Abstract
In this paper we study the application of the Affine Morphological Scale Space (AMSS) to the analysis of singularities (corners or multiple junctions) of the shapes present in a 2-D image. We introduce a new family of travelling wave solutions of AMSS which determines the evolution of the initial shapes given by conics. We characterize the evolution of corners accross the scales according to their angle. We develop a numerical algorithm to compute AMSS accross the scales and we present some experimental results about corners and multiple junction detection.
Similar content being viewed by others
References
Alvarez, L., Guichard, F., Lions, P. L., and Morel, J. M. 1992. Axiomatisation et nouveaux opérateurs de la morphologie mathematique. C. R. Acad. Sci. Paris, t.315, Série I, pp. 265-268.
Alvarez, L., Lions, P. L., and Morel, J. M. 1992. Image selective smoothing and edge detection by nonlinear diffusion (II). SIAM Journal on Numerical Analysis, 29:845-866.
Alvarez, L., Guichard, F., Lions, P. L., and Morel, J. M. 1993. Axioms and fundamental equations of image processing. Arch. for Rat. Mech., t.123. 3:199-257.
Alvarez, L. and Mazorra, L. 1994. Signal and image restoration using shock filters and anisotropic diffusion. SIAM Journal on Numerical Analysis, 31(2):590-605.
Alvarez, L. and Morel, J. M. 1994. Formalization and computational aspects of image analysis. Acta Numerica, pp. 1-59.
Asada, H. and Brady, M. 1986. The curvature primal sketch. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(1).
Beaudet, P. R. 1978. Rotational invariant image operators. 4th Intern. Conf. Patt. Recog., Tokio, pp. 579-583.
Caselles, V. and Sbert, C. 1996. What is the best causal scale space for 3-D images?. In SIAM Journal on Applied Math, 56:1199-1246.
Catté, F., Dibos, F., and Koepfler, G. 1993. A morphological approach of mean curvature motion. Report 9310, CEREMADE. Université Paris Dauphine.
Cohignac, T., Eve, F., Guichard, F., Lopez, C., and Morel, J. M. 1993. Numerical analysis of the fundamental equation of image processing (to appear).
Crandall, M. G., Ishii, H., and Lions, P. L. 1992. User's guide to viscosity solution of second order partial differential equation. Bull. AMS, 27/1:1-67.
Deriche, R. and Giraudon, G. 1993. A computational approach for corner and vertex detection. International Journal of Computer Vision, 10(2):101-124.
Deriche, R. and Blaszka, T. 1993. Recovering and characterizing image features using an efficient model based approach. Proc. IEEE Conference on Computer Vision and Pattern Recognition, New York, pp. 14-17.
Deriche, R. 1994. Private communication.
Dreschler, L. and Nagel, H. H. 1982. On the selection of critical points and local curvature extrema of region boundaries for interframe matching. Intern. Conf. Patt. Recog., pp. 542-544.
Faugeras, O. 1993. On the evolution of simple curves of the real projective plane. C. R. Acad. Sci. Paris, T.317, Série I (6):565- 570.
Gage, M. and Hamilton, R. S. 1986. The heat equation shrinking convex plane curves. J. Differential Geometry, 23:69-96.
Grayson, M. 1987. The heat equation shrinks embedded plane curves to round points. J. Differential Geometry, 26:285-314.
Hummel, R. 1986. Representations based on zero-crossing in scale-space. Proc. IEEE Computer Vision and Pattern Recognition Conference, pp. 204-209.
Julesz, B. 1981. Textons, the elements of texture perception, and their interactions. Nature, Vol. 290.
Kitchen, L. and Rosenfeld, A. 1982. Gray-level corner detection. Patt. Recog. Letter, (1):95-102.
Koenderink, J. J. 1984. The structure of images. Biol. Cybern., 50:363-370.
Lindeberg, T. 1993. Detecting salient blob-like image structures and their scales with a scale-space primal sketch: A method for focus-of-attention. Intl. J. of Computer Vision, 11(3):283-318.
Lopez, C. and Morel, J. M. 1992. Axiomatisation of shape analysis and application to texture hyperdiscrimination. Proc. of the Trento Conference on Surface Tension and Movement by Mean Curvature, De Gruyter Publishers: Berlin.
Mackworth, A. and Mokhtarian, F. 1986. Scale-based description and recognition of planar curves and two-dimensional shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(1).
Mackworth, A. and Mokhtarian, F. 1992. A theory of multiscale, curvature-based shape representation for planar curves. IEEE Trans. Pattern Anal. Machine Intell., 14:789-805.
Maragos, P. 1987. Tutorial on advances in morphological image processing and analysis. Optical Engineering, 26(7).
Marr, D. 1982. Vision. Freeman and Co.
Matheron, G. 1975. Random Sets and Integral Geometry. John Wiley: N. Y.
Merriman, B., Bence, J., and Osher, S. 1992. Diffusion generated motion by mean curvature. CAM Report 92-18, Dept. of Mathematics, University of California, Los Angeles CA 90024. 1555.
Merriman, B., Bence, J., and Osher, S. 1994. Motion of multiple junctions: A level set approach. Journal of Computational Physics, 112:334-363.
Morel, J. M. and Solimini, S. 1994. Variational Methods in Image Segmentation. Birkhauser.
Osher, S. and Sethian, J. 1988. Fronts propagating with curvature dependent speed: Algorithms based on the Hamilton-Jacobi formulation. J. Comp. Physics, 79:12-49.
Perona, P. and Malik, J. 1987. A scale space and edge detection using anisotropic diffusion. Proc. IEEE Computer Soc. Workshop on Computer Vision.
Rohr, K. 1994. Localization properties of direct corner detectors. Journal of Mathematical Imaging and Vision, 4:139-150.
Sapiro, G. and Tannenbaum, A. 1993. Affine invariant scale-space. International Journal of Computer Vision, 11(1):25-44.
Sapiro, G. and Tannenbaum, A. 1993. On invariant curve evolution and image analysis. Indiana University Mathematics Journal, 42(3):985-1009.
Sapiro, G. and Tannenbaum, A. 1994. On affine plane curve evolution. Journal of Functional Analysis, 119(1):79-120.
Serra, J. 1982. Image Analysis and Mathematical Morphology, Academic Press, Vol. 1.
Witkin, A. P. 1983. Scale-space filtering. Proc. of IJCAI, Karlsruhe, pp. 1019-1021.
Yuille, A. and Poggio, T. 1986. Scaling theorems for zero crossings. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 8.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alvarez, L., Morales, F. Affine Morphological Multiscale Analysis of Corners and Multiple Junctions. International Journal of Computer Vision 25, 95–107 (1997). https://doi.org/10.1023/A:1007959616598
Issue Date:
DOI: https://doi.org/10.1023/A:1007959616598