrt( r * r + t * t + 2.f * r * mu * t );
float muSi = ( nu * t + muS * r ) / ri;
ri = max( Rg, ri );
// ???? ??? ??? ???? ??? ??? ??? ??? ??? ??
// = ??? ??? ???? ?? ???? ?? 0
if ( muSi >= -sqrt( 1.f - Rg * Rg / ( ri * ri ) ) )
{
float3 ti = Transmittance( r, mu, t ) * Transmittance( ri, muSi );
ray = exp( -( ri - Rg ) / HR ) * ti;
mie = exp( -( ri - Rg ) / HM ) * ti;
}
}
t
30

???? ???? ??? ?? ??, ??? ?????. ??? ???? ??? ???
? ???? ??? ????? ??????? ??.
?? ?? ????? ???? ?? ????.
? ?[? ???] : ???? ??? ??? ?? ( ?? ?? ?? X )
? ?? : ?? ?? ( ???, ? ?? ?? ?? ?? �� )
?? : ??? ?? ??
? ?,? : ? ??? red ?? ? ( ?? ?? ?? X )
Multiple Inscatter Table
R G B A
?? = ? ? ? ??? +
? ??
??
? ?,?
[numthreads( RES_MU_S, RES_MU / INSCATTER1_GROUP_Y, 1)]
{
float3 uvw = ( DTid.xyz + 0.5f ) / float3( RES_MU_S * RES_NU, RES_MU, RES_R );
float4 ray = deltaSRTex.SampleLevel( deltaSRSampler, uvw, 0 );
float4 mie = deltaSMTex.SampleLevel( deltaSMSampler, uvw, 0 );
float pitch = RES_MU_S * RES_NU;
float slicePitch = pitch * RES_MU;
inscatterBuffer[( DTid.z * slicePitch ) + ( DTid.y * pitch ) + DTid.x] = float4( ray.rgb, mie.r );
}
? ? ? ??? ? ? ?,?? ?? ???? ??? ???.
? ??
? ?
?
? ?? ???? ??? ???.
31

??? ???? ??? LUT? ?? ???? ?? ??? ?? ???? ??? ? ??
??? ? ??? ??? ??? ??? ?? ????.
??? ??? ?? ? ??? red ?? ?? ???? ???? ?? ??? ?? ???
?? ????.
?? ??? ? ??? ??? ????? ?? ?? ?? y?? ?? ?? v? ????
?? ?? ????.
? ? ? ??
? ?,?
??,?
? ?,?
?
? ?,?
?
? ?
?
? ?
?
[numthreads( RES_MU_S * RES_NU / INSCATTERS_GROUP_X, RES_MU / INSCATTERS_GROUP_Y, 1 )]
{
float r;
float4 dhdH;
GetRdhdH( g_threadGroupZ, r, dhdH );
float mu, muS, nu;
GetMuMuSNu( float3( DTid.xy, g_threadGroupZ ), r, dhdH, mu, muS, nu );
float3 raymie;
Inscatter( r, mu, muS, nu, raymie );
deltaJBuffer[DTid] = float4( raymie, 0.f );
}
y
32

?? ?? y? ???? ?? ?? ???? ??? Inscatter ??? ? ??? ?? ??
? ???? ??.
??? ?? ??? ?? ???? ??? ????. ?? ???? ?? ???? ???.
?? ?? ?? ??? ???? ???? ? ??? ???? ????.
void Inscatter( float r, float mu, float muS, float nu, out float3 raymie )
{
r = clamp( r, Rg, Rt );
mu = clamp( mu, -1.f, 1.f );
muS = clamp( muS, -1.f, 1.f );
// Using formula sum and differences of cosine
float var = sqrt( 1.f - mu * mu ) * sqrt( 1.f - muS * muS );
nu = clamp( nu, mu * muS - var, mu * muS + var );
// cos( theta to the horizon )
float cthetamin = -sqrt( 1.f - ( Rg * Rg ) / ( r * r ) );
float3 v = float3( sqrt( 1.0 - mu * mu ), 0.f, mu );
float sx = v.x == 0.f ? 0.f : ( nu - muS * mu ) / v.x;
float3 s = float3( sx, sqrt( max( 0.f, 1.f - sx * sx - muS * muS ) ), muS );
raymie = 0.f;
cthetamin
33

