High-Quality Rendering of Quartic Spline Surfaces on the GPU

We present a novel GPU-based algorithm for high-quality rendering of bivariate spline surfaces. An essential difference to the known methods for rendering graph surfaces is that we use quartic smooth splines on triangulations rather than triangular meshes. Our rendering approach is direct since we do not use an intermediate tessellation but rather compute ray-surface intersections (by solving quartic equations numerically) as well as surface normals (by using Bernstein-Bezier techniques) for Phong illumination on the GPU. Inaccurate shading and artifacts appearing for triangular tesselated surfaces are completely avoided. Level of detail is automatic since all computations are done on a per fragment basis. We compare three different (quasi-) interpolating schemes for uniformly sampled gridded data, which differ in the smoothness and the approximation properties of the splines. The results show that our hardware-based renderer leads to visualizations (including texturing, multiple light sources, environment mapping, and so forth) of highest quality.

[1]  Philipp Slusallek,et al.  Interactive ray tracing of free-form surfaces , 2004, AFRIGRAPH '04.

[2]  Hans-Peter Seidel,et al.  An introduction to polar forms , 1993, IEEE Computer Graphics and Applications.

[3]  Stefan Guthe,et al.  Hardware Accelerated Per-Pixel Displacement Mapping , 2004, Graphics Interface.

[4]  Christian Rössl,et al.  Reconstruction of volume data with quadratic super splines , 2004, IEEE Transactions on Visualization and Computer Graphics.

[5]  Frank Losasso,et al.  Geometry clipmaps: terrain rendering using nested regular grids , 2004, ACM Trans. Graph..

[6]  M. Sabin.  The use of piecewise forms for the numerical representation of shape , 1976 .

[7]  Jan Bruijns.  Quadratic Bezier triangles as drawing primitives , 1998, Workshop on Graphics Hardware.

[8]  Mark A. Duchaineau,et al.  ROAMing terrain: Real-time Optimally Adapting Meshes , 1997, Proceedings. Visualization '97 (Cat. No. 97CB36155).

[9]  C. D. Boor,et al.  Box splines , 1993 .

[10]  Ingo Wald,et al.  State of the Art in Ray Tracing Animated Scenes , 2009, Comput. Graph. Forum.

[11]  A. Knoll,et al.  Fast and Robust Ray Tracing of General Implicits on the GPU , 2007 .

[12]  N. England,et al.  Graphics Hardware , 2019, IEEE Computer Graphics and Applications.

[13]  Tamy Boubekeur,et al.  Generic mesh refinement on GPU , 2005, HWWS '05.

[14]  Hans-Peter Seidel,et al.  Ray tracing of spline surfaces: Bézier clipping, Chebyshev boxing, and bounding volume hierarchy – a critical comparison with new results , 1997, The Visual Computer.

[15]  Valerio Pascucci,et al.  Terrain Simplification Simplified: A General Framework for View-Dependent Out-of-Core Visualization , 2002, IEEE Trans. Vis. Comput. Graph..

[16]  Kenneth I. Joy,et al.  Adaptive 4-8 texture hierarchies , 2004, IEEE Visualization 2004.

[17]  Philipp Slusallek,et al.  Realtime Ray Tracing and its use for Interactive Global Illumination , 2003, Eurographics.

[18]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[19]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[20]  Anselmo Lastra,et al.  A shading language on graphics hardware: the pixelflow shading system , 1998, SIGGRAPH.

[21]  Marc Stamminger,et al.  GPU-Based Rendering of PN Triangle Meshes with Adaptive Tessellation , 2006 .

[22]  P. Schröder,et al.  Evaluation of Subdivision Surfaces on Programmable Graphics Hardware , 2003 .

[23]  Frank Zeilfelder,et al.  Construction of cubic 3D spline surfaces by Lagrange interpolation at selected points , 2003 .

[24]  Frank Zeilfelder,et al.  Developments in bivariate spline interpolation , 2000 .

[25]  H.-P. Seidel,et al.  Incremental Raycasting of Piecewise Quadratic Surfaces on the GPU , 2006, 2006 IEEE Symposium on Interactive Ray Tracing.

[26]  Jörg Peters,et al.  A realtime GPU subdivision kernel , 2005, SIGGRAPH 2005.

[27]  C. D. Boor,et al.  B-Form Basics. , 1986 .

[28]  O. Abert,et al.  Direct and Fast Ray Tracing of NURBS Surfaces , 2006, 2006 IEEE Symposium on Interactive Ray Tracing.

[29]  Bruce R. Piper,et al.  An explicit basis for C 1 quartic by various bivariate splines , 1987 .

[30]  Paolo Cignoni,et al.  Planet-sized batched dynamic adaptive meshes (P-BDAM) , 2003, IEEE Visualization, 2003. VIS 2003..

[31]  Tomoyuki Nishita,et al.  Ray tracing trimmed rational surface patches , 1990, SIGGRAPH.

[32]  NishitaTomoyuki,et al.  Ray tracing trimmed rational surface patches , 1990 .

[33]  Gerald Farin,et al.  Triangular Bernstein-Bézier patches , 1986, Comput. Aided Geom. Des..

[34]  Charles T. Loop,et al.  Real-time GPU rendering of piecewise algebraic surfaces , 2006, ACM Trans. Graph..

[35]  Frank Zeilfelder,et al.  An explicit quasi-interpolation scheme based on C1 quartic splines on type-1 triangulations , 2008, Comput. Aided Geom. Des..

[36]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[37]  Jörg Peters,et al.  Curved PN triangles , 2001, I3D '01.

[38]  Markus Geimer,et al.  Interactive Ray Tracing of Trimmed Bicubic Bézier Surfaces without Triangulation , 2005, WSCG.

[39]  Frank Zeilfelder,et al.  Smooth approximation and rendering of large scattered data sets , 2001, Proceedings Visualization, 2001. VIS '01..

[40]  C. D. Boor,et al.  A sharp upper bound on the approximation order of smooth bivariate PP functions , 1993 .

[41]  James F. Blinn,et al.  Real-time GPU rendering of piecewise algebraic surfaces , 2006, SIGGRAPH 2006.

[42]  W. Boehm,et al.  Bezier and B-Spline Techniques , 2002 .

[43]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[44]  C. Micchelli,et al.  On multivariate -splines , 1989 .

[45]  Reinhard Klein,et al.  GPU-based trimming and tessellation of NURBS and T-Spline surfaces , 2005, SIGGRAPH '05.

[46]  Tim Weyrich,et al.  Eurographics Symposium on Point-based Graphics (2006) Gpu-based Ray-casting of Quadratic Surfaces , 2022 .

[47]  Wolfgang Straßer,et al.  A meshing scheme for efficient hardware implementation of butterfly subdivision using displacement mapping , 2005, IEEE Computer Graphics and Applications.

[48]  Don Herbison-Evans,et al.  Solving Quartics and Cubics for Graphics , 1995 .

[49]  Jens Schneider,et al.  GPU-Friendly High-Quality Terrain Rendering , 2006, J. WSCG.

[50]  H. Hagen,et al.  Interactive Ray Tracing of Arbitrary Implicits with SIMD Interval Arithmetic , 2007, 2007 IEEE Symposium on Interactive Ray Tracing.