CS380: Introduction to Computer Graphics Light Transport ...vclab.kaist.ac.kr/cs380/slide23-lighttransport.pdf · 18/06/11 2 Min H. Kim (KAIST) Foundations of 3D Computer Graphics,

Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler, MIT Press, 2012

CS380: Introduction to Computer GraphicsLight Transport

Chapter 21

Min H. KimKAIST School of Computing


SUMMARYColor and Ray-Tracing

2


Ray tracing

3


Ray-plane• Plane described by the equation• We start by representing every point along the

ray using a single parameter λ

• Plugging this into the plane equation, we get

4

Ax+By+Cz+D=0

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

pxpypz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

+ λdxdydz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

0 = A(px + λdx )+ B(py + λdy )+C(pz + λdz )+ D = λ(Adx + Bdy +Cdz )+ (Apx + Bpy +Cpz + D)

pd

Eye origin Pixel ray vector


Ray-triangle• Suppose we wish to test if a point is inside or

outside of a triangle in 2D • Consider the three “sub” triangle

• When is inside of , then all three sub-triangles will agree on their clockwisedness. When is outside, then they all disagree.

5

q

Δ( p1 p2 p3)

Δ( p1 p2 q),Δ( p1 qp3)

and Δ( qp2 p3)

q Δ( p1 p2 p3)

q


Ray-sphere• Suppose we have a sphere with radius R and

center c modeled as the set of points that satisfy the equation

• Plugging the below into the above:

• We get

6

[x, y, z]t

(x − cx )2 + (y − cy )

2 + (z − cz )2 − r2 = 0

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

pxpypz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

+ λdxdydz

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

0 = (px + λdx − cx )2 + (py + λdy − cy )

2 + (pz + λdz − cz )2 − r2

0 = (dx2 + dy

2 + dz2 )λ 2 + (2dx (px − cx )+ 2dy(py − cy )+ 2dz (pz − cz ))λ +

(px − cx )2 + (py − cy )

2 + (pz − cz )2 − r2


Three sources of light together

7Courtesy of Doug A. Bowman


Bounding volume hierarchy (BVH)

8


LIGHT TRANSPORTChapter 21

9


RENDERING EQUATIONHemispherical formulation

10


Solid angle• Steradian (sr)– The solid angle subtended at the center of a sphere

by an area on its surface numerically equal to the square of the radius.

11


Solid angle• 2D vs. 3D angle

12


Solid angle• A measure on the hemisphere to integrate functions over the

hemisphere [unit: steradian (sr)]

• A finite solid angle subtended by an area on the hemisphere is defined as the total area divided by the squared radius of the hemisphere:

• The area of the hemisphereis:

13

Ω

Ω = Ar2

2πr2

1 steradian ⇔ A = r2


Solid angle• The solid angle area covered by a complete hemisphere

equals:• The solid angle covered by a complete sphere equals:• Note that the solid angle is not dependent on the

shape of surface ,but is only dependent on the total area .

14

2π

4π

A


Solid angle• We can project the surface or object on the

hemisphere and compute the solid angle of the projection.

• The two objects different in shape can still subtend the same sold angle.

15


Radiant power• A bunch of photons flying through space in a

direction.• Radiant power: Each photon carries energy

[unit: joules]

16


Radiant flux• Radiant flux: radiant power measured in

seconds, measured in watts [joules/sec]• Radiant flux does not specify the size of the light

source or receiver.• Flux does not include a specification of the

distance between the light source and the receivers.

17


Radiant flux• A bunch of photons flying through

space in a direction.• We have a sensor (W, X) out in

space– W is a wedge of directions– Pass through the surface X

• Each photon carries energy in units of joules

• Radiant flux: energy measured in seconds, measured in watts:

18

Φ(W ,X)


Radiant power vs. distance• d2 law

19


Irradiance• Irradiance: Measured radiant flux by the area of

the sensor (measured in square meters)

• Smaller and smaller X around a single point(under continuity assumptions about )à irradiance

• Note E at a different normal is different

20

E(W ,X) := Φ(W ,X)X

xΦ

En (W , x)

!ʹn

E !n(W , "x)≠E !ʹn (W , "x)


Irradiance• Suppose the sensor surface X broken up into a bunch of

smaller surfaces, Xi.• Radiant flux over the entire sensor as the sum:

• Under reasonable continuity assumption about , radiance flux from pointwise (pixelwise) irradiance:

21

Φ(W ,X )= Φ(W ,Xi )=

i∑ Xi E(W ,Xi )

i∑

Φ

Φ(W ,X )= E !n( "x )X∫ (W , "x)dA


Radiosity (radiant exitance)• Exitant radiant power per unit surface area [unit:

W/sqm]

• E.g., consider a light source, of area 0.1 sqm, that emits 100 watts. à The radiant exitance of the light is 1,000 W/sqm at each point of its source.