???? ?? ????? ?? ?? ?? ???? ? ??? ?? ??? ???? ???
?? ???? ???? ??? ??? ?? ??? ?(greflectance)? ???
(gtransp)? ??? ??.
// Compute equation(7)
// ?(0 ~ ?)? ?? ??
for ( int itheta = 0; itheta < INSCATTER_SPHERICAL_INTEGRAL_SAMPLES; ++itheta )
{
float theta = ( float( itheta ) + 0.5f ) * dtheta;
float ctheta = cos( theta );
float greflectance = 0.f;
float dground = 0.f;
float3 gtransp = 0.f;
// if ground visible in direction w compute transparency gtransp between x and ground
if ( ctheta < cthetamin )
{
greflectance = AVERAGE_GROUND_REFLECTANCE / M_PI;
dground = -r * ctheta - sqrt( r * r * ( ctheta * ctheta - 1.f ) + Rg * Rg );
gtransp = Transmittance( Rg, -(r * ctheta + dground) / Rg , dground );
}
34

// ?(0 ~ 2?)? ?? ??
for ( int iphi = 0; iphi < 2 * INSCATTER_SPHERICAL_INTEGRAL_SAMPLES; ++iphi )
{
float phi = ( float( iphi ) + 0.5f ) * dphi;
float dw = dtheta * dphi * sin( theta );
float3 w = float3( cos( phi ) * sin( theta ), sin( phi ) * sin( theta ), ctheta );
float nu1 = dot( s, w );
float nu2 = dot( v, w );
float pr2 = PhaseFunctionR( nu2 );
float pm2 = PhaseFunctionM( nu2 );
float3 gnormal = ( float3( 0.f, 0.f, r ) + dground * w ) / Rg;
float3 girradiance = Irradiance( Rg, dot( gnormal, s ) );
// first term = light reflected from the ground and attenuated before reaching x
float3 raymie1 = greflectance * girradiance * gtransp;
???? ??? ?? ?? ??? ???? ????.
?? ???? ? ?? ??? ?? ?? ??? ? ? ??.
??? ???? ??? ?? ??.
? ? = ? ? + ?
��
? ??, ? ? ?? ? ? ? ??
? ? = ?
��
? ??, ? ? ?? ? ? ? ?? = ? ??, ? ? ?
��
?? ? ? ? ??
? ? = ? ??, ? ? ?
???? ???? ? ??. Lambertian BRDF ??? ?? ??? ? ? ??.
35

????? ?? ? ? ??? ??? ( 1. ??? ?? ???? ?? ?? y? ????
2. ?? ?? y?? ?? ?? v? ???? ) ?? ???? 2? ????.
?? ??? ???? ??? ?? ??? ????. ??? ? ??? ??? ?? ??
????.
g_order? ???? ?? ?? ??? ?? ??? ???? ???? ?? ??? ??
?? ??? ?? ??? ??? ?? ????.
// second term = inscattered light = deltaS
// ???? ??? ?? ??? ??? ??? 1??? ??? ??? ?? ??? ?? ??? ?? 2? ??.
if ( g_order == 2 )
{
// ?? ?? ?? ???? ?? ??? ???? ????. ??? ? ?? ??? ???? ????? ??? ???? ?????.
float pr1 = PhaseFunctionR( nu1 );
float pm1 = PhaseFunctionM( nu1 );
float3 ray1 = Sample4D( deltaSRTex, deltaSRSampler, r, w.z, muS, nu1 ).rgb;
float3 mie1 = Sample4D( deltaSMTex, deltaSMSampler, r, w.z, muS, nu1 ).rgb;
raymie1 += ray1 * pr1 + mie1 * pm1;
}
else
{
// ?? ??? ?? ?? ??? ?? ??? ?? ????.
raymie1 += Sample4D( deltaSRTex, deltaSRSampler, r, w.z, muS, nu1 ).rgb;
}
// light coming from direction w and scattered in direction v
// Equation 7
raymie += raymie1 * ( betaR * exp( -( r - Rg ) / HR ) * pr2 + betaMSca * exp( -( r - Rg ) / HM ) * pm2 ) * dw;
}
}
}
1
2
36

