Surface distance maps

We present a new parameterized representation called surface distance maps for distance computations on piecewise 2-manifold primitives. Given a set of orientable 2-manifold primitives, the surface distance map represents the (non-zero) signed distance-to-closest-primitive mapping at each point on a 2-manifold. The distance mapping is computed from each primitive to the set of remaining primitives. We present an interactive algorithm for computing the surface distance map of triangulated meshes using graphics hardware. We precompute a surface parameterization and use the it to define an affine transformation for each mesh primitive. Our algorithm efficiently computes the distance field by applying this affine transformation to the distance functions of the primitives and evaluating these functions using texture mapping hardware. In practice, our algorithm can compute very high resolution surface distance maps at interactive rates and provides tight error bounds on their accuracy. We use surface distance maps for path planning and proximity query computation among complex models in dynamic environments. Our approach can perform planning and proximity queries in a dynamic environment with hundreds of objects at interactive rates and offer significant speedups over prior algorithms.

[1]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[2]  László Szirmay-Kalos,et al.  Approximate Ray‐Tracing on the GPU with Distance Impostors , 2005, Comput. Graph. Forum.

[3]  David G. Kirkpatrick,et al.  Linear Time Euclidean Distance Algorithms , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[4]  Dinesh Manocha,et al.  A Voronoi-based hybrid motion planner , 2001, Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180).

[5]  Kai Hormann,et al.  Surface Parameterization: a Tutorial and Survey , 2005, Advances in Multiresolution for Geometric Modelling.

[6]  Calvin R. Maurer,et al.  A Linear Time Algorithm for Computing Exact Euclidean Distance Transforms of Binary Images in Arbitrary Dimensions , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  James F. Blinn,et al.  Simulation of wrinkled surfaces , 1978, SIGGRAPH.

[8]  Mie Sato,et al.  Penalized-Distance Volumetric Skeleton Algorithm , 2001, IEEE Trans. Vis. Comput. Graph..

[9]  Marek Teichmann,et al.  Polygonal Approximation of Voronoi Diagrams of a Set of Triangles in Three Dimensions , 1997 .

[10]  Markus H. Gross,et al.  Signed distance transform using graphics hardware , 2003, IEEE Visualization, 2003. VIS 2003..

[11]  Markus Denny,et al.  Solving Geometric Optimization Problems using Graphics Hardware , 2003, Comput. Graph. Forum.

[12]  P. Danielsson.  Euclidean distance mapping , 1980 .

[13]  Craig Gotsman,et al.  Fast Approximation of High-Order Voronoi Diagrams and Distance Transforms on the GPU , 2006, J. Graph. Tools.

[14]  Ronald Peikert,et al.  Optimized Bounding Polyhedra for GPU-Based Distance Transform , 2006 .

[15]  Ari Rappoport,et al.  Computing Voronoi skeletons of a 3-D polyhedron by space subdivision , 2002, Comput. Geom..

[16]  Adam Finkelstein,et al.  Stylized video cubes , 2002, SCA '02.

[17]  Robert L. Cook,et al.  Shade trees , 1984, SIGGRAPH.

[18]  D. Manocha,et al.  Fast proximity computation among deformable models using discrete Voronoi diagrams , 2006, ACM Trans. Graph..

[19]  Howie Choset,et al.  Sensor based motion planning: the hierarchical generalized Voronoi graph , 1996 .

[20]  Marko Subasic,et al.  Level Set Methods and Fast Marching Methods , 2003 .

[21]  Olivier D. Faugeras,et al.  The Vector Distance Functions , 2003, International Journal of Computer Vision.

[22]  Edwin Earl Catmull,et al.  A subdivision algorithm for computer display of curved surfaces. , 1974 .

[23]  Mark H. Overmars,et al.  Approximating Voronoi Diagrams of Convex Sites in any Dimension , 1998, Int. J. Comput. Geom. Appl..

[24]  Dinesh Manocha,et al.  Fast proximity computation among deformable models using discrete voronoi diagrams: implementation details , 2006, International Conference on Computer Graphics and Interactive Techniques.

[25]  Dinesh Manocha,et al.  DiFi: Fast 3D Distance Field Computation Using Graphics Hardware , 2004, Comput. Graph. Forum.

[26]  D. Meiron,et al.  Efficient algorithms for solving static hamilton-jacobi equations , 2003 .

[27]  Dinesh Manocha,et al.  Interactive motion planning using hardware-accelerated computation of generalized Voronoi diagrams , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[28]  Ross T. Whitaker,et al.  Interactive deformation and visualization of level set surfaces using graphics hardware , 2003, IEEE Visualization, 2003. VIS 2003..

[29]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[30]  Ming C. Lin,et al.  Efficient collision detection for animation and robotics , 1993 .

[31]  O. Cuisenaire.  Distance transformations: fast algorithms and applications to medical image processing , 1999 .

[32]  Dinesh Manocha,et al.  Fast computation of generalized Voronoi diagrams using graphics hardware , 1999, SIGGRAPH.

[33]  S.F.F. Gibson,et al.  Using distance maps for accurate surface representation in sampled volumes , 1998, IEEE Symposium on Volume Visualization (Cat. No.989EX300).

[34]  Dinesh Manocha,et al.  Fast computation of generalized Voronoi diagrams using graphics hardware , 1999, SIGGRAPH.

[35]  Dinesh Manocha,et al.  Fast and simple 2D geometric proximity queries using graphics hardware , 2001, I3D '01.

[36]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[37]  Ronald N. Perry,et al.  Kizamu: a system for sculpting digital characters , 2001, SIGGRAPH.

[38]  Ronald N. Perry,et al.  Adaptively sampled distance fields: a general representation of shape for computer graphics , 2000, SIGGRAPH.

[39]  Mario Botsch,et al.  Feature sensitive surface extraction from volume data , 2001, SIGGRAPH.

[40]  Dinesh Manocha,et al.  Interactive 3D distance field computation using linear factorization , 2006, I3D '06.