Determining the geometry of boundaries of objects from medial data

International Journal of Computer Vision

Volumn 63, Issue 1, 2005, Pages 45-64

  Damon, James ^a  

^a University of North Carolina at Chapel Hill   (United States)

Author keywords

Blum medial axis; Geometric medial map; Grassfire flow; Intrinsic geometry; Radial flow; Radial shape operator; Relative critical set; Relative geometry; Skeletal structures

Indexed keywords

COMPUTATIONAL GEOMETRY; COMPUTATIONAL METHODS; HAMILTONIANS; RADIAL FLOW; STATISTICAL METHODS; THEOREM PROVING; VECTORS;

BLUM MEDIAL AXIS; GEOMETRIC MEDIAL MAP; GRASSFIRE FLOW; INTRINSIC GEOMETRY; RADIAL SHAPE OPERATOR; RELATIVE CRITICAL SET; RELATIVE GEOMETRY; SKELETAL STRUCTURES;

OBJECT RECOGNITION;

EID: 14844348522     PISSN: 09205691     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11263-005-4946-5     Document Type: Article

References (35)

1. Bogaevski, I. 2002. Perestroikas of shocks and singularities of minimum functions. Physica D, 173:1-28.
2. Blum, H. and Nagel, R. 1978. Shape description using weighted symmetric axis features. Pattern Recognition, 10:167-180.
3. Bruce, J.W. Giblin, P.J., and Gibson, C.G. 1983. Symmetry sets. Proc. Roy. Soc. Edinburgh, 101A: 163-186.
4. Bruce, J.W. and Giblin, P.J. 1986. Growth, motion, and 1-parameter families of symmetry sets. Proc. Roy. Soc. Edinburgh, 104A:179-204.
5. Bruce, J.W., Giblin, P.J., and Tari, F. 1986. Ridges, crests, and subparabolic lines of evolving surfaces. Int. Jour. Comp. Vision, 18(3):195-210.
6. Damon, J. 2003. Smoothness and geometry of boundaries associated to skeletal structures I: Sufficient conditions for smoothness. Annales Inst. Fourier, 53:1941-1985.
7. Damon, J. 2004. Smoothness and Geometry of Boundaries Associated to Skeletal Structures II: Geometry in the Blum case. Compositio Mathematica, 140(6):1657-1674.
8. Damon, J. 1998. Generic structure of two dimensional images under Gaussian blurring. SIAM Jour. Appl. Math, 59:97-138
9. Damon, J. 1999. Properties of ridges and cores for two dimensional images. Jour. Math. Imag. and Vision, 10:163-174.
10. Damon, J. Generic Geometry of Functions. In preparation.
11. Damon, J. Global Geometry of Regions and Boundaries via Skeletal and Medial Integrals. Submitted for publ.
12. Eberly, D. 1996. Ridges in Image and Data Analysis. Series in Computational Imaging and Vision Series. Kluwer Academic Publishers Doordrecht, the Netherlands.
13. Giblin, P.J. 2000. Symmetry sets and medial axes in two and three dimensions. The Mathematics of Surfaces, Roberto Cipolla and Ralph Martin (eds.), Springer-Verlag, pp. 306-321.
14. Gibson, C.G. et al. 1976. Topological stability of smooth mappings. Springer Lecture notes in Math. 552:Springer-Verlag.
15. Golubitsky, M. and Guillemin, V. 1974. Stable Mappings and their Singularities. Springer Graduate Texts in Mathematics. Springer-Verlag, Berlin-Heidelberg.
16. Keller R. 1999. Generic transitions of relative critical sets in parametrized families with applications to image analysis. Ph.D. Thesis, Dept. Math., Univ. of North Carolina.
17. Kimia, B.B., Tannenbaum, A., and Zucker, S. 1990. Toward a computational theory of shape: An overview. In Three Dimensional Computer Vision, O. Faugeras (ed.), MIT Press.
18. Mather, J. 1973. Stratifications and mappings. In Dynamical Systems, M. Peixoto (ed.), Academic Press: New York.
19. Mather, J. 1983. Distance from a manifold in Euclidean space, in Proc. Symp. Pure Math, vol 40 Pt 2, 199-216.
20. n and applications to image analysis. Ph.D. Thesis, Dept. Math., Univ. of North Carolina.
21. Nackman L.R. 1982. Three dimensional shape description using the symmetric axis transform. Ph.D. Thesis, Department of Computer Science, University of North Carolina.
22. Nackman, L.R. and Pizer, S. 1985. Three dimensional shape description using the symmetric axis transform I; Theory. IEEE PAMI, 7(2):187-202.
23. Pizer, S., Eberly, D. et al. 1998. Zoom-invariant vision of figural shape: The mathematics of cores comp. Vision and Image Understanding, 69:55-71.
24. Pizer, S. et al. 2003. Deformable M-reps for 3D medical image segmentation. Int. Jour. Comp. Vision, 55(2/3):85-106.
25. Pizer, S. et al. 1999. Multiscale medial shape-based analysis of image objects. IEEE Trans. Med. Imaging, 18:851-865.
26. Pizer, S. et al. 2003b. Multiscale medial loci and their properties. Int. Jour. Comp. Vision, 55(2/3):155-179.
27. Siddiqi, K., Bouix, S., Tannenbaum, A., and Zucker, S. 2002. The Hamilton-Jacobi skeleton. Int. Jour. Comp. Vision, 48(3):215-231
28. Siersma, D. 1999. Properties of conflict sets in the plane. Geometry and topology of caustics-caustics 1998. Banach Center Publ., 50:267-276.
29. Sotomayor, J., Siersma, D., and Garcia R. 1999. Curvatures of conflict surfaces in euclidean 3-space geometry and topology of caustics-caustics 1998. Banach Center Publ., 50:277-285.
30. Szekely, G., Naf, M., Brechbuhler, Ch., and Kubler, O. 1994. Calculating 3d Voronoi diagrams of large unrestricted point sets for skeleton generation of complex 3d shapes. In Proc. 2nd Int. Workshop on Visual Form, World Scientific Publ., pp. 532-541.
31. Van Manen, M. 2003a. Curvature and Torsion Formulas for Conflict Sets. Caustics '02 (Warsaw). Polish Acad. Sciences Warsaw.
32. Van Manen, M. 2003b. The geometry of conflict sets. Ph.D. Thesis, Dept. of Mathematics, Univ. of Utrecht.
33. Yomdin, J. 1981. On the local structure of the generic central set. Compositio. Math., 43:225-238.
34. Yushkevich, P., Fletcher, T. et al 2002. Continuous medial representations for geometric object modeling in 2D and 3D. In Proc. First Generative Model Based Workshop, Copenhagen, pp. 11-19 and Image and Vision Computing, 21:17-27 (2003).
35. Yushkevich, P., Fletcher, T. et al 2002. Continuous medial representations for geometric object modeling in 2D and 3D. In Proc. First Generative Model Based Workshop, Copenhagen, pp. 11-19 and Image and Vision Computing, 21:17-27 (2003).