2IV60 Computer graphics
set 11: Ray-tracing
Jack van Wijk
TU/e
John Tsiombikas
Caustic Graphics
Overview
Ray-tracing:
• Based on geometric optics;
• Mirroring, transparency, cast shadows;
• Many other effects…
• And very time consuming…
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Ray casting (reprise)
Normalized view volume
z
Algorithm:
Per pixel:
- Calculate intersections
- Determine first
yv
xv
front =
visible
zv
pixel
View plane
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Ray-tracing principle 1
light source
Ray-tracing:
opaque
projection point
mirroring
view plane
Mirroring and
transparent
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Ray-tracing principle 2
light source
Ray-tracing:
projection point
pixel
view plane
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Ray-tracing algorithm 1
var fbuf: array[N,N] of color;
{ frame-buffer }
For all 1<= i, j <=N do { for all pixels }
fbuf [i,j] := RayTrace((0,0,0),(i,j,d), 1)
projection point
pixel
d
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Ray-tracing algorithm 2
function RayTrace(P: point; u:vector; level: integer): color;
if level > MaxLevel or GetIntersections (P, u, t, N) = 0
then return BkColor;
level := level+1;
P
col := famb*Iamb;
u
Q := P + ut;
if (L Q).N > 0 and GetIntersections (Q, L  Q) = 0
then col := col + fdif*I
+ fspec*I*power(N.H(u,N), nspec);
col := col + fspec*Raytrace(Q, R(u,N), level);
col:= (1–ftra)*col + ftra*Raytrace(Q, T(u,N), level);
return col;
N
H
R
L
Q
T
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Calculation intersections
function GetIntersections (P: point; u: vector; t: array of real;
N: array of vector): integer;
{ Get the intersections of a line with the scene }
For all objects do:
Calculate intersection points with surface;
Store the results in the arrays t and N,
sorted for t.
Return the number of intersection points.
Equation line:
P
u
t
X  P  ut
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Line/Sphere intersections
Equation line:
X  P  ut (1)
Equation sphere:
(X  C)  (X  C)  r
2
(2)
Substitute (1) in (2):
P
u
t
Q
(P  ut  C)  ( P  ut  C)  r 2 , with P  P  C :
(P  ut )  (P  ut )  r 2  0, or
C
(u  u)t 2  ( 2P  u)t  ( P  P  r 2 )  0, or
t1,2
 P  u  ( P  u) 2  (u  u)(P  P  r 2 )

u u
r
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Line/polygon intersection
Equation line:
X  P  ut (1)
Equation plane polygon:
N  X   D (2)
Substitute (1) in (2):
N
P
u
t
Q
N  (P  ut )   D, or
N  P  N  ut   D, hence
t
 D NP
Nu
Check if Q is inside polygon
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Ray-tracing acceleration 1
Limit intersection calculations:
Hierarchy of Bounding Volumes
Bounding Boxes
Bounding Spheres
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Ray-tracing acceleration 2
Limit intersection calculations:
Space-subdivision
Regular
Adaptive (Octtree, quadtree)
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Aliasing 1
• Generic problem of raster graphics
• Shapes are continuous, pixels are discrete
• Jagged edges, Moiré patterns, flickering textures, …
• Especially annoying during animation
• More pixels?
• More samples!
Aliased
Anti-aliased
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Aliasing 2
ray
pixel
1 ray/pixel: blocky patterns
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Anti-aliasing 1
ray
pixel
average 5 rays/pixel:
smoother transitions
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Anti-aliasing 2
ray
pixel
adaptive sampling:
more efficient, not fail save
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Anti-aliasing 3
ray
pixel
dense patterns
give problems
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Anti-aliasing 3
ray
pixel
dense patterns
give problems
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Anti-aliasing 3
ray
pixel
dense patterns
give problems
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Anti-aliasing 4
ray
pixel
random sampling gives
a better result
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Anti-aliasing 4
ray
pixel
random sampling gives
a better result
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Anti-aliasing 4
ray
pixel
random sampling gives
a better result
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More optical effects…
Imperfect reflection
Non-point light source
Cook, Porter, Carpenter, 1984
Distributed ray tracing 1
Motion blur
Cook, Porter, Carpenter, 1984
Distributed ray tracing 1
• Monte Carlo sampling
• Typically 16 rays/pixel;
• Each ray is shifted along a small
random vector (jitter)
• rough reflecting surfaces, nonpoint light sources, depth of field,
motion blur
ray
pixel
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Distributed ray tracing 2
R
L
P
• rough reflecting surfaces, nonpoint light sources, depth of
field, motion blur
u
T
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Radiosity
• Surface-surface
interreflection
• Physically based simulation
• Calculate for each polygon
the balance between incoming
and outgoing light
• Important for architecture
and indoor scenes
Cornell box
Cohen et al., 1988
Finally
• That concludes our tour through the world
of 2D and 3D graphics