arXiv subjects

Carlos Cabrelli

Publications and source records attributed to Carlos Cabrelli.

42 records · Page 3Linked to original sources

arXiv2006

Optimal Shift Invariant Spaces and Their Parseval Generators

Given a set of functions F={f_1,...,f_m} of L2(Rd), we study the problem of finding the shift-invariant space V with n generators {phi_1,...,phi_n} that is ``closest'' to the functions of F in the sense that V minimize the least square distance from the data F to V over the set of all shift-invariant spaces that can be generated by n or less generators. The Eckart-Young Theorem uses the singular value decomposition to provide a solution to a related problem in finite dimension. We transform the problem under study into an uncountable set of finite dimensional problems each of which can be solved using an extension of the Eckart-Young Theorem. We prove that the finite dimensional solutions can be patched together and transformed to obtain the optimal shift-invariant space solution to the original problem, and we produce a Parseval frame for the optimal space. A typical application is the problem of finding a shift-invariant space model that describes a given class of signals or images (e.g., the class of chest X-Rays), from the observation of a set of m signals or images f_1,...f_m, which may be theoretical samples, or experimental data.

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arXiv2006

Density of the Set of Generators of Wavelet Systems

Given a function $ψ$ in $ \LL^2(\R^d)$, the affine (wavelet) system generated by $ψ$, associated to an invertible matrix $a$ and a lattice $\zG$, is the collection of functions $\{|\det a|^{j/2} ψ(a^jx-γ): j \in \Z, γ\in \zG\}$. In this article we prove that the set of functions generating affine systems that are a Riesz basis of $ \LL^2(\R^d)$ is dense in $ \LL^2(\R^d)$. We also prove that a stronger result is true for affine systems that are a frame of $ \LL^2(\R^d)$. In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of $ \LL^2(\R^d)$ with the induced metric. As a byproduct we introduce the $p$-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems.

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arXiv2005

Refinable shift invariant spaces in R^d

Let $ϕ: \R^d \longrightarrow \C$ be a compactly supported function which satisfies a refinement equation of the form $ϕ(x) = \sum_{k\inΛ} c_k ϕ(Ax - k),\quad c_k\in\C$, where $Γ\subset\R^d$ is a lattice, $Λ$ is a finite subset of $Γ$, and $A$ is a dilation matrix. We prove, under the hypothesis of linear independence of the $Γ$-translates of $ϕ$, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of $L=[c_{Ai-j}]_{i,j\inΓ}$ and a finite dimensional subspace $\mathcal H$ in the shift invariant space generated by $ϕ$. We provide a basis of $\mathcal H$ and show that its elements satisfy a property of homogeneity associated to the eigenvalues of $L$. If the function $ϕ$ has accuracy $κ$, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than $κ$. These latter functions are associated to eigenvalues that are powers of the eigenvalues of $A^{-1}$. Further we show that the dimension of $\mathcal H$ coincides with the local dimension of $ϕ$, and hence, every function in the shift invariant space generated by $ϕ$ can be written locally as a linear combination of translates of the homogeneous functions.

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arXiv2005

Functions in Sampling Spaces

Sampling theory in spaces other than the space of band-limited functions has recently received considerable attention. This is in part because the band-limitedness assumption is not very realistic in many applications. In addition, band-limited functions have very slow decay which translates in poor reconstruction. In this article we study the sampling problem in general shift invariant spaces. We characterize the functions in these spaces and provide necessary and sufficient conditions for a function in $L^2(\R)$ to belong to a sampling space. Furthermore we obtain decompositions of a sampling space in sampling subspaces. These decompositions are related with determining sets. Some examples are provided.

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arXiv2005

Local bases for refinable spaces

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular these functions include all the homogeneous polynomials that are reproducible by the generator, what links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension than the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling.

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arXiv2005

Variations on the Theme of Journe's Lemma

Journe's Lemma is a critical component of many questions related to the product ${BMO}$ theory of S.-Y. Chang and R. Fefferman. This article presents several different variants of the Lemma, some known, some implicit in the literature, and some new.

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