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Spherical Harmonics 1
Spherical Harmonics
??
1. ??
2. Irradiance Map
3. ?????????(Associated Legendre Polynomials)
4. ??????
5. ??(Projection)
6. ??????????Irradiance Map
7. ???
8. Reference
??
Spherical Harmonics(????????)?????????????????. ???????????????
??????????????????????.
??????????????(球面調和函數, ??: spherical harmonics)???????
???????????????. ???????????????????????
??.
?????????. ?????????????????????????????????????????
??????????.
????Irradiance Map? Direct3D 11/12???????????????????????????????
????.
Irradiance Map
Irradiance Map???????????Image Based Lighting(??IBL)??????. ????Diffuse ?????
???????????Irradiance Map?????. ????????360?????????????????
???.
??: https://learnopengl.com/PBR/IBL/Diffuse-irradiance

Irradiance Map?360???????????????????????????????????.
?: ????radiance
: ????
?: ???????, ????????
?: ????????radiance
: ????
?: brdf
?: ????radiance
Diffuse ????brdf????Lambert BRDF?????Lambert BRDF????????brdf?????????
?????. ?????????????????????????????????????.
???Lambert BRDF?????Irradiance????.
???????Irradiance Map???????????????. ?????????????????????
??????????. ?????????(2048x2048)???????(32x32)?????????????
Irradiance Map??????. ??????Diffuse ????????????????????????????
????????.
??Irradiance Map????????????????.
agl::TextureTrait trait = cubeMap->GetTrait();
trait.m_width = trait.m_height = 32; // ?? ????? ?? ??
trait.m_format = agl::ResourceFormat::R8G8B8A8_UNORM_SRGB;
trait.m_bindType |= agl::ResourceBindType::RenderTarget; // ?? ???? ??
auto irradianceMap = agl::Texture::Create( trait );
EnqueueRenderTask( [irradianceMap]()
{
irradianceMap->Init();
} );
?????????????????????????????????????????????Irradiance
Map??????. ?????????????????????????.
?????????. ??????????????????????????SV_VertexID?????????
????????????????????. ??????????????????????????????
?????6?????????.
struct VS_OUTPUT
{
uint vertexId : VERTEXID;
};
VS_OUTPUT main( uint vertexId : SV_VertexID )
{
VS_OUTPUT output = (VS_OUTPUT)0;
L ?
(x,ω ?
) = L ?
(x,ω ?
) + ? f ?
(x,ω ?
,ω )L ?
(x,ω ?
)(ω ? ? n)d ω ?
r o e o ∫
Ω
r i o i i i i
L ?
(x,ω ?
)
r o
x
w ?
,w ?
o i
L ?
(x,ω ?
)
r o
Ω
f ?
(x,ω ?
,ω ?
)
r i o
L ?
(x,ω ?
)
i i
L ?
(x,ω ?
) = ? ?
L ?
(x,ω ?
)(ω ?
? n)d ω ?
r o
π
σ
∫
Ω
i i i i
E(x) = ?
L ?
(x,ω ?
)(ω ?
? n)d ω ?
∫
Ω
i i i i

output.vertexId = vertexId;
return output;
}
???????????. ???????????????????????????????. ??????
GS_OUTPUT?SV_RenderTargetArrayIndex??????????. ????????????????????
??.
struct GS_INPUT
{
uint vertexId : VERTEXID;
};
struct GS_OUTPUT
{
float4 position : SV_POSITION;
float3 localPosition : POSITION0;
uint rtIndex : SV_RenderTargetArrayIndex;
};
static const float4 projectedPos[] =
{
{ -1, -1, 0, 1 },
{ -1, 1, 0, 1 },
{ 1, -1, 0, 1 },
{ 1, 1, 0, 1 }
};
static const float3 vertices[] =
{
{ -1, -1, -1 },
{ -1, 1, -1 },
{ 1, -1, -1 },
{ 1, 1, -1 },
{ -1, -1, 1 },
{ -1, 1, 1 },
{ 1, -1, 1 },
{ 1, 1, 1 }
};
static const int4 indices[] =
{
{ 6, 7, 2, 3 },
{ 0, 1, 4, 5 },
{ 5, 1, 7, 3 },
{ 0, 4, 2, 6 },
{ 4, 5, 6, 7 },
{ 2, 3, 0, 1 }
};
[maxvertexcount(4)]
void main( point GS_INPUT input[1], inout TriangleStream<GS_OUTPUT> triStream )
{
GS_OUTPUT output = (GS_OUTPUT)0;
output.rtIndex = input[0].vertexId;
for ( int i = 0; i < 4; ++i )
{
output.position = projectedPos[i];
int index = indices[input[0].vertexId][i];
output.localPosition = vertices[index];
triStream.Append( output );
}
triStream.RestartStrip();
}
????????????????????????????.
????????radiance???????????.

