Tessellating and Rendering Bezier/B-Spline/NURBS Curves and Surfaces using Geometry Shader in GPU

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1 Tessellating and Rendering Bezier/B-Spline/NURBS Curves and Surfaces using Geometry Shader in GPU CIS 665 PROJECT FINAL REPORT 5/2/07 Dongsoo Han University of Pennsylvania PROJECT ABSTRACT NURBS(Non Uniform Rational B-Splines) are nearly ubiquitous for computer-aided design (CAD), manufacturing (CAM), and 3D modelers and part of numerous industry wide 3D data exchange standards such as IGES and STEP. With a rapidly growing hardware system, NURBS is getting more attentions in real-time systems or 3D games. In this new trend, the biggest obstacle is that tessellating NURBS into triangles is still slower than loading pre-triangulated data. However NURBS can give lots of freedom to 3D application developers such as unlimited control over LOD (Level of Detail) or high rendering quality to the curved surface. In this project, geometry shader is used to tessellate and render curves and surfaces. The benefits of using geometry shader are to reduce the rendering data transferring from CPU to GPU as well as tessellate and render curves and surfaces as a parallel process. 1. INTROUDUCTION 1.1. Technology A geometry shader can generate new primitives from existing primitives like pixels, lines and triangles. It is executed after vertex shader and its input is the whole primitive or primitive with adjacency information. Then it can emit zero or more primitives, which are rasterized and their fragments passed to fragment shader. With the input of control vertices and other information such as number of input control vertices, geometry shader can generate line or triangle primitives to render Bezier, B-Spline or NURBS curves and surfaces Target Platforms Hardware 1. Video card: NVidia GeForce 8800 GTX 2. Hardware Platform: x86 Intel or AMD 3. Hard disk: approximately 10 MB Software 1. Operating System: Window XP 2. Programming Tool: Visual Studio 2005 Professional 1

2 3. Programming Language: C/C++, Cg 2.0 (beta) 4. Major Programming Library: MFC (User Interface), Win32 API, OpenGL/GLU 5. 3 rd Party Programming Library: N/A 2. PROJECT DEVELOPMENT 2.1. Technical Background What is Bezier curve Bezier curve is a weighted average of control points (vertices). These weights are Bernstein polynomials and also called blending function. where the polynomials Equation 1 Equation 2 Degree three Bezier curve is most common and can be represented as below. It is usually called Cubic Bezier curve. Below is matrix form. Equation 3 2

3 t p0x p1x p2x p3x () = t Bt p0y p1y p2y p3y t p p p p 0z 1z 2z 3z Bt () = G B T Bez Bez Equation 4 In this project matrix form was used since we can take advantage of matrix vector multiplication inside GPU. Below is the screenshot that shows Bezier curve generated by geometry shader. Figure What is B-Spline curve B-Spline curve is a set of piecewise polynomial curves represented as below. n () () X t = PkBk, d t k = 0 Equation 5 3

4 B 1 t t t = 0 otherwise k k+ 1 k,1 () t t t t t Bkd, t Bkd, 1 t Bk+ 1, d 1 t tk+ d 1 tk tk+ d tk+ 1 k k+ d () = () + () Equation 6 Particularly, degree three uniform B-Spline curve can be represented as matrix form as below t 1 x() t = P P P P t t Equation 7 In this project, matrix form was used to tessellate and render unform B-Spline curves and surfaces in geometry shader. Figure 2 4

5 2.1.3 What is NURBS curve NURBS stands for Non Uniform Rational B-Splines. NURBS curve can be evaluated using Coxde Boor algorithm (Equation 6). With weights associated to each control point, conic sections can be created What is NURBS surface Equation Implementation Details CPU GPU Input (uniform textures, parameters) Control Vertices(Points) Number of CV LOD value Output (Geometry Shader) Line or Triangle primitives Figure 3 5

6 Figure 3 explains that control vertices (points) are transferred to GPU (geometry shader) along with other uniform information such as number of cv or LOD(Level of Detail) value. The output of geometry shader is either lines or triangles corresponding to curves and surfaces. The nice thing of geometry shader is that it can control LOD automatically based on the distance between camera (eye) position to the object (curve or surface). Figure 5 is B-Spline surface geometry shader program. In Figure 6, 7 and 8, red surface is tessellated by CPU and triangle list is transferred to GPU. On the other hand yellow surface is done by GPU. Figure 6, 7 and 8 shows different LODs which are automatically done by geometry shader. Figure 4 There are two projects in visual studio To compare implementation in CPU vs. GPU, Brep project contains curve and surface CPU implementations. Cg folder in GLModeler project contains geometry shader codes. GLModeler project also contains user interface and interaction implementations. LINE_ADJ TRIANGLE_OUT void bspline_surface_txtr_gp( state.matrix.mvp.inverse, AttribArray<float4> pos : POSITION, AttribArray<float4> color : COLOR0, uniform float4x4 modelviewproj : state.matrix.mvp, uniform float4x4 inversemvp : uniform int segments = 10, uniform int numofucv = 8, uniform int numofvcv = 5, { uniform samplerrect datax, uniform samplerrect datay, uniform samplerrect dataz ) 6

