A Bibliography on Digital and Computational Convexity (1961-1988)
IEEE Transactions on Pattern Analysis and Machine Intelligence
Fractals everywhere
The algebraic basis of mathematical morphology. I. dilations and erosions
Computer Vision, Graphics, and Image Processing
CVGIP: Image Understanding
Digitizations preserving topological and differential geometric properties
Computer Vision and Image Understanding
Fundamentals of surface voxelization
Graphical Models and Image Processing
A realistic digitization model of straight lines
Computer Vision and Image Understanding
Preserving Topology by a Digitization Process
Journal of Mathematical Imaging and Vision
An approach to discretization based on the Hausdorff metric
ISMM '98 Proceedings of the fourth international symposium on Mathematical morphology and its applications to image and signal processing
ISMM '98 Proceedings of the fourth international symposium on Mathematical morphology and its applications to image and signal processing
Algorithms for Graphics and Imag
Algorithms for Graphics and Imag
Digital Image Processing
Supercover of Straight Lines, Planes and Triangles
DGCI '97 Proceedings of the 7th International Workshop on Discrete Geometry for Computer Imagery
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Topological properties of Hausdorff discretization, and comparison to other discretization schemes
Theoretical Computer Science
Hausdorff Sampling of Closed Sets into a Boundedly Compact Space
Digital and Image Geometry, Advanced Lectures [based on a winter school held at Dagstuhl Castle, Germany in December 2000]
Hausdorff Discretizations of Algebraic Sets and Diophantine Sets
DGCI '00 Proceedings of the 9th International Conference on Discrete Geometry for Computer Imagery
Parallel Line Grouping Based on Interval Graphs
DGCI '00 Proceedings of the 9th International Conference on Discrete Geometry for Computer Imagery
Hausdorff sampling of closed sets into a boundedly compact space
Digital and image geometry
Graph-theoretical properties of parallelism in the digital plane
Discrete Applied Mathematics
Continuous digitization in Khalimsky spaces
Journal of Approximation Theory
A topological sampling theorem for Robust boundary reconstruction and image segmentation
Discrete Applied Mathematics
Towards a general sampling theory for shape preservation
Image and Vision Computing
Digital segments and Hausdorff discretization
IWCIA'08 Proceedings of the 12th international conference on Combinatorial image analysis
How to find a khalimsky-continuous approximation of a real-valued function
IWCIA'04 Proceedings of the 10th international conference on Combinatorial Image Analysis
Topologically correct image segmentation using alpha shapes
DGCI'06 Proceedings of the 13th international conference on Discrete Geometry for Computer Imagery
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In this paper, a new approach to thediscretization of n-dimensional Euclidean figures is studied: thediscretization of a compact Euclidean set K is a discrete set Swhose Hausdorff distance to K is minimal; in particular such adiscretization depends on the choice of a metric in the Euclideanspace, for example the Euclidean or a chamfer distance. We call sucha set S a Hausdorff discretizing set of K. The set ofHausdorff discretizing sets of K is nonvoid, finite, and closedunder union; we consider thus in particular the greatest one amongsuch sets, which we call the maximal Hausdorff discretization ofK. We give a mathematical description of Hausdorff discretizingsets: it is related to the discretization by dilationconsidered by Heijmans and Toet and the cover discretizationstudied by Andrès. We have a bound on the Hausdorff distancebetween a compact set and its maximal Hausdorff discretization, andthe latter converges (for the Hausdorff metric) to the compact setwhen the spacing of the discrete grid tends to zero. Such aconvergence result holds also for the discretization by dilation whenthe structuring element satisfies the covering assumption. Ourapproach is here the most general possible. In a next paper we willconsider the case where the underlying metric on points satisfiessome general constraints in relation to the cells associated to thediscrete points, and we will then see that these constraintsguarantee that the usual supercover and coverdiscretizations give indeed Hausdorff discretizing sets.