?? ?? ??? ?? ??? ??? ? ???? ??? ?? ??? ?? ?? ???
?? ??? ???? ???.
?? ?? ??? ??? ?? ???? ????? ??? ???? ??.
y
[numthreads( RES_MU_S, RES_MU / INSCATTERN_GROUP_Y, 1 )]
{
float r;
float4 dhdH;
float mu, muS, nu;
deltaSRBuffer[DTid] = float4( Inscatter( r, mu, muS, nu ), 0.f );
}
37

??? ???? ???? ?? ???? ???? ?? ???? ???? ????.
float3 Integrand( float r, float mu, float muS, float nu, float t )
{
float ri = sqrt( r * r + t * t + 2.f * r * t * mu );
float mui = ( r * mu + t ) / ri;
float muSi = ( nu * t + muS * r ) / ri;
return Sample4D( deltaJTex, deltaJSampler, ri, mui, muSi, nu ).rgb * Transmittance( r, mu, t );
}
float3 Inscatter( float r, float mu, float muS, float nu )
{
float3 raymie = 0.f;
float dx = limit( r, mu ) / float( INSCATTER_INTEGRAL_SAMPLES );
float xi = 0.f;
float3 raymiei = Integrand( r, mu, muS, nu, 0.f );
for ( int i = 0; i < INSCATTER_INTEGRAL_SAMPLES; ++i )
{
float3 raymiej = Integrand( r, mu, muS, nu, xj );
raymie += ( raymiei + raymiej ) * 0.5f * dx;
xi = xj;
raymiei = raymiej;
}
return raymie;
}
38

?? ?? ????? 31? ?????? ???? ? ?? ??? ?? ??? ?? ??
????.
???? ???? 2? ??? ???? ? ?? ??? ??? ?? ??. ? ? ????
??? ?? ??? ? ??? ?? ?? ????.
[numthreads( RES_MU_S, RES_MU / INSCATTERN_GROUP_Y, 1 )]
{
float r;
float4 dhdH;
float mu, muS, nu;
float3 uvw = ( DTid.xyz + 0.5f ) / float3( RES_MU_S * RES_NU, RES_MU, RES_R );
float pitch = RES_MU_S * RES_NU;
float slicePitch = pitch * RES_MU;
inscatterBuffer[( DTid.z * slicePitch ) + ( DTid.y * pitch ) + DTid.x] +=
float4( deltaSRTex.SampleLevel( deltaSRSampler, uvw, 0 ).rgb / PhaseFunctionR( nu ), 0.f );
}
39

// 4?? ?? ??? ?????
for ( int order = 2; order <= 4; ++order )
{
int* gpuScatteringOrder = static_cast<int*>( renderer.LockBuffer( precomputeBuf ) );
gpuScatteringOrder[0] = order;
renderer.UnLockBuffer( precomputeBuf );
renderer.BindConstantBuffer( SHADER_TYPE::CS, 0, 1, &precomputeBuf );
// Compute deltaJ
renderer.BindShader( SHADER_TYPE::CS, csInscatterS );
renderer.BindShaderResource( SHADER_TYPE::CS, 1, 1, &deltaESrv );
renderer.BindRandomAccessResource( 0, 1, &deltaJRav );
// ??? ??? ?? ????? ???? Device Lost? ??. ??? ??? Dispatch.
for ( int i = 0; i < INSCATTERS_GROUP_Z; ++i )
{
gpuScatteringOrder = static_cast<int*>( renderer.LockBuffer( precomputeBuf ) );
gpuScatteringOrder[0] = order;
gpuScatteringOrder[1] = i;
renderer.UnLockBuffer( precomputeBuf );
renderer.BindConstantBuffer( SHADER_TYPE::CS, 0, 1, &precomputeBuf );
renderer.Dispatch( INSCATTERS_GROUP_X, INSCATTERS_GROUP_Y );
renderer.WaitGPU( );
}
��
// Compute deltaS
renderer.BindShader( SHADER_TYPE::CS, csInscatterN );
renderer.BindShaderResource( SHADER_TYPE::CS, 4, 1, &deltaJSrv );
renderer.BindShaderResource( SHADER_TYPE::CS, 2, 1, nullptr );
renderer.BindRandomAccessResource( 0, 1, &deltaSRRav );
renderer.Dispatch( INSCATTERN_GROUP_X, INSCATTERN_GROUP_Y, INSCATTERN_GROUP_Z );
��
} 40

