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  • 1.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Continuous digitization in Khalimsky spaces2006Report (Other (popular scientific, debate etc.))
  • 2.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Continuous extension in topological digital spaces2004Report (Other scientific)
  • 3.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Digital straight lines in the Khalimsky plane2005In: Mathematica Scandinavica, no 96, p. 49-64Article in journal (Refereed)
  • 4.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Digital straight lines in the Khalimsky plane2003Report (Other scientific)
  • 5.
    Melin, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Digital surfaces and boundaries in Khalimsky spaces2007In: Journal of Mathematical Imaging and Vision, ISSN 0924-9907, E-ISSN 1573-7683, Vol. 28, no 2, p. 169-177Article in journal (Refereed)
    Abstract [en]

    Let X be a smallest-neighborhood space, sometimes called an Alexandrov space. We demonstrate that the graph of a Khalimsky-continuous mapping X->Z is a surface having a Jordan--Brouwer type separation property. We study infima and suprema of families of such continuous mappings, a study that naturally leads to the introduction of an extended Khalimsky line. Moreover, we show that the boundary of a connected subset, U, of the Khalimsky plane is connected precisely when the complement of U is connected.

  • 6.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Digital surfaces in Khalimsky spaces2005Report (Other scientific)
  • 7.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Digitization in Khalimsky spaces2004Licentiate thesis, monograph (Other scientific)
  • 8.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Extension of continuous functions in digital spaces with the Khalimsky topology2005In: Topology Appl., no 153, p. 52-65Article in journal (Refereed)
  • 9.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Extension of continuous functions in digital spaces with the Khalimsky topology2003Report (Other scientific)
  • 10.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Extensions of continous functions in digital spaces with the Khalimsky topology2003Report (Other scientific)
    Abstract [en]

    The digital space Z^n equipped with Efim Khalimsky's topology is a connected space. We study continuous functions Z^n -> Z, from a subset of Khalimsky n-space to the Khalimsky line. We give necessary and sufficient conditions for such a function to be extendable to a continuous function Z^n -> Z. We classify the subsets A of the digital plane such that every continuous function A -> Z can be extended to a continuous function on the whole plane.

  • 11.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Flerdimensionell analys i bildbehandling2006Other (Other (popular scientific, debate etc.))
    Abstract [en]

    En uppsats om hur idéer från flerdimensionell analys kan användas för bildbehandling på datorer. Något om digital geometri.

  • 12.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    How to find a Khalimsky-continuous approximation of a real-valued function2004Report (Other scientific)
  • 13.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    How to find a Khalimsky-continuous approximation of a real-valued function. In Combinatorial Image Analysis2004Conference paper (Refereed)
    Abstract [en]

    Given a real-valued continuous function defined on n-dimensional Euclidean space, we construct a Khalimsky-continuous integer-valued approximation. From a geometrical point of view, this digitization takes a hypersurface that is the graph of a function and produces a digital hypersurface—the graph of the digitized function.

  • 14.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Introduktionskurs i matematik för DVP.2004Other (Other (popular scientific, debate etc.))
  • 15.
    Melin, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Locally finite spaces and the join operator2007In: Mathematical Morphology and its Applications to Signal and Image Processing, 2007, p. 63–74-Conference paper (Refereed)
    Abstract [en]

    The importance of digital geometry in image processing is well documented. To understand global properties of digital spaces and manifolds we need a solid understanding of local properties. We shall study the join operator, which combines two topological spaces into a new space. Under the natural assumption of local finiteness, we show that spaces can be uniquely decomposed as a join of indecomposable spaces.

1 - 15 of 15
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