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Generalized Fractal Transforms and Self-Similar Objects in Cone Metric Spaces
H. Kunze∗ , D. La Torre∗∗ and F. Mendivil∗∗∗ , E. R. Vrscay∗∗∗∗ January 17, 2012
Department of Mathematics and Statistics, University of Guelph, Canada Department of Economics, Business and Statistics, University of Milan, Italy ∗∗∗ Department of Mathematics and Statistics, Acadia University, Canada ∗∗∗∗ Department ofApplied Mathematics, University of Waterloo, Canada Abstract We use the idea of a scalarization of a cone metric to prove that the topology generated by any cone metric is equivalent to a topology generated by a related metric. We then analyze the case of an ordering cone with empty interior and we provide alternative definitions based on the notion of interior cone. Finally we discuss theimplications of such cone metrics in the theory of iterated function systems and generalized fractal transforms and suggest some applications in fractal-based image analysis.


Keywords: Cone metric space, Completeness, Contractivity, Self-similarity, Digital image analysis.



Properties of contractive mappings are used throughout mathematics, usually to invoke Banach’stheorem on fixed points of contractions. In the classical case of iterated function systems (IFSs), the existence of self-similar objects relies on this same theorem. Fundamental ingredients of the theory of IFS are the use of complete metric spaces and the notion of contractivity, which both depend on the definition of the underlying distance. Much recent work has focused on the extension of thenotion of metric spaces and the related notion of contractivity. One such extension is the idea of cone metric spaces. In this context, the distance is no longer a positive number but a vector, in general an element of a Banach space which has been equipped with an ordering cone. In applications, it is often useful to compare two objects in multiple ways and combine these various comparisons together.Using a cone metric allows this and thus allows a better description of the complexity of the problem. Of course, this implies that many results of the classical theory of metric spaces needed to be adapted. 1

For us, a motivating application of cone metric spaces is image process, in particular when studying the structural similarity of images. This is a natural situation in which two imagesare compared using several different criteria, leading to vector-valued distances. The paper is structured as follows: Section 2 provides the basic definitions and results of cone metric space, Section 3 shows by scalarization techniques that the topology generated by any cone metric is equivalent to a topology generated by a related metric. Section 3.2 presents the case of an ordering cone withempty interior. Section 4 presents a construction of a Hausdorff cone-metric between compact subsets. Finally Section 5 illustrates a relevant application of cone metric spaces in fractal image analysis.


Cone metric space: preliminary properties

We will use E to denote a Banach space and P ⊂ E will be a pointed cone in E. This means that P satisfies 1. 0 ∈ P , 2. α, β ∈ R with α, β ≥ 0 andx, y ∈ P implies αx + βy ∈ P , 3. P ∩ −P = {0}. The cone P induces an order in E by x ≤ y if y − x ∈ P or, said another way, there is some p ∈ P so that x + p = y. The elements of P are said to be positive and the elements of the interior of P are strictly positive. We assume that P is closed and will also usually assume that int(P ) = ∅. However, in Section 3.2 we are particularly interested inthe case when int(P ) is empty. It is easy to show that p + int(P ) ⊆ int(P ) for every p ∈ P . We say that x y if y − x ∈ int(P ), so 0 x means x ∈ int(P ). A pointed wedge satisfies properties 1 and 2, so every cone is a wedge but not conversely. A cone metric on a set X is a function d : X × X → P so that 1. d(x, y) = 0 iff x = y, 2. d(x, y) = d(y, x), 3. d(x, y) ≤ d(x, z) + d(z, y) for all x,...
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