?? Inscatter Table
? �� ??
? = ???��
(�� : ??? ??????
???( zenith angle ))
? ? = ???�� ??? (�� ??? : ??? ???( zenith angle ))
? = ???? (? : ??? ??? ?? ?)
41

????? ?? ??? ?? ?? ??? ???? ??.
Multiple Irrdiance Table
[numthreads( IRRADIANCE_W / IRRADIANCE_GROUP_X, IRRADIANCE_H / IRRADIANCE_GROUP_Y, 1 )]
{
float r, muS;
GetIrradianceRMuS( DTid, r, muS );
float3 s = float3( max( sqrt( 1.0 - muS * muS ), 0.f ), 0.f, muS );
float3 result = 0.f;
// ????? ??? ??
for ( int iphi = 0; iphi < 2 * IRRADIANCE_INTEGRAL_SAMPLES; ++iphi )
{
float phi = ( float( iphi ) + 0.5f ) * dphi;
for ( int itheta = 0; itheta < IRRADIANCE_INTEGRAL_SAMPLES / 2; ++itheta )
{
float theta = ( float( itheta ) + 0.5f ) * dtheta;
float dw = sin( theta ) * dtheta * dphi;
float3 w = float3( cos( phi ) * sin( theta ), sin( phi ) * sin( theta ), cos( theta ) );
float nu = dot( s, w );
// ?? ?? ??? ????? ?????? ?? ??? ????? ???? ??? ??.
if ( g_order == 2 )
{
float pr = PhaseFunctionR( nu );
float pm = PhaseFunctionM( nu );
float3 ray = Sample4D( deltaSRTex, deltaSRSampler, r, w.z, muS, nu ).rgb;
float3 mie = Sample4D( deltaSMTex, deltaSMSampler, r, w.z, muS, nu ).rgb;
result += ( ray * pr + mie * pm ) * w.z * dw;
}
else
{
result += Sample4D( deltaSRTex, deltaSRSampler, r, w.z, muS, nu ).rgb * w.z * dw;
}
}
}
irradianceBuffer[DTid.xy] = float4( result, 0.f );
} 42

?? Irradiance Table
Multiple Irradiance Table
? = ???��
(�� : ??? ????? ???( zenith angle ))
? �� ??
43

?? ???? ??? ?? ?, ?? ?, ??? ?? ?? ??? ?? ??? ????.
Rendering
VS_OUTPUT main( uint vertexID : SV_VertexID )
{
VS_OUTPUT output = (VS_OUTPUT)0;
float2 uv = float2( ( vertexID & 1 ) << 1, vertexID & 2 );
float4 pos = float4( float2( -1.f, 1.f ) + uv * float2( 2.f, -2.f ), 0.f, 1.f );
output.position = pos;
float4x4 invVewProjection = mul( g_invProjectionMatrix, g_invViewMatrix );
float4 worldPos = mul( float4( pos.xy, 1.f, 1.f ), invVewProjection );
output.worldPos = worldPos.xyz / worldPos.w;
return output;
}
float4 main( PS_INPUT input ) : SV_Target
{
float3 viewRay = normalize( input.wolrdPos - g_cameraPos.xyz );
float scale = 0.00001f; // ?? ??? ??? ?? ??? ?
float3 viewPos = g_cameraPos.xyz * scale;
viewPos.y += Rg + HeightOffset;
float r = length( viewPos );
float mu = dot( viewPos, viewRay ) / r;
float t = -r * mu - sqrt( r * r * ( mu * mu - 1.f ) + Rg * Rg );
float3 attenuation;
float3 inscatterColor = InscatterColor( viewPos, t, viewRay, r, mu, attenuation );
float3 groundColor = GroundColor( viewPos, t, viewRay, r, mu, attenuation );
float3 sunColor = SunColor( viewPos, t, viewRay, r, mu );
return float4( HDR( sunColor + inscatterColor + groundColor ), 1.f );
}
44

