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Dimensions and projections
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2006 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns dimensions and projections of sets that could be described as fractals. The background is applied problems regarding analysis of human tissue. One way to characterize such complicated structures is to estimate the dimension. The existence of different types of dimensions makes it important to know about their properties and relations to each other. Furthermore, since medical images often are constructed by x-ray, it is natural to study projections.

This thesis consists of an introduction and a summary, followed by three papers.

Paper I, Anders Nilsson, Dimensions and Projections: An Overview and Relevant Examples, 2006. Manuscript.

Paper II, Anders Nilsson and Peter Wingren, Homogeneity and Non-coincidence of Hausdorff- and Box Dimensions for Subsets of ℝn, 2006. Submitted.

Paper III, Anders Nilsson and Fredrik Georgsson, Projective Properties of Fractal Sets, 2006. To be published in Chaos, Solitons and Fractals.

The first paper is an overview of dimensions and projections, together with illustrative examples constructed by the author. Some of the most frequently used types of dimensions are defined, i.e. Hausdorff dimension, lower and upper box dimension, and packing dimension. Some of their properties are shown, and how they are related to each other. Furthermore, theoretical results concerning projections are presented, as well as a computer experiment involving projections and estimations of box dimension.

The second paper concerns sets for which different types of dimensions give different values. Given three arbitrary and different numbers in (0,n), a compact set in ℝn is constructed with these numbers as its Hausdorff dimension, lower box dimension and upper box dimension. Most important in this construction, is that the resulted set is homogeneous in the sense that these dimension properties also hold for every non-empty and relatively open subset.

The third paper is about sets in space and their projections onto planes. Connections between the dimensions of the orthogonal projections and the dimension of the original set are discussed, as well as the connection between orthogonal projection and the type of projection corresponding to realistic x-ray. It is shown that the estimated box dimension of the orthogonal projected set and the realistic projected set can, for all practical purposes, be considered equal.

Place, publisher, year, edition, pages
Umeå: Matematik och matematisk statistik , 2006. , 86 p.
Keyword [en]
Fractals, Hausdorff dimension, box dimension, packing dimension, projections
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-939ISBN: 91-7264-113-4 (print)OAI: oai:DiVA.org:umu-939DiVA: diva2:145127
Supervisors
Available from: 2006-11-20 Created: 2006-11-20 Last updated: 2016-08-31Bibliographically approved
List of papers
1. Dimensions and projections: an overview and relevant examples
Open this publication in new window or tab >>Dimensions and projections: an overview and relevant examples
Manuscript (Other academic)
Identifiers
urn:nbn:se:umu:diva-5570 (URN)
Available from: 2006-11-20 Created: 2006-11-20 Last updated: 2010-01-13Bibliographically approved
2. Homogeneity and non-coincidence of Hausdorff- and box dimensions for subsets of ℝn
Open this publication in new window or tab >>Homogeneity and non-coincidence of Hausdorff- and box dimensions for subsets of ℝn
In: Studia Mathematica, ISSN 0039-3223Article in journal (Refereed) Submitted
Identifiers
urn:nbn:se:umu:diva-5571 (URN)
Available from: 2006-11-20 Created: 2006-11-20Bibliographically approved
3. Projective properties of fractal sets
Open this publication in new window or tab >>Projective properties of fractal sets
2008 (English)In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 35, no 4, 786-794 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, it is shown that a bound on the box dimension of a set in 3D can be established by estimating the box dimension of the discrete image of the projected set i.e. from an image in 2D. It is possible to impose limits on the Hausdorff dimension of the set by estimating the box dimension of the projected set. Furthermore, it is shown how a realistic X-ray projection can be viewed as equivalent to an ideal projection when regarding estimates of fractal dimensions.

Place, publisher, year, edition, pages
Oxford: Pergamon Press, 2008
National Category
Mathematics Physical Sciences
Identifiers
urn:nbn:se:umu:diva-41613 (URN)10.1016/j.chaos.2006.05.091 (DOI)000251006300019 ()
External cooperation:
Available from: 2011-03-30 Created: 2011-03-30 Last updated: 2017-12-11Bibliographically approved

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Citation style
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