7 float4 newpt = mul(inversemvp, pos[0]); int v = round(newpt.x); float4 cv[4]; float4 cv1[4]; float step = 1; // dynamic LOD based on the view distance float3 pt; pt.x = texrect(datax, float2(0, v)).r; pt.y = texrect(datay, float2(0, v)).r; pt.z = texrect(dataz, float2(0, v)).r; float3 eyepos = (mul(inversemvp, float4(0, 0, 0, 1))).xyz; float dist = distance(eyepos, pt); segments = ( 60 / dist ); if ( segments < 2 ) segments = 2; else if ( segments > 13 ) segments = 13; for ( int i = 0; i < numofucv - 3; i++ ) { cv[0].x = texrect(datax, float2(i, v)).r; cv[0].y = texrect(datay, float2(i, v)).r; cv[0].z = texrect(dataz, float2(i, v)).r; cv[1].x = texrect(datax, float2(i + 1, v)).r; cv[1].y = texrect(datay, float2(i + 1, v)).r; cv[1].z = texrect(dataz, float2(i + 1, v)).r; cv[2].x = texrect(datax, float2(i + 2, v)).r; cv[2].y = texrect(datay, float2(i + 2, v)).r; cv[2].z = texrect(dataz, float2(i + 2, v)).r; cv[3].x = texrect(datax, float2(i + 3, v)).r; cv[3].y = texrect(datay, float2(i + 3, v)).r; cv[3].z = texrect(dataz, float2(i + 3, v)).r; cv1[0].x = texrect(datax, float2(i, v + step)).r; cv1[0].y = texrect(datay, float2(i, v + step)).r; cv1[0].z = texrect(dataz, float2(i, v + step)).r; cv1[1].x = texrect(datax, float2(i + 1, v + step)).r; cv1[1].y = texrect(datay, float2(i + 1, v + step)).r; cv1[1].z = texrect(dataz, float2(i + 1, v + step)).r; cv1[2].x = texrect(datax, float2(i + 2, v + step)).r; cv1[2].y = texrect(datay, float2(i + 2, v + step)).r; cv1[2].z = texrect(dataz, float2(i + 2, v + step)).r; cv1[3].x = texrect(datax, float2(i + 3, v + step)).r; cv1[3].y = texrect(datay, float2(i + 3, v + step)).r; cv1[3].z = texrect(dataz, float2(i + 3, v + step)).r; for( int j = 0; j < segments; j++ ) { float t = j / (float) (segments-1); float4 tvec = float4(1, t, t*t, t*t*t); 7

8 float4 b = mul(bsplinebasis, tvec); float4 p = cv[0]*b.x + cv[1]*b.y + cv[2]*b.z + cv[3]*b.w; float4 p1 = cv1[0]*b.x + cv1[1]*b.y + cv1[2]*b.z + cv1[3]*b.w; } } } emitvertextransform(p, color[0], modelviewproj); emitvertextransform(p1, color[0], modelviewproj); Figure 5 To render a full surface, each geometry shader renders only one strip. If there are mutiple strips to tessellate and render, geometry shaders work as a parallel process and it can achieve a big performance increase. Figure 6 8

9 Figure 7 Figure 8 9

10 3. Conclusion Using geometry shader to tessellate and render Bezier/B-Spline/Nurbs curves and surfaces has three benifits: 1. The volume of transferring data to GPU is much less than using triangles. Also it is independent to the LOD (Level of Detail). Therefore without suffering from limited bandwidth, high quality curves and surfaces can be displayed. 2. Tessellation and rendering can be processed in parallel way. 3. LOD can be achieved easily and doesn t cause any overload to CPU. Also adaptive tessellation can be also achieved along with LOD. The limitation of geometry shader: 1. The number of emit primitives are limited. -> We can overcome this by subdividing input curves or surfaces. 2. The size of geometry shader and loop iteration are limited. -> Sensitive coding and tuning are necessary. 4. REFERENCES David F. Rogers 2001 An Introduction to NURBS with Historical Perspective Gerald Farin 2002 CURVES AND SURFACES FOR CAGD A Practical Guide Fifth Edition Les Piegl and Wayne Tiller 1996 The NURBS Book 2 nd Edition Georg Glaeser and Hellmuth Stachel 1998 Open Geometry OpenGL + Advanced Geometry SAMUEL R. BUSS University of California, San Diego 3-D Computer Graphics A Mathematical Introduction with OpenGL Michael Guthe, Akos Balazs, Reinhard Klein GPU-based trimming and tessellation of NURBS and T-Spline surfaces Hans-Friedrich Pabst, Jan P. Springer, Andr e Schollmeyer, Robert Lenhardt, Christian Lessig, Bernd Fr ohlich Efficient Rendering of Trimmed NURBS Subodh Kumar & Dinesh Manocha Department of Computer Science University of North Carolina Efficient Rendering of Trimmed NURBS Surfaces 10

11 Thomas W. Sederberg An Introduction to B-Spline Curves Brian A. Barsky Berkeley Computer Graphics Laboratory A STUDY OF PARAMETRIC UNIFORM B-SPLINE CURVE AND SURFACE REPRESENTATIONS Christopher K. Ingram University of Waterloo A Geometric B-Spline Over the Triangular Domain Center for Visual Information Technology Sample Geometry Shader SIGGRAPH 2005 Course 37 Notes GPU Shading and Rendering Ronen Barzel University of California, San Diego Introduction to Computer Graphics Lecture Notes Jared Hoberock Geometry Shader Hello World Denis Zorin Computer Graphics Lecture Notes NVidia GeForce 8800 OpenGL Extensions 5. CREDITES Some images are from Wikipedia( 11