Frustum Culling in OpenGL
Ref: MarkMoley.com
Fall 2006 revised
1
Culling Techniques
Fall 2006 revised
2
Outline
Introduction


View Frustum
Plane Equation
Frustum Plane Extraction
Frustum/Point/Sphere Inclusion Tests
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Definition (View Frustum)
the volume of space that includes everything
visible from a given viewpoint
defined by six planes arranged in the shape
of a pyramid with the top chopped off
If a point is inside this volume then it's in the
frustum and it's visible, and vice versa

Visible here mean potentially visible. It might be
behind another point that obscures it, but it's still
in the frustum. … the scope of occlusion culling
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View Frustum (cont)
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5
Definition (Plane)
Divide the space into two halves;
extends to infinity
any given point is (in front of | behind |
on) the plane.
In R3, a plane is defined by four
numbers:

A plane equation: Ax + By + Cz +D = 0
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Plane Equation
 : Ax  By  Cz  D  0
Signed distancefrom( x0 , y0 , z0 ) to
d
Ax0  By0  Cz0  D
A2  B 2  C 2
normalvector:  A B C 
T
 0 if ( x0 , y0 , z0 ) is on normalside

d  0
if ( x0 , y0 , z0 ) is on 
 0 if ( x , y , z ) is behind normal
0
0
0

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Normalized Plane Equation
 : Ax  By  Cz  D  0
normalvector:  A B C  , A2  B 2  C 2  1
Signed distancefrom( x0 , y0 , z0 ) to
T
d  Ax0  By0  Cz 0  D
Signed distance to origin : D
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Frustum Culling
Idea:


If object is not in frustum, do not send it through
pipeline (it is not visible)
Wrap the object in a bounding volume; test the
bounding volume against frustum (in world
coordinate system)
 Sphere, bounding boxes, …
Tasks:


Frustum plane extraction
Inclusion tests for different bounding volumes
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Frustum Plane Extraction
The rendering pipeline:
s
M
P
t
u
s : object coordinate
M : model view matrix u : normalizeddevice coordinate
t  PMs
In frustum culling, s is
usually a point on BV, in
P : projectionmatrix
world coordinate
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Rendering Pipeline (cont)
s
M

Object coordinate: s  s x
 t x  c0
t   c
 y   1
 t z   c2
  
t w  c3
c4
c5
c8
c9
c6
c10
c7
c11
Clip coordinate:

tx
tw
sy
u
t
P
sz
sw

T
c : product
of PM
c12   s x 
c13   s y 
c14   s z 
 
c15   sw 
ty
tw
  u
tz T
tw
In OpenGL,clip
volume is [-1,1]3
x
uy
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uz

T
11
Frustum Plane Extraction (Left)
 1  u x  1 
tx
tw
uy
 t w  0  t w  t x  t x  t w  0
 t x  c0
t   c
 y   1
 t z  c2
  
t w  c3
c
0,
c4
c8
c5 c9
c6 c10
c7
c11
c12   s x 
c13   s y 
c14   s z 
 
c15   sw 
 s   c3,  s   0  c0,  c3,  s  0
-1
1
ux
Left
where ci , is t hei th row in C
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Left Plane (cont)
uy
The left plane in
clip coordinate:
1  ux
The corresponding
equation in world
coordinate:
all s  x
c
y
c0 c4 c8
c c c
9
C 1 5
c2 c6 c10

c3 c7 c11
c12 
c13 
c14 

c15 
z 1 thatsatisfy the equation
0 ,  c3,  s  0  c0 ,  c3,   x
y
z 1  0
-1
1
Left
T
c0  c3 x  c4  c7  y  c8  c11 z  c12  c15   0
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ux
Frustum Plane Extraction (Right)
ux  1 
uy
tx
1
tw
 tw  0 t x  tw  tw  t x  0
 t x  c0
t   c
 y   1
 t z  c2
  
t w  c3
c4
c8
c5 c9
c6 c10
c7
c11
c12   s x 
c13   s y 
c14   s z 
 
c15   sw 
c
3,  s   c0 ,  s   0  c3,  c0 ,  s  0
-1
1
ux
Right
where ci , is t hei th row in C
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Right Plane (cont)
uy
The right plane in
clip coordinate:
ux  1
The corresponding
equation in world
coordinate:
all s  x
c
y
-1
Right
ux
1
z 1 thatsatisfy the equation
3,  c0 ,  s  0  c3,  c0 ,   x
y
z 1  0
T
c3  c0 x  c7  c4  y  c11  c8 z  c15  c12   0
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Summary
c3  c0 x  c7  c4  y  c11  c8 x  c15  c12   0 [left ]
c3  c0 x  c7  c4  y  c11  c8 x  c15  c12   0 [right ]
c3  c1 x  c7  c5  y  c11  c9 x  c15  c13   0 [bot t om]
c3  c1 x  c7  c5  y  c11  c9 x  c15  c13   0 [t op]
c3  c2 x  c7  c6  y  c11  c10 x  c15  c14   0 [near]
c3  c2 x  c7  c6  y  c11  c10 x  c15  c14   0 [far]
c0
c
C 1
c2

