Topological properties of Hausdorff discretization, and comparison to other discretization schemes

• Authors: Mohamed Tajine;Christian Ronse
• Venue: Theoretical Computer Science
• Year: 2002

We study a new framework for the discretization of closed setsand operators based on Hausdorff metric: a Hausdorff discretizationof an n-dimensional Euclidean figure F of Rn, in the discrete spaceD=Zn, is a subset S of D whose Hausdorff distance to F is minimal (can be considered as the resolution of the discrete space D); inparticular such a discretization depends on the choice of a metricon Rn. This paper is a continuation of our works (Ronse and Tajine,J. Math. Imaging Vision 12 (3) (2000) 219; Hausdorff discretizationfor cellular distances, and its relation to cover and supercoverdiscretization (to be revised for JVCIR), 2000, Wagner et al., AnApproach to Discretization Based on the Hausdorff Metric. I.ISMM'98, Kluwer Academic Publishers, Dordrecht, 1998, pp. 67-74),in which we have studied some properties of Hausdorffdiscretizations of compact sets. In this paper, we study theproperties of Hausdorff discretization for metrics induced by anorm and we refine this study for the class of homogeneous metrics.We prove that for such metrics the popular covering discretizationsare Hausdorff discretizations. We also compare the Hausdorffdiscretization with the Bresenham discretization (Bresenham, IBMSystems J. 4 (1) (1965) 25). Actually, we prove that the Bresenhamdiscretization of a straight line of R2 is not always a gooddiscretization relatively to the Hausdorff metric. This result isan extension of Tajine et al. (Hausdorff Discretization and itsComparison with other Discretization Schemes, DGCI'99, Paris,Lecture Notes in Computer Sciences Vol. 1568, Springer, Berlin,1999, pp. 399-410), in which we prove the same result for a segmentof R2. Finally, we study how some topological properties of theEuclidean plane R2 are translated in discrete space for Hausdorffdiscretizations. Actually, we prove that a Hausdorff discretizationof a connected closed set is 8-connected and its maximal Hausdorffdiscretization is 4-connected for homogeneous metrics. We study anew framework for the discretization of closed sets and operatorsbased on Hausdorff metric: a Hausdorff discretization of ann-dimensional Euclidean figure F of Rn, in the discrete space D=Zn,is a subset S of D whose Hausdorff distance to F is minimal ( canbe considered as the resolution of the discrete space D); inparticular such a discretization depends on the choice of a metricon Rn. This paper is a continuation of our works (Ronse and Tajine,J. Math. Imaging Vision 12 (3) (2000) 219; Hausdorff discretizationfor cellular distances, and its relation to cover and supercoverdiscretization (to be revised for JVCIR), 2000, Wagner et al., AnApproach to Discretization Based on the Hausdorff Metric. I.ISMM'98, Kluwer Academic Publishers, Dordrecht, 1998, pp. 67-74),in which we have studied some properties of Hausdorffdiscretizations of compact sets. In this paper, we study theproperties of Hausdorff discretization for metrics induced by anorm and we refine this study for the class of homogeneous metrics.We prove that for such metrics the popular covering discretizationsare Hausdorff discretizations. We also compare the Hausdorffdiscretization with the Bresenham discretization (Bresenham, IBMSystems J. 4 (1) (1965) 25). Actually, we prove that the Bresenhamdiscretization of a straight line of R2 is not always a gooddiscretization relatively to the Hausdorff metric. This result isan extension of Tajine et al. (Hausdorff Discretization and itsComparison with other Discretization Schemes, DGCI'99, Paris,Lecture Notes in Computer Sciences Vol. 1568, Springer, Berlin,1999, pp. 399-410), in which we prove the same result for a segmentof R2. Finally, we study how some topological properties of theEuclidean plane R2 are translated in discrete space for Hausdorffdiscretizations. Actually, we prove that a Hausdorff discretizationof a connected closed set is 8-connected and its maximal Hausdorffdiscretization is 4-connected for homogeneous metrics. We study anew framework for the discretization of closed sets and operatorsbased on Hausdorff metric: a Hausdorff discretization of ann-dimensional Euclidean figure F of Rn, in the discrete space D=Zn,is a subset S of D whose Hausdorff distance to F is minimal ( canbe considered as the resolution of the discrete space D); inparticular such a discretization depends on the choice of a metricon Rn. This paper is a continuation of our works (Ronse and Tajine,J. Math. Imaging Vision 12 (3) (2000) 219; Hausdorff discretizationfor cellular distances, and its relation to cover and supercoverdiscretization (to be revised for JVCIR), 2000, Wagner et al., AnApproach to Discretization Based on the Hausdorff Metric. I.ISMM'98, Kluwer Academic Publishers, Dordrecht, 1998, pp. 67-74),in which we have studied some properties of Hausdorffdiscretizations of compact sets. In this paper, we study theproperties of Hausdorff discretization for metrics induced by anorm and we refine this study for the class of homogeneous metrics.We prove that for such metrics the popular covering discretizationsare Hausdorff discretizations. We also compare the Hausdorffdiscretization with the Bresenham discretization (Bresenham, IBMSystems J. 4 (1) (1965) 25). Actually, we prove that the Bresenhamdiscretization of a straight line of R2 is not always a gooddiscretization relatively to the Hausdorff metric. This result isan extension of Tajine et al. (Hausdorff Discretization and itsComparison with other Discretization Schemes, DGCI'99, Paris,Lecture Notes in Computer Sciences Vol. 1568, Springer, Berlin,1999, pp. 399-410), in which we prove the same result for a segmentof R2. Finally, we study how some topological properties of theEuclidean plane R2 are translated in discrete space for Hausdorffdiscretizations. Actually, we prove that a Hausdorff discretizationof a connected closed set is 8-connected and its maximal Hausdorffdiscretization is 4-connected for homogeneous metrics.