Computer Graphics 2
Lecture 13:
Ray-Tracing Techniques
Dr. Benjamin Mora
Benjamin Mora
1
University of Wales
Swansea
Content
• What is Ray-Tracing?
• Mathematics for ray-tracing intersections.
• Common Ray-Tracing effects.
• Other Ray-Tracing Algorithms.
• Current Ray-Tracing Technology.
Benjamin Mora
2
University of Wales
Swansea
What Is Ray-Tracing ?
Benjamin Mora
3
University of Wales
Swansea
What is Ray Tracing ?
• The alternative to rasterization-based
methods (like OpenGL or DirectX).
– Image-Order algorithm.
– Every pixel is processed separately.
• Slower than rasterization methods in the
case of few primitives to render.
• Rarely used in games, widely used for realistic
computer graphics.
• www.povray.org.
Benjamin Mora
4
University of Wales
Swansea
What is Ray Tracing ?
Jaime Vives Piqueres (2004), http://www.povray.org/i/hof/office-13.jpg
Benjamin Mora
5
University of Wales
Swansea
What is Ray Tracing ?
Jaime Vives Piqueres (2001), http://www.povray.org/i/hof/office-13.jpg
Benjamin Mora
6
University of Wales
Swansea
What is Ray Tracing ?
image plane
primary
ray
view
location
secondary
rays
Benjamin Mora
7
University of Wales
Swansea
What is Ray Tracing ?
• Basic (or Naïve )Algorithm (Whitted 1980):
FOR every pixel DO
FOR every primitive in the scene DO
IF there is a ray/primitive intersection
test if this is the closest intersection computed so far.
If this is the case,
keep this primitive as the closest
primitive for the next tests.
• Basic Complexity:
– O(number of rays*number of primitives).
– Optimized algorithms can do much better (See the next lectures)!
Benjamin Mora
8
University of Wales
Swansea
Mathematics for
Ray-Tracing intersections
Benjamin Mora
9
University of Wales
Swansea
Parametric Ray/Plane Formulation
• A ray can be described with a parametric
formulation from a (starting) point and a direction
Pt   P0  t Vdirection
• A parametric plane can be described from one
point and two vectors:
Px, y   P0  x Vx  y Vy
• A plane can also be defined from a point and a
normal vector (here P0 and Vz).
Vz
Vy
Vx
P0
Benjamin Mora
10
University of Wales
Swansea
Ray/Plane Intersection
• We are looking for the point p(t) such that p(t)
belong to the plane h.
• h is defined with a point o and a normal n.
p0
vdir
n
p(t)
h
o
• If vdir perpendicular to n, then either 0 solution, or
an infinite number of solutions if p0 belongs to h
(the vector p0o perpendicular to n).
Benjamin Mora
11
University of Wales
Swansea
Ray/Plane Intersection
• Otherwise, there is only one point that is solution.
We have to find out t.
• The solution comes from the fact that the vector
p(t)o is perpendicular to n:
p(t )o  n  0 (1)
o  pt .  n  0 (2)
p0
vdir
pt   n  o  n
n
p(t)
 p0  t  vdir   n  o  n
h
o
(4)
t  vdir  n  o  n  p0  n  p0 o  n
t
Benjamin Mora
(3)
12
p0 o  n
vdir  n
(5)
(6)
University of Wales
Swansea
Ray/Cube Intersection
txmin txmax
p(t) = p0 +v•t
p0
tymax
tymin
v
tymin
tymax
t
txmin txmax
The ray intersects the cell only if max(txmin, tymin)<min(txmax,
tymax).
Finding out the exit face consists of finding out
minimum of (txmax, tymax).
Benjamin Mora
13
the
University of Wales
Swansea
Ray/Cube Intersection
• A cube can be defined with 6 planes (faces).
• 3 pairs of opposite faces (let’s reference them as
x,y,z) .
• A parametric ray p(t) intersects 2 opposite faces at
2 locations, with 2 respective parameters tmin and
tmax. (let’s suppose tmin<=tmax).
• The 6 faces are intersected with 6 different
parameters (txmin,txmax, tymin,tymax, tzmin,tzmax).
• The ray intersects the cube only if:
Max (txmin,tymin,tzmin)< Min(txmax,tymax,tzmax).
Benjamin Mora
14
University of Wales
Swansea
Ray/Sphere Intersection
• A sphere can be defined by a center o and a
radius r.
• Looking for the points p(t) such as:
norm(p(t)o)=r.
p0
vdir
p(t)
r
o
(1) p(t )o 2  r 2
(2) o  p0  t  vdir    p0 o  t  vdir   r 2
2
2
 
