1. View-dependent Adaptive Tessellation of Spline Surfaces
Jatin Chhugani & Subodh Kumar
Johns Hopkins University

2. Motivation Why use Splines? CAD/CAM , Entertainment Industry
Medical Visualization
Examples
Submarines
Animation Characters
Human body especially the heart and brain

3. Garden Model ( 38,646 patches)

4. Splines Non-Uniform Rational B-Spline (NURBS) Bezier patch (rational)
Degree m x n
Domain space (u, v)  [0,1] x [0,1]
For 0  i  m, 0  j  n,
Control points : pij
Weights: wij

5.   wijpij Bmi(u) Bnj (v) i=0 j=0
m n
  wijpij Bmi(u) Bnj (v)
i=0 j=0
F(u,v) =
  wij Bmi(u) Bnj (v)
where
Bernstein function Bni (t) = t i (1- t)n-i

6. Rendering Splines Ray tracing Pixel level surface subdivision
J. Kajiya, T. Nishita et al., J. Whitted
Pixel level surface subdivision
E.Catmull, M. Shantz et al.
Scan-line based
J. Blinn, J. Lane et al., J. Whitted
Polygonal Approximations
Forward (static)
Backward (dynamic)

7. Forward Technique Backward Technique View-dependent tessellation
Incremental triangulation
Backward Technique
Pre-tessellate patches densely
Apply polygon simplification

8. Forward Technique Backward Technique Advantage
Allows arbitrary precision
Disadvantage
Significant computation overhead
Backward Technique
Advantage
Low run-time computation
Disadvantages
Large storage requirement
Upper limit on detail of the model

9. Our Approach Hybrid of backward and forward techniques
Pre-compute domain samples
Select samples and triangulate dynamically
Generate additional detail when necessary

10. Sampling the Domain Space

11. Domain Space Tessellation
Uniform Tessellation
Adaptive Tessellation

12. Uniform Tessellation
Domain Space

13. Uniform Tessellation Fast Over-tessellation Step size computation
Triangulation
Over-tessellation
Triangle Rendering Bottleneck

14. Adaptive Tessellation
Domain Space

15. Adaptive Tessellation
Generates triangles only where needed
Inefficient
Many ‘stopping test’ computations
Triangulation algorithm not simple

16. Basic Idea Pre-sample adaptively to some precision At rendering time
Maintain samples in ‘order of importance’
More importance in the areas of high curvature
At rendering time
Select samples
Triangulate
If higher precision needed
Perform uniform tessellation

17. Pre-Sampling
Choose samples on each patch

18. Computationally Intractable
Questions: #1
For a predefined deviation (between the surface and its triangulation) threshold, what is the minimum number of points required on a patch for a given position and orientation of the patch? And where?
Computationally Intractable

19. Questions: #2
Is there a correlation between these points as the viewing parameters change?
Not Always.

20. Deviation = Δ0 Deviation = Δ1 (< Δ0)
Figure 1
Figure 2
No common points (except the end points)

21. Heuristic
Given a triangulation of a surface that deviates by more than Δ, add samples at the points of highest deviation until the resulting deviation is less than Δ.

22. Pre-Processing Algorithm

23. Pre-Sampling 1 2 3 4
Domain Space

24. Pre-Sampling 1 2 Point A Point B 3 4
Point of maximum object space deviation for a triangle

25. Pre-Sampling 1 2 5 3 4
Domain Space

26. Pre-Sampling 1 2 C 5 D B A 3 4
Point of maximum object space deviation for a triangle

27. Pre-Sampling 1 2 5 6 3 4
Domain Space

28. What is stored ? Ordered set of (u,v) pairs Deviation in object space
by decreasing deviation
Deviation in object space
i.e., deviation after the sample is added
3-d Vertex
optional

29. Rendering Time Algorithm
Given screen space deviation bounds

30. Scaling Factor for a patch
Scaling Factor for a vector at point P =
Minimum ratio of the length of the vector to its projected length on the image plane. Q
Ratio = Q P P
Eye
Image Plane

31. Scaling Factor for a patch
Pre-processing
Partition space
For each patch, use the partition containing it
If too many partitions for a patch, subdivide patch
Run-time (for each frame)
Compute the scaling factor for each partition
Scaling factor a patch is that of its partition

32. Runtime Algorithm Compute max allowable deviation (say c)
Let p = max deviation in current triangulation
if (c > p)
delete domain samples having deviation less than c
update triangulation

33. Example P1 P2 P3 P4 P5 P6 P7 P8 UV Values 26 24 21 19 14 13 6 3
Deviation
Here p = 3
Let c = 20
UV Values P1 P2 P3 P4 26 24 21 19
Deviation

34. Runtime Algorithm if (c < p)
add domain samples having deviation greater than c
update triangulation
if pre-computed set is exhausted
Uniformly tessellate triangles having deviation greater than c

35. Adaptive + Uniform Tessellation
Pre-computed Samples
Computed at run time to uniformly tessellate triangles having deviation greater than threshold

36. Potential Problem
Cracks in the model

37. Crack

38. Different samples on adjacent boundary curves
Crack
Boundary Curve v u
If Extra time
Samples on the boundary curve for left patch
Samples on the boundary curve for the right patch
Different samples on adjacent boundary curves

39. Crack Prevention
Sample the boundary curves separately from the interior, to prevent cracks in adjacent patches.
Modify the interior patch sampling by deleting points too close to the boundary.

40. Same samples on adjacent boundary curves
Crack Prevention
Boundary Curve v u
If Extra time
Samples on the boundary curve for the two adjacent patches
Same samples on adjacent boundary curves

41. Results
Model details

42. Pre-sampling Performance
Results
Pre-sampling Performance

43. Comparison for average number of triangles generated per frame
Results
Comparison for average number of triangles generated per frame
[20]: “Interactive display of large NURBS models” by S. Kumar, D. Manocha and A. Lastra

44. Run-time behavior of our algorithm
Results
Run-time behavior of our algorithm

45. Conclusions Combines forward and backward techniques
Uses adaptive and uniform tessellation
Low triangle count, small memory footprint
Applicable to class of parametric surfaces
Towards real-time spline surface rendering

46. Acknowledgements Shankar Krishnan Lifeng Wang UBC Modeling group
Alpha 1 Modeling system
National Science Foundation

47. The End.