Rendering - Inscatter Color
float3 InscatterColor( float3 x, float t, float3 viewRay, float r, float mu, out float3 attenuation )
{
attenuation = 1.f;
// ?? ??? ?? ?? ? ??? ???? ??? ??
float d = -r * mu - sqrt( r * r * ( mu * mu - 1.f ) + Rt * Rt );
if ( d > 0.f )
{
x += d * viewRay;
t -= d;
mu = ( r * mu + d ) / Rt;
r = Rt;
}
static const float EPS = 0.005f;
// ?? ??? ??? ???? ??? ??
if ( r < Rg + HeightOffset + EPS )
{
float diff = ( Rg + HeightOffset + EPS ) - r;
x -= diff * viewRay;
t -= diff;
mu = dot( x, viewRay ) / r;
r = Rg + HeightOffset + EPS;
}
if ( r <= Rt )
{
float nu = dot( viewRay, g_sunDir.xyz );
float muS = dot( x, g_sunDir.xyz ) / r;
float4 inscatter = max( Sample4D( inscatterTex, inscatterSampler, r, mu, muS, nu ), 0.f );
if ( t > 0.f )
{
float3 x0 = x + t * viewRay;
float altitude0 = length( x0 );
float mu0 = dot( x0, viewRay ) / altitude0;
float muS0 = dot( x0, g_sunDir.xyz ) / altitude0;
// ??? ??? ??? ??? ???? ????? ??
attenuation = AnalyticTransmittance( r, mu, t );
45

if ( altitude0 > Rg + HeightOffset )
{
// ??? ??? ??? ??? ??
inscatter = max( inscatter - attenuation.rgbr * Sample4D( inscatterTex, inscatterSampler, altitude0,
mu0, muS0, nu ), 0.f );
// ???? ??? ? ???? ??? ??? ???? ??? ???? ? ??? ?? ??? ??
float muHoriz = -sqrt( 1.f - ( Rg / r ) * ( Rg / r ) );
if ( abs( mu - muHoriz ) < EPS )
{
float a = ( ( mu - muHoriz ) + EPS ) / ( 2.f * EPS );
mu = muHoriz - EPS;
altitude0 = sqrt( r * r + t * t + 2.f * r * t * mu );
mu0 = ( r * mu + t ) / altitude0;
float4 inscatter0 = Sample4D( inscatterTex, inscatterSampler, r, mu, muS, nu );
float4 inscatter1 = Sample4D( inscatterTex, inscatterSampler, altitude0, mu0, muS0, nu );
float4 inscatterA = max( inscatter0 - attenuation.rgbr * inscatter1, 0.f );
mu = muHoriz + EPS;
altitude0 = sqrt( r * r + t * t + 2.f * r * t * mu );
mu0 = ( r * mu + t ) / altitude0;
inscatter0 = Sample4D( inscatterTex, inscatterSampler, r, mu, muS, nu );
inscatter1 = Sample4D( inscatterTex, inscatterSampler, altitude0, mu0, muS0, nu );
float4 inscatterB = max( inscatter0 - attenuation.rgbr * inscatter1, 0.f );
inscatter = lerp( inscatterA, inscatterB, a );
}
}
}
inscatter.w *= smoothstep( 0.f, 0.02f, muS );
float phaseR = PhaseFunctionR( nu );
float phaseM = PhaseFunctionM( nu );
// Mie ??? ??
result = max( inscatter.rgb * phaseR + getMie( inscatter ) * phaseM, 0.f );
}
return result * ISun;
} 46

Transmittance Table? ????? Inscatter Table? ?? ???? ??? ??
???? ?? ? ?? ?? ???? ???? ?? ?? ?? ???? ?? ????
??? ???.
?? ???? ??? ?? ???? ?? ???? ??? ??? ??? ?? ???
??.
? ??? a ~ c ?? ?? ???? S a �� c = ????
?
??[?0]? ?? ?? a ~ b? b ~ c ??
? ?? ????? ???? ??? ?? ??? ? ??.
a
b
c
?
?
?
?? ?0 = ?
?
?
??[?0] + ?
?
?
??[?0]
47