(3) t 2  vdir  t  2  p0 o  vdir   p0 o 2  r 2  0
2
Benjamin Mora
15
University of Wales
Swansea
Ray/Triangle Intersections
• Several ways to achieve this goal.
– Badouel’s Algorithm.
– Moller’s Algorithm (not shown).
– Other Algorithm (close to Segura et al. 98).
p0
vdir
p(t)
Benjamin Mora
16
University of Wales
Swansea
Badouel’s Algorithm
• First, find the intersection p(t) between the ray and
the plane containing the triangle.
• Then use the barycentric coordinates of the point to
sort the intersection out (2D algorithm).
p2
p(t)
p0 pt    . p0 p1   . p0 p2   0,   0,     1
p1
xt  x0   x1  x0    ( x2  x0 )
yt  y0    y1  y0    ( y2  y0 )
zt  z0   z1  z0    ( z2  z0 )
p0
Benjamin Mora
17
University of Wales
Swansea
Other Ray/Triangle Algorithms
• The edges of the triangles can be considered as
oriented.
• The Ray should always be “on the same side” of
the edges
• => Three triple products with the same sign.
p
vdir
p0
p1
p(t)
p2
Segura, R.J., Feito, F.R. Segura, R.J., Feito, F.R.: An algorithm for determining intersection
segment-polygon in 3D. Computer & Graphics, Vol.22, No.5, pp.587-592, 1998.
Benjamin Mora
18
University of Wales
Swansea
Other Ray/Triangle Algorithms
• There is an intersection Ray/Triangle only if:

 

 

 
Sign pp0  pp1  v  Sign pp1  pp2  v  Sign pp2  pp0  v
p
vdir
p0
• Fast algorithm.
p2
p1
p(t)
– Can take advantage of SIMD instructions on processor.
• But:
– Only in order to detect an intersection.
– Extra work must be done in order to get the tri-linear interpolating
weights (including a div) and the real intersection location if needed
(but not always required).
Benjamin Mora
19
University of Wales
Swansea
Triangle/Box Intersection
• Why?
– Bounding boxes are used much to speedup intersections.
• Standalone boxes.
• Spatial subdivision techniques.
– Therefore needed to construct such
acceleration structures.
20
Benjamin Mora
University of Wales
Swansea
Triangle/Box Intersection
• Solution 1: Working in the triangle plane.
• For every face of the cube:
– Find the face/triangle plane intersection.
• A line!
– Clip the triangle/polygon with this line. (Sutherland ).
• Intersection occurs if result not NULL!
• Complex and Costly.
Benjamin Mora
21
University of Wales
Swansea
Triangle/Box Intersection
• Solution 2: Use the Separating Axis
Theorem. (Akenine-Moller)
• No intersection occurs between 2
convex objects if there exists a
separating plane.
Benjamin Mora
22
University of Wales
Swansea
Triangle/Box Intersection (2D)
Benjamin Mora
23
University of Wales
Swansea
Triangle/Box Intersection (2D)
Benjamin Mora
24
University of Wales
Swansea
Triangle/Box Intersection (3D)
• 13 planes to be tested in 3D.
• For each plane:
– Compute distances to the triangle vertices
• Using the plane equation.
– Compute distances to the box vertices.
– No intersection if MIN of one distance set greater
than MAX of the other distance set.
• If condition is false for all 13 cases, an
intersection exists.
Benjamin Mora
25
University of Wales
Swansea
Triangle/Box Intersection (3D)
• Plane directions are determined from:
–
Ei X Tj
Where Ei are the directions of the edges of
the box, and Tj the directions of the edges
of the triangle. (9 cases)
– The faces of the box. (3 cases)
– The triangle plane normal. (1 case)
Benjamin Mora
26
University of Wales
Swansea
Simple Ray-Tracing Effects
Benjamin Mora
27
University of Wales
Swansea
Simple Ray-Tracing Effects
• The main operation is to find an intersection from a
given ray.
• This operation can be used (by casting secondary
rays) to compute simple effects that are more
difficult to implement with OpenGL like (but fog):
– Reflection.
– Refraction.
– Shadows.
Source: Min Chen
– Fog.
Benjamin Mora
28
University of Wales
Swansea
Simple Ray-Tracing Effects
• Basic Lighting model:
object
colour
I  I object
+
result of
tracing
reflective ray
result of
tracing
+
refractive ray
Rin
 km  I reflection
 kt  I refraction
Rfl
Benjamin Mora
km
29
kt
Rfr
University of Wales
Swansea
Simple Ray-Tracing Effects
• Shading can be done in the same way as to
OpenGL.
– But better methods exist.
N
V (-Rin)
R
L
Source: Min Chen

