In the last decade it has become clear that one of the central themes within Gabor analysis (with respect to general timefrequency lattices) is a duality theory for Gabor frames, including the Wexler-Raz biorthogonality condition, the Ron-Shen’s duality principle or Janssen’s representation of a Gabor frame operator. All these results are closely connected with the so-called Fundamental Identit...

This work revolves around Gabor filters. It involves theoretical aspects such as the uncertainty principle as well as practical issues such as filter bank design and application of Gabor filters to texture segmentation. This thesis is composed of three main parts. In the first part, we explore the role of the metric of the Gabor feature space, and its usefulness in capturing what we refer to as...

Beamlet migration based on local perturbation theory is proposed. The method is formulated with a local background velocity and local perturbations for each window of the wave eld decomposition using Gabor-Daubechies frame and local cosine basis. The prop-agators and phase-correction operators are obtained analytically for the G-D tight-frame, and numerically for the local cosine basis. The num...

In [11], three of the authors obtained a characterization of certain types of reproducing systems. In this work, we apply these results and methods to various affine–like, wave packets and Gabor systems to determine their frame properties. In particular, we study how oversampled systems inherit properties (like the frame bounds) of the original systems. Moreover, our approach allows us to study...

We consider the frame property of the Gabor system G(g, α, β) = {eg(t − αm) : m,n ∈ Z} for the case of rational oversampling, i.e. α, β ∈ Q. A ’rational’ analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, α, β) does not generate a frame if αβ = n−1 n . Special attention is paid to the first Hermite function h1(t) = te −πt .

We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjöstrand’s class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard analysis” techniques are not available on locally compact abelian groups, a new time-frequency appro...

Motivated by a recent generalization of the Balian-Low theorem and by new research in wireless communications we analyze the construction of Wilson bases for general time-frequency lattices. We show that orthonormal Wilson bases for L(R) can be constructed for any time-frequency lattice whose volume is 1 2 . While this may be expected, our second result may come as a surprise. Namely we prove t...

In this paper we consider the construction of multiresolution Gabor dictionaries appropriate for audio signal analysis. Motivated by a desire for parsimony and efficiency, we propose and formalise the idea of reduced multi-Gabor systems, showing that they constitute a frame for L2(R) and other Hilbert spaces of interest. In order to demonstrate the practicality of such a scheme, we apply it to ...

We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjöstrand’s class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard analysis” techniques are not available on locally compact abelian groups, a new time-frequency appro...

The metaplectic representation describes a class of automorphisms of the Heisenberg group H = H(G), defined for a locally compact abelian groupG. ForG = R ,H is the usual Heisenberg group. For the case when G is the finite cyclic group Zn, only partial constructions are known. Here we present new results for this case and we obtain an explicit construction of the metaplectic operators on C. We ...