Direct Rendering of Trimmed NURBS Surfaces


 Clifford Bond
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1 Direct Rendering of Trimmed NURBS Surfaces
2 Hardware Graphics Pipeline 2/ 81 Hardware Graphics Pipeline GPU Video Memory CPU Vertex Processor Raster Unit Fragment Processor Render Target Screen
3 Extended Hardware Graphics Pipeline 3/ 81 Extended Hardware Graphics Pipeline GPU Video Memory CPU Vertex Processor Raster Unit Intersection Shader Fragment Processor Render Target
4 Intersection Shader 4/ 81 Intersection Shader Conceptual extension of the hardware rendering pipeline Optional stage after the rasterization unit for nontriangular geometry Specifically for ray object intersection tests  Analytically, e.g. for quadrics  Numerically, e.g. for NURBS surfaces Can perform early ray termination test
5 Intersection Shader 5/ 81 Intersection Shader CPU Convex hull
6 Intersection Shader 6/ 81 Intersection Shader CPU Parametric object description GPU Convex hull vertices Convex hull
7 Intersection Shader 7/ 81 Intersection Shader CPU Parametric object description Convex hull vertices... GPU Convex hull Candidate rays, and ray directions are generated
8 Intersection Shader 8/ 81 Intersection Shader CPU Parametric object description Convex hull vertices GPU Convex hull Candidate rays, and ray directions are generated Intersection test
9 Intersection Shader 9/ 81 Intersection Shader Each eye ray corresponds to exactly one fragment Current hardware  Ray generation is performed in the vertex shader and rasterization unit  Intersection shader is part of the fragment shader program
10 Intersection Shader 10 / 81 Convex Hulls Used to reduce the number of ray object intersection tests Rays which are generated by the projection of the convex hull form a candidate set of rays Inverse acceleration structure Tightness of convex hulls can be used to trade CPU vs. GPU computation time
11 Intersection Shader 11 / 81 Intersection Shader Pros No tessellation and minimal storage Pixelaccurate  silhouettes,  intersection curves,  normals Seamless integration into standard graphics pipeline Cons Ray casting is expensive Artifact free intersection test can be difficult to perform
12 Intersection Shader 12 / 81 Intersection Shader Pros No tessellation and minimal storage Pixelaccurate  silhouettes,  intersection curves,  normals Seamless integration into standard graphics pipeline Cons Ray casting is expensive Artifact free intersection test can be difficult to perform
13 Intersection Shader 13 / 81 Intersection Shader Pros No tessellation and minimal storage Pixelaccurate  silhouettes,  intersection curves,  normals Seamless integration into standard graphics pipeline Cons Ray casting is expensive Artifact free intersection test can be difficult to perform
14 Intersection Shader 14 / 81 Intersection Shader Quadrics NURBS Nodebased Implicit Surface
15 Ray Casting of NURBS Surfaces on the GPU 15 / 81 Ray Casting of NURBS Surfaces on the GPU
16 Ray Casting of NURBS Surfaces on the GPU 16 / 81 Overview 1. Patch is subdivided in a set of subpatches 2. A Convex hull is computed for each subpatch and its parameter range in (u,v) space is determined 3. Candidate rays are generated by rendering the convex hulls 4. Intersection test is performed using Newton Iteration 5. Trimming is performed using Bézier Clipping and PointInPolygon test
17 Preprocessing 17 / 81 Preprocessing 1. Generation of convex hulls  Conversion of NURBS surfaces into Bézier patches  Subdivision of Bézier patches  Userprovided level, a flatness criteria or a minimum size of a convex hull primitive determines the number of subdivision steps  Convex hull consists of a set of polygonal primitives
18 Preprocessing 18 / 81 Preprocessing 2. Conversion of trimming curves into Bézier curves  Necessary for Bézier clipping algorithm
19 Preprocessing 19 / 81 Preprocessing 2. Conversion of trimming curves into Bézier curves  Necessary for Bézier clipping algorithm 3. Generation of fragment programs  Customised for surface properties
20 Preprocessing 20 / 81 Preprocessing 2. Conversion of trimming curves into Bézier curves  Necessary for Bézier clipping algorithm 3. Generation of fragment programs  Customised for surface properties 4. Parameters are stored in textures  Control points and knot vectors of surfaces  Control points of trimming curves