Irrdiance?????????Lambert BRDF????????Lambert BRDF????????????????
?????. ?????????? ?
???????????????????????????????.
??????????????????????????.
????????????????????????????????????.
???????????????.
#include "Common/Constants.fxh"
TextureCube CubeMap : register( t0 );
SamplerState LinearSampler : register( s0 );
static const float SampleDelta = 0.025f;
struct PS_INPUT
{
float4 position : SV_POSITION;
float3 localPosition : POSITION0;
uint rtIndex : SV_RenderTargetArrayIndex;
};
float4 main(PS_INPUT input) : SV_TARGET
{
float3 normal = normalize( input.localPosition );
float3 up = ( abs( normal.y ) < 0.999 ) ? float3( 0.f, 1.f, 0.f ) : float3( 0.f, 0.f, 1.f );
float3 right = normalize( cross( up, normal ) );
up = normalize( cross( normal, right ) );
float3x3 toWorld = float3x3( right, up, normal );
float3 irradiance = 0.f;
float numSample = 0.f;
for ( float phi = 0.f; phi < 2.f * PI; phi += SampleDelta )
{
for ( float theta = 0.f; theta < 0.5f * PI; theta += SampleDelta )
{
float3 tangentSample = float3( sin( theta ) * cos( phi ), sin( theta ) * sin( phi ), cos( theta ) );
float3 worldSample = normalize( mul( tangentSample, toWorld ) );
irradiance += CubeMap.Sample( LinearSampler, worldSample ).rgb * cos( theta ) * sin( theta );
++numSample;
}
}
irradiance = PI * irradiance / numSample;
L ?
(x,ω ?
) = ? ? L ?
(x,ω ?
)(ω ? ? n)d ω ?
r o
π
σ
∫
Ω
i i i i
σ
? ?
L ?
(x,ω ?
)(ω ? ?
π
1
∫
Ω
i i i n)d ω ?
i
? ? ? L(θ,?)cos(θ)sin(θ)dθd?
π
1
∫
?=0
2π
∫
θ=0
?
2
π
? ? ? ? ? L ?
(x,ω ?
)cos(θ)sin(θ)
π
1
n1
2π
n2
?
2
π
?=0
∑
n1
θ=0
∑
n2
i i
= ? ? ? L ?
(x,ω ?
)cos(θ)sin(θ)
n1n2
π
?=0
∑
n1
θ=0
∑
n2
i i

return float4( irradiance, 1.f );
}
????????Irradiance Map????????.
?????Irradiance Map????????????????.
float3 ImageBasedLight( float3 normal )
{
return IrradianceMap.Sample( LinearSampler, normal ).rgb;
}
// ...
float4 lightColor = float4( ImageBasedLight( normal ), 1.f ) * MoveLinearSpace( Diffuse );
??????????????????Irradiance Map???. ??????Irradiance Map????24KB( 32 *
32 * 6 * 4Byte )????????????. ?????????????????108Byte( 3 * 9 * 4Byte )????
??????????????????. ???????????????????????.
?????????(Associated Legendre Polynomials)
?????????????????????????????????????. ????????????
????????????????.
??????????????? ?
?????????? ?
? ?
?????????????????.
??? ?
?-1 ~ 1????????????????????????.
? ????????????????????????????????.
????? ?
? ?
??????????(Bands)???????? ?
?Band Index???0??????????
??????. ??? ?
?0 ~ ???????????????.
??????? Irradiance Map
P l m
P ?
(x)
l
m
x
l m l
m l
P ?
0
0
P ?
P ?
1
0
1
1
P ?
P ?
P ?
2
0
2
1
2
2

