Marcela Fabio

Undecimated Wavelet Transform from Orthogonal Spline Wavelets

The decimated discrete wavelet transform (DWT) gives us a powerful tool in many signal processing applications. It provides stable time—scale representations for any square integrable function as well as a suitable structure of the available information. In connection with this choice, well known families of biorthogonal or orthogonal wavelets are available.

Wavelet projection methods for solving pseudodifferential inverse problems

We consider the Inverse Problem (IP) associated to an equation of the form Af = g, where A is a pseudodifferential operator with symbol. It consists in finding a solution f for given data g. When the operator A is not strongly invertible and the data is perturbed with noise, the IP may be ill-posed and the solution must be approximate carefully. For the present application we regard a particular orthonormal wavelet basis and perform a wavelet projection method to construct a solution to the Forward Problem (FP). The approximate solution to the IP is achieved based on the decomposition of the perturbed data calculating the elementary solutions that are nearly the preimages of the wavelets. Based on properties of both, the basis and the operator, and taking into account the energy of the data, we can handle the error that arises from the partial knowledge of the data and from the non-exact inversion of each element of the wavelet basis. We estimate the error of the approximation and discuss the advantages of the proposed scheme.

Solving deconvolution type problems by wavelet decomposition methods

The paper considers an inverse problem associated with equations of the form Kf = g, where K is a convolution-type operator. The aim is to find a solution f for given function g. We construct approximate solutions by applying a wavelet basis that is well adapted to this problem. For this basis we calculate the elementary solutions that are the approximate preimages of the wavelets. The solution for the inverse problem is then constructed as an appropriate finite linear combination of the elementary solutions. Under certain assumptions we estimate the approximation error and discuss the advantages of the proposed scheme.

Self-similar and multiscaling functions of dimension r

In this work we will generalize results linking multiresolution analysis structures and vectorial spaces generated from integer shifts of self-similar or radial basis functions. This connection results of a remarkable relation between causal scaling and causal radial functions, recently exposed by T. Blu and M. Unser for the unidimensional case. Here, we will detail some definitions and will enunciate the main theorems for the r dimensional case.