Connectivity on Complete Lattices

Classically, connectivity is a topological notion for sets, often introduced by means of arcs. A nontopological axiomatics has been proposed by Matheron and Serra. The present paper extends it to complete sup-generated lattices. A connection turns out to be characterized by a family of openings labelled by the sup-generators, which partition each element of the lattice into maximal terms, of zero infima. When combined with partition closings, these openings generate strong sequential alternating filters. Starting from a first connection several others may be designed by acting on some dilations or symmetrical operators. When applying this theory to function lattices, one interprets the so-called connected operators in terms of actual connections, as well as the watershed mappings. But the theory encompasses the numerical functions and extends, among others, to multivariate lattices.

[1]  Laurent Najman,et al.  Definition and some properties of the watershed of a continuous function , 1993 .

[2]  Henk J. A. M. Heijmans,et al.  The algebraic basis of mathematical morphology. I Dilations and erosions , 1990, Comput. Vis. Graph. Image Process..

[3]  Luc Vincent,et al.  Morphological grayscale reconstruction in image analysis: applications and efficient algorithms , 1993, IEEE Trans. Image Process..

[4]  C. Lantuéjoul,et al.  On the use of the geodesic metric in image analysis , 1981 .

[5]  Laurent Najman,et al.  Watershed of a continuous function , 1994, Signal Process..

[6]  H. Heijmans,et al.  The algebraic basis of mathematical morphology , 1988 .

[7]  Fernand Meyer,et al.  Topographic distance and watershed lines , 1994, Signal Process..

[8]  H.J.A.M. Heijmans,et al.  Introduction to connected operators , 1999 .

[9]  Beatriz Marcotegui,et al.  Morphological Segmentation of Image Sequences , 1994, ISMM.

[10]  José Crespo,et al.  Theoretical aspects of morphological filters by reconstruction , 1995, Signal Process..

[11]  J. Serra,et al.  An overview of morphological filtering , 1992 .

[12]  Henk J. A. M. Heijmans,et al.  Convergence, continuity, and iteration in mathematical morphology , 1992, J. Vis. Commun. Image Represent..

[13]  Montse Pardàs,et al.  Joint Region and Motion Estimation with Morphological Tools , 1994, ISMM.

[14]  J. Crespo.  Morphological connected filters and intra-region smoothing for image segmentation , 1993 .

[15]  Fernand Meyer,et al.  Minimum Spanning Forests for Morphological Segmentation , 1994, ISMM.

[16]  Luis Torres,et al.  Connectivity filters for image sequences , 1992, Optics & Photonics.

[17]  Philippe Salembier,et al.  Morphological multiscale image segmentation , 1992, Other Conferences.

[18]  Philippe Salembier,et al.  Practical Extensions of Connected Operators , 1996, ISMM.

[19]  Philippe Jean Salembier Clairon,et al.  Practical Extensions of Connected Operators , 1996 .

[20]  Jean C. Serra.  Equicontinuous functions: a model for mathematical morphology (Invited Paper) , 1992, Optics & Photonics.

[21]  Jean Serra,et al.  Morphological pyramids for image coding , 1993, Other Conferences.

[22]  Christian Ronse,et al.  Set-Theoretical Algebraic Approaches to Connectivity in Continuous or Digital Spaces , 2004, Journal of Mathematical Imaging and Vision.

[23]  Beatriz Marcotegui.  Segmentation de séquences d'images en vue du codage. (Segmentation of image sequences for coding) , 1996 .

[24]  Ronald Jones,et al.  An Attribute-Based Approach to Mathematical Morphology , 1996, ISMM.

[25]  Henk J. A. M. Heijmans,et al.  The algebraic basis of mathematical morphology : II. Openings and closings , 1991, CVGIP Image Underst..

[26]  Pierre Soille,et al.  Mathematical Morphology and Its Applications to Image Processing , 1994, Computational Imaging and Vision.

[27]  Azriel Rosenfeld,et al.  Connectivity in Digital Pictures , 1970, JACM.

[28]  Linda G. Shapiro,et al.  Computer and Robot Vision , 1991 .

[29]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[30]  R. P. Dilworth.  Review: G. Birkhoff, Lattice theory , 1950 .

[31]  Jean Paul Frédéric Serra.  Morphological Image Operators (Henk J. A. M. Heijmans) , 1996, SIAM Rev..

[32]  G. Matheron.  Random Sets and Integral Geometry , 1976 .

[33]  Philippe Salembier,et al.  Connected operators and pyramids , 1993, Optics & Photonics.