Fall 2006 revised 1

Frustum Culling in OpenGL

Ref: MarkMoley.com

Fall 2006 revised 2

Culling Techniques

Fall 2006 revised 3

Outline

Introduction View Frustum Plane Equation

Frustum Plane ExtractionFrustum/Point/Sphere Inclusion Tests

Fall 2006 revised 4

Definition (View Frustum)

the volume of space that includes everything visible from a given viewpointdefined by six planes arranged in the shape of a pyramid with the top chopped offIf 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

Fall 2006 revised 5

View Frustum (cont)

Fall 2006 revised 6

Definition (Plane)

Divide the space into two halves; extends to infinityany 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

Fall 2006 revised 7

Plane Equation

normal behind is ),,( if0

on is ),,( if0

side normalon is ),,( if0

: vectornormal

to),,( from distance

0:

000

000

000

222

000

000

zyx

zyx

zyx

d

CBA

CBA

DCzByAxd

zyxSigned

DCzByAx

T

Fall 2006 revised 8

Normalized Plane Equation

D

DCzByAxd

zyxSigned

CBACBA

DCzByAxT

:origin todistance Signed

to),,( from distance

1, : vectornormal

0:

000

000

222

Fall 2006 revised 9

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

Fall 2006 revised 10

Frustum Plane Extraction

The rendering pipeline:

PMst

s

coordinate device normalized :u

coordinateobject :

matrix viewmodel :

matrix projection:

M

P

PMs ut

In frustum culling, s is usually a point on BV, in world coordinate

In frustum culling, s is usually a point on BV, in world coordinate

Fall 2006 revised 11

Rendering Pipeline (cont)

Tzyx

T

tt

t

t

tt

w

z

y

x

w

z

y

x

Twzyx

uuu

s

s

s

s

cccc

cccc

cccc

cccc

t

t

t

t

sssss

w

z

w

y

w

x

:coordinate Clip

:coordinateObject

151173

141062

13951

12840

PMs ut

In OpenGL,clip volume is [-

1,1]3

c : product of

PM

Fall 2006 revised 12

Frustum Plane Extraction (Left)

00

11

wxxww

w

xx

ttttt

t

tu

w

z

y

x

w

z

y

x

s

s

s

s

cccc

cccc

cccc

cccc

t

t

t

t

151173

141062

13951

12840 ux

uy

-1 1

Left Cic

sccscscth

i in row theis where

00

,

,3,0,3,0

Fall 2006 revised 13

Left Plane (cont)The left plane in clip coordinate:

The corresponding equation in world coordinate:

0

010

equation esatisfy th that 1

15121187430

,3,0,3,0

cczccyccxcc

zyxccscc

zyxsallT

xu 1 ux

uy

-1 1

Left

151173

141062

13951

12840

cccc

cccc

cccc

cccc

C

Fall 2006 revised 14

Frustum Plane Extraction (Right)

00

11

xwwxw

w

xx

ttttt

t

tu

ux

uy

-1 1

Right Cin row theis where

00

,

,0,3,0,3

thi ic

sccscsc

w

z

y

x

w

z

y

x

s

s

s

s

cccc

cccc

cccc

cccc

t

t

t

t

151173

141062

13951

12840

Fall 2006 revised 15

Right Plane (cont)

The right plane in clip coordinate:

The corresponding equation in world coordinate:

1xu ux

uy

-1 1

Right

0

010

equation esatisfy th that 1

12158114703

,0,3,0,3

cczccyccxcc

zyxccscc

zyxsallT

Fall 2006 revised 16

Summary [far] 0

[near] 0

[top] 0

[bottom] 0

[right] 0

[left] 0

141510116723

141510116723

13159115713

13159115713

12158114703

12158114703

ccxccyccxcc

ccxccyccxcc

ccxccyccxcc

ccxccyccxcc

ccxccyccxcc

ccxccyccxcc

MV][[Proj]

151173

141062

13951

12840

cccc

cccc

cccc

cccc

C

Fall 2006 revised 17

PointInFrustum Test

A point in frustumin all 6 halfspaces

The valid sides of all halfspaces

are > 0

Fall 2006 revised 18

SphereInFrustum Test

p p

PointInOffsetFrustum Test

Minkowski sum of sphere and frustum

Minkowski sum of sphere and frustum

Fall 2006 revised 19

Minkowski Sum Coordinate

dependent!

Fall 2006 revised 20

Signed Distance to Left Plane

Plane equations are further normalized to facilitate distance computation

ux

uy

-1 1

Left

frustum outside ispoint if 0

frustum[3]frustum[2]frustum[1]frustum[0]),,( 000000

d

zyxzyxd

d<0

Fall 2006 revised 21

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

Fall 2006 revised 22

How can bounding volume hierarchies help?

View-frustum culling

Ray-tracing

Collision detection

Fall 2006 revised 23

View-frustum culling

Ray-tracing

Collision detection

How can bounding volume hierarchies help?

Fall 2006 revised 24

How can bounding volume hierarchies help?

View-frustum culling

Ray-tracing

Collision detection

Fall 2006 revised 25

How can bounding volume hierarchies help?

View-frustum culling

Ray-tracing

Collision detection

Fall 2006 revised 26

Example (Culling Off: FPS 15.5)

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Example (Culling On: FPS 31.0)

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Epilogue

Portal renderingWorld coordinates?!Plane representation in R3

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Indoor Scenes

Similar to building walkthroughOcclusions in scene are common (culling important)Geometric database can be huge; preprocessing is required to facilitate smooth viewing

Fall 2006 revised 30

Preprocessing for Indoor Scenes

Potential visible set (PVS) only load the rooms that are visible from the

current room

Fall 2006 revised 31

M

RK

L

Ki

B

C

E1

23

4

5

6

7

8

M

R

B

C

L

K

E

Ki

1

6

2

3

4

5

7

8

Portal RenderingPortal Rendering

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Portal Rendering

only render the other rooms in PVS if the “portal surfaces” are in sight

Fall 2006 revised 33

Earlier, we said…

PMst

s

coordinate clip :u

coordinate world:

PMs ut

But aren’t s local

coordinates?

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Recall Pipeline…

Model Transform

Viewing Transform

ModelviewMatrix

worldcoordinates

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Frustum Culling Scenario

the scene is (mostly) static and specified in their world coordinate (the modeling transform is identity)

the viewer navigates around them, changing the viewing transform only

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Therefore …

Model Transform

Viewing Transform

ModelviewMatrix

worldcoordinates

I

s

t u

t = PMs M : viewing transform s : world coordinate

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VerificationL R

X

Z

x–z+10 –x–z+10

Fall 2006 revised 38

Supplement: Representing a Plane in R3

Rdcdc

z

y

x

zyx

bAx

zyx

,,

1

0

1

0

1

1

0

solution space null

ofsolution Complete

02

Rdcdc

z

y

x

z

y

x

,,

1

0

1

0

1

1

0

0

2

solution complete

0

0

2

solution particular

x: pivot variable

Fall 2006 revised 39

Plane in R3 (cont)

0

1

1

1

,

0

0

2

,,

1

0

1

0

1

1

0

0

2

solution complete

02

npxnp

Rdcdc

z

y

x

zyx

p

n

x

Fall 2006 revised 40

Summary

Algebraic equationParametric equationVector equationDepending on the application, select the most suitable equation In/out test Projection …

Fall 2006 revised 41

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)

Useful in GPU-assisted collision detection applications

Fall 2006 revised 42

Display Viewing Frustum

Useful for projective texturing (showing where the projector is)Useful for illustrating frustum cullingGiven [-1,1]3 clip coordinates, determine their corresponding world coordinates to render