Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness

Damon, James

Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1941-1985.

Nous introduisons une structure squelette $(M,U)$ dans ${ℝ}^{n+1}$, qui consiste en un ensemble stratifié de Whitney de dimension $n$ sur lequel est défini un “champ radial de vecteurs” multiformes $U$. C’est une extension de la notion du “Blum medial axis” d’une région dans ${ℝ}^{n+1}$ avec un bord lisse générique. Puis, pour de telles structures squelettes, on peut définir “un bord associé” $ℬ$. Nous introduisons des invariants géométriques du champ radial de vecteurs $U$ et un “flot radial” de $M$ à $ℬ$. Ils nous permettent d’obtenir des conditions numériques suffisantes pour que le bord soit lisse, et de déterminer sa géométrie. Nous établissons en même temps l’existence d’un voisinage tubulaire d’un tel ensemble stratifié de Whitney.

We introduce a skeletal structure $(M,U)$ in ${ℝ}^{n+1}$, which is an $n$- dimensional Whitney stratified set $M$ on which is defined a multivalued “radial vector field” $U$. This is an extension of notion of the Blum medial axis of a region in ${ℝ}^{n+1}$ with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” $ℬ$. We introduce geometric invariants of the radial vector field $U$ on $M$ and a “radial flow” from $M$ to $ℬ$. Together these allow us to provide sufficient numerical conditions for the smoothness of the boundary $ℬ$ as well as allowing us to determine its geometry. In the course of the proof, we establish the existence of a tubular neighborhood for such a Whitney stratified set.

[BA] M. Brady; H. Asada Smoothed Local Symmetries and their Implementation, Intern. J. Robotics Research, Volume 3 (1984), pp. 36-61 | DOI

[BG] J. W. Bruce; P.J. Giblin Growth, motion, and 1-parameter families of symmetry sets, Proc. Royal Soc. Edinburgh, Volume 104A (1986), pp. 179-204 | DOI | MR | Zbl

[BGG] J.W. Bruce; P.J. Giblin; C.G. Gibson Symmetry sets, Proc. Royal Soc. Edinburgh, Volume 101A (1983), pp. 163-186 | MR | Zbl

[BGT] J.W. Bruce; P.J. Giblin; F. Tari Ridges, crests, and subparabolic lines of evolving surfaces, Int. J. Comp. Vision, Volume 18 (1996) no. 3, pp. 195-210 | DOI

[BN] H. Blum; R. Nagel Shape description using weighted symmetric axis features, Pattern Recognition, Volume 10 (1978), pp. 167-180 | DOI | Zbl

[Brz] L.N. Bryzgalova Singularities of the maximum of a function that depends on the parameters, Funct. Anal. Appl, Volume 11 (1977), pp. 49-51 | DOI | MR | Zbl

[D1] J. Damon Smoothness and Geometry of Boundaries Associated to Skeletal Structures II : Geometry in the Blum Case (to appear in Compositio Math) | MR | Zbl

[D2] J. Damon Determining the Geometry of Boundaries of Objects from Medical Data (submitted)

[Gb] P.J. Giblin; Roberto Cipolla and Ralph Martin (eds.) Symmetry Sets and Medial Axes in Two and Three Dimensions, The Mathematics of Surfaces (2000), pp. 306-321 | Zbl

[GG] M. Golubitsky; V. Guillemin Stable Mappings and their Singularities, Graduate Texts in Math., Springer, 1974 | MR | Zbl

[Gi] C.G. Gibson et al. Topological stability of smooth mappings, Lecture Notes in Math., 552, Springer, 1976 | MR | Zbl

[Go] M. Goresky Triangulation of Stratified Objects, Proc. Amer. Math. Soc., Volume 72 (1978), pp. 193-200 | DOI | MR | Zbl

[Hi] M. Hirsch Differential Topology, Graduate Texts in Mathematics, Springer, 1976 | MR | Zbl

[KTZ] B.B. Kimia; A. Tannenbaum; S. Zucker; O. Faugeras (ed.) Toward a computational theory of shape: An overview, Three Dimensional Computer Vision (1990)

[Le] M. Leyton A Process Grammar for Shape, Art. Intelligence, Volume 34 (1988), pp. 213-247 | DOI

[M1] J. Mather; M. Peixoto (ed.) Stratifications and mappings, Dynamical Systems (1973) | Zbl

[M2] J. Mather Distance from a manifold in Euclidean space, Proc. Symp. Pure Math., Volume 40 (1983) no. 2, pp. 199-216 | MR | Zbl

[Mu] J. Munkres Elementary Differential Topology, Annals Math. Studies, 54, Princeton University Press, 1961 | MR | Zbl

[P1] S. Pizer et al. Deformable $M$-reps for 3D Medical Image Segmentation (to appear), Int. J. Comp. Vision, Volume 55 (2003) no. 2-3

[P2] S. Pizer et al. Segmentation, Registration, and Shape Measurement of Variation via Image Object Shape, IEEE Trans. Med. Imaging, Volume 18 (1999), pp. 851-865 | DOI

[P3] S. Pizer et al. Multiscale Medial Loci and Their Properties (to appear), Int. J. Comp. Vision, Volume 55 (2003) no. 2-3

[SB] K. Siddiqi; S. Bouix; A. Tannenbaum; S. Zucker The Hamilton-Jacobi Skeleton, Int. J. Comp. Vision, Volume 48 (2002), pp. 215-231 | DOI | Zbl

[SN] G. Szekely; M. Naf; Ch. Brechbuhler; O. Kubler Calculating 3d Voronoi diagrams of large unrestricted point sets for skeleton generation of complex 3d shapes, Proc. 2nd Int. Workshop on Visual Form (1994), pp. 532-541

[V] J. Verona Stratified Mappings-Structure and Triangulability, Lecture Notes, 1102, Springer, 1984 | MR | Zbl

[Y] J. Yomdin On the local structure of the generic central set, Comp. Math., Volume 43 (1981), pp. 225-238 | Numdam | MR | Zbl