22

M = B = dΦdA


Radiance• In , shrink W down to a single vector,

dividing out by |W|, the solid angle measure of W (unit: steradians) à radiance

23

En (W , x)

L!n(W , "x)=

E !n(W , "x)W

(the wedge of all directioncovers 4π steradians)

L!n(!w , "x)= lim

W→ !wL!n(W , "x)

L!n(!w , "x): L( !w , "x)withgiven

surfaceorientation!n

L(!w , "x):=L

−!w(!w , "x)

L−!w(!w , "x)= 1cos(θ )L!n(

!w , "x)


Radiance• Smaller and smaller wedges around a vector

pointing toward • Under reasonable continuity assumption for

radiance:unit:

24

L( !w , "x):=L

−!w(!w , "x)= L!n(

!w , "x)cos(θ )

!w

x

Φis the angle

between and n −

wθ

Watt / (sr ⋅m2 ⋅nm)


Incoming radiance• Incoming radiance:

• Given a spatially and angularly varyingwe can compute the radiant flux over a large sensor (W,X) as:

25

L( !w, "x) := 1

cos(θ )limW→ !w

1W

limX→ "x

Φ(W ,X)X

⎛⎝⎜

⎞⎠⎟

L(w, x)

Φ(W ,X )= L(!w , "x)cos(θ )dw

W∫( )dAX∫


Outgoing radiance• In fact, radiance remains constant along a ray in

free space:• We can define outgoing radiance from

the point in the direction . • Reverse the order of the arguments to L

• where is the radiant flux of photons leaving a finite surface X along vectors in the wedge W

26

L(!w, "x) = L( !w, "x + !w)

L( x,w)

x w

L( !x, "w) := 1

cos(θ )limW→ "w

1W

limX→ !x

Φ(X,W )X

⎛⎝⎜

⎞⎠⎟

Φ(X,W )


Radiance in computer graphics• We calculate the color of some 3D point in an

OpenGL fragment shader as the outgoing radiance , where is the “view vector”.

• In the fragment shader, this outgoing radiance value is also the incoming radiance value at the corresponding sample location on the image plane and is thus used to color the pixel.

• In ray tracing, when we trace a ray along the ray, we can interpret this as calculating the

incoming radiance value, .27

!x

L( x,v)

v

( x,d )

L(−d, x)


Radiant flux along a ray• Slide our sensor (W,X) along the vector to

obtain the shifted sensor (W,X’), where

• Radiant fluxes will not agree (due to the d2 law)

28

w

X ' = X +w

Φ(W ,X) ≠ Φ(W , ′X )


Radiance along a ray• Radiance will agree

29

L( !w, "x) := 1cos(θ )

limW→ !w

1W

limX→ "x

Φ(W ,X)X

⎛⎝⎜

⎞⎠⎟

= 1cos(θ )

limX→ "x

1X

limW→ !w

Φ(W ,X)W

⎛⎝⎜

⎞⎠⎟

= 1cos(θ )

limX→ "x

1X

limW→ !w

Φ(W ,X + !w)W

⎛⎝⎜

⎞⎠⎟

= 1cos(θ )

lim′X → "x+ !w

1′X

limW→ !w

Φ(W , ′X )W

⎛⎝⎜

⎞⎠⎟

= 1cos(θ )

limW→ !w

1W

lim′X → "x+ !w

Φ(W , ′X )′X

⎛⎝⎜

⎞⎠⎟= L( !w, "x + !w)

∵ lim

W→ "w

Φ(W , ʹX )Φ(W ,X ) =1 → X '≈ X +

"w⎛

⎝⎜

⎞

⎠⎟

In fact, in the limit for very small wedges, our senor and shifted sensor measure the same set of photons.


Reflection• Bidirectional reflectance distribution function (BRDF) is

the reflection ratio that converges and does not actually depend on the size of the wedges.

• Making the incoming wedges W smaller around a fixed , this function f converges to

• The simplest BRDF is the constant BRDF (for diffuse material):

30

f !x ,"n(W ,

"v)= L1( !x , "v)

E "ne(W , !x)

= L1( !x , "v)

L"ne (W , !x)W

!w f !x ,"n(

"w , "v)

f !x ,"n("w , "v)=1

f !x ,"n("w , "v)= lim

W→ "w L

1( !x , "v)L"ne (W , !x)W


Reflection• On a diffuse surface, the amount of scattered

photons drops by a cosine factor to zero, while the outgoing radiance does not depend on outgoing angle in the BRDF definition. (don’t get confused with the viewing direction.)

