Papers by Carlos Cabrelli
A shift-invariant space is a space of functions that is invariant under integer translations. Suc... more A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are characterized in terms
Bookmarks Related papers MentionsView impact
Given a set of functions F={f_1,...,f_m} of L2(Rd), we study the problem of finding the shift-inv... more 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
Bookmarks Related papers MentionsView impact
Journal of Fourier Analysis and Applications, Oct 3, 2008
Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal coll... more Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for the given data, in a sense defined in the paper, and provides an optimal model for sampling in union of subspaces. The results are proved in a general setting and then applied to the case of low dimensional subspaces of R^N and to infinite dimensional shift-invariant spaces in L^2(R^d). We also present an iterative search algorithm for finding the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary design, signal models for signals with finite rate of innovation, and the subspace segmentation problem.
Bookmarks Related papers MentionsView impact
Sampling Theory in Signal and Image Processing, Apr 1, 2009
A new paradigm in Sampling theory has been developed recently by Lu and Do. In this new approach ... more A new paradigm in Sampling theory has been developed recently by Lu and Do. In this new approach the classical linear model is replaced by a non-linear, but structured model consisting of a union of subspaces. This is the natural approach for the new theory of compressed sampling, representation of sparse signals and signals with finite rate of innovation. In this article we extend the theory of Lu and Do, for the case that the subspaces in the union are shift-invariant spaces. We describe the subspaces by means of frame generators instead of orthonormal bases. We show that, the one to one and stability conditions for the sampling operator, are valid for this more general case.
Bookmarks Related papers MentionsView impact
We prove the existence of sampling sets and interpolation sets near the critical density, in Pale... more We prove the existence of sampling sets and interpolation sets near the critical density, in Paley Wiener spaces of a locally compact abelian (LCA) group G . This solves a problem left by Gr\"ochenig, Kutyniok, and Seip in the article: `Landau's density conditions for LCA groups ' (J. of Funct. Anal. 255 (2008) 1831-1850). To achieve this result, we prove the existence of universal Riesz bases of characters for L2(Omega), provided that the relatively compact subset Omega of the dual group of G satisfies a multi-tiling condition. This last result generalizes Fuglede's theorem, and extends to LCA groups setting recent constructions of Riesz bases of exponentials in bounded sets of Rd.
Bookmarks Related papers MentionsView impact
In this note we study frame-related properties of a sequence of functions multiplied by another f... more In this note we study frame-related properties of a sequence of functions multiplied by another function. In particular we study frame and Riesz basis properties. We apply these results to sets of irregular translates of a bandlimited function $h$ in $L^2(\R^d)$. This is achieved by looking at a set of exponentials restricted to a set $E \subset \R^d$ with frequencies in a countable set $\Lambda$ and multiplying it by the Fourier transform of a fixed function $h \in L^2(E)$. Using density results due to Beurling, we prove the existence and give ways to construct frames by irregular translates.
Bookmarks Related papers MentionsView impact
Given a set of points \F in a high dimensional space, the problem of finding a union of subspaces... more Given a set of points \F in a high dimensional space, the problem of finding a union of subspaces \cup_i V_i\subset \R^N that best explains the data \F increases dramatically with the dimension of \R^N. In this article, we study a class of transformations that map the problem into another one in lower dimension. We use the best model in the low dimensional space to approximate the best solution in the original high dimensional space. We then estimate the error produced between this solution and the optimal solution in the high dimensional space.
Bookmarks Related papers MentionsView impact
Bookmarks Related papers MentionsView impact
Eprint Arxiv Math 0703438, Mar 1, 2007
In this article, we develop a general method for constructing wavelets {|det A_j|^{1/2} g(A_jx-x_... more In this article, we develop a general method for constructing wavelets {|det A_j|^{1/2} g(A_jx-x_{j,k}): j in J, k in K}, on irregular lattices of the form X={x_{j,k} in R^d: j in J, k in K}, and with an arbitrary countable family of invertible dxd matrices {A_j in GL_d(R): j in J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L^2(R^d) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R^d. Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.
Bookmarks Related papers MentionsView impact
Applied and Computational Harmonic Analysis, 2015
Bookmarks Related papers MentionsView impact
Information Processing Letters, 1991
Bookmarks Related papers MentionsView impact
... identified as such, is not to be taken as an expression of opinion as to whether or not they ... more ... identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to Cover design: Alex Gerasev Printed on ... An analysis of the behaviour of the product of a function in some Hardy space with a function in the dual (Lip-schitz space) is made in ...
Bookmarks Related papers MentionsView impact
Appl Comput Harmonic Anal, 2004
Bookmarks Related papers MentionsView impact
Ipl, 1992
Bookmarks Related papers MentionsView impact
Advances in Mathematics, 2015
Bookmarks Related papers MentionsView impact
Bookmarks Related papers MentionsView impact
ABSTRACT
Bookmarks Related papers MentionsView impact
Bookmarks Related papers MentionsView impact
Houston journal of mathematics
Bookmarks Related papers MentionsView impact
Revista de la Unión Matemática Argentina
This paper is a survey about recent results on sparse representa- tions and optimal models in die... more This paper is a survey about recent results on sparse representa- tions and optimal models in dierent settings. Given a set of functions, we show that there exists an optimal collection of subspaces minimizing the sum of the square of the distances between each function and its closest subspace in the collection. Further, this collection of subspaces gives the best sparse representation for the given data, in a sense dened later, and provides an optimal model for sampling in a union of subspaces.
Bookmarks Related papers MentionsView impact
Uploads
Papers by Carlos Cabrelli