???????????????????????????????????????????. ????3??
????????????.
?????????????????????????????????????????. ??? ?
?????
?????????????????????.
????????????????.
?
???1?????????????????3?????????????????????????.
???3??????????????????????????.
double P(int l, int m, double x)
{
// evaluate an Associated Legendre Polynomial P(l,m,x) at x
double pmm = 1.0;
if (m > 0)
{
double somx2 = sqrt((1.0 - x) * (1.0 + x))
double fact = 1.0;
for (int i = 1; i <= m; i++)
{
pmm *= (-fact) * somx2;
fact += 2.0;
}
}
if (l == m) return pmm;
double pmmp1 = x * (2.0 * m + 1.0) * pmm;
if (l == m + 1) return pmmp1;
double pll = 0.0;
for (int ll = m + 2; ll <= l; ++ll)
{
pll = ((2.0 * ll - 1.0) * x * pmmp1 - (ll + m - 1.0) * pmm) / (ll - m);
pmm = pmmp1;
pmmp1 = pll;
}
return pll;
}
??????
Spherical Harmonics???????????????????????????.
???????????????????????????????????????????????????
?. ??????????????????????????.
????????????????.
1 P ? =
m
m
(?1) (2m ?
m
1)!!(1 ? x )
2 m/2
!!
n!! = { ?
n ? (n ? 2)...5 ? 3 ? 1, n > 0 odd
n ? (n ? 2)...6 ? 4 ? 2, n > 0 even
2 P ? =
m+1
m
x(2m + 1)P ?
m
m
3 (l ? m)P ?
=
l
m
x(2l ? 1)P ?
?
l?1
m
(l + m ? 1)P ?
l?2
m
P ?
(x)
0
0
Y ?
(θ,?) :=
l
m
AP ?
(cosθ)e
l
m im?

??? ?
?????????????????? ?
??????????????.
?????????? ?
? ?
?????????????????????. ?
????0???????????
?? ?
? ?
? ?
????????.
?????????????????.
??????????????????????????????????????????????????
????.
??????????????????.
double K(int l, int m)
{
// renormalisation constant for SH function
double temp = ((2.0 * l + 1.0) * factorial(l - m)) / (4.0 * PI * factorial(l + m));
return sqrt(temp);
}
double SH(int l, int m, double theta, double phi)
{
// return a point sample of a Spherical Harmonic basis function
// l is the band, range [0..N]
// m in the range [-l..l]
// theta in the range [0..Pi]
y ?
(θ,φ) =
l
m
{ ?
?
K ?
cos(mφ)P ?
(cosθ), m > 0
2 l
m
l
m
?
K ?
sin(?mφ)P ?
(cosθ), m < 0
2 l
m
l
m
K ?
P ?
(cosθ), m = 0
l
0
l
0
P K
K ? =
l
m
?
?
4π
(2l + 1)
(l + ∣m∣)!
(l ? ∣m∣)!
l m l
m ?l l
y ?
0
0
y y ? y ?
1
?1
1
0
1
1
y ? y ? y ? y ? y ?
2
?2
2
?1
2
0
2
1
2
2
??: https://users.soe.ucsc.edu/~pang/160/s13/projects/bgabin/Final/report/Spherical Harmonic Lighting Comparison.htm

// phi in the range [0..2*Pi]
const double sqrt2 = sqrt(2.0);
if (m == 0) return K(l, 0) * P(l, m, cos(theta));
else if (m > 0) return sqrt2 * K(l, m) * cos(m * phi) * P(l, m, cos(theta));
else return sqrt2 * K(l, -m) * sin(-m * phi) * P(l, -m, cos(theta));
}
???????????????????????????????????. ???????????????
????????????????????????????????. ?????????????????
??????????????.
???????????????????????????????. ??? ?
????????????.
??(Projection)
???????????????????. ??????????????????Irradiance Map?????
????????????????? ???????(Basis Function)???(Projection)???????????
??.
????????????????????????????????. ?????????????????
???????????????????.
?????????????????????????. ????????????????????????
?????????????????.
????????????????????.
(x,y,z) = (sinθcos?,sinθsin?,cosθ)
l = 2
y ?
( ) =
0
0
n 0.282095
y ?
( ) =
1
?1
n 0.488603y
y ?
( ) =
1
0
n 0.488603z
y ?
( ) =
1
1
n 0.488603x
y ?
( ) =
2
?2
n 1.092548xy
y ?
( ) =
2
?1
n 1.092548yz
y ?
( ) =
2
0
n 0.315392(3z ?
2
1)
y ?
( ) =
2
1
n 1.092548xz
y ?
( ) =
2
2
n 0.546274(x ?
2
y )
2
f(x) = c ?
b ?
(x) +
1 1 c ?
b ?
(x) +
2 2 c ?
b ?
(x) +
3 3 c ?
b ?
(x) +
4 4 ...
c ? =
i f(x)b ?
(x)
∫ i

??????????????????????????????.
???????????????????Irradiance Map??????????.
??????????Irradiance Map
??????????Irradiance Map ??????????.
??: Spherical Harmonic Lighting: The Gritty Details
? ? L ?
(x,ω ?
)cos(θ)sin(θ)dθd?
π
1
∫
?=0
2π
∫
θ=0
?
2
π
i i

