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Dimensions and projections
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
2006 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Umeå: Matematik och matematisk statistik , 2006. , s. 86
Nyckelord [en]
Fractals, Hausdorff dimension, box dimension, packing dimension, projections
Nationell ämneskategori
Matematik
Identifikatorer
URN: urn:nbn:se:umu:diva-939ISBN: 91-7264-113-4 (tryckt)OAI: oai:DiVA.org:umu-939DiVA, id: diva2:145127
Handledare
Tillgänglig från: 2006-11-20 Skapad: 2006-11-20 Senast uppdaterad: 2016-08-31Bibliografiskt granskad
Delarbeten
1. Dimensions and projections: an overview and relevant examples
Öppna denna publikation i ny flik eller fönster >>Dimensions and projections: an overview and relevant examples
Manuskript (Övrigt vetenskapligt)
Identifikatorer
urn:nbn:se:umu:diva-5570 (URN)
Tillgänglig från: 2006-11-20 Skapad: 2006-11-20 Senast uppdaterad: 2010-01-13Bibliografiskt granskad
2. Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of Rn
Öppna denna publikation i ny flik eller fönster >>Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of Rn
2007 (Engelska)Ingår i: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 181, nr 3, s. 285-296Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

A class of subsets of Rnis constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. Foreach triple (r, s, t) of numbers in the interval (0, n] with r < s < t, a compact setK is constructed so that for any non-empty subset U relatively open in K, we have(dimH(U), dimB(U), dimB(U)) = (r, s, t). Moreover, 2−n ≤ Hr(K) ≤ 2nr/2.

Ort, förlag, år, upplaga, sidor
Polish Academy of Sciences, 2007
Nationell ämneskategori
Matematisk analys
Identifikatorer
urn:nbn:se:umu:diva-5571 (URN)10.4064/sm181-3-5 (DOI)000249896500005 ()2-s2.0-34548543814 (Scopus ID)
Anmärkning

Originally included in thesis in manuscript form.

Tillgänglig från: 2006-11-20 Skapad: 2006-11-20 Senast uppdaterad: 2022-03-22Bibliografiskt granskad
3. Projective properties of fractal sets
Öppna denna publikation i ny flik eller fönster >>Projective properties of fractal sets
2008 (Engelska)Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 35, nr 4, s. 786-794Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
Oxford: Pergamon Press, 2008
Nationell ämneskategori
Matematik Fysik
Identifikatorer
urn:nbn:se:umu:diva-41613 (URN)10.1016/j.chaos.2006.05.091 (DOI)000251006300019 ()2-s2.0-34748926923 (Scopus ID)
Externt samarbete:
Tillgänglig från: 2011-03-30 Skapad: 2011-03-30 Senast uppdaterad: 2023-03-23Bibliografiskt granskad

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