We discuss an elementary procedure that allows us to construct dual pairs of wavelet frames based on certain dual pairs of Gabor frames and vice versa. The construction preserves tightness of the involved frames. Starting with Gabor frames generated by characteristic functions the construction leads to a class of tight wavelet frames that include the Shannon (orthonormal) wavelet, and applying ...

The analysis sparsity model is a very effective approach in modern Compressed Sensing applications. Specifically, redundant operators can lead to fewer measurements needed for reconstruction when employing the \(l_1\)-minimization Sensing. In this paper, we pick an eigenvector of Zauner unitary matrix and –under certain assumptions on ambient dimension– build spark deficient Gabor frame. operat...

In this paper we investigate the computational aspects of some recently proposed iterative methods for approximating the canonical tight and canonical dual window of a Gabor frame (g, a, b). The iterations start with the window g while the iteration steps comprise the window g, the k iterand γk, the frame operators S and Sk corresponding to (g, a, b) and (γk, a, b), respectively, and a number o...

We point out a connection between Gabor analysis and noncommutative analysis. Especially, the strong Morita equivalence of noncommutative tori appears as underlying setting for Gabor analysis, since the construction of equivalence bimodules for noncommutative tori has a natural formulation in the notions of Gabor analysis. As an application we show that Feichtinger’s algebra is such an equivale...

The well-known density theorem for one-dimensional Gabor systems of the form (e2Jl'imbx g(x-na)}m,nEZ, where g E L2(JR), states that a necessary and sufficient condition for the existence of such a systemwhQse linear span is dense in L2(R), or which forms a frame for L2(R), is that the density condition a b :s I is satisfied. The main goal of this paper is to study the analogous problem for Gab...

Natural images are typically a composition of cartoon and texture structures. One common task is to separate such an image into two single images, one containing the cartoon part and the other containing the texture part. Recently, a powerful class of algorithms using sparse approximation and l1 minimization has been introduced to resolve this problem, and numerous inspiring empirical results h...

In this paper we propose a method for automatic local time adaptation of the spectrogram of an audio signal, based on its decomposition within a Gabor multi-frame. The sparsity of the analyses within each individual frame is evaluated through the Rényi entropies measures. According to the sparsity of the decompositions, an optimal resolution and a reduced multi-frame are determined, defining an...

Several concepts for the localization of a frame are studied. The intrinsic localization of a frame is defined by the decay properties of its Gramian matrix. Our main result asserts that the canonical dual frame possesses the same intrinsic localization as the original frame. The proof relies heavily on Banach algebra techniques, in particular on recent spectral invariance properties for certai...