Mark Foskey
ABSTRACT
Applications of of the medial axis have been limited because of its instability and algebraic complexity. In this paper, we use a simplification of the medial axis, the θ-SMA, that is parameterized by a separation angle (θ) formed by the vectors connecting a point on the medial axis to the closest points on the boundary. We present a formal characterization of the degree of simplification of the θ-SMA as a function of θ, and we quantify the degree to which the simplified medial axis retains the features of the original polyhedron.We present a fast algorithm to compute an approximation of the θ-SMA. It is based on a spatial subdivision scheme, and uses fast computation of a distance field and its gradient using graphics hardware. The complexity of the algorithm varies based on the error threshold that is used, and is a linear function of the input size. We have applied this algorithm to approximate the SMA of models with tens or hundreds of thousands of triangles. Its running time varies from a few seconds, for a model consisting of hundreds of triangles, to minutes for highly complex models.
- N. Amenta, S. Choi, and R. Kolluri. The power crust. In Proc. ACM Solid Modeling, pages 249--260, 2001. Google Scholar
Digital Library
- D. Attali and J. Boissonnat. A linear bound on the complexity of the Delaunay triangulation of points on polyhedral surfaces. In Proc. ACM Solid Modeling, pages 139--146, 2002. Google ScholarDigital Library
- D. Attali and A. Montanvert. Computing and simplifying 2d and 3d continuous skeletons. Computer Vision and Image Understanding, 67(3):261--273, 1997. Google ScholarDigital Library
- J. August, A. Tannenbaum, and S. Zucker. On the evolution of the skeleton. In Proc. Int. Conf. on Computer Vision, 1999.Google ScholarCross Ref
- H. Blum. A transformation for extracting new descriptors of shape. In W. Wathen-Dunn, editor, Models for the Perception of Speech and Visual Form, pages 362--380. MIT Press, 1967.Google Scholar
- J.-D. Boissonnat. Geometric structures for three-dimensional shape representation. ACM Trans. Graph., 3(4):266--286, 1984. Google ScholarDigital Library
- J.-D. Boissonnat and F. Cazals. Smooth surface reconstruction via natural neighbour interpolation of distance functions. In ACM Symp. Computational Geometry, pages 223--232, 2000. Google ScholarDigital Library
- C.-S. Chiang. The Euclidean distance transform. Ph.D. thesis, Dept. Comput. Sci., Purdue Univ., West Lafayette, IN, Aug. 1992. Report CSD-TR 92-050. Google ScholarDigital Library
- H. I. Choi, S. W. Choi, and H. P. Moon. Mathematical theory of medial axis transform. Pacific J. Math., 181(1):56--88, 1997.Google ScholarCross Ref
- S. W. Choi and H.-P. Seidel. Linear onesided stability of MAT for weakly injective 3D domain. In Proc. ACM Solid Modeling, pages 344--355, 2002. Google ScholarDigital Library
- T. Culver, J. Keyser, and D. Manocha. Accurate computation of the medial axis of a polyhedron. In Proc. ACM Solid Modeling, pages 179--190, 1999. Google ScholarDigital Library
- P.-E. Danielsson. Euclidean distance mapping. Computer Graphics and Image Processing, 14:227--248, 1980.Google ScholarCross Ref
- T. K. Dey and W. Zhao. Approximate medial axis as a Voronoi subcomplex. In Proc. ACM Solid Modeling, pages 356--366, 2002. Google ScholarDigital Library
- T. K. Dey and W. Zhao. Approximating the medial axis from the Voronoi diagram with a convergence guarantee. In European Symp. Algorithms, pages 387--398, 2002. Google ScholarDigital Library
- D. Dutta and C. M. Hoffmann. A geometric investigation of the skeleton of CSG objects. In Proc. ASME Conf. Design Automation, 1990.Google ScholarCross Ref
- M. Etzion and A. Rappoport. Computing the Voronoi diagram of a 3-D polyhedron by separate computation of its symbolic and geometric parts. In Proc. ACM Solid Modeling, pages 167--178, 1999. Google ScholarDigital Library