I  I a  ka  f d   I p  kd   L  N   k s  V  R n
Benjamin Mora
30

University of Wales
Swansea
Simple Ray-Tracing Effects
• Reflection:
Benjamin Mora
31
University of Wales
Swansea
Simple Ray-Tracing Effects
• Reflection:
N
Vincident
Vreflected
i
i
L
Vreflected  Vincident  2  N  N Vincident , N  1
Benjamin Mora
32
University of Wales
Swansea
Simple Ray-Tracing Effects
• Refraction:
R fr  (ri cos i  cos r )  N  riV
N
V (-Rin)
R
Snell’s Law
sin  r i
  ir
sin  i r
Benjamin Mora
Rfl
i  i
r
33
L
Rfr
University of Wales
Swansea
Simple Ray-Tracing Effects
• Total Internal Reflection:
N
V (-Rin)
critical
angle
critical
Rfr
Benjamin Mora
34
University of Wales
Swansea
Simple Ray-Tracing Effects
• Taking into account reflection and refraction several
times (recursively) requires an exponential number
of rays!
• The algorithm should stop after reaching a
maximum number of consecutive
reflections/refractions.
• Only reduced to some specific lighting effects.
Benjamin Mora
35
University of Wales
Swansea
Simple Ray-Tracing Effects
Benjamin Mora
36
University of Wales
Swansea
Simple Ray-Tracing Effects
• Fog effect (also possible with OpenGL).
– Depth Cue:
– Fog:
I out  I in  1  f   I fog  f
0
 d  Dmin
f 
 Dmax  Dmin
1
d  Dmin
Dmin  d  Dmax
d  Dmax
or
f  1
Benjamin Mora
1
 dD
e 