c3
c4
c8
c5 c9
c6 c10
c7
c11
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c12 
c13 
 [Proj][MV]
c14 

c15 
16
PointInFrustum Test
A point in frustumin all 6 halfspaces
The valid sides
of all halfspaces
are > 0
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SphereInFrustum Test
p
p
PointInOffsetFrustum Test
Minkowski sum of sphere
and frustum
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Minkowski Sum
Fall 2006 revised
Coordinate
dependent!
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Signed Distance to Left Plane
Plane equations are further normalized
to facilitate distance computation
d ( x0 , y0 , z0 )  frustum[0]x0  frustum[1]y0  frustum[2]z0  frustum[3]
d  0 if pointis outside frustum
uy
d<0
-1
Fall 2006 revised
Left
1
ux
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Remarks
Use bounding spheres

Can be conservative if the object is slender
Use hierarchical bounding volume
where appropriate (see next page)
BoxInFrustum:



complicated and expensive
AABB: already so
OBB: more so
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How can bounding volume
hierarchies help?
View-frustum
culling
Ray-tracing
Collision
detection
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How can bounding volume
hierarchies help?
View-frustum
culling
Ray-tracing
Collision
detection
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How can bounding volume
hierarchies help?
View-frustum
culling
Ray-tracing
Collision
detection
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How can bounding volume
hierarchies help?
View-frustum
culling
Ray-tracing
Collision
detection
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Example (Culling Off: FPS 15.5)
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Example (Culling On: FPS 31.0)
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Epilogue
Portal rendering
World coordinates?!
Plane representation in R3
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Indoor Scenes
Similar to building
walkthrough
Occlusions in scene
are common (culling
important)
Geometric database
can be huge;
preprocessing is
required to facilitate
smooth viewing
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Preprocessing for Indoor Scenes
Potential visible set (PVS)

only load the rooms that are visible from the current
room
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1
M
B
2
4
3
C
5
K
M
8
Ki
Portal Rendering
B
1
E
L
6
R
7
4
2
C
3
R
E
7
6
5
L
K
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Ki 31
Portal Rendering
only render the other rooms in PVS if
the “portal surfaces” are in sight
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Earlier, we said…
s
M
But aren’t s local
coordinates?
P
t
u
s : world coordinate
u : clip coordinate
t  PMs
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Recall Pipeline…
Modelview
Matrix
Model
Transform
Viewing
Transform
world
coordinates
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Frustum Culling Scenario
the scene is (mostly)
static and specified
in their world
coordinate (the
modeling transform
is identity)
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the viewer navigates
around them,
changing the
viewing transform
only
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Therefore …
Modelview
Matrix
Viewing
Transform
Model
Transform
I
world
coordinates
s
t
u
t = PMs
M : viewing transform
s : world coordinate
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Verification
x–z+10
R –x–z+10
L
X
Z
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Supplement: Representing a Plane in R3
x yz20
Complet esolut ionof Ax  b
null space solut ion
x yz 0
particularsolution
 x    2
 y   0 
 x
1 
 1
   
 y   c 1   d  0  , c , d  R  z   0 
   
 
 
 
 z 
0
 1 
completesolution
x: pivot variable
 x   2 1
 1
 y    0   c 1  d  0 , c, d  R
     
 
 z   0  0
 1 
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Plane in R3 (cont)
x yz20
completesolution
 x   2 1
 1
 y      c 1  d  0 , c, d  R
  0  
 
 z   0  0
 1 
  2
1
p   0 , n   1   x  p   n  0
 0 
 1 
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n
x
p
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Summary
Algebraic equation
Parametric equation
Vector equation
Depending on the application, select
the most suitable equation



In/out test
Projection
…
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Useful in GPU-assisted collision
detection applications
Screen-Space Bounding Box (SSBB)
Compute and display
the screen-space
bounding box of an
AABB (axis-aligned
bounding box)
Convert world
coordinates to clip
coordinates,
perspective-divide to
get normalized device
coordinate (and then
window coordinates)
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Display Viewing Frustum
Useful for projective
texturing (showing
where the projector is)
Useful for illustrating
frustum culling
Given [-1,1]3 clip
coordinates, determine
their corresponding
world coordinates to
render
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