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Digital Geometry, Combinatorics, and Discrete Optimization
Stockholm University, Faculty of Science, Department of Mathematics.
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 [en]
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: urn:nbn:se:su:diva-47399ISBN: 978-91-7447-196-0 (print)OAI: oai:DiVA.org:su-47399DiVA: diva2:373896
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
List of papers
1. Chord properties of digital straight linesegments
Open this publication in new window or tab >>Chord properties of digital straight linesegments
2010 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 106, no 2, 169-195 p.Article in journal (Refereed) Published
Abstract [en]

We exhibit the structure of digital straight line segments in the 8-connected plane and in the Khalimsky plane by considering vertical distances and unions of two segments.

National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-51812 (URN)
Available from: 2011-01-12 Created: 2011-01-12 Last updated: 2017-12-11Bibliographically approved
2. The number of Khalimsky-continuous functions on intervals
Open this publication in new window or tab >>The number of Khalimsky-continuous functions on intervals
2010 (English)In: Rocky Mountain Journal of Mathematics, ISSN 0035-7596, Vol. 40, no 5, 1667-1687 p.Article in journal (Refereed) Published
Abstract [en]

We determine the number of Khalimsky-continuous functions defined on an interval and with values in an interval.

National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-47775 (URN)10.1216/RMJ-2010-40-5-1667 (DOI)000284480100014 ()
Available from: 2010-12-07 Created: 2010-12-03 Last updated: 2011-11-23Bibliographically approved
3. The number of continuous curves in digitalgeometry
Open this publication in new window or tab >>The number of continuous curves in digitalgeometry
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.

Keyword
Digital geometry, Khalimsky topology, Khalimsky plane, Khalimsky-continuous function
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-51816 (URN)10.4171/PM/1858 (DOI)
Available from: 2011-01-12 Created: 2011-01-12 Last updated: 2011-01-12Bibliographically approved
4. The number of Khalimsky-continuous functions between two points
Open this publication in new window or tab >>The number of Khalimsky-continuous functions between two points
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We shall investigate the number of digital straight line segments between two points. For a fixed length, we conjecture that the maximum for these values is attained at height 2, and the minimum at height which is equal to the half of length. We shall prove that a local maximum is attained at height 2 and that this is true for all heights in an asymptotic sense. For the latter, we transform digital straight line segments to chords of the unit circle.

Keyword
Digital geometry, digital straight line segments
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-47767 (URN)
Available from: 2010-12-02 Created: 2010-12-02 Last updated: 2011-01-12Bibliographically approved
5. Discrete convexity built on differences
Open this publication in new window or tab >>Discrete convexity built on differences
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-51817 (URN)
Available from: 2011-01-12 Created: 2011-01-12 Last updated: 2011-01-12Bibliographically approved
6. Convexity of marginal functions in the discrete case
Open this publication in new window or tab >>Convexity of marginal functions in the discrete case
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-51820 (URN)
Available from: 2011-01-12 Created: 2011-01-12 Last updated: 2011-01-12Bibliographically approved

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