37
University of Wales
Swansea
Simple Ray-Tracing Effects
• Super Sampling can be used in order to improve
Anti-Aliasing.
non-adaptive
supersampling
Benjamin Mora
adaptive
supersampling
38
stochastic
sampling
University of Wales
Swansea
Simple Ray-Tracing Effects
Benjamin Mora
39
University of Wales
Swansea
Simple Ray-Tracing Effects
• The final color is the average of all the sub-sampled rays.
• Uniform Super-Sampling:
– Nice but not always needed.
• Adaptive Super-Sampling:
– Useful for the regions with large color transitions (high frequencies),
like object boundaries.
– Trade-off between quality and speed.
• Stochastic Super-Sampling:
– Take advantage of the visual system limitations.
Benjamin Mora
40
University of Wales
Swansea
Simple Ray-Tracing Effects
• Stochastic Super-Sampling:
– Proposed by Robert L. Cook.
• Stochastic Sampling in Computer Graphics, ACM
transaction on Graphics No. 5(1), January 1986, pp.
51-72.
– The ray location(s) are randomly chosen inside a pixel.
– Our visual system is less sensitive to noise than to
contours.
Benjamin Mora
41
University of Wales
Swansea
Other Ray-Tracing
Algorithms.
Benjamin Mora
42
University of Wales
Swansea
Other Ray-Tracing Algorithms.
• Pyramidal Ray-Tracing.
• Cone Tracing.
• Beam Tracing.
Benjamin Mora
43
University of Wales
Swansea
Pyramidal Ray-Tracing.
• Proposed by:
Whitted, Turner. An Improved Illumination Model for Shaded Display. C.A.C.M.
vol. 23, no. 6, June 1980, pp. 343-3~19.
• And improved by:
J. Genetti, D. Gordon and G. Williams. Adaptive supersampling in object space
using pyramidal rays. Computer Graphics Forum 17 (1998) 29-54.
• Main Idea: Replacing Rays by Pyramids to improve
aliasing.
image plane
primary
ray
view
location
Benjamin Mora
44
University of Wales
Swansea
Pyramidal Ray-Tracing.
J. Genetti, D. Gordon and G. Williams.
Adaptive supersampling in object space using
pyramidal rays. Computer Graphics Forum 17
(1998) 29-54.
The triangle will contribute here to
7/64 of the final pixel colour.
Benjamin Mora
45
University of Wales
Swansea
Pyramidal Ray-Tracing.
• If a pyramid only partially intersects a graphics
primitive (e.g., triangle), then the pyramid is
subdivided until a given depth is reached.
– The final pixel value is the (weighted) average of all the
samples inside the triangle.
– Non-intersecting pyramids keep on computing
intersections with the rest of the scene.
Benjamin Mora
46
University of Wales
Swansea
Cone Tracing.
• Proposed by:
John Amanatides. Ray Tracing with Cones. Siggraph 1984, pp. 129-135.
image plane
primary
ray
view
location
Partial
intersection
Benjamin Mora
47
University of Wales
Swansea
Cone Tracing.
• Similar to pyramidal ray-tracing, but the pyramid is
now replaced by a cone.
– Allows easier intersection computations.
– 1 cone per pixel, no subdivision.
• For a ray-triangle intersection:
– The cone is projected onto the triangle plane.
• Circular projection.
– If there is a partial intersection detected, the colour of the
triangle is weighted by the corresponding covering
percentage, and computing intersections will carry on.
Benjamin Mora
48
University of Wales
Swansea
Cone Tracing: Result
Benjamin Mora
49
University of Wales
Swansea
Beam Tracing.
• Proposed by:
Paul S. Heckbert and Pat Hanrahan. BEAM TRACING POLYGONAL
OBJECTS. Siggraph 1984, pp. 119-127.
• Takes advantages of spatial coherency between rays to
group them and save computations.
• Rays are distributed after Reflections and
Refractions according to the
reflection/refraction coefficients.
• Constructs a beam tree.
• Outdated.
Benjamin Mora
50
University of Wales
Swansea
Current Ray-Tracing Technology
Benjamin Mora
51
University of Wales
Swansea
Current Ray-Tracing Technology
• Ray-Tracing is now used a lot for realistic rendering
of scenes.
– Not just Reflections/Refractions.
• Has a better complexity than OpenGL based
renderings (when optimized).
– Scalable.
• A simple ray representation (no cone or pyramid) is
often used.
• Therefore, researchers have recently tried to
hardware accelerate it.
Benjamin Mora
52
University of Wales
Swansea
Current Ray-Tracing Technology
• 3 different approaches:
– Cluster based CPU approach.
• Use SIMD instructions with packed rays.
• A lot of threads in parallel.
Ingo Wald, Timothy J. Purcell, Jörg Schmittler, Carsten Benthin and Philipp Slusallek.
Realtime Ray Tracing and its use for Interactive Global Illumination. Eurographics
2003 State of the Art Report.
– GPU Based.
• Not very efficient because not really designed for RT.
Tim Foley, Jeremy Sugerman. KD-Tree Acceleration Structures for a GPU Raytracer.
In Proceedings of Graphics Hardware 2005.
– Hardware (FPGA) accelerated ray-tracing.
• Seems to be efficient.
• Need to compute a spatial subdivision tree first, which is not
really appropriate to dynamic scenes.
University of Wales
Benjamin Mora
53
Swansea
Current Ray-Tracing Technology
• The RPU unit. See Video !!!
Sven Woop, Jorg Schmittler, and Philipp Slusallek. RPU: A Programmable Ray
Processing Unit for Realtime Ray Tracing. Siggraph 2005.
Benjamin Mora
54
University of Wales
Swansea