Properties of Ridges and Cores for TwoDimensional Images
 James Damon
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
Pizer and Eberly introduced the “core” as the analogue of the medial axis for greyscale images. For twodimensional images, it is obtained as the “ridge” of a “medial function” defined on 2 + 1dimensional scale space. The medial function is defined using Gaussian blurring and measures the extent to which a point is in the center of the object measured at a scale. Numerical calculations indicate the core has properties quite different from the medial axis. In this paper we give the generic properties of ridges and cores for twodimensional images and explain the discrepancy between core and medial axis properties. We place cores in a larger “relative critical set structure”, which coherently relates disjoint pieces of core. We also give the generic transitions which occur for sequences of images varying with a parameter such as time. The genericity implies the stability of the full structure in any compact viewing area of scale space under sufficiently small L2 perturbations of the image intensity function. We indicate consequences for finding cores and also for adding “markings” to completely determine the structure of the medial function.
 H. Blum and R. Nagel, “Shape description using weighted symmetric axis features,” Pattern Recognition, Vol. 10, pp. 167–180, 1978.
 J. Canny, “A computational approach to edge detection,” IEEE PAMI, Vol. 8, No. 6, pp. 679–698, 1987.
 J.R. Crowley, “A representation for shape based on peaks and ridges in the difference of lowpass transform,” IEEE PAMI, Vol. 6, No. 2, pp. 156–170, 1984.
 J. Damon, “Local Morse theory for solutions to the heat equation and Gaussian blurring,” Jour. Diff. Eqtns, Vol. 115, pp. 368–401, 1995.
 J. Damon, “Generic properties of solutions to partial differential equations,” Arch. Rat. Mech. Anal., Vol. 140, pp. 353–403, 1997.
 J. Damon, “Deformations of sections of singularities and Gorenstein surface singularities,” Amer. J. Math., Vol. 109, pp. 695–722, 1987.
 J. Damon, “Singularities with scale threshold and discrete functions exhibiting generic properties,” in Proc. Third Int.Workshop on Real and Complex Singularities, Sao Carlos, M. Ruas (Eds.), Mat. Contemp. 12, Soc. Bras. Mat., pp. 45–65, 1997.
 J. Damon, “Generic structure of two dimensional images under Gaussian blurring,” SIAM Jour. Appl. Math.,Vol. 59, pp. 97–138, 1999.
 I. Daubechies, “Ten lectures on wavelets,” CBMSNSF Conference Series 61, SIAM, 1992.
 D. Eberly et al., “Ridges for image analysis,” Jour. Math. Imaging and Vision, Vol. 4, pp. 351–371, 1994.
 D. Eberly, “Ridges in image and data analysis,” Series in computational imaging and vision series, Kluwer Academic Publishers: Doordrecht, the Netherlands, 1996.
 J. Furst, J. Miller, R. Keller, and S. Pizer, “Image loci are ridges in geometric spaces,” in Scale Space Theory in Computer Vision, B.M. ter Haar Romeny et al. (Eds.), Springer Lecture Notes in Computer Science, 1997, Vol. 1252, pp. 176–187.
 J. Gauch, “The multiresolution intensity axis of symmetry and its application to image segmentation,” Ph.D. thesis, Dept. Comp. Science, Univ. of North Carolina, 1989.
 M. Golubitsky and V. Guillemin, “Stable mappings and their singularities,” Springer Graduate Texts in Mathematics, SpringerVerlag, 1974.
 P. Johansen, “On the classification of toppoints in scalespace,” Jour. Math. Imag. and Vision, Vol. 4, pp. 57–68, 1994.
 R. Keller, Ph.D. thesis, Dept. Math., Univ. of North Carolina, 1998, in preparation.
 J.J. Koenderink, “The structure of images,” Biological Cybernetics, Vol. 50, pp. 363–370, 1984.
 T. Lindeberg, “Scale space theory in computer vision,” Series in Engineering and Computer Science, Kluwer Academic Publishers: Doordrecht, the Netherlands, 1994.
 L. Lifshitz, “Image segmentation using global knowledge and a priori information,” Ph.D. thesis, Univ. of North Carolina, 1987, Technical Report 87–012.
 J. Miller, “Relative critical sets in ℝℝ and applications to image analysis,” Ph.D. thesis, Dept. Math., Univ. of North Carolina, 1998.
 B. Morse, S. Pizer, and A. Liu, “Multiscale medial axis in medical images,” Image and Vision Computing, Vol. 12, pp. 112–131, 1993.
 B. Morse, et al. “Zoominvariant vision of figural shape: Effects on cores of image disturbances,” Comp. Vision and Image Understanding, to appear.
 S. Pizer, et al., “Object shape before boundary shape: Scale space medial axis,” Jour. Math. Imag. andVision,Vol. 4, pp. 303–313, 1994.
 S. Pizer, et al., “Zoominvariant vision of figural shape: The mathematics of cores,” Comp. Vision and Image Understanding, to appear.
 S. Pizer, J. Gauch, and L. Lifshitz, “Interactive 2D and 3D object definition in medical images based on multiresolution image descriptions,” SPIE Proceedings, Vol. 914 (Part B), pp. 438–449, 1988.
 I. Porteous, “The normal singularities of a submanifold,” Jour. Diff. Geom., Vol. 5, pp. 543–564, 1971.
 J. Rieger, “Generic evolution of edges on families of diffused greyvalue surfaces,” Jour. Math. Imaging and Vision, Vol. 5, No. 95, pp. 207–217
 B.M. ter Haar Romeny (Ed.), Geometry Driven Diffusion in Computer Vision, Kluwer Academic Publishers: Doordrecht, the Netherlands, 1994.
 A. Witkin, “Scale space filtering,” in Proc. Int. Joint Conf. on Artificial Intelligence, Karlsruhe, 1983, pp. 1019–1021.
 Title
 Properties of Ridges and Cores for TwoDimensional Images
 Journal

Journal of Mathematical Imaging and Vision
Volume 10, Issue 2 , pp 163174
 Cover Date
 19990301
 DOI
 10.1023/A:1008379107611
 Print ISSN
 09249907
 Online ISSN
 15737683
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 ridges and cores
 relative critical set
 Gaussian blurring
 medial functions
 genericity
 Industry Sectors
 Authors

 James Damon ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA