Franklin Mendivil

Fractal vector measures and vector calculus on planar fractal domains

Abstract We define an abstract framework for self-similar vector-valued Borel measures on a compact space X based upon a formulation of Iterated Function Systems (IFS) on such measures. This IFS method permits the construction of tangent and normal vector measures to planar fractal curves. Line integrals of smooth vector fields over planar fractal curves may then be defined. These line integrals then lead to a formulation of Greens Theorem and the Divergence Theorem for planar regions bounded by fractal curves. The abstract setting also naturally leads to “probability measure”-valued measures. These measures may be used to color the geometric attractor of an IFS in a self-similar way.

Correspondence Between Fractal-Wavelet Transforms and Iterated Function Systems With Grey Level Maps

It is well known that the action of a “Fractal Transform” or (Local) Iterated Function System with Grey Level Maps (IFSM) on a function f(x) induces a very simple mapping on its expansion coefficients c ij in the Haar wavelet basis. This is the basis of the “discrete fractal-wavelet transform”: subtrees of the wavelet coefficient tree are scaled and copied to lower subtrees. Such transforms, which we shall also refer to as IFSW—IFS on wavelet coefficients—have been introduced into image processing with other (compactly supported) wavelet basis sets in an attempt to remove the blocking artifacts in the standard IFS block encoding algorithms. Although not as straightforward as in the Haar case, we show that there is a relationship between such wavelet transforms and IFSM. In fact, for most such transforms, there is an equivalent IFSM, which provides a further mathematical basis for their use in image processing. We also present results for the case of periodized wavelets, a common implementation in image processing. Finally, we prove some results on the fractal dimension of the graph of an attractor of IFSM or IFSW operators.

Generalized fractal transforms and self-similar objects in cone metric spaces

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 quasi-interior points. Finally we discuss the implications of such cone metrics in the theory of iterated function systems and generalized fractal transforms and suggest some applications in fractal-based image analysis.

Iterated Function Systems on Multifunctions

We introduce a method of iterated function systems (IFS) over the space of set-valued mappings (multifunctions). This is done by first considering a couple of useful metrics over the space of multifunctions F(X,Y). Some appropriate IFS-type fractal transform operators T:F(X,Y)->F(X,Y) are then defined which combine spatially-contracted and range-modified copies of a multifunction u to produce a new multifunction v = Tu. Under suitable conditions, the fractal transform T is contractive, implying the existence of a fixed-point set-valued mapping u. Some simple examples are then presented. We then consider the inverse problem of approximation of set-valued mappings by fixed points of fractal transform operators T and present some preliminary results.

Function algebras and the lattice of compactifications

We provide some conditions as to when K(X) _ K(Y) for two locally compact spaces X and Y (where K(X) is the lattice of all Hausdorff compactifications of X). More specifically, we prove that K(X) C K(Y) if and only if C*(X)/Co(X) C*(Y)/Co(Y). Using this result, we prove several extensions to the case where K(X) is embedded as a sub-lattice of K(Y) and to where X and Y are not locally compact. One major contribution is in the use of function algebra techniques. The use of these techniques makes the extensions simple and clean and brings new tools to the subject.

The Sizes of Rearrangements of Cantor Sets

A linear Cantor setC withzero Lebesgue measure is associated withthe countablecollection of the bounded complementary open intervals. A rearrangment ofC has the same lengths of its com- plementary intervals, but with different locations. We study the Hausdorff and packing h-measures and dimensional properties of the set of all rearrangments of some given C for general dimension functions h. For each set of complementary lengths, we construct a Cantor set rearrangement which has the maximal Hausdorff and the minimal packing h-premeasure, up to a constant. We also show that if the packing measure of this Cantor set is positive, then there is a rearrangement which has infinite packing measure.

Two Algorithms for Non-Separable Wavelet Transforms and Applications to Image Compression

Previously, the use of non-separable wavelets in image processing has been hindered by the lack of a fast algorithm to perform a non-separable wavelet transform. We present two such algorithms in this paper. The first algorithm implements a periodic wavelet transform for any valid wavelet filter sequence and dilation matrices satisfying a trace condition. We discuss some of the complicating issues unique to the non-separable case and how to overcome them. The second algorithm links Haar wavelets and complex bases and uses this link to drive the algorithm. For the complex bases case, the asymmetry of the wavelet trees produced leads to a discussion of the complexities in implementing zero-tree and other wavelet compression methods. We describe some preliminary attempts at using this algorithm, with non-separable Haar wavelets, for reducing the blocking artifacts in fractal image compression.

Self-Affine Vector Measures and Vector Calculus on Fractals

In this paper, we construct an IFS framework for studying self-similar vector measures. These measures have several applications, including the tangent and normal vector measure “fields” to fractal curves. Using the tangent vector measure, we define a line integral of a smooth vector field over a fractal curve. This then leads to a formulation of Green’s Theorem (and the Divergence Theorem) for planar regions bounded by fractal curves. The general IFS setting also leads to “probability measure” — valued measures, which give one way to coloring the geometric attractor of an IFS in a self-similar way.

A Chaos game algorithm for generalized iterated function systems

In this paper we provide an extension of the classical Chaos game for IFSP. The paper is divided into two parts: in the first one, we discuss how to determine the integral with respect to a measure which is a combination of a self-similar measure from an IFSP along with a density given by an IFSM. In the second part, we prove a version of the Ergodic Theorem for the integration of a continuous multifunction with respect to the invariant measure of an IFSP. These results are in line with some recent extensions of IFS theory to multifunctions.

Annealing a Genetic Algorithm for Constrained Optimization

In this paper, we adapt a genetic algorithm for constrained optimization problems. We use a dynamic penalty approach along with some form of annealing, thus forcing the search to concentrate on feasible solutions as the algorithm progresses. We suggest two different general-purpose methods for guaranteeing convergence to a globally optimal (feasible) solution, neither of which makes any assumptions on the structure of the optimization problem. The former involves modifying the GA evolution operators to yield a Boltzmann-type distribution on populations. The latter incorporates a dynamic penalty along with a slow annealing of acceptance probabilities. We prove that, with probability one, both of these methods will converge to a globally optimal feasible state.