• Outgoing radiance is• Using ,

the reflected radiance is

31

L1( !x , "v)= Wi f !x ,"n(Wi ,

"v)L"ne (Wi , !x)i∑

f x,n (w, v) and Ln

e ( w, x)

L1( !x , "v)= f !x ,"n("w , "v)L"ne (

"w , !x)dwH∫

= f !x ,"n("w , "v)Le( "w , !x)cosθdw

H∫


Mirror reflection• In a mirror surface, the bounced

radiance along a ray, depends only on where

• Mirror reflection: where is some material coefficient.

• Mirror reflected light is

• Note that no integration is required in perfect mirror reflection.

32

Le −B(v), x[ ]

B(v) = 2(v ⋅ n)n − v

k x,n (v) = L

1( x, v)Le[−B(v), x]

k x,n (v)

L1( x, v) = k x,n (

v)Le[−B(v), x]


Rendering Equation• Reflection equation in shorthand

where B is the bounce operator of light.• Recursive light transport is• We can say

33

L1 =BLe

Li+1 = BLi

Lt =Le +L1+L2+L3+ ... =Le +B(Le +L1+L2+L3+ ...) =Le +BLt


Rendering Equation• The rendering equation (integral equation) is

34

Lt( !x , "v)=Le( !x , "v)+ f !x ,"n("w , "v)Lt( "w , !x)cos(θ )dw

H∫Lt( !x , "v):receivedlightLe( !x , "v): emittinglightf !x ,"n("w , "v):reflectancefunction

Lt( "w , !x): incidentlightcos(θ ): thecosinelawH : theentire(hemi-)sphericalsolidangledw:differentialsolidangle


Global Illumination (2nd bounce)

where is the point first hit along the ray

35

Le : light source emissionL1 : the first bounceL2 : the second bounceLn : the nth bounce

L2( !x , "v)= f !x ,"n("w , "v)cos(θ )L1( "w , "x)dw

H∫= f !x ,"n(

"w , "v)cos(θ ) f !ʹx ,"ʹn ("ʹw , "w)cos( ʹθ )Le( " ʹw , "ʹx )d ʹw

ʹH∫( )dwH∫= f !x ,"n(

"w , "v)cos(θ ) f !ʹx ,"ʹn ("ʹw , "w)cos( ʹθ )Le( " ʹw , "ʹx )d ʹw

ʹH∫ dwH∫

ʹ!x ( !x,−"w)


Path tracing• Adding one more bounce• Geometric path of light transport:• Not as a nested integral over two hemispheres,

but as an integral over an appropriate space of pathsà Path tracing

36

Le( !x , "v)+L1( !x , "v)+L2( !x , "v)

( !x , !ʹx , ! ʹ́x )

light


• is less important than direct lighting.• It is needed to simulate blurry reflections of the

surrounding environment• Caustic effects can also be seen due to• Total light:• (higher bounces account for the overall distributions of

light and darkness in the environment)

Path tracing

37

L2

L2

Lt = Le + L1 + L2 + L3 + ...

Caustic effects


Path tracing• Reflection equation in shorthand

where B is the bounce operator of light.• Recursive light transport is• We can say

• This expresses an equation that must hold for the total equilibrium distribution .

38

L1 =BLe

Li+1 =BLi

Lt =Le +L1+L2+L3+ ...Lt =Le +B(Le +L1+L2+L3+ ...)Lt =Le +BLt

Lt( !x , "v)=Le( !x , "v)+ f !x ,"n(

"w , "v)Lt( "w , !x)cos(θ )dwH∫

Lt

(∵L1 =BLe )


Camera Sensor• Photon count at pixel (i,j) in a physical camera

model:

where T is the duration of the shutter, Ωi,j is the spatial support of pixel (i,j)at the sensor pointand W is the wedge of the incoming vectorfrom the aperture toward the film point.

39

dt dA dwFi , j( !x)Lt(

"w , !x)cos(θ )W∫Ωi , j∫T∫

x


Sensor• Other effects– Motion blur: due to integration over the shutter

time– Depth of field: due to integration over the solid

angle of the aperture(focus and blur effects)

40


Sensor• Motion blur & Depth-of-field effect

• Subsurface scattering

41


REFLECTION EQUATION (SUMMARY)

42

BRDF

n

o

df

h

hq

dq

i


Lt( !x , "v)=Le( !x , "v)+ f !x ,"n("w , "v;m,η)Lt( "w , !x)cos(θ )dw

H∫

Lt( !x , "v):receivedlightLe( !x , "v): emittinglightf !x ,"n("w , "v):reflectancefunction