?????????2???????.
1. ?→ ??????????????????????????.
2. ?→ ???????????. ??????Radiance???????????????0??
??????Radiance??????.
???????????????????????????.
??? ?
????(Zenith Angle)?????????? ?
???0????. ???????? ? ?
??
?Irradiance?????????. Irradiance?????????? ?
????Irradiance?????????
??.
?? ?
? ?
? ?
??????????????.
??? ?
?????????? ?
?????????????????????????????. ?
???????????4??4.A??????????.
?????????? ?
???????????.
?
????????????????????.
???????????????????????????????????????.
? ? L ?
(x,ω ?
)sin(θ)dθd?
∫?=0
2π
∫θ=0
π
i i
max(cos(θ),0)
L ? =
lm ? ? L(θ,?)y ?
(θ,?)sin(θ)dθd?
∫
?=0
2π
∫
θ=0
π
l
m
A ? =
l ? max(cos(θ),0)y ?
(θ,0)dθ
∫
θ=0
π
l
0
cos(θ) m Llm A ?
l
E ?
lm
E(θ,?) = ? E ?
y ?
(θ,?)
l,m
∑ lm l
m
E ?
lm L ?
lm A ?
l
E ?
=
lm ?
A ?
L ?
?
2l + 1
4π
l lm
?
?
2l+1
4π
cos(θ)
?
A ?
l
^
?
=
A ?
l
^ ?
A ?
?
2l + 1
4π
l
?
A ?
l
^
? =
A ?
0
^ 3.1415
? =
A ?
1
^ 2.0943
? =
A ?
2
^ 0.7853
? =
A ?
3
^ 0
?
=
A ?
4
^ ?0.1309
? =
A ?
5
^ 0
?
=
A ?
6
^ 0.0490

??? ?
??????????????????????27????(RGB ??3?* SH ??9?)???
Irradiance Map?????????. ???????????????? ?
???????????????
??.
void ShFunctionL2( float3 v, out float Y[9] )
{
// L0
Y[0] = 0.282095f; // Y_00
// L1
Y[1] = 0.488603f * v.y; // Y_1-1
Y[2] = 0.488603f * v.z; // Y_10
Y[3] = 0.488603f * v.x; // Y_11
// L2
Y[4] = 1.092548f * v.x * v.y; // Y_2-2
Y[5] = 1.092548f * v.y * v.z; // Y_2-1
Y[6] = 0.315392f * ( 3.f * v.z * v.z - 1.f ) ; // Y_20
Y[7] = 1.092548f * v.x * v.z; // Y_21
Y[8] = 0.546274f * ( v.x * v.x - v.y * v.y ); // Y_22
}
??? ?
????????????????????. ?????????????????????.
#include "Common/Constants.fxh"
#include "SH/SphericalHarmonics.fxh"
??: On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object
l ≤ 2
y ?
( )
l
m
n
L ?
lm
L ?
=
lm ? ? ? L(θ,?)y ?
(θ,?)sin(θ)dθd?
π
1
∫
?=0
2π
∫
θ=0
π
l
m
= ? ? ? ? ? L(θ,?)y ?
(θ,?)sin(θ)
π
1
n1
2π
n2
π
?=0
∑
n1
θ=0
∑
n2
l
m
= ? ? ?
L(θ,?)y ?
(θ,?)sin(θ)
n1n2
2π
?=0
∑
n1
θ=0
∑
n2
l
m

