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Abstract
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| In a recent paper we have introduced a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach, which uses multivalued maps, provides a better framework to de ne topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In particular, we characterized the deletion of simple points, one of the most important processing operations in digital topology, as a particular kind of retraction. A simple point of a binary image is defined as a point whose deletion does not alter the topology of the image. However, it is well known that the parallel deletion of simple points needs not to preserve topology (the simple set being the middle points in a 3x2 rectangle). A set whose deletion does not change the topology is called a deletable set. In this work we deepen into the properties of this family of continuous maps, now concentrating on parallel deletion of simple points and thinning algorithms, seeing them as digital (deformation) retractions. We show that if D is deletable, then there exists a multivalued retraction from X to X-D). Although the converse is not true, in general, we give conditions for it to hold. In order to guarantee that the parallel deletion of simple points preserve the topology, several strategies have been developed in the literature. We show how some of the more extended ones are also related or even can be characterized as retractions in terms of our notion of continuity. | |
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International
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Si |
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Congress
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Algebraic Topological Methods in Computer Science (ATMCS 2008) |
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960 |
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Place
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Paris (Francia) |
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Reviewers
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Si |
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ISBN/ISSN
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1111111111 |
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Start Date
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07/07/2008 |
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End Date
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11/07/2008 |
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From page
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13 |
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To page
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14 |
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Algebraic Topological Methods in Computer Science (ATMCS) III |