ゲームグラフィックス特論

10
2
3
•  l
dωi
t, b, n
dE
Li (l) =
d!i cos✓i
• 
dE = Li (l)d!i cos✓i
• 
E=
Z
n
Li (l) cos ✓i d!i
θi
⌦
Ω
l
dωi
4
•  BRDF
n
dLo (v)
f (l, v) =
dE(l)
Lo (v) =
Z
Lo(v)
⌦
φ
φo φi
f (l, v) ⌦ Li (l) cos ✓i d!i
t
• 
Lo (✓o ,
o)
θo
θi
dE = Li (l)d!i cos✓i
•  v
v
l
=
Z
2⇡
i =0
Z
⇡/2
f (✓i ,
✓i =0
i , ✓o ,
o)
⌦ Li (✓i ,
i ) cos ✓i
sin ✓i d✓i d
i
(d!i = sin ✓i d✓i d i )
5
• 
ωL n
v
LL
Lo (v) =
EL
Z
!L
n
lL θ
i
L
f (l, v) ⌦ LL cos ✓i d!i ⇡ f (lL , v) ⌦ EL cos ✓iL
v
6
• 
cdif f
Lo (v) =
E
⇡
• 
E
=
Z
cdiff
LL cos✓i d!i
!L
⇡ EL cos✓iL
7
e(p)
l
n
p
p
Li(p, l)
Θ
p
Li (p, l) : p
Θ
p
e(p) =
Z
e
l
Li (p, l)ld!i
⇥
8
e(p)
p
E(p, n)
E(p, n) = n · e(p)
E(p, n) = 0
E(p, n) = n · e(p)
(n
)
n
p
9
• 
• 
• 
• 
• 
• 
e(p)
l=
, EL = cL |e(p)|
|e(p)|
• 
pL
pL
l=
|pL
rL
p
, EL =
p|
(
2
⇡rL
LL
2
r
rL
pL
L L)
r
l
p
10
• 
• 
→ LA
• 
• 
LA
11
12
• 
Lo (v)
=
Z
n
⌦
v
f (l, v) ⌦ Li (l) cos ✓i d!i
cdif f
=
⌦ LA
⇡
Z
cos ✓i d!i
⌦
= cdif f ⌦ LA
• 
Lo (v)
cdif f
=
⌦
⇡
⇡LA +
n
X
k=1
ELk cos✓ik
!
13
• 
BRDF
Z
n
X
Lo (v) = LA
f (l, v) cos ✓i d!i +
f (lk , v)ELk cos✓ik
⌦
RA(v)
• 
RA (v) =
Z
f (l, v) cos ✓i d!i
⌦
k=1
14
• 
camb
cdiff
cdiff
cspec
Lo (v) = camb ⌦ LA +
n
X
k=1
f (lk , v)ELk cos✓ik
15
16
(Environment Mapping)
• 
• 
• 
(Reflection Mapping)
• 
• 
• 
• 
•  Blinn
Newell
•  Sphere Mapping
•  Cube Mapping
•  Dual Paraboloid Mapping
17
Environment Mapping
v
n
r
r = 2 (n ⋅ v) n − v
18
Blinn
Newell
• 
• 
• 
r = (rx , ry , rz ) = 2(n · v)n
v
n
y
v
⇢ = acos( rz )
= atan2(ry , rx )
r = (rx, ry, rz)
/2
x
0
/2
0
/2
19
#version 150 core!
in vec4 pv;
//
!
in vec3 nv;
//
!
uniform mat4 mw;
//
!
uniform mat3 mg;
//
!
out vec2 t;
//
!
const float PI = 3.141593;!
void main(void)!
{!
vec3 v = -normalize(mw * pv).xyz;
//
vec3 r = reflect(v, normalize(mg * nv)); //
float s = -acos(r.z) / PI;!
float t = atan(r.y, r.x) * 0.5 / PI + 0.5;!
t = vec2(s, t);!
…
!
!
20
Sphere Mapping
• 
• 
• 
• 
• 
21
Sphere Mapping
v = (vx,vx,vx)
u = (ux,ux,ux)
y
Sphere Mapping
0
n
hx
B ux
Ms = B
@ vx
0
h = (hx,hx,hx)
hy
uy
vy
0
hz
uz
vz
0
y'
O
r
x
1
0
0 C
C
0 A
1
n'
z
v0
n0
r
v'
= (0, 0, 1)
= Ms n
= 2(n0 · v0 )n0
z'
v0
22
y'
r
n' = (n'x, n'y, n'z)
r = (rx , ry , rz )
(n'x, n'y)
h
v
O
v' = (0, 0, 1)
v0 = (0, 0, 1)
z'
0
q
m = rx2 + ry2 + (rz + 1)2
h =q
0
n =
(rx , ry , rz + 1)
rx2 + ry2 + (rz + 1)2
✓
r x ry r z + 1
, ,
m m
m
◆
= n0
23
1
y'
v
1
n'y
v
n'
-1
O
-1
1
n'x
x'
O
q
m = rx2 + ry2 + (rz + 1)2
1
rx
1
n0x
+ =
+
u=
2
2
2m 2
n0y
1
ry
1
u=
+ =
+
2
2
2m 2
u
1 u
24
#version 150 core!
