We introduce a skeletal structure in , which is an - dimensional Whitney stratified set on which is defined a multivalued “radial vector field” . This is an extension of notion of the Blum medial axis of a region in with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” . We introduce geometric invariants of the radial vector field on and a “radial flow” from to . Together these allow us to provide sufficient numerical conditions for the smoothness of the boundary 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.
Nous introduisons une structure squelette dans , qui consiste en un ensemble stratifié de Whitney de dimension sur lequel est défini un “champ radial de vecteurs” multiformes . C’est une extension de la notion du “Blum medial axis” d’une région dans avec un bord lisse générique. Puis, pour de telles structures squelettes, on peut définir “un bord associé” . Nous introduisons des invariants géométriques du champ radial de vecteurs et un “flot radial” de à . 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.
Keywords: skeletal structures, Whitney stratified sets, Blum medial axis, shock set, radial shape operator, grassfire flow, radial flow
Mot clés : structure squelette, ensemble stratifié de Whitney, axe moyen de Blum, ensemble de choc, opérateur de forme, flot radial
Damon, James 1
@article{AIF_2003__53_6_1941_0, author = {Damon, James}, title = {Smoothness and geometry of boundaries associated to skeletal structures {I:} sufficient conditions for smoothness}, journal = {Annales de l'Institut Fourier}, pages = {1941--1985}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {6}, year = {2003}, doi = {10.5802/aif.1997}, zbl = {1047.57014}, mrnumber = {2038785}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1997/} }
TY - JOUR AU - Damon, James TI - Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness JO - Annales de l'Institut Fourier PY - 2003 SP - 1941 EP - 1985 VL - 53 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1997/ DO - 10.5802/aif.1997 LA - en ID - AIF_2003__53_6_1941_0 ER -
%0 Journal Article %A Damon, James %T Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness %J Annales de l'Institut Fourier %D 2003 %P 1941-1985 %V 53 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1997/ %R 10.5802/aif.1997 %G en %F AIF_2003__53_6_1941_0
Damon, James. Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness. Annales de l'Institut Fourier, Volume 53 (2003) no. 6, pp. 1941-1985. doi : 10.5802/aif.1997. https://aif.centre-mersenne.org/articles/10.5802/aif.1997/
[BA] Smoothed Local Symmetries and their Implementation, Intern. J. Robotics Research, Volume 3 (1984), pp. 36-61 | DOI
[BG] Growth, motion, and 1-parameter families of symmetry sets, Proc. Royal Soc. Edinburgh, Volume 104A (1986), pp. 179-204 | DOI | MR | Zbl
[BGG] Symmetry sets, Proc. Royal Soc. Edinburgh, Volume 101A (1983), pp. 163-186 | MR | Zbl
[BGT] Ridges, crests, and subparabolic lines of evolving surfaces, Int. J. Comp. Vision, Volume 18 (1996) no. 3, pp. 195-210 | DOI
[BN] Shape description using weighted symmetric axis features, Pattern Recognition, Volume 10 (1978), pp. 167-180 | DOI | Zbl
[Brz] 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] Smoothness and Geometry of Boundaries Associated to Skeletal Structures II : Geometry in the Blum Case (to appear in Compositio Math) | MR | Zbl
[D2] Determining the Geometry of Boundaries of Objects from Medical Data (submitted)
[Gb] Symmetry Sets and Medial Axes in Two and Three Dimensions, The Mathematics of Surfaces (2000), pp. 306-321 | Zbl
[GG] Stable Mappings and their Singularities, Graduate Texts in Math., Springer, 1974 | MR | Zbl
[Gi] Topological stability of smooth mappings, Lecture Notes in Math., 552, Springer, 1976 | MR | Zbl
[Go] Triangulation of Stratified Objects, Proc. Amer. Math. Soc., Volume 72 (1978), pp. 193-200 | DOI | MR | Zbl
[Hi] Differential Topology, Graduate Texts in Mathematics, Springer, 1976 | MR | Zbl
[KTZ] Toward a computational theory of shape: An overview, Three Dimensional Computer Vision (1990)
[Le] A Process Grammar for Shape, Art. Intelligence, Volume 34 (1988), pp. 213-247 | DOI
[M1] Stratifications and mappings, Dynamical Systems (1973) | Zbl
[M2] Distance from a manifold in Euclidean space, Proc. Symp. Pure Math., Volume 40 (1983) no. 2, pp. 199-216 | MR | Zbl
[Mu] Elementary Differential Topology, Annals Math. Studies, 54, Princeton University Press, 1961 | MR | Zbl
[P1] Deformable -reps for 3D Medical Image Segmentation (to appear), Int. J. Comp. Vision, Volume 55 (2003) no. 2-3
[P2] Segmentation, Registration, and Shape Measurement of Variation via Image Object Shape, IEEE Trans. Med. Imaging, Volume 18 (1999), pp. 851-865 | DOI
[P3] Multiscale Medial Loci and Their Properties (to appear), Int. J. Comp. Vision, Volume 55 (2003) no. 2-3
[SB] The Hamilton-Jacobi Skeleton, Int. J. Comp. Vision, Volume 48 (2002), pp. 215-231 | DOI | Zbl
[SN] 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] Stratified Mappings-Structure and Triangulability, Lecture Notes, 1102, Springer, 1984 | MR | Zbl
[Y] On the local structure of the generic central set, Comp. Math., Volume 43 (1981), pp. 225-238 | Numdam | MR | Zbl
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