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

James Damon[1]

  • [1] University of North Carolina, Department of Mathematics, Chapel Hill NC 27599 (USA)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 6, page 1941-1985
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Damon, James. "Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness." Annales de l’institut Fourier 53.6 (2003): 1941-1985. <http://eudml.org/doc/116090>.

@article{Damon2003,
abstract = {We introduce a skeletal structure $(M,U)$ in $\{\mathbb \{R\}\}^\{n+1\}$, 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 $\{\mathbb \{R\}\}^\{n+1\}$ with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” $\{\mathcal \{B\}\}$. We introduce geometric invariants of the radial vector field $U$ on $M$ and a “radial flow” from $M$ to $\{\mathcal \{B\}\}$. Together these allow us to provide sufficient numerical conditions for the smoothness of the boundary $\{\mathcal \{B\}\}$ 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.},
affiliation = {University of North Carolina, Department of Mathematics, Chapel Hill NC 27599 (USA)},
author = {Damon, James},
journal = {Annales de l’institut Fourier},
keywords = {skeletal structures; Whitney stratified sets; Blum medial axis; shock set; radial shape operator; grassfire flow; radial flow; Whitney stratified set; skeletal strucuture; shape operator; tubular neighborhood},
language = {eng},
number = {6},
pages = {1941-1985},
publisher = {Association des Annales de l'Institut Fourier},
title = {Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness},
url = {http://eudml.org/doc/116090},
volume = {53},
year = {2003},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 6
SP - 1941
EP - 1985
AB - We introduce a skeletal structure $(M,U)$ in ${\mathbb {R}}^{n+1}$, 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 ${\mathbb {R}}^{n+1}$ with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” ${\mathcal {B}}$. We introduce geometric invariants of the radial vector field $U$ on $M$ and a “radial flow” from $M$ to ${\mathcal {B}}$. Together these allow us to provide sufficient numerical conditions for the smoothness of the boundary ${\mathcal {B}}$ 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.
LA - eng
KW - skeletal structures; Whitney stratified sets; Blum medial axis; shock set; radial shape operator; grassfire flow; radial flow; Whitney stratified set; skeletal strucuture; shape operator; tubular neighborhood
UR - http://eudml.org/doc/116090
ER -

References

top
  1. H. Blum, R. Nagel, Shape description using weighted symmetric axis features, Pattern Recognition 10 (1978), 167-180 Zbl0379.68067
  2. M. Brady, H. Asada, Smoothed Local Symmetries and their Implementation, Intern. J. Robotics Research 3 (1984), 36-61 
  3. J.W. Bruce, P.J. Giblin, C.G. Gibson, Symmetry sets, Proc. Royal Soc. Edinburgh 101A (1983), 163-186 Zbl0593.58012MR824218
  4. J. W. Bruce, P.J. Giblin, Growth, motion, and 1-parameter families of symmetry sets, Proc. Royal Soc. Edinburgh 104A (1986), 179-204 Zbl0656.58022MR877901
  5. J.W. Bruce, P.J. Giblin, F. Tari, Ridges, crests, and subparabolic lines of evolving surfaces, Int. J. Comp. Vision 18 (1996), 195-210 
  6. L.N. Bryzgalova, Singularities of the maximum of a function that depends on the parameters, Funct. Anal. Appl 11 (1977), 49-51 Zbl0369.58010MR482807
  7. J. Damon, Smoothness and Geometry of Boundaries Associated to Skeletal Structures II : Geometry in the Blum Case Zbl1071.57022MR2098407
  8. J. Damon, Determining the Geometry of Boundaries of Objects from Medical Data (submitted) 
  9. P.J. Giblin, Symmetry Sets and Medial Axes in Two and Three Dimensions, The Mathematics of Surfaces (2000), 306-321, Springer-Verlag Zbl0979.53004
  10. C.G. Gibson et al., Topological stability of smooth mappings, 552 (1976), Springer Zbl0377.58006MR436203
  11. M. Golubitsky, V. Guillemin, Stable Mappings and their Singularities, (1974), Springer Zbl0294.58004MR341518
  12. M. Goresky, Triangulation of Stratified Objects, Proc. Amer. Math. Soc. 72 (1978), 193-200 Zbl0392.57001MR500991
  13. M. Hirsch, Differential Topology, (1976), Springer Zbl0356.57001MR448362
  14. B.B. Kimia, A. Tannenbaum, S. Zucker, Toward a computational theory of shape: An overview, Three Dimensional Computer Vision (1990), MIT Press 
  15. M. Leyton, A Process Grammar for Shape, Art. Intelligence 34 (1988), 213-247 
  16. J. Mather, Stratifications and mappings, Dynamical Systems (1973), Academic Press, New York Zbl0286.58003
  17. J. Mather, Distance from a manifold in Euclidean space, Proc. Symp. Pure Math. 40 (1983), 199-216 Zbl0519.58015MR713249
  18. J. Munkres, Elementary Differential Topology, 54 (1961), Princeton University Press Zbl0107.17201MR163320
  19. S. Pizer et al., Deformable M -reps for 3D Medical Image Segmentation (to appear), Int. J. Comp. Vision 55 (2003) 
  20. S. Pizer et al., Segmentation, Registration, and Shape Measurement of Variation via Image Object Shape, IEEE Trans. Med. Imaging 18 (1999), 851-865 
  21. S. Pizer et al., Multiscale Medial Loci and Their Properties (to appear), Int. J. Comp. Vision 55 (2003) 
  22. K. Siddiqi, S. Bouix, A. Tannenbaum, S. Zucker, The Hamilton-Jacobi Skeleton, Int. J. Comp. Vision 48 (2002), 215-231 Zbl1012.68757
  23. 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), 532-541, World Scientific Publ. 
  24. J. Verona, Stratified Mappings-Structure and Triangulability, 1102 (1984), Springer Zbl0543.57002MR771120
  25. J. Yomdin, On the local structure of the generic central set, Comp. Math. 43 (1981), 225-238 Zbl0465.58008MR622449

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.