in vec4 pv;
in vec3 nv;
uniform mat4 mw;
uniform mat3 mg;
uniform mat3 ms;
out vec2 t;
void main(void)!
{!
t = normalize(ms
…!
//
!
//
!
//
//
!
// Sphere Mapping
//
!
!
* n).xy * 0.5 + 0.5;!
!
25
Cubic Environment Mapping
• 
• 
• 
26
27
glTexImage2D(GL_TEXTURE_CUBE_MAP_NEGATIVE_X, 0, internal,
WIDTH, HEIGHT, 0, GL_RGB, GL_UNSIGNED_BYTE, textureNX);!
glTexImage2D(GL_TEXTURE_CUBE_MAP_POSITIVE_X, 0, internal,
WIDTH, HEIGHT, 0, GL_RGB, GL_UNSIGNED_BYTE, texturePX);
glTexImage2D(GL_TEXTURE_CUBE_MAP_NEGATIVE_Y, 0, internal,
WIDTH, HEIGHT, 0, GL_RGB, GL_UNSIGNED_BYTE, textureNY);
glTexImage2D(GL_TEXTURE_CUBE_MAP_POSITIVE_Y, 0, internal,
WIDTH, HEIGHT, 0, GL_RGB, GL_UNSIGNED_BYTE, texturePY);
glTexImage2D(GL_TEXTURE_CUBE_MAP_NEGATIVE_Z, 0, internal,
WIDTH, HEIGHT, 0, GL_RGB, GL_UNSIGNED_BYTE, textureNZ);
glTexImage2D(GL_TEXTURE_CUBE_MAP_POSITIVE_Z, 0, internal,
WIDTH, HEIGHT, 0, GL_RGB, GL_UNSIGNED_BYTE, texturePZ);
!
!
!
!
!
!
28
#version 150 core!
…!
!
uniform samplerCube cubemap;
…!
// Cube Map
sampler!
!
in vec3 r;
…!
//
!
!
void main(void)!
{!
vec4 rcolor = texture(cubemap, r);
…!
//
!
29
Dual Paraboloid Mapping
• 
• 
• 
• 
p
d = u2 +✓v 2
1
d
0.25 d2
✓ = arctan
1.5
0
0.25
1
1
θ$
0.5
-0.4
y = x2
-1
-0.2
d
0.2
-0.5
-1
-1.5
◆
0.4
30
Paraboloid Mapping
r = (rx , ry , rz )
• 
0
8
>
< u=
rz < 0
8
>
< u=
rz
rx
+ 0.5
2(1 + rz )
ry
>
v
=
+ 0.5
:
2(1 + rz )
>
: v=
rx
2(1
2(1
ry
rz )
rz )
+ 0.5
+ 0.5
0
1
B 0
)B
@ 0
0
0
1
B 0
)B
@ 0
0
0
1
0
0
0
1
0
0
1
1
0
2
1
1
0
2
10
1
1
rx
B
C
1 C
C B ry C
0 A @ rz A
2
1
10
1
1
rx
B
C
1 C
C B ry C
0 A @ rz A
2
1
31
32
OpenGL
(rx, ry, rz, 1)T
• 
• 
• 
• 
• 
• 
• 
• 
33
GLSL
(
)
#version 150 core!
…!
!
uniform sampler2D front; //
uniform sampler2D back; //
…!
!
!
!
in vec3 r;
…!
//
!
!
main()!
{!
vec2 uv = r.xy * 0.5 / (1.0 + abs(r.z)) + 0.5;!
vec4 rcolor = texture(r.z >= 0.0 ? front : back, uv);!
…!