?? a ~ b ??? ??? ????
? ?? ?????? ??? ?? ??? ? ??.
????? ??? ??.
32? ppt?? ???? Mie ??? ??? getMie ???? ????.
getMie ??? ? ? ? ??
? ?,?
??,?
? ?,?
?
? ?,?
?
? ?
?
? ?
? ? ??? ?? ??? ???? ?????.
?
?
?
??[?0] = ?
?
?
?? ?0 ? ?
?
?
??[?0]
S a �� ? = S a �� c ? ?(? �� ?)S b �� c �� ? = ? ? �� ? = ?(? �� ?)?(? �� ?)
inscatter = max( inscatter - attenuation.rgbr * Sample4D( inscatterTex, inscatterSampler, altitude0, mu0, muS0, nu ), 0.f );
float4 inscatterA = max( inscatter0 - attenuation.rgbr * inscatter1, 0.f );
float4 inscatterB = max( inscatter0 - attenuation.rgbr * inscatter1, 0.f );
? ? ? ??
? ?,?
??,?
? ?,?
?
? ?
? (�� ? ?,?
?
= ? ?,?
?
= ? ?,?
?
)
return rayMie.rgb * rayMie.w / max( rayMie.r, 1e-4 ) * ( betaR.r / betaR ); 48

Rendering �C Ground Color
float3 GroundColor( float3 x, float t, float3 viewRay, float r, float mu, float3 attenuation )
{
if ( t > 0.f )
{
float3 x0 = x + t * viewRay;
float altitude0 = length( x0 );
float3 n = x0 / altitude0;
float muS = dot( n, g_sunDir.xyz );
// ???? ???? ??? ???? 0??.
float3 sunLight = TransmittanceWithShadow( altitude0, muS );
float3 groundSkyLight = Irradiance( altitude0, muS );
float3 sceneColor = float3( 0.35f, 0.35f, 0.35f );
float3 reflectance = sceneColor * float3( 0.2f, 0.2f, 0.2f );
if ( altitude0 > Rg + HeightOffset )
{
reflectance = float3( 0.4f, 0.4f, 0.4f );
}
// ??? ??? ??
float3 groundColor = reflectance * ISun / M_PI * ( max( muS, 0.f ) ) * ( sunLight + groundSkyLight );
result = attenuation * groundColor;
}
return result;
}
?? ?? ??? ??? ????. ?????? ???? ?? ?? ?? ?? ????.
49

Rendering �C Sun Color
float3 SunColor( float3 x, float t, float3 viewRay, float r, float mu )
{
if ( t > 0.f )
{
return 0.f;
}
else
{
float3 transmittance = r <= Rt ? TransmittanceWithShadow( r, mu ) : 1.f;
float isun = step( cos( M_PI / 180 ), dot( viewRay, g_sunDir.xyz ) ) * ISun;
return transmittance * isun;
}
}
??? ?? ?? ??? ??? ?? ??? ??? ?? 1? ?? ?? ? ??? ??
??( Radiant Intensity )? ???? ???? ????.
50
+ + =
Inscatter color Ground color Sun color Final result

Reference
? ??? ??? : https://www.flaticon.com/authors/simpleicon
? https://www.alanzucconi.com/2017/10/10/atmospheric-scattering-1/
? https://ebruneton.github.io/precomputed_atmospheric_scattering/
51

Appendix
?? ???? ??? ?? ??? ???
? = ?? , ? = ??
? =
?? ? ??
??
, ? ? =
?? ? ??
?
, ? =
?? ? ??
?
?? = ?? = ?2 + 2??? + ?2
? ?,? =
?? ? ??
??
=
?? + ?? ? ??
??
=
?? ? + ??
??
52
?
?d
?
?
?
?
? ?
?
? ?,?
?
float muSi = ( nu * t + muS * r ) / ri; // 38? ppt

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Precomputed atmospheric scattering(?? ?? ?? ??)

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