TextureCube CubeMap : register( t0 );
SamplerState LinearSampler : register( s0 );
RWStructuredBuffer<float3> Coeffs : register( u0 );
static const int ThreadGroupX = 16;
static const int ThreadGroupY = 16;
static const float3 Black = (float3)0;
static const float SampleDelta = 0.025f;
static const float DeltaPhi = SampleDelta * ThreadGroupX;
static const float DeltaTheta = SampleDelta * ThreadGroupY;
groupshared float3 SharedCoeffs[ThreadGroupX * ThreadGroupY][9];
groupshared int TotalSample;
[numthreads(ThreadGroupX, ThreadGroupY, 1)]
void main( uint3 GTid: SV_GroupThreadID, uint GI : SV_GroupIndex)
{
if ( GI == 0 )
{
TotalSample = 0;
}
GroupMemoryBarrierWithGroupSync();
float3 coeffs[9] = { Black, Black, Black, Black, Black, Black, Black, Black, Black };
int numSample = 0;
for ( float phi = GTid.x * SampleDelta; phi < 2.f * PI; phi += DeltaPhi )
{
for ( float theta = GTid.y * SampleDelta; theta < PI; theta += DeltaTheta )
{
float3 sampleDir = normalize( float3( sin( theta ) * cos( phi ), sin( theta ) * sin( phi ), cos( theta ) ) );
float3 radiance = CubeMap.SampleLevel( LinearSampler, sampleDir, 0 ).rgb;
float y[9];
ShFunctionL2( sampleDir, y );
[unroll]
for ( int i = 0; i < 9; ++i )
{
coeffs[i] += radiance * y[i] * sin( theta );
}
++numSample;
}
}
int sharedIndex = GTid.y * ThreadGroupX + GTid.x;
[unroll]
for ( int i = 0; i < 9; ++i )
{
SharedCoeffs[sharedIndex][i] = coeffs[i];
coeffs[i] = Black;
}
InterlockedAdd( TotalSample, numSample );
GroupMemoryBarrierWithGroupSync();
if ( GI == 0 )
{
for ( int i = 0; i < ThreadGroupX * ThreadGroupY; ++i )
{
[unroll]
for ( int j = 0; j < 9; ++j )
{
coeffs[j] += SharedCoeffs[i][j];
}
}
float dOmega = 2.f * PI / float( TotalSample );
[unroll]
for ( int i = 0; i < 9; ++i )
{

Coeffs[i] = coeffs[i] * dOmega;
}
}
}
?????? ?
?????????????????.
float3 ImageBasedLight( float3 normal )
{
float3 l00 = { IrradianceMapSH[0].x, IrradianceMapSH[0].y, IrradianceMapSH[0].z }; // L00
float3 l1_1 = { IrradianceMapSH[0].w, IrradianceMapSH[1].x, IrradianceMapSH[1].y }; // L1-1
float3 l10 = { IrradianceMapSH[1].z, IrradianceMapSH[1].w, IrradianceMapSH[2].x }; // L10
float3 l11 = { IrradianceMapSH[2].y, IrradianceMapSH[2].z, IrradianceMapSH[2].w }; // L11
float3 l2_2 = { IrradianceMapSH[3].x, IrradianceMapSH[3].y, IrradianceMapSH[3].z }; // L2-2
float3 l2_1 = { IrradianceMapSH[3].w, IrradianceMapSH[4].x, IrradianceMapSH[4].y }; // L2-1
float3 l20 = { IrradianceMapSH[4].z, IrradianceMapSH[4].w, IrradianceMapSH[5].x }; // L20
float3 l21 = { IrradianceMapSH[5].y, IrradianceMapSH[5].z, IrradianceMapSH[5].w }; // L21
float3 l22 = { IrradianceMapSH[6].x, IrradianceMapSH[6].y, IrradianceMapSH[6].z }; // L22
static const float c1 = 0.429043f;
return c1 * l22 * ( normal.x * normal.x - normal.y * normal.y ) + c3 * l20 * normal.z * normal.z + c4 * l00 - c5 * l20
+ 2.f * c1 * ( l2_2 * normal.x * normal.y + l21 * normal.x * normal.z + l2_1 * normal.y * normal.z )
+ 2.f * c2 * ( l11 * normal.x + l1_1 * normal.y + l10 * normal.z );
}
????????3?3.2????????????????????????.
????????? ????????
L ?
lm
E(n) = c ?
L ?
(x ?
1 22
2
y ) +
2
c ?
L ?
z +
3 20
2
c ?
L ? ?
4 00 c ?
L ?
5 20
+2c ?
(L ?
xy +
1 2?2 L21xz + L ?
yz)
2?1
+2c ?
(L ?
x +
2 11 L ?
y +
1?1 L ?
z)
10
c ? =
1 0.429043
c ?
=
2 0.511664
c ? =
3 0.743125
c ? =
4 0.886227
c ? =
5 0.247708
?? ??

???
?????????????.
???????????????????.
GitHub - xtozero/SSR at irradiance_map
Screen Space Reflection. Contribute to xtozero/SSR development by creating an account on
GitHub.
https://github.com/xtozero/ssr/tree/irradiance_map
Reference
1. Diffuse irradiance
2. Spherical Harmonic Lighting: The Gritty Details
3. An Efficient Representation for Irradiance Environment Maps
4. On the relationship between radiance and irradiance: determining the illumination from images of a convex
Lambertian object
5. Diffuse IrradianceMap?Spherical harmonics??????
?
??
?
??

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Spherical Harmonics.pdf