In this paper, we investigate extreme singular values of the analysis matrix of a Gabor frame @math with a random window @math . Columns of such matrices are time and frequency shifts of @math , and @math is the set of time-frequency shift indices. Our aim is to obtain bounds on the singular values of such random time-frequency structured matrices for various choices of the frame set @math , and to investigate their dependence on the structure of @math , as well as on its cardinality. We also compare the results obtained for Gabor frame analysis matrices with the respective results for matrices with independent identically distributed entries.
To the best of our knowledge, singular values of the analysis matrix @math of a Gabor frame with random window and general @math with @math were not studied before. At the same time, the following bounds on the singular values of the synthesis matrix @math of a Gabor system @math with a Steinhaus window @math and @math have been established in @cite_6 .
@cite_6
2963293266 2160051394 2138584058 2953076353
Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal l2-norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good time–frequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler–Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found. Consider a Bernoulli-Gaussian complex n-vector whose components are Vi = XiBi, with Xi C N(0, Px) and binary Bi mutually independent and iid across i. This random q-sparse vector is multiplied by a square random matrix U, and a randomly chosen subset, of average size n p, p ∈ [0,1], of the resulting vector components is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where U is typically a matrix with iid components, to allow U satisfying a certain freeness condition. This class of matrices encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verdu, as well as a number of information-theoretic bounds, to study the input-output mutual information and the support recovery error rate in the limit of n → ∞. We also extend the scope of the large deviation approach of Rangan and characterize the performance of a class of estimators encompassing thresholded linear MMSE and l1 relaxation. 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 neurophysiological evidence from the visual cortex of mammalian brains suggests that the filter response profiles of the main class of linearly-responding cortical neurons (called simple cells) are best modeled as a family of self-similar 2D Gabor wavelets. We therefore derive the conditions under which a set of continuous 2D Gabor wavelets will provide a complete representation of any image, and we also find self-similar wavelet parametrization which allow stable reconstruction by summation as though the wavelets formed an orthonormal basis. Approximating a "tight frame" generates redundancy which allows low-resolution neural responses to represent high-resolution images. This paper provides novel results for the recovery of signals from undersampled measurements based on analysis @math -minimization, when the analysis operator is given by a frame. We both provide so-called uniform and nonuniform recovery guarantees for cosparse (analysis-sparse) signals using Gaussian random measurement matrices. The nonuniform result relies on a recovery condition via tangent cones and the uniform recovery guarantee is based on an analysis version of the null space property. Examining these conditions for Gaussian random matrices leads to precise bounds on the number of measurements required for successful recovery. In the special case of standard sparsity, our result improves a bound due to Rudelson and Vershynin concerning the exact reconstruction of sparse signals from Gaussian measurements with respect to the constant and extends it to stability under passing to approximately sparse signals and to robustness under noise on the measurements.