34
35
• 
Z (0,0,1)
•  1 ≤ x, y ≤ 1
•  z2 > 0
(x, y, z)
• 
• 
•  Sphere Mapping
z2 = 1 x2 y2
(0.5x + 0.5, 0.5y + 0.5)
36
37
• 
• 
(x, y, z)
• 
x2 + y2
1
xy
38
• 
• 
• 
BRDF
39
• 
• 
• 
40
41
EL
lL θ
i
Li(l)
n41
L
n
l
v
v
Ω
E = EL cos✓iL
E=
Z
Li (l) cos ✓i d!i
⌦
42
En
Ω
n
l
Li(l)
:p
:p
:n
:
n
p
l
θi
p
θi
l
Li(l)
n
Ω
En =
Z
En
n
En
Li (l) cos ✓i d!i
⌦
43
×(1
Rspec)×
+ Rspec×
44
Cube Map
47
48
49
• 
fi (x), fj (x) =
∫ f (x) f (x) dx
i
j
• 
fi (n), fj (n) =
∫
f (n) fj (n) dω
Θ i
• 
n
dω
Θ
dω$
" 0 i≠ j
fi ( ), fj ( ) = #
$ 1 i= j
n
Θ#
50
ftarget()
• 
kj = ftarget ( ), fj ( )
• 
ftarget()
n
ftarget ( ) ≈ ∑ kj fj ( )
j=1
fj()
51
• 
n
1 dn 2
Pn ( x ) = n
x −1)
n (
2 n! dx
• 
m
n
P
( x ) = (−1)
m
(1− x )
2
m
2
dm
P x
m n( )
dx
• 
m
m
(n − m) Pnm ( x ) = x (2n −1) Pn−1
( x ) − (n + m −1) Pn−2
( x)
m
Pmm ( x ) = (−1) ( 2m −1)!!(1− x
m
m
Pm+1
x
=
x
2m
+1
P
( ) (
) m ( x)
m
2 2
)
P00 ( x ) = 1
P10 ( x ) = x
52
n=0
m
n=1
0
P00 ( x ) = 1
1
P10 ( x ) = x
P11 ( x ) = − 1− x 2
2
1
P ( x ) = (3x 2 −1)
2
P21 ( x ) = −3x 1− x 2
0
2
n=2
P22 ( x ) = 3 (1− x 2 )
53
3
l = 0, m = 0
Plm(x)
l = 1, m = 0
l = 1, m = 1
2
l = 2, m = 0
l = 2, m = 1
l = 2, m = 2
1
-5
-4
-3
-2
-1
0
-1
-2
-3
1
2
3
54
$
0
i≠ j
&
+1 m
m
∫ −1 Pi (x)Pj (x) dx = % 2 (n + m)! i = j = n
& 2n +1 (n − m)!
'
55
Yl,m (✓, ) = ( 1)
m+|m|
2
Klm eim
|m|
Pl
2l +1 (l − m )!
K =
4π (l + m )!
m
l
(x, y, z) = (sinθ cos φ, sinθ sin φ, cosθ)
(cos (✓))
56
m=0
|Ylm (θ, φ)|
l=0
l=1
l=2
l=3
m=1
m=2
m=3
57
Ylm (θ, φ)
m = -3
l=0
l=1
l=2
l=3
m = -2
m = -1
m=0
m=1
m=2
m=3
58
+π
+1
−π
−1
∫ ∫
=
Yl m (θ , φ) Yl!m! (θ , φ) d (cosθ) d φ
+π
π
−π
0
∫ ∫
Yl m (θ , φ) Yl!m! (θ , φ) sin θ dθ d φ =
{
1 l = l!, m = m!
0
59
sin θ = x2 + y2
cosθ = z = 1− x2 − y2
∞
+l
f (θ , φ) = ∑ ∑ clmYl m (θ , φ)
(x, y)
l=0 m=−l
m
l
+π
+1
−π
−1
+π
π
−π
0
∫ ∫
=∫ ∫
c =
φ
f (θ , φ) Yl m (θ , φ)d (cosθ) dφ
f (θ , φ) Yl m (θ , φ) sin θ dθ d φ
light probe
62
E
• 
2
1 2
2
2
0 2
3 2
0
4 0
0
5 2
E = k c (x − y ) + k c z + k c − k c
+2k1 (c2−2 xy + c12 xz + c2−1 yz)
+2k2 (c11 x + c1−1 y + c10 z)
• 
(x, y, z)
n
n = (x, y, z) = (sin θ cos φ, sin θ sin φ, cosθ)
• 
Idiff
Idiff = (Kdiff ⊗ E) fscale
k1 = 0.429043
k2 = 0.511664
k3 = 0.743125
k4 = 0.886227
k5 = 0.247708
63
• 
Lambert
•  https://github.com/tokoik/ggsample10
• 
uniform
• 
sh
•  sh[0] = c00, sh[1] = c1-1, sh[2] = c10, sh[3] = c11, sh[4] =
c2-2, sh[5] = c2-1, sh[6] = c20, sh[7] = c21, sh[8] = c22
• 
•  simple.vert
• 
[email protected]
64
Lambert