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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)
xv
yv
zv
Normalized view volume
View planepixel
front =
visible
Algorithm:
Per pixel:
- Calculate intersections
- Determine firstz
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Ray-tracing principle 1
Ray-tracing:
view plane
projection point
light source
Mirroring and
transparent
mirroring
opaque
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Ray-tracing principle 2
Ray-tracing:
view plane
projection point
pixel
light source
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Ray-tracing algorithm 1
var fbuf: array[N,N] of color; { frame-buffer }
For all 1 0 and GetIntersections (Q, L − Q) = 0
then col := col + fdif*I
+ fspec*I*power(N.H(u,N), nspec);
Ray-tracing algorithm 2
R
LPu
H
T
N
Q
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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:
Calculation intersections
Pu
tuPX +=
t
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Line/Sphere intersections
Pu
Q
C
r
(1) tuPX +=
Equation line:
t
(2) )()( 2r=−⋅− CXCX
Equation sphere:
uu
PPuuuPuP
PPuPuu
uPuP
CPPCuPCuP
⋅
−∆⋅∆⋅−⋅∆±⋅∆−=
=−∆⋅∆+⋅∆+⋅
=−+∆⋅+∆
−=∆=−+⋅−+
))(()(
or ,0)()2()(
or ,0)()(
:with ,)()(
22
1,2
22
2
2
rt
rtt
rtt
rtt
Substitute (1) in (2):
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Line/polygon intersection
Pu
N
(1) tuPX +=
Equation line:
t
(2) D−=⋅ XN
Equation plane polygon:
uN
PN
uNPN
uPN
⋅
⋅−−=
−=⋅+⋅
−=+⋅
Dt
Dt
Dt
hence ,
or ,)(
Substitute (1) in (2):
Check if Q is inside polygon
Q
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Ray-tracing acceleration 1
Limit intersection calculations:
Hierarchy of Bounding Volumes
Bounding SpheresBounding Boxes
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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!
H&B 21-1:639-647Aliased Anti-aliased
Aliasing 2
pixelray
1 ray/pixel: blocky patterns
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Anti-aliasing 1
pixelray
average 5 rays/pixel:
smoother transitions
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Anti-aliasing 2
pixelray
adaptive sampling:
more efficient, not fail save
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Anti-aliasing 3
pixelray
dense patterns
give problems
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Anti-aliasing 3
pixelray
dense patterns
give problems
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Anti-aliasing 3
pixelray
dense patterns
give problems
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Anti-aliasing 4
pixelray
random sampling gives
a better result
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Anti-aliasing 4
pixelray
random sampling gives
a better result
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Anti-aliasing 4
pixelray
random sampling gives
a better result
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More optical effects…
Cook, Porter, Carpenter, 1984
Imperfect reflection
Non-point light source
Distributed ray tracing 1
Cook, Porter, Carpenter, 1984
Motion blur
Distributed ray tracing 1
• Monte Carlo sampling
• Typically 16 rays/pixel;
• Each ray is shifted along a small
random vector (jitter)
• rough reflecting surfaces, non-
point light sources, depth of field,
motion blur
pixelray
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Distributed ray tracing 2
• rough reflecting surfaces, non-
point light sources, depth of
field, motion blur
R
Pu
T
L
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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
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Cohen et al., 1988
Finally
• That concludes our tour through the world
of 2D and 3D graphics
