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

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

Garden Model ( 38,646 patches)

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

  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

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)

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

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

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

Sampling the Domain Space

Domain Space Tessellation Uniform Tessellation Adaptive Tessellation

Uniform Tessellation Domain Space

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

Adaptive Tessellation Domain Space

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

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

Pre-Sampling Choose samples on each patch

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

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

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

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.

Pre-Processing Algorithm

Pre-Sampling 1 2 3 4 Domain Space

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

Pre-Sampling 1 2 5 3 4 Domain Space

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

Pre-Sampling 1 2 5 6 3 4 Domain Space

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

Rendering Time Algorithm Given screen space deviation bounds

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

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

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

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

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

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

Potential Problem Cracks in the model

Crack

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

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.

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

Results Model details

Pre-sampling Performance Results Pre-sampling Performance

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

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

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

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

The End.