1. Frustum Culling in OpenGL
Ref: MarkMoley.com
Fall 2006 revised

2. Culling Techniques
Fall 2006 revised

3. Outline Introduction Frustum Plane Extraction
View Frustum
Plane Equation
Frustum Plane Extraction
Frustum/Point/Sphere Inclusion Tests
Fall 2006 revised

4. 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
Fall 2006 revised

5. View Frustum (cont)
Fall 2006 revised

6. 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
Fall 2006 revised

7. Plane Equation
Fall 2006 revised

8. Normalized Plane Equation
Fall 2006 revised

9. Frustum Culling Idea: Tasks:
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: P M s u t
In frustum culling, s is usually a point on BV, in world coordinate
Fall 2006 revised

11. Rendering Pipeline (cont)M s u t
c : product of PM
In OpenGL,clip volume is [-1,1]3
Fall 2006 revised

12. Frustum Plane Extraction (Left)ux uy -1 1
Left
Fall 2006 revised

13. Left Plane (cont) The left plane in clip coordinate:ux uy -1 1
Left
The left plane in clip coordinate:
The corresponding equation in world coordinate:
Fall 2006 revised

14. Frustum Plane Extraction (Right)ux uy -1 1
Right
Fall 2006 revised

15. Right Plane (cont) The right plane in clip coordinate:ux uy -1 1
Right
The right plane in clip coordinate:
The corresponding equation in world coordinate:
Fall 2006 revised

16. Summary
Fall 2006 revised

17. The valid sides of all halfspaces are > 0
PointInFrustum Test
A point in frustumin all 6 halfspaces
The valid sides of all halfspaces are > 0
Fall 2006 revised

18. SphereInFrustum Test PointInOffsetFrustum Test
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
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. How can bounding volume hierarchies help?
View-frustum culling
Ray-tracing
Collision detection
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)
Fall 2006 revised

27. Example (Culling On: FPS 31.0)
Fall 2006 revised

28. Portal rendering World coordinates?! Plane representation in R3
Epilogue
Portal rendering
World coordinates?!
Plane representation in R3
Fall 2006 revised

29. 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
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. Portal Rendering M R K L Ki B C E 1 2 3 4 5 6 7 8 M B 1 4 E 2 C 7 3 6
Fall 2006 revised Ki

32. Portal Rendering
only render the other rooms in PVS if the “portal surfaces” are in sight
Fall 2006 revised

33. But aren’t s local coordinates?
Earlier, we said… P M s u t
But aren’t s local coordinates?
Fall 2006 revised

34. Recall Pipeline… Modelview Matrix Fall 2006 revised world coordinates
Transform
Viewing
Modelview
Matrix
world
coordinates
Fall 2006 revised

35. 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
Fall 2006 revised

36. Therefore … I t u s t = PMs M : viewing transform s : world coordinate
Model
Transform
Viewing
Modelview
Matrix
world
coordinates I t u s
t = PMs
M : viewing transform
s : world coordinate
Fall 2006 revised

37. Verification L R X Z
x–z+10
–x–z+10
Fall 2006 revised

38. Supplement: Representing a Plane in R3
x: pivot variable
Fall 2006 revised

39. Plane in R3 (cont) n x p
Fall 2006 revised

40. Summary Algebraic equation Parametric equation Vector equation
Depending on the application, select the most suitable equation
In/out test
Projection …
Fall 2006 revised

41. Screen-Space Bounding Box (SSBB)
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)
Fall 2006 revised

42. 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
Fall 2006 revised