Lt( "w , !x): incidentlightcos(θ ): thecosinelawH : theentire(hemi-)sphericalsolidangledw:differentialsolidangle

!n

!w

df

!h

hq

dq

!w

Rendering Equation

43


Microfacet-based Reflectance Model

44

Lt( !x , "v)=Le( !x , "v)+ f !x ,"n("w , "v;m,η)Lt( "w , !x)cos(θ )dw

H∫f !x ,"n("w , "v;η)=kd f !xdiffuse +ks f !x ,"nspecular , wherekd +ks =1

f !xdiffuse:diffusetermf !x ,"nspecular( "w , "v;m,η): specularterm


Diffuse Reflection

• Diffuse reflection– Internal scattering– Multiple reflection within microfacets

• It is a constant per material

45

f !x ,"ndiffuse( "w , "v)= ρ

π,

whereρ:diffusealbedo


Specular Reflection (D)-1

46

f !x ,"nspecular( "w , "v;m,η)= D(

"h;m)G( "w , "v ,

"h)F( "w ,

"h;η)

4("n ⋅ "w)("n ⋅ "v)D : Microfacetnormaldistributionfunction(probabilitydensityfunction),1= D(

"h)("h ⋅ "n)dwhH∫

Multiplespecularlobes: D= wjD(mj )j∑ , wj

j∑ =1

(1)Beckmanndistribution[Cook1982]:

D θh;m( ) =exp(−tan2(θh)/m2)

πm2cos4θh,

θh = cos−1(!h ⋅ !n),m:roughnessparameter



47

(2)GGXdistribution[Walter2007]:

D(θh;m)=m2

π cos4θh m2+ tan2θh( )2 ,

θh = cos−1(!h ⋅ !n),m:roughnessparameter


"h;m)G( "w , "v ,

"h)F( "w ,

"h;η)

4("n ⋅ "w)("n ⋅ "v)D : Microfacetnormaldistributionfunction(probabilitydensityfunction),1= D(

"h)("h ⋅ "n)dwhH∫

Multiplespecularlobes: D= wjD(mj )j∑ , wj

j∑ =1



48

With a roughness of 0.5

https://mynameismjp.wordpress.com/2016/10/09/sg-series-part-4-specular-lighting-from-an-sg-light-source/

With a roughness of 0.25

• Facet distribution


Specular Reflection (G)-1

49

(1)V-groovecavitymodel[Cook1982]:

Gshadowing(!w ,!h)=min 1,2|

!h ⋅ !n|| !w ⋅ !n|| !w ⋅!h|

⎛

⎝⎜

⎞

⎠⎟,

Gmasking(!v ,!h)=min 1,2|

!h ⋅ !n|| !v ⋅ !n|| !v ⋅!h|

⎛

⎝⎜

⎞

⎠⎟,

G( !w , !v ,!h)=min Gshadowing(

!w ,!h),Gmasking(

!v ,!h)( ).


"h;m)G( "w , "v ,

"h)F( "w ,

"h;η)

4("n ⋅ "w)("n ⋅ "v)G : Geometricattenuationfunction(shadowing/masking)


Specular Reflection (G)-2

50

(2)SmithGmodel[Heitz2014]:

G( !w , !v;m)= 21+ 1+m2 tan2 !w⎛

⎝⎜⎜

⎞

⎠⎟⎟

21+ 1+m2 tan2 !v⎛

⎝⎜⎜

⎞

⎠⎟⎟

m:roughnessparameter


"h;m)G( "w , "v ,

"h)F( "w ,

"h;η)

4("n ⋅ "w)("n ⋅ "v)G : Geometricattenuationfunction(shadowing/masking)


!n = !w= !v (0)F

Specular Reflection (F)

51

(1)Schlickapproximation:F(!h, !w;η)=F(0;η)+(1−cosθh)5(1−F(0;η))

θh = cos−1!h ⋅ !w( )

F(0;η): Fresnelcolordependsontherefractiveindexη ofeachmaterial


"h;m)G( "w , "v ,

"h)F( "w ,

"h;η)

4("n ⋅ "w)("n ⋅ "v)F : Fresnelpolarizationeffect

(0) 0.5F =


CS580: Computer Graphics• If you want to learn more about advanced

rendering for global illumination, there is a course (CS580) delivered by Prof. Kim.

• http://vclab.kaist.ac.kr/cs580

52
http://vclab.kaist.ac.kr/cs580

Documents

CS380: Introduction to Computer Graphics Light Transport ...vclab.kaist.ac.kr/cs380/slide23-lighttransport.pdf · 18/06/11 2 Min H. Kim (KAIST) Foundations of 3D Computer Graphics,