Double preconditioning for Gabor frames Peter Balazs We present an application of the general idea of preconditioning in the context of Gabor frames. While most (iterative) algorithms aim at a more or less costly exact numerical calculation of the inverse Gabor frame matrix, we propose here the use of “very cheap methods” to find an approximation for it, based on (double) preconditioning. We th...

We consider three different versions of the Zak transform (ZT) for discrete-time signals, namely, the discretetime ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex z-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory. We discuss ...

Given certain compactly supported functions g ∈ L2(Rd) whose Zd-translates form a partition of unity, and real invertible d×d matrices B, C for which ||CT B|| is sufficiently small, we prove that the Gabor system {EBmTCng}m,n∈Zd forms a frame, with a (non-canonical) dual Gabor frame generated by an explicitly given finite linear combination of shifts of g. For functions g of the above type and ...

Let g be a continuous, compactly supported function on R such that the integer translates of g constitute a partition of unity. We show that the Gabor system (g, a, b), with window g and time-shift and frequency-shift parameters a, b > 0 has no lower frame bound larger than 0 if b = 2, 3, . . . and a > 0. In particular, (g, a, b) is not a Gabor frame if g is a continuous, compactly supported wa...

| The Gabor transform yields a discrete representation of a signal in the phase space 1, 2]. Since the Gabor transform is non-orthogonal, eecient reconstruction of a signal from its phase space samples is not straightforward and involves the computation of the so-called dual Gabor function. We present a unifying approach to the derivation of numerical algorithms for discrete Gabor analysis, bas...

The theory of localized frames is a recently introduced concept with broad implications to frame theory in general, as well as to the special cases of Gabor and wavelet frames. Using the new notion of a R-dual sequence associated with a Bessel sequence, we derive several duality principles concerning localization in abstract frame theory. As applications of our results we prove a duality princi...

This paper extends to two dimensions the frame criterion developed by Daubechies for one-dimensional wavelets, and it computes the frame bounds for the particular case of 2D Gabor wavelets. Completeness criteria for 2D Gabor image representations are important because of their increasing role in many computer vision applications and also in modeling biological vision, since recent neurophysiolo...