Given a set of functions F = {f1, . . . , fm} ⊂ L2(R), we study the problem of finding the shift-invariant space V with n generators {φ1, . . . , φn} that is “closest” to the functions of F in the sense that V = argminV ′∈Vn m X i=1 wi‖fi − PV ′fi‖, where wis are positive weights, and Vn is the set of all shift-invariant spaces that can be generated by n or less generators. The Eckart-Young The...

Curvelets provide a new multiresolution representation with several features that set them apart from existing representations such as wavelets, multiwavelets, steerable pyramids, and so on. They are based on an anisotropic notion of scaling. The frame elements exhibit very high direction sensitivity and are highly anisotropic. In this paper we describe these properties and indicate why they ma...

This paper deals with multiwavelets which are a recent generalization of wavelets in the context of multirate lter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are t for processing multi-channel signals. First, we will recall some general results on multi lters by looking at them as time-varying lters. Then, we wil...

This paper deals with multiwavelets which are a recent generalization of wavelets in the context of multirate lter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are t for processing multi-channel signals. First, we will recall some general results on multiilters by looking at them as time-varying lters. Then, we wil...

In this paper, we introduce a unified theory of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start from defining multiresolution analysis (MRA) generated by a single generator on manifolds, ...

The deepest and most difficult question in Fusion Frame Theory is the construction of fusion frames with added properties for specific applications. In frame theory we have a powerful tool introduced by Benedetto and Fickus [1] called frame potentials (See [5] for a deep analysis of frame potentials). Frame potentials are a valuable tool for showing the existence of frames with certain specifie...

This paper presents an evaluation of different types and families of multiwavelets in stereo correspondence matching. Different multiwavelet families with different filter types such as balanced versus unbalanced, symmetricsymmetric versus symmetric-antisymmetric are used. Normalized cross correlation is employed to find the best correspondence points and generate a disparity map. In the case o...

Based on multiresolution analysis (MRA) structures combined with the unitary extension principle (UEP), many frame wavelets were constructed, which are called UEP framelets. The aim of this letter is to derive general properties of UEP framelets based on the spectrum of the center space of the underlying MRA structures. We first give the existence theorem, that is, we give a necessary and suffi...

The affine systems generated by Ψ ⊂ L(R) are the systems AA(Ψ) = {D A Tk Ψ : j ∈ Z, k ∈ Zn}, where Tk are the translations, and DA the dilations with respect to an invertible matrix A. As shown in [5], there is a simple characterization for those affine systems that are a Parseval frame for L(R). In this paper, we correct an error in the proof of the characterization result from [5], by redefin...