21 Curve and Surface Evaluation 21 / 81 Curve and Surface Evaluation Bézier curves / surfaces  de Casteljau algorithm is used for evaluation  Efficient because linear interpolation is supported directly in hardware
22 Curve and Surface Evaluation 22 / 81 Curve and Surface Evaluation Bézier curves / surfaces  de Casteljau algorithm is used for evaluation  Efficient because linear interpolation is directly supported in hardware NURBS curves / surfaces  de Boor algorithm is used for evaluation  Only knot spans which are relevant to evaluate the curve at a certain point are copied into registers
23 Ray Patch Intersection 23 / 81 Ray Patch Intersection Ray is represented as intersection of two planes P 1 P 2 : : r n r n 1 2 x x d 1 d 2 = 0 = 0
24 Ray Patch Intersection 24 / 81 Ray Patch Intersection Ray is represented as intersection of two planes NURBS surface P 1 P 2 : : r n r n 1 2 x x d 1 d 2 = 0 = 0 n m S = i j, i= 0 j= 0 p q ( u, v) N () u N () u pi j
25 Ray Patch Intersection 25 / 81 Ray Patch Intersection Ray is represented as intersection of two planes NURBS surface Objective function system 0 : 0 : = = d x n P d x n P r r ( ) () () j i q j n i m j p i p u N u N v u S, 0 0, = = = ( ) ( ) = ,, ), ( d v u S n d v u S n v u F
26 Ray Patch Intersection 26 / 81 Newton Iteration Each step takes the form where the Jacobian J is given by ) ), (( ), ( ), ( 1 1 k k k v u F J v u v u + = ( ) ( ) ( ) ( ) = v v u S n u v u S n v v u S n u v u S n J,,,,
27 Ray Patch Intersection 27 / 81 Newton Iteration Newton Iteration terminates if  an approximation for a root has been found F( u, v) k ) < ε,  the maximum number of iterations has been exceeded,  the algorithm diverges, i.e. ( u v) ) F( u, v) ) F >, k + 1 k
28 Ray Patch Intersection 28 / 81 Newton Iteration Newton Iteration has to be implemented in one rendering pass
29 Ray Patch Intersection 29 / 81 Newton Iteration Newton Iteration has to be implemented in one rendering pass
30 Ray Patch Intersection 30 / 81 Initial Values 1. Fixed initial guess for each primitive of the convex hull  Center of (u,v) parameter space enclosed by primitive of the convex hull
31 Ray Patch Intersection 31 / 81 Initial Values 1. Fixed initial guess for each primitive of the convex hull  Center of (u,v) parameter space enclosed by primitive of the convex hull 2. (u,v) Texturing  (u,v) parameter stored for each vertex during preprocessing  Parameters are linearly interpolated along the surfaces of the convex hull
32 Ray Patch Intersection 32 / 81 Initial Values 2.1. Viewindependent (u,v) Texturing  The (u,v) parameters as computed during the preprocessing are interpolated
33 Ray Patch Intersection 33 / 81 Initial Values 2.1. Viewindependent (u,v) Texturing  The (u,v) parameters as computed during the preprocessing are interpolated 2.2. Viewdependent (u,v) Texturing  Vertex shader: Intersection test between the ray from the camera to the vertex and the surface patch  (u,v) parameters of the intersection points are interpolated
34 Trimming 34 / 81 Trimming Based on pointinpolygon test A ray is shot along the parametric surface and the number of intersections with the trimming curves is counted  The parity of the number of intersections determines if the fragment is inside or outside the trimming region Bézier clipping is used to determine the intersection of the ray with the trimming curves
35 Bézier Clipping 35 / 81 Bézier Clipping Mixture between subdivisionbased and numerical intersection algorithm Can be used for ray Bézier curve and ray Bézier patch intersection test Guaranteed to find all intersection points  Some special cases for ray surface patch intersection
36 Bézier Clipping 36 / 81 Bézier Clipping Ray is given in implicit form ax + by c = 0
37 Bézier Clipping 37 / 81 Bézier Clipping Ray is given in implicit form ax + by c = 0 Bézier curve is given by n () n () u p B () u, p = ( x y ) C = i i i i, i= 0 i
38 Bézier Clipping 38 / 81 Bézier Clipping Ray is given in implicit form ax + by c = 0 Bézier curve is given by () n () u p B () u, p = ( x y ) Objective function (distance field) d n C = i i i i, n i= 0 () n () u d B () u, d = ax + by c = i= 0 i i i i i i