- K. Hoff, T. Culver, J. Keyser, M. Lin, and D. Manocha. Fast computation of generalized Voronoi diagrams using graphics hardware. Proc. ACM SIGGRAPH, pages 277--286, 1999. Google ScholarDigital Library
- C. M. Hoffmann. How to construct the skeleton of CSG objects. In Proc. 4th IMA Conf. on The Mathematics of Surfaces, Bath, UK, 1990. Oxford University Press.Google Scholar
- L. Lam, S.-W. Lee, and C. Y. Chen. Thinning methodologies---A comprehensive survey. IEEE Trans. PAMI, 14(9):869--885, 1992. Google ScholarDigital Library
- E. Larsen, S. Gottschalk, M. Lin, and D. Manocha. Fast proximity queries with swept sphere volumes. Technical Report TR99-018, Dept. Comput. Sci., University of North Carolina, 1999.Google Scholar
- V. Milenkovic. Robust construction of the Voronoi diagram of a polyhedron. In Proc. 5th Canad. Conf. Comput. Geom., pages 473--478, 1993.Google Scholar
- I. Ragnemalm. The euclidean distance transformation in arbitrary dimensions. Pattern Recognition Letters, 14:883--888, 1993. Google ScholarDigital Library
- J. Reddy and G. Turkiyyah. Computation of 3D skeletons using a generalized Delaunay triangulation technique. Computer-Aided Design, 27:677--694, 1995.Google ScholarCross Ref
- E. C. Sherbrooke, N. M. Patrikalakis, and E. Brisson. Computation of the medial axis transform of 3-D polyhedra. In Proc. ACM Solid Modeling, pages 187--199. ACM, 1995. Google ScholarDigital Library
- K. Siddiqi, S. Bouix, A. Tannenbaum, and S. W. Zucker. The Hamilton-Jacobi skeleton. In International Conf. Computer Vision, pages 828--834, 1999. Google ScholarDigital Library
- D. W. Storti, G. M. Turkiyyah, M. A. Ganter, C. T. Lim, and D. M. Stal. Skeleton-based modeling operations on solids. In Proc. ACM Solid Modeling, pages 141--154, 1997. Google ScholarDigital Library
- M. Styner, G. Gerig, S. Joshi, and S. Pizer. Automatic and robust computation of 3D medial models incorporating object variability. International Journal of Computer Vision, to appear. Google ScholarDigital Library
- G. Taubin. A signal processing approach to fair surface design. In Proc. ACM SIGGRAPH, pages 351--358, 1995. Google ScholarDigital Library
- G. M. Turkiyyah, D. W. Storti, M. Ganter, H. Chen, and M. Vimawala. An accelerated triangulation method for computing the skeletons of free-form solid models. Computer-Aided Design, 29(1):5--19, Jan. 1997.Google ScholarCross Ref
- J. Vleugels and M. Overmars. Approximating generalized Voronoi diagrams in any dimension. Technical Report UU-CS-1995-14, Dept. Comput. Sci., Utrecht University, 1995.Google Scholar
- S. A. Wilmarth, N. M. Amato, and P. F. Stiller. Motion planning for a rigid body using random networks on the medial axis of the free space. ACM Symp. Computational Geometry, pages 173--180, 1999. Google ScholarDigital Library
- Y. Y. Zhang and P. S. P. Wang. Analytical comparison of thinning algorithms. Int. J. Pattern Recognit. Artif. Intell., 7:1227--1246, 1993.Google ScholarCross Ref
Index Terms
- Efficient computation of a simplified medial axis
Recommendations
Homotopy-preserving medial axis simplification
SPM '05: Proceedings of the 2005 ACM symposium on Solid and physical modelingWe present a novel algorithm to compute a simplified medial axis of a polyhedron. Our simplification algorithm tends to remove unstable features of Blum's medial axis. Moreover, our algorithm preserves the topological structure of the original medial ...
Approximate medial axis as a voronoi subcomplex
SMA '02: Proceedings of the seventh ACM symposium on Solid modeling and applicationsMedial axis as a compact representation of shapes has evolved as an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to ...
Progressive medial axis filtration
SA '13: SIGGRAPH Asia 2013 Technical BriefsThe Scale Axis Transform provides a parametric simplification of the Medial Axis of a 3D shape which can be seen as a hierarchical description. However, this powerful shape analysis method has a significant computational cost, requiring several minutes ...
Comments