Carlos Cabrelli

Self-similarity and multiwavelets in higher dimensions

Introduction Matrices, tiles, and the joint spectral radius Generalized self-similarity and the refinement equation Multiresolution analysis Examples Bibliography Appendix A. Index of symbols.

Automatic representation of binary images

A method of representing black and white images through the description of the boundaries of the objects that define such images is proposed. In order to obtain such a representation, this method uses several algorithms which perform boundary extraction, contour following, segmentation, pattern classification, and curve fitting. One of the advantages of this method is that the image can be reconstructed at any size. It can also be rotated or translated without losing any quality. In addition to achieving a good data-compression rate, the coding-decoding process is computationally very efficient. Also shown is the application of these algorithms to characters in order to obtain fonts that may be downloaded for modern laser printers. >

Iterated fuzzy set systems : a new approach to the inverse problem for fractals and other sets

Abstract Images with grey or colour levels admit a natural representation in terms of fuzzy sets, but without the usual probabilistic interpretation of the latter. We introduce a fuzzy set approach which incorporates, in part, the technique of iterated function systems (IFS) for the construction, analysis, and/or approximation of typically fractal sets and images. The method represents a significant departure from IFS, especially in the interpretation of the resulting image. The introduction of “grey-level maps,” ϑ i : [0, 1] → [0, 1] associated with the contractive maps w i of the IFS affords much greater flexibility in the generation of images as well as in the inverse problem.

Minimum entropy deconvolution and simplicity; a noniterative algorithm

Minimum entropy deconvolution (MED) is a technique developed by Wiggins (1978) with the purpose of separating the components of a signal, as the convolution model of a smooth wavelet with a series of impulses. The advantage of this method, as compared with traditional methods, is that it obviates strong hypotheses over the components, which require only the simplicity of the output. The degree of simplicity is measured with the Varimax norm for factor analysis. An iterative algorithm for computation of the filter is derived from this norm, having as an outstanding characteristic its stability in presence of noise. Geometrical analysis of the Varimax norm suggests the definition of a new criterion for simplicity: the D norm. In case of multiple inputs, the D norm is obtained through modification of the kurtosis norm. One of the most outstanding characteristics of the new criterion, by comparison with the Varimax norm, is that a noniterative algorithm for computation of the deconvolution filter can be deriv...

Sums of Cantor sets

We find conditions on the ratios of dissection of a Cantor set so that the group it generates under addition has positive Lebesgue measure. In particular, we answer affirmatively a special case of a conjecture posed by J. Palis.

Existence of multiwavelets in R^n

For a q-regular Multiresolution Analysis of multiplicity r with arbitrary dilation matrix A for a general lattice Γ in R n , we give necessary and sufficient conditions in terms of the mask and the symbol of the vector scaling function in order that an associated wavelet basis exists. We also show that if 2r(m - 1) > n where m is the absolute value of the determinant of A, then these conditions are always met, and therefore an associated wavelet basis of q-regular functions always exists. This extends known results to the case of multiwavelets in several variables with an arbitrary dilation matrix A for a lattice Γ.

Accuracy of Several Multidimensional Refinable Distributions

Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σk∈Λ ck f(Ax−k), where Λ is a finite subset of Γ, the ck are r×r matrices, and f=(f1,...,fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced from linear combinations of translates of f1,...,fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα=Σi=1rΣk∈Γyα,i(k) fi(x+k). These coefficients are multivariate polynomials yα,i(x) of degree |α| evaluated at lattice points k∈Γ.

CLASSIFYING CANTOR SETS BY THEIR FRACTAL DIMENSIONS

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovich and Taylor. We classify these Cantor sets in terms of their -Hausdorff and -packing measures, for the family of dimension functions , and characterize this classification in terms of the underlying sequences.

The Kantorovich metric for probability measures on the circle

In this paper we show that there exists an analytic expression for the Kantorovich distance between probability measures on the circle. Previously such an expression was only known for measures supported on the real line. In the case that the measures are discrete, this formula enables us to show that the Kantorovich distance can be computed in linear time. This is important for applications, in particular in pattern recognition where this distance is used for texture analysis. As another application we see that the analytic expression found allows us to solve a Minimal Matching Problem in linear time, for which so far only n log n algorithms were known.

Sums of Cantor sets yielding an interval

In this paper we prove that if a Cantor set has ratios of dissection bounded away from zero, then there is a natural number N, such that its N-fold sum is an interval. Moreover, for each element z of this interval, we explicitly construct the N elements of C whose sum yields z. We also extend a result of Mendes and Oliveira showing that when s is irrational Ca + Ca* is an interval if and only ifa/(l— 2a)a /(\ — 2a) > 1. 2000 Mathematics subject classification: primary 28A80, 26A30.