39 Bézier Clipping 39 / 81 Bézier Clipping Objective function as explicit, nonparametric Bézier curve n () n () u ( u, d() u ) = D B () u, D = ( u d ) D = i i i i, i= 0 Because D(u) is a Bézier curve, the control points are evenly spaced, i.e. i u i = i n
40 Bézier Clipping 40 / 81 Bézier Clipping ray 2 Bézier curve
41 Bézier Clipping 41 / 81 Bézier Clipping 4 2 d (1/3,3) (2/3,4) (1,4) 3 ray u 2 (0,2) Bézier curve Explicit Bézier curve
42 Bézier Clipping 42 / 81 Bézier Clipping 4 2 d (1/3,3) (2/3,4) (1,4) 3 ray u min u max u 2 (0,2) Bézier curve Explicit Bézier curve
43 Bézier Clipping 43 / 81 Bézier Clipping If the convex hull DOES NOT intersect the d=0 axis then no intersection point exists If the convex hull DOES intersect the d=0 axis then the convex hull property for Bézier curve guarantees that an intersection lies between u min and u max The de Casteljau algorithm can be used to subdivide the curve into three segments  Two of the segments, [0, u min ) and ( u max,1], do not contain a ray curve intersection
44 Bézier Clipping 44 / 81 Bézier Clipping If the [u min, u max ] is bigger than some predefined threshold then the interval contains two intersections and is itself subdivided  The search is continued in each subinterval separately The interval contraction is applied until the interval [u min, u max ] k is smaller than a predefined threshold  The intersection point is the center of the final interval [u min, u max ] k
45 Bézier Clipping 45 / 81 Bézier Clipping on the GPU Algorithm has been reformulated to be iterative  For efficiency reasons implemented as a state machine on the GPU The convex hull is reconstructed only locally  Only the edges which contain u min and u max are found and the corresponding control points are stored  Reduces the number of registers necessary
46 Bézier Clipping 46 / 81 Bézier Clipping on the GPU If a subdivision of [u min, u max ] k is necessary, the search is continued only in the left subinterval  The limited number of register on the GPU does not allow to store different right subintervals  A global right subinterval is used which requires only one scalar for its representation After the search terminated in the left subinterval it is continued in the global right subinterval
47 Bézier Clipping 47 / 81 Bézier Clipping on the GPU 0 1 u min,0 u max,0
48 Bézier Clipping 48 / 81 Bézier Clipping on the GPU 0 1 u min,0 u min,1 subdivision u max,1 u max,0
49 Bézier Clipping 49 / 81 Bézier Clipping on the GPU 0 1 u min,0 u min,1 subdivision u max,1 u max,0 u min,2 u max,2
50 Bézier Clipping 50 / 81 Bézier Clipping on the GPU 0 1 u min,0 u max,0 subdivision u min,1 u max,1 u min,2 u max,2 subdivision u min,3 u max,3
51 Bézier Clipping 51 / 81 Bézier Clipping on the GPU 0 1 u min,0 u max,0 subdivision u min,1 u max,1 u min,2 u max,2 subdivision u min,3 u max,3 u min,4 u max,4
52 Bézier Clipping 52 / 81 Bézier Clipping on the GPU 0 1 u min,0 u max,0 subdivision u min,1 u max,1 u min,2 u max,2 subdivision u min,3 u max,3 u min,4 u max,4 ε intersection
53 Early Ray Termination 53 / 81 Early Ray Termination Reduces the number of intersection tests Intersection shader program is not executed for a ray if it can be guaranteed that any intersection which would be found is behind the one that has already been encountered for the fragment Convex hull property for parametric patches enables early ray termination based on minimal depth value of bounding primitive for subpatch
54 Early Ray Termination 54 / 81 Early Ray Termination
55 Early Ray Termination 55 / 81 Early Ray Termination
56 Early Ray Termination 56 / 81 Early Ray Termination
57 Early Ray Termination 57 / 81 Early Ray Termination
58 Early Ray Termination 58 / 81 Early Ray Termination d cur d min depth
59 Early Ray Termination 59 / 81 Early Ray Termination GPU provides EarlyZ  BUT cannot be used if depth value is changed in the fragment shader program  Depth value has to be written for consistent depth buffering
60 Early Ray Termination 60 / 81 Early Ray Termination GPU provides EarlyZ  BUT cannot be used if depth value is changed in the fragment shader program  Depth value has to be written for consistent depth buffering Manual EarlyZ can be implemented by branching in the fragment shader  Current depth value from depth buffer is compared to minimal depth value of convex hull
