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The number of continuous curves in digitalgeometry
Stockholm University, Faculty of Science, Department of Mathematics.
2010 (English)In: Portugaliae Mathematica, ISSN 0032-5155, Vol. 67, no 1, 75-89 p.Article in journal (Refereed) Published
Abstract [en]

As a model for continuous curves in digital geometry, we study the Khalimsky-continuous functions defined on the integers and with values in the set of integers or the set of natural numbers. We determine the number of such functions on a given interval. It turns out that these numbers are related to the Delannoy and Schröder arrays, and a relation between these numbers is established.

Place, publisher, year, edition, pages
2010. Vol. 67, no 1, 75-89 p.
Keyword [en]
Digital geometry, Khalimsky topology, Khalimsky plane, Khalimsky-continuous function
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-51816DOI: 10.4171/PM/1858OAI: oai:DiVA.org:su-51816DiVA: diva2:386167
Available from: 2011-01-12 Created: 2011-01-12 Last updated: 2011-01-12Bibliographically approved
In thesis
1. Digital Geometry, Combinatorics, and Discrete Optimization
Open this publication in new window or tab >>Digital Geometry, Combinatorics, and Discrete Optimization
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two parts: digital geometry and discrete optimization.

In the first part we study the structure of digital straight line segments. We also study digital curves from a combinatorial point of view.

In Paper I we study the straightness in the 8-connected plane and in the Khalimsky plane by considering vertical distances and unions of two segments. We show that we can investigate the straightness of Khalimsky arcs by using our knowledge from the 8-connected plane.

In Paper II we determine the number of Khalimsky-continuous functions with 2, 3 and 4 points in their codomain. These enumerations yield examples of known sequences as well as new ones. We also study the asymptotic behavior of each of them.

In Paper III we study the number of Khalimsky-continuous functions with codomain Z and N. This gives us examples of Schröder and Delannoy numbers. As a byproduct we get some relations between these numbers.

In Paper IV we study the number of Khalimsky-continuous functions between two points in a rectangle. Using a generating function we get a recurrence formula yielding this numbers.

 

In the second part we study an analogue of discrete convexity, namely lateral convexity.

In Paper V we define by means of difference operators the class of lateral convexity. The functions have plus infinity in their codomain. For the real-valued functions we need to check the difference operators for a smaller number of points. We study the relation between this class and integral convexity.

In Paper VI we study the marginal function of real-valued functions in this class and its generalization. We show that for two points with a certain distance we have a Lipschitz property for the points where the infimum is attained. We show that if a function is in this class, the marginal function is also in the same class.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2010. 42 p.
Keyword
Digital geometry, Khalimsky topology, Khalimsky plane, Khalimsky-continuous function, digital straight line segments, discrete optimization, discrete convexity, integral convexity, lateral convexity, marginal function
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-47399 (URN)978-91-7447-196-0 (ISBN)
Public defence
2011-01-21, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Submitted. Paper 5: Manuscript. Paper 6: Manuscript.Available from: 2010-12-29 Created: 2010-12-01 Last updated: 2011-01-12Bibliographically approved

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