distribution ray-tracing
1
distribution ray-tracing
use many rays to compute average values over pixel
areas, time, area lights, reflected directions, ...
2
antialiasing origin
compute average color subtended by a pixel
pixel center
C = raytrace(E, Q − E)
pixel average
C =
E
1
Ap
d
Q
BP
raytrace(E, Q − E)
˝d A
Q
: camera origin; Q : point on image plane; P : pixel of area
A_p
3
antialiasing by deterministic int.
subdivide the pixel in squares
cast rays through square centers
average result
[Shirley]
4
deterministic antialias pseudocode
[Shirley]
//antial
iasingpixel(i,j)-deterministic
c=0
fo
rsx=0tons
fors
y=0tons
u=(i+(sx+
0.5)/ns)/width
v=(j+(sy+
0.5)/ns)/height
Q=imagePlan
ePoint(u,v)
c+=raytrace
(E,Q-E)
c/
=ns*n
s
5
antialiasing by monte carlo estimation
pick random points in pixel area
cast rays trough them
average result
[Shirley]
6
monte carlo antialiasing pseudocode
[Shirley]
//antial
iasingpixel(i,j)-montecarlo
c=0
fo
rs=0tons*ns
(rx,r
y)=random2
d()
u=(
i+rx)/wi
dth
v=(
i+ry)/he
ight
Q=i
magePoint(u,
v)
c+=raytrace(E,Q
-E)
c/
=ns*n
s
7
monte carlo antialiasing pseudocode
[Shirley]
//antial
iasingpixel(i,j)-stratifiedmontecarlo
c=0
fo
rsx=0tons
fors
y=0tons
(
rx,ry)=ran
dom2d()
u=(i+(sx+
rx)/ns)/width
v=(j+(sy+
ry)/ns)/height
Q=imagePoin
t(u,v)
c+=raytrace
(E,Q-E)/ns*ns
8
raytraced images are too "clean"
soft shadows come from area light
raytracing only supports point ligts
[Jason Waltman / jasonwaltman.com]
9
raytraced images are too "clean"
blurry reflections come from rough materials
raytracing only supports perfectly sharp mirror
[Jensen]
10
raytraced images are too "clean"
depth of field come from lens system
raytracing only support pinhole camera
[Jason Waltman / jasonwaltman.com]
11
raytraced images are too "clean"
motion blur come from shutter speed
raytracing only support infinitely fast shutter speed
[Jason Waltman / jasonwaltman.com]
12
soft shadows origin
area ligts create penumbras
light is only partially visible from a given point
want to compute how much light hits the point
[Shirley]
13
approximate soft shadows principle
point light
C = Cl
˝V (P, S) ˝shading(P, S)
area light
C =
P
Cl
AL
˝d B
S
L
V (P, S)
˝shading(P, S) ˝dA
S
: point on the surface; S : point on the light; V : visibility
function (0 or 1); L : light of area AL ; Cl : total light intensity
14
soft shadows by deterministic int.
approximate area light as a set of point lights
equivalent to quadrature rule
for each point, compute shadows and lighting
average results
[Shirley]
15
soft shadows by monte carlo est.
use Monte Carlo integration
pick random points on the light
easy for quad lights, hard (but possible) for others
compute shadows and lighting
average results
[Shirley]
16
soft shadows by monte carlo est.
for quads
Si = Sc + (0.5 − ri,1 )lu + (0.5 − ri,2 )lv
Ë Ì
C
Sc
=
Cl
N
˝K
N
i
V (P, Si )
˝shading(P, S )
i
: light source center; u, v : light source tangent vectors; l :
light source size; ri : uniformly sampled random 2d vector in
[0, 1]
2
; N : total number of samples
17
how many samples?
1 sample
9 samples
36 samples
[Bala]
100 samples
18
blurry reflections origin
real materials often have blurred reflections
light is scattered around a set of directions
want to compute how much light reaches the surface
along these directions
[Shirley]
19
approx. blurry reflections principle
mirror reflection
C = kr
˝raytrace(P, R)
blurry reflection
C =
P
kr
A°
˝d B°
R
raytrace(P, R)
˝d A
R
: point on the surface; R : reflected direction; ° : solid angle
where object is reflecting; kr : reflection coefficient
20
blurred reflection by monte carlo est.
compute reflected direction R
pick random direction R around R
′
compute reflected color along R
′
average results
[Shirley]
21
blurred reflection by monte carlo est.
for quads
Ri =
Ë Ì
C
R
R+(0.5−ri,1 )lu+(0.5−ri,2 )lv
=
kr
N
˝K
| Ri |
N
i
raytrace(P, Ri )
: reflected direction; u, v : vectors orthogonal to R; l : blur
size; ri : uniformly sampled random 2d vector in [0, 1]2 ; N :
total number of samples
22
depth-of-field origin
images in eyes and cameras are formed by lenses
not all rays converge to a point
so only some object appear sharp (in focus)
while all other objects do not (out of focus)
[Shirley]
23
approximate depth-of-field principle
pinhole
C =
1
AP
˝d
Q
BP
raytrace(E, Q − E)
˝d A
Q
dof
C =
E
1
AP AL
˝d
Q
BP
dB
F
L
rt(F, Q − F)
˝d A ˝d A
F
Q
: camera origin; Q : point on image plane; F : point on "lens"
plane around E; P : pixel of area AP ; L : lens of area AL
24
depth-of-field by monte carlo est.
pick random points on film and image plane
compute color from aperture point toward image
average results
[Shirley]
25
depth-of-field by monte carlo est.
for quads
Fi = E + (0.5 − si,1 )la u + (0.5 − si,2 )la v
Qi = E + (x + 0.5 − ri,1 )lp u + (y + 0.5 − ri,2 )lp v − nz
Ë Ì
C
u, v
=
1
N
˝K
N
i
raytrace(Fi , Qi − Fi )
: vectors parallel to image plane; lp , la : pixel / aperture
size; n : distance to image plane; ri , si : uniformly sampled
random 2d vectors in [0, 1]2 ; N : total number of samples
26
motion blur origin
takes time for images to form on camera/eye
during that time the camera/eye is open
sensors take the "average" of what is happening
but objects move around in that time
27
approximate motion blur principle
no blur
C(t) = raytrace(E, d, St )
motion blur
C(t,
E
š t)
=
1
št
˝d B
s
[t,t+š t]
raytrace(E, d, S s )
˝d l
s
: any ray origin; d : any ray direction; S t : scene at time t ; t :
time when shutter opens; š t : time that shutter stays open;
št
= tk+1 − tk
: for animation, time between frames
28
motion blur by monte carlo estimation
for any ray (camera, shadow, reflection), pick random time in
the shutter interval
update object transformations for picked time
compute intersection
average results
29
motion blur by monte carlo estimation
for aliased camera rays
Ë Ì
C
E
=
1
N
˝K
N
i
raytrace(E, Q − E, S t )
i
: camera origin; Q : point on image plane; ti : uniformly
sampled random variable in [0, 1]; N : total number of
samples
30
combining estimations
e.g. compute soft shadows and antialias the pixel
C =
Cl
AP AL
˝
∫
Q
∫
V (P(Q), S)
˝sh(P(Q), S) ˝dA ˝dA
S
Q
S
Q
B
AP , S
[Bala]
B
AL
31
combining estimations: path tracing
idea: randomize the whole path
compute all the integrals at once!
[Bala]
32