61 Early Ray Termination 61 / 81 Early Ray Termination OpenGL: Simultaneous read and write of buffer results in undefined behaviour  No memorycache synchronisation  Parallel processing of rays / fragments In practice reading from the currently bound depth buffer is possible (Nvidia boards)  Depth buffer is accessed as texture, similar to shadow mapping
62 Early Ray Termination 62 / 81 Early Ray Termination Reading from depth buffer is conservative  Depth buffer values are monotonically decreasing within one render pass  Values obtained in the intersection shader are always equal or greater than the correct value  Depth buffering is performed after fragment shader Some unnecessary intersection tests are performed but images are artifact free
63 Early Ray Termination 63 / 81 Early Ray Termination Reading from depth buffer is conservative  Depth buffer values are monotonically decreasing within one render pass  Values obtained in the intersection shader are always equal or greater than the correct value  Depth buffering is performed after fragment shader Some unnecessary intersection tests are performed but images are artifact free Requires fronttoback sorting of convex hull primitives
64 Results 64 / 81 Results All tests performed at 1280x % of screen covered by objects AMD Athlon Single Nvidia Quadro FX SLI usually nearly doubles the performance Results are draw times in ms
65 Results 65 / 81 No trimming curve Different number of subpatches Patches Sub Patches Number of Iterations
66 Results 66 / 81 One trimming curve Different number of subpatches Patches Sub Patches Number of Iterations
67 67 / 81 Results Different number of trimming curves 2048 subpatches
68 Results 68 / 81 Newton Iteration  1, 2, 3 and 4 Newton Iteration steps (ltr, ttb)
69 Results 69 / 81 Newton Iteration  1, 2, 3 and 4 Newton Iteration steps (ltr, ttb)
70 Results 70 / 81 Newton Iteration  1, 2, 3 and 4 Newton Iteration steps (ltr, ttb)
71 Results 71 / 81 Newton Iteration  1, 2, 3 and 4 Newton Iteration steps (ltr, ttb)
72 Results 72 / 81 Early ray termination Quads NURBS
73 Comparison 73 / 81 Comparison with Guthe et al. [2005] Integration into hardware graphics pipeline Pixel error Trimming Guthe et al. Seamless Estimator for screen space error Texturebased, often high resolutions necessary GPUCast Seamless Pixelaccurate (some artifacts through nonconvergence) Pixelaccurate
74 Comparison 74 / 81 Comparison with Guthe et al. [2005] Guthe et al. GPUCast Number of trims Unlimited Currently max. 10 MultiGPU setups Does not scale Scales well Quality at surface intersections Possible model complexity Appropriate tessellation factor hard to determine Complex models possible but realtime not guaranteed in these cases Pixel accurate without additional computations Limited to medium complex models on current hardware
75 Limitations 75 / 81 Limitations At the very limit of GPU resources  Especially registers  Adding functionality or improving the current one is (very) hard  Many GLSL compiler bugs  Got better for NV4X over the years  Alternative SH was used
76 Limitations 76 / 81 Limitations Usually one fragment program per surface Limited degree of NURBS surfaces (max. degree 7) Images are not artifact free  Newton Iteration does not always converge No acceleration structure for trimming  Currently for each fragment all trimming curves have to be tested  About 10 trimming curves are supported efficiently Manual EarlyZ is not officially supported
77 Future Work 77 / 81 Future Work Requires DirectX10 graphics hardware Reliable intersection tests  Bezier Clipping  Subdivisionbased methods  Resultantbased methods Acceleration structure for trimming  First experiments with a quad tree are promising NURBS with arbitrary degree
78 Conclusions 78 / 81 Conclusions Pixelaccurate rendering of NURBS surfaces without tessellation Realtime performance for medium size scenes Number of trimming curves limited Some image artifacts because of nonconverging intersection tests Next generation of GPUs provides processing power and feature to overcome current limitations
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81 Demo 81 